KILLED proof of input_dvCxkb8j5l.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 319 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 60 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 383 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 1394 ms] (30) CpxRNTS (31) CompletionProof [UPPER BOUND(ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (36) CdtProblem (37) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 178 ms] (42) CdtProblem (43) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 53 ms] (44) CdtProblem (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 55 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 48 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 67 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 79 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms] (84) CdtProblem (85) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 122 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtLeafRemovalProof [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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 153 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (164) 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(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x)) g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x'))))) h(0, S(x)) -> h(0, x) h(0, 0) -> 0 g(S(x), 0) -> 0 f(S(x), 0) -> 0 h(S(x), x2) -> h(x, x2) g(0, x2) -> 0 f(0, x2) -> 0 S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x)) g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x'))))) h(0', S(x)) -> h(0', x) h(0', 0') -> 0' g(S(x), 0') -> 0' f(S(x), 0') -> 0' h(S(x), x2) -> h(x, x2) g(0', x2) -> 0' f(0', x2) -> 0' S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x)) g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x'))))) h(0, S(x)) -> h(0, x) h(0, 0) -> 0 g(S(x), 0) -> 0 f(S(x), 0) -> 0 h(S(x), x2) -> h(x, x2) g(0, x2) -> 0 f(0, x2) -> 0 S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x)) [1] g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x'))))) [1] h(0, S(x)) -> h(0, x) [1] h(0, 0) -> 0 [1] g(S(x), 0) -> 0 [1] f(S(x), 0) -> 0 [1] h(S(x), x2) -> h(x, x2) [1] g(0, x2) -> 0 [1] f(0, x2) -> 0 [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x)) [1] g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x'))))) [1] h(0, S(x)) -> h(0, x) [1] h(0, 0) -> 0 [1] g(S(x), 0) -> 0 [1] f(S(x), 0) -> 0 [1] h(S(x), x2) -> h(x, x2) [1] g(0, x2) -> 0 [1] f(0, x2) -> 0 [1] The TRS has the following type information: f :: S:0 -> S:0 -> S:0 S :: S:0 -> S:0 h :: S:0 -> S:0 -> S:0 g :: S:0 -> S:0 -> S:0 0 :: S:0 Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: g_2 f_2 h_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x)) [1] g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x'))))) [1] h(0, S(x)) -> h(0, x) [1] h(0, 0) -> 0 [1] g(S(x), 0) -> 0 [1] f(S(x), 0) -> 0 [1] h(S(x), x2) -> h(x, x2) [1] g(0, x2) -> 0 [1] f(0, x2) -> 0 [1] The TRS has the following type information: f :: S:0 -> S:0 -> S:0 S :: S:0 -> S:0 h :: S:0 -> S:0 -> S:0 g :: S:0 -> S:0 -> S:0 0 :: S:0 Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(S(S(x'')), S(S(x1))) -> h(h(f(S(x''), S(S(x1))), g(x'', S(S(S(S(x1)))))), h(g(S(S(S(x''))), S(x1)), f(S(S(S(S(S(S(x'')))))), x1))) [3] f(S(S(x'')), S(0)) -> h(h(f(S(x''), S(0)), g(x'', S(S(S(0))))), 0) [3] f(S(0), S(S(x3))) -> h(0, h(g(S(S(0)), S(x3)), f(S(S(S(S(S(0))))), x3))) [3] f(S(0), S(0)) -> h(0, 0) [3] g(S(S(x4)), S(x')) -> h(h(g(S(x4), S(x')), f(S(S(S(S(x4)))), x')), h(f(S(x4), S(S(S(x')))), g(x4, S(S(S(S(S(x')))))))) [3] g(S(0), S(x')) -> h(h(g(0, S(x')), f(S(S(S(0))), x')), 0) [3] h(0, S(x)) -> h(0, x) [1] h(0, 0) -> 0 [1] g(S(x), 0) -> 0 [1] f(S(x), 0) -> 0 [1] h(S(x), x2) -> h(x, x2) [1] g(0, x2) -> 0 [1] f(0, x2) -> 0 [1] The TRS has the following type information: f :: S:0 -> S:0 -> S:0 S :: S:0 -> S:0 h :: S:0 -> S:0 -> S:0 g :: S:0 -> S:0 -> S:0 0 :: S:0 Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> h(h(f(1 + x'', 1 + 0), g(x'', 1 + (1 + (1 + 0)))), 0) :|: z' = 1 + 0, x'' >= 0, z = 1 + (1 + x'') f(z, z') -{ 3 }-> h(h(f(1 + x'', 1 + (1 + x1)), g(x'', 1 + (1 + (1 + (1 + x1))))), h(g(1 + (1 + (1 + x'')), 1 + x1), f(1 + (1 + (1 + (1 + (1 + (1 + x''))))), x1))) :|: x1 >= 0, z' = 1 + (1 + x1), x'' >= 0, z = 1 + (1 + x'') f(z, z') -{ 3 }-> h(0, h(g(1 + (1 + 0), 1 + x3), f(1 + (1 + (1 + (1 + (1 + 0)))), x3))) :|: z' = 1 + (1 + x3), z = 1 + 0, x3 >= 0 f(z, z') -{ 3 }-> h(0, 0) :|: z = 1 + 0, z' = 1 + 0 f(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 f(z, z') -{ 1 }-> 0 :|: z' = x2, z = 0, x2 >= 0 g(z, z') -{ 3 }-> h(h(g(0, 1 + x'), f(1 + (1 + (1 + 0)), x')), 0) :|: z' = 1 + x', z = 1 + 0, x' >= 0 g(z, z') -{ 3 }-> h(h(g(1 + x4, 1 + x'), f(1 + (1 + (1 + (1 + x4))), x')), h(f(1 + x4, 1 + (1 + (1 + x'))), g(x4, 1 + (1 + (1 + (1 + (1 + x'))))))) :|: x4 >= 0, z = 1 + (1 + x4), z' = 1 + x', x' >= 0 g(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 g(z, z') -{ 1 }-> 0 :|: z' = x2, z = 0, x2 >= 0 h(z, z') -{ 1 }-> h(x, x2) :|: z' = x2, x >= 0, z = 1 + x, x2 >= 0 h(z, z') -{ 1 }-> h(0, x) :|: z' = 1 + x, x >= 0, z = 0 h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z' = 1 + 0, z - 2 >= 0 f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 f(z, z') -{ 3 }-> h(0, 0) :|: z = 1 + 0, z' = 1 + 0 f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 f(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 g(z, z') -{ 3 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z = 1 + 0, z' - 1 >= 0 g(z, z') -{ 3 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z - 2 >= 0, z' - 1 >= 0 g(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 g(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 h(z, z') -{ 1 }-> h(0, z' - 1) :|: z' - 1 >= 0, z = 0 h(z, z') -{ 1 }-> h(z - 1, z') :|: z - 1 >= 0, z' >= 0 h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { h } { f, g } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z' = 1 + 0, z - 2 >= 0 f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 f(z, z') -{ 3 }-> h(0, 0) :|: z = 1 + 0, z' = 1 + 0 f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 f(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 g(z, z') -{ 3 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z = 1 + 0, z' - 1 >= 0 g(z, z') -{ 3 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z - 2 >= 0, z' - 1 >= 0 g(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 g(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 h(z, z') -{ 1 }-> h(0, z' - 1) :|: z' - 1 >= 0, z = 0 h(z, z') -{ 1 }-> h(z - 1, z') :|: z - 1 >= 0, z' >= 0 h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 Function symbols to be analyzed: {h}, {f,g} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z' = 1 + 0, z - 2 >= 0 f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 f(z, z') -{ 3 }-> h(0, 0) :|: z = 1 + 0, z' = 1 + 0 f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 f(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 g(z, z') -{ 3 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z = 1 + 0, z' - 1 >= 0 g(z, z') -{ 3 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z - 2 >= 0, z' - 1 >= 0 g(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 g(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 h(z, z') -{ 1 }-> h(0, z' - 1) :|: z' - 1 >= 0, z = 0 h(z, z') -{ 1 }-> h(z - 1, z') :|: z - 1 >= 0, z' >= 0 h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 Function symbols to be analyzed: {h}, {f,g} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: h after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z' = 1 + 0, z - 2 >= 0 f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 f(z, z') -{ 3 }-> h(0, 0) :|: z = 1 + 0, z' = 1 + 0 f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 f(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 g(z, z') -{ 3 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z = 1 + 0, z' - 1 >= 0 g(z, z') -{ 3 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z - 2 >= 0, z' - 1 >= 0 g(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 g(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 h(z, z') -{ 1 }-> h(0, z' - 1) :|: z' - 1 >= 0, z = 0 h(z, z') -{ 1 }-> h(z - 1, z') :|: z - 1 >= 0, z' >= 0 h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 Function symbols to be analyzed: {h}, {f,g} Previous analysis results are: h: runtime: ?, size: O(1) [0] ---------------------------------------- (23) 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: 1 + z + z' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z' = 1 + 0, z - 2 >= 0 f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 f(z, z') -{ 3 }-> h(0, 0) :|: z = 1 + 0, z' = 1 + 0 f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 f(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 g(z, z') -{ 3 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z = 1 + 0, z' - 1 >= 0 g(z, z') -{ 3 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z - 2 >= 0, z' - 1 >= 0 g(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 g(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 h(z, z') -{ 1 }-> h(0, z' - 1) :|: z' - 1 >= 0, z = 0 h(z, z') -{ 1 }-> h(z - 1, z') :|: z - 1 >= 0, z' >= 0 h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 Function symbols to be analyzed: {f,g} Previous analysis results are: h: runtime: O(n^1) [1 + z + z'], size: O(1) [0] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> s :|: s >= 0, s <= 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z' = 1 + 0, z - 2 >= 0 f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 f(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 g(z, z') -{ 3 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z = 1 + 0, z' - 1 >= 0 g(z, z') -{ 3 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z - 2 >= 0, z' - 1 >= 0 g(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 g(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 h(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z = 0 h(z, z') -{ 1 + z + z' }-> s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 Function symbols to be analyzed: {f,g} Previous analysis results are: h: runtime: O(n^1) [1 + z + z'], size: O(1) [0] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed SIZE bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> s :|: s >= 0, s <= 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z' = 1 + 0, z - 2 >= 0 f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 f(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 g(z, z') -{ 3 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z = 1 + 0, z' - 1 >= 0 g(z, z') -{ 3 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z - 2 >= 0, z' - 1 >= 0 g(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 g(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 h(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z = 0 h(z, z') -{ 1 + z + z' }-> s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 Function symbols to be analyzed: {f,g} Previous analysis results are: h: runtime: O(n^1) [1 + z + z'], size: O(1) [0] f: runtime: ?, size: O(1) [0] g: runtime: ?, size: O(1) [0] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> s :|: s >= 0, s <= 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z' = 1 + 0, z - 2 >= 0 f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 f(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 g(z, z') -{ 3 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z = 1 + 0, z' - 1 >= 0 g(z, z') -{ 3 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z - 2 >= 0, z' - 1 >= 0 g(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 g(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 h(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z = 0 h(z, z') -{ 1 + z + z' }-> s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 Function symbols to be analyzed: {f,g} Previous analysis results are: h: runtime: O(n^1) [1 + z + z'], size: O(1) [0] f: runtime: INF, size: O(1) [0] g: runtime: ?, size: O(1) [0] ---------------------------------------- (31) 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: none And the following fresh constants: none ---------------------------------------- (32) 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(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x)) [1] g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x'))))) [1] h(0, S(x)) -> h(0, x) [1] h(0, 0) -> 0 [1] g(S(x), 0) -> 0 [1] f(S(x), 0) -> 0 [1] h(S(x), x2) -> h(x, x2) [1] g(0, x2) -> 0 [1] f(0, x2) -> 0 [1] The TRS has the following type information: f :: S:0 -> S:0 -> S:0 S :: S:0 -> S:0 h :: S:0 -> S:0 -> S:0 g :: S:0 -> S:0 -> S:0 0 :: S:0 Rewrite Strategy: INNERMOST ---------------------------------------- (33) 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 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> h(g(x', 1 + x), f(1 + (1 + (1 + x')), x)) :|: z = 1 + x', z' = 1 + x, x' >= 0, x >= 0 f(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 f(z, z') -{ 1 }-> 0 :|: z' = x2, z = 0, x2 >= 0 g(z, z') -{ 1 }-> h(f(1 + x, 1 + x'), g(x, 1 + (1 + (1 + x')))) :|: z' = 1 + x', x >= 0, x' >= 0, z = 1 + x g(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 g(z, z') -{ 1 }-> 0 :|: z' = x2, z = 0, x2 >= 0 h(z, z') -{ 1 }-> h(x, x2) :|: z' = x2, x >= 0, z = 1 + x, x2 >= 0 h(z, z') -{ 1 }-> h(0, x) :|: z' = 1 + x, x >= 0, z = 0 h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (35) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 f(0, z0) -> 0 g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(S(z0), 0) -> 0 g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) Tuples: F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1))) F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) F(S(z0), 0) -> c2 F(0, z0) -> c3 G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(z0), 0) -> c6 G(0, z0) -> c7 H(0, S(z0)) -> c8(H(0, z0)) H(0, 0) -> c9 H(S(z0), z1) -> c10(H(z0, z1)) S tuples: F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1))) F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) F(S(z0), 0) -> c2 F(0, z0) -> c3 G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(z0), 0) -> c6 G(0, z0) -> c7 H(0, S(z0)) -> c8(H(0, z0)) H(0, 0) -> c9 H(S(z0), z1) -> c10(H(z0, z1)) K tuples:none Defined Rule Symbols: f_2, g_2, h_2 Defined Pair Symbols: F_2, G_2, H_2 Compound Symbols: c_2, c1_2, c2, c3, c4_2, c5_2, c6, c7, c8_1, c9, c10_1 ---------------------------------------- (37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing nodes: G(S(z0), 0) -> c6 F(0, z0) -> c3 F(S(z0), 0) -> c2 G(0, z0) -> c7 H(0, 0) -> c9 ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 f(0, z0) -> 0 g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(S(z0), 0) -> 0 g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) Tuples: F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1))) F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) S tuples: F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1))) F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) K tuples:none Defined Rule Symbols: f_2, g_2, h_2 Defined Pair Symbols: F_2, G_2, H_2 Compound Symbols: c_2, c1_2, c4_2, c5_2, c8_1, c10_1 ---------------------------------------- (39) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: f(0, z0) -> 0 g(S(z0), 0) -> 0 ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1))) F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) S tuples: F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1))) F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) K tuples:none Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: F_2, G_2, H_2 Compound Symbols: c_2, c1_2, c4_2, c5_2, c8_1, c10_1 ---------------------------------------- (41) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. H(0, S(z0)) -> c8(H(0, z0)) We considered the (Usable) Rules: f(S(z0), 0) -> 0 g(0, z0) -> 0 f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) And the Tuples: F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1))) F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(F(x_1, x_2)) = 0 POL(G(x_1, x_2)) = 0 POL(H(x_1, x_2)) = x_2 POL(S(x_1)) = [1] + x_1 POL(c(x_1, x_2)) = x_1 + x_2 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(f(x_1, x_2)) = 0 POL(g(x_1, x_2)) = 0 POL(h(x_1, x_2)) = 0 ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1))) F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) S tuples: F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1))) F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) H(S(z0), z1) -> c10(H(z0, z1)) K tuples: H(0, S(z0)) -> c8(H(0, z0)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: F_2, G_2, H_2 Compound Symbols: c_2, c1_2, c4_2, c5_2, c8_1, c10_1 ---------------------------------------- (43) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. H(S(z0), z1) -> c10(H(z0, z1)) We considered the (Usable) Rules: g(0, z0) -> 0 f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) And the Tuples: F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1))) F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(F(x_1, x_2)) = 0 POL(G(x_1, x_2)) = 0 POL(H(x_1, x_2)) = x_1 POL(S(x_1)) = [1] + x_1 POL(c(x_1, x_2)) = x_1 + x_2 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(f(x_1, x_2)) = 0 POL(g(x_1, x_2)) = 0 POL(h(x_1, x_2)) = 0 ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1))) F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) S tuples: F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1))) F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: F_2, G_2, H_2 Compound Symbols: c_2, c1_2, c4_2, c5_2, c8_1, c10_1 ---------------------------------------- (45) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1))) by F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)), G(0, S(x1))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)), G(0, S(x1))) S tuples: F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)), G(0, S(x1))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: F_2, G_2, H_2 Compound Symbols: c1_2, c4_2, c5_2, c8_1, c10_1, c_2 ---------------------------------------- (47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) S tuples: F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: F_2, G_2, H_2 Compound Symbols: c1_2, c4_2, c5_2, c8_1, c10_1, c_2, c_1 ---------------------------------------- (49) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) We considered the (Usable) Rules:none And the Tuples: F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(F(x_1, x_2)) = [1] + x_1 POL(G(x_1, x_2)) = [1] + x_2 POL(H(x_1, x_2)) = 0 POL(S(x_1)) = [1] POL(c(x_1)) = x_1 POL(c(x_1, x_2)) = x_1 + x_2 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(f(x_1, x_2)) = 0 POL(g(x_1, x_2)) = 0 POL(h(x_1, x_2)) = 0 ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) S tuples: F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: F_2, G_2, H_2 Compound Symbols: c1_2, c4_2, c5_2, c8_1, c10_1, c_2, c_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) by F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0), F(S(S(S(x0))), 0)) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0), F(S(S(S(x0))), 0)) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) S tuples: G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0), F(S(S(S(x0))), 0)) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: G_2, H_2, F_2 Compound Symbols: c4_2, c5_2, c8_1, c10_1, c_2, c_1, c1_2, c1_1 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) S tuples: G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: G_2, H_2, F_2 Compound Symbols: c4_2, c5_2, c8_1, c10_1, c_2, c_1, c1_2, c1_1 ---------------------------------------- (55) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) We considered the (Usable) Rules:none And the Tuples: G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(F(x_1, x_2)) = [1] + x_1 POL(G(x_1, x_2)) = [1] + x_2 POL(H(x_1, x_2)) = 0 POL(S(x_1)) = 0 POL(c(x_1)) = x_1 POL(c(x_1, x_2)) = x_1 + x_2 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(f(x_1, x_2)) = 0 POL(g(x_1, x_2)) = 0 POL(h(x_1, x_2)) = 0 ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) S tuples: G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: G_2, H_2, F_2 Compound Symbols: c4_2, c5_2, c8_1, c10_1, c_2, c_1, c1_2, c1_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) by G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) S tuples: G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: G_2, H_2, F_2 Compound Symbols: c5_2, c8_1, c10_1, c_2, c_1, c1_2, c1_1, c4_2 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) by G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0), G(0, S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0), G(0, S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) S tuples: F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0), G(0, S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c_2, c_1, c1_2, c1_1, c4_2, c5_2, c5_1 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) S tuples: F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c_2, c_1, c1_2, c1_1, c4_2, c5_2, c5_1 ---------------------------------------- (63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) We considered the (Usable) Rules:none And the Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(F(x_1, x_2)) = x_1 POL(G(x_1, x_2)) = [1] POL(H(x_1, x_2)) = 0 POL(S(x_1)) = [1] POL(c(x_1)) = x_1 POL(c(x_1, x_2)) = x_1 + x_2 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(f(x_1, x_2)) = 0 POL(g(x_1, x_2)) = 0 POL(h(x_1, x_2)) = 0 ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) S tuples: F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c_2, c_1, c1_2, c1_1, c4_2, c5_2, c5_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) by F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), G(0, S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), G(0, S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) S tuples: F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), G(0, S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c_2, c_1, c1_2, c1_1, c4_2, c5_2, c5_1 ---------------------------------------- (67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) S tuples: F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c_2, c_1, c1_2, c1_1, c4_2, c5_2, c5_1 ---------------------------------------- (69) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) We considered the (Usable) Rules:none And the Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(F(x_1, x_2)) = [1] + x_1 POL(G(x_1, x_2)) = [1] + x_2 POL(H(x_1, x_2)) = 0 POL(S(x_1)) = 0 POL(c(x_1)) = x_1 POL(c(x_1, x_2)) = x_1 + x_2 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(f(x_1, x_2)) = 0 POL(g(x_1, x_2)) = 0 POL(h(x_1, x_2)) = 0 ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) S tuples: F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c_2, c_1, c1_2, c1_1, c4_2, c5_2, c5_1 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) by F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(0), S(0)) -> c(H(0, 0), G(0, S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(0), S(0)) -> c(H(0, 0), G(0, S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) S tuples: F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(0), S(0)) -> c(H(0, 0), G(0, S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c_2, c_1, c1_2, c1_1, c4_2, c5_2, c5_1 ---------------------------------------- (73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(S(0), S(0)) -> c(H(0, 0), G(0, S(0))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) S tuples: F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c_2, c_1, c1_2, c1_1, c4_2, c5_2, c5_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) by F(S(S(x0)), S(S(z1))) -> c(H(h(f(S(x0), S(S(z1))), g(x0, S(S(S(S(z1)))))), h(g(S(S(S(x0))), S(z1)), f(S(S(S(S(S(S(x0)))))), z1))), G(S(x0), S(S(z1)))) F(S(S(x0)), S(0)) -> c(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G(S(x0), S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) S tuples: F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c_1, c1_2, c1_1, c4_2, c5_2, c5_1, c_2 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) by F(S(0), S(S(z1))) -> c(H(0, h(g(S(S(0)), S(z1)), f(S(S(S(S(S(0))))), z1)))) F(S(0), S(0)) -> c(H(0, 0)) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(0)) -> c(H(0, 0)) S tuples: F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_2, c1_1, c4_2, c5_2, c5_1, c_2, c_1 ---------------------------------------- (79) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(S(0), S(0)) -> c(H(0, 0)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) S tuples: F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_2, c1_1, c4_2, c5_2, c5_1, c_2, c_1 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) by F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) S tuples: F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_2, c1_1, c4_2, c5_2, c5_1, c_2, c_1 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) by F(S(S(x0)), S(S(z1))) -> c1(H(h(f(S(x0), S(S(z1))), g(x0, S(S(S(S(z1)))))), h(g(S(S(S(x0))), S(z1)), f(S(S(S(S(S(S(x0)))))), z1))), F(S(S(S(S(x0)))), S(z1))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), F(S(S(S(S(x0)))), 0)) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), F(S(S(S(S(x0)))), 0)) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) S tuples: F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), F(S(S(S(S(x0)))), 0)) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_2, c1_1, c4_2, c5_2, c5_1, c_2, c_1 ---------------------------------------- (85) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) S tuples: F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_2, c1_1, c4_2, c5_2, c5_1, c_2, c_1 ---------------------------------------- (87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) We considered the (Usable) Rules:none And the Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(F(x_1, x_2)) = [1] + x_1 POL(G(x_1, x_2)) = [1] + x_2 POL(H(x_1, x_2)) = 0 POL(S(x_1)) = 0 POL(c(x_1)) = x_1 POL(c(x_1, x_2)) = x_1 + x_2 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(f(x_1, x_2)) = 0 POL(g(x_1, x_2)) = 0 POL(h(x_1, x_2)) = 0 ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) S tuples: F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_2, c1_1, c4_2, c5_2, c5_1, c_2, c_1 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) by F(S(0), S(S(z1))) -> c1(H(0, h(g(S(S(0)), S(z1)), f(S(S(S(S(S(0))))), z1))), F(S(S(S(0))), S(z1))) F(S(0), S(0)) -> c1(H(0, 0), F(S(S(S(0))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(0)) -> c1(H(0, 0), F(S(S(S(0))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(0)) -> c1(H(0, 0), F(S(S(S(0))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c4_2, c5_2, c5_1, c_2, c_1, c1_2 ---------------------------------------- (91) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(S(0), S(0)) -> c1(H(0, 0), F(S(S(S(0))), 0)) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c4_2, c5_2, c5_1, c_2, c_1, c1_2 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) by F(S(S(z0)), S(0)) -> c1(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0)) F(S(0), S(0)) -> c1(H(0, 0)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) F(S(0), S(0)) -> c1(H(0, 0)) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c4_2, c5_2, c5_1, c_2, c_1, c1_2 ---------------------------------------- (95) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(S(0), S(0)) -> c1(H(0, 0)) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c4_2, c5_2, c5_1, c_2, c_1, c1_2 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) by G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c4_2, c5_2, c5_1, c_2, c_1, c1_2, c4_1 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) by G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c4_2, c5_2, c5_1, c_2, c_1, c1_2, c4_1 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) by G(S(S(z0)), S(x1)) -> c4(H(h(g(S(z0), S(x1)), f(S(S(S(S(z0)))), x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(0), S(x1)) -> c4(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F(S(0), S(x1))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_2, c5_1, c_2, c_1, c1_2, c4_2, c4_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) by G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_2, c5_1, c_2, c_1, c1_2, c4_2, c4_1 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) by G(S(S(z0)), S(x1)) -> c5(H(h(g(S(z0), S(x1)), f(S(S(S(S(z0)))), x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), G(0, S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), G(0, S(S(S(x1))))) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), G(0, S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), G(0, S(S(S(x1))))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), G(0, S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), G(0, S(S(S(x1))))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (107) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (109) 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(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) We considered the (Usable) Rules:none And the Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(F(x_1, x_2)) = [1] + x_1 POL(G(x_1, x_2)) = [1] + x_2 POL(H(x_1, x_2)) = 0 POL(S(x_1)) = 0 POL(c(x_1)) = x_1 POL(c(x_1, x_2)) = x_1 + x_2 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(f(x_1, x_2)) = 0 POL(g(x_1, x_2)) = 0 POL(h(x_1, x_2)) = 0 ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) by G(S(0), S(z1)) -> c5(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (113) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) by F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) by F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (117) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) by F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (119) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) by F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) by F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (123) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) by F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (125) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(S(x0)), S(S(z1))) -> c(H(h(f(S(x0), S(S(z1))), g(x0, S(S(S(S(z1)))))), h(g(S(S(S(x0))), S(z1)), f(S(S(S(S(S(S(x0)))))), z1))), G(S(x0), S(S(z1)))) by F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (127) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(S(x0)), S(0)) -> c(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G(S(x0), S(0))) by F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) by F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (131) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) by F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (133) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(0), S(S(z1))) -> c(H(0, h(g(S(S(0)), S(z1)), f(S(S(S(S(S(0))))), z1)))) by F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (135) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) by F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (137) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) by F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (139) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) by F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (141) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) by F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (143) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) by F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (145) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(S(x0)), S(S(z1))) -> c1(H(h(f(S(x0), S(S(z1))), g(x0, S(S(S(S(z1)))))), h(g(S(S(S(x0))), S(z1)), f(S(S(S(S(S(S(x0)))))), z1))), F(S(S(S(S(x0)))), S(z1))) by F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (147) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) by F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1)) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1)) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1)) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (149) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) by F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0)) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1)) F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0)) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1)) F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0)) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (151) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) by F(S(S(z0)), S(0)) -> c1(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0)) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1)) F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0)) F(S(S(z0)), S(0)) -> c1(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0)) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1)) F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0)) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (153) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(0), S(S(z1))) -> c1(H(0, h(g(S(S(0)), S(z1)), f(S(S(S(S(S(0))))), z1))), F(S(S(S(0))), S(z1))) by F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1)) F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0)) F(S(S(z0)), S(0)) -> c1(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0)) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1)) F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0)) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (155) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(S(S(z0)), S(0)) -> c1(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0)) by F(S(S(z0)), S(0)) -> c1(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0)) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1)) F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0)) F(S(S(z0)), S(0)) -> c1(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0)) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1)) F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0)) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (157) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) by G(S(S(S(z0))), S(z1)) -> c4(H(f(S(S(S(z0))), S(z1)), h(f(S(S(z0)), S(S(S(z1)))), h(h(g(z0, S(S(S(S(S(z1)))))), f(S(S(S(z0))), S(S(S(S(z1)))))), g(z0, S(S(S(S(S(S(S(z1))))))))))), F(S(S(S(z0))), S(z1))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1)) F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0)) F(S(S(z0)), S(0)) -> c1(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0)) G(S(S(S(z0))), S(z1)) -> c4(H(f(S(S(S(z0))), S(z1)), h(f(S(S(z0)), S(S(S(z1)))), h(h(g(z0, S(S(S(S(S(z1)))))), f(S(S(S(z0))), S(S(S(S(z1)))))), g(z0, S(S(S(S(S(S(S(z1))))))))))), F(S(S(S(z0))), S(z1))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1)) F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0)) G(S(S(S(z0))), S(z1)) -> c4(H(f(S(S(S(z0))), S(z1)), h(f(S(S(z0)), S(S(S(z1)))), h(h(g(z0, S(S(S(S(S(z1)))))), f(S(S(S(z0))), S(S(S(S(z1)))))), g(z0, S(S(S(S(S(S(S(z1))))))))))), F(S(S(S(z0))), S(z1))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (159) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) by G(S(S(0)), S(z0)) -> c4(H(f(S(S(0)), S(z0)), h(h(g(0, S(S(S(z0)))), f(S(S(S(0))), S(S(z0)))), 0)), F(S(S(0)), S(z0))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1)) F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0)) F(S(S(z0)), S(0)) -> c1(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0)) G(S(S(S(z0))), S(z1)) -> c4(H(f(S(S(S(z0))), S(z1)), h(f(S(S(z0)), S(S(S(z1)))), h(h(g(z0, S(S(S(S(S(z1)))))), f(S(S(S(z0))), S(S(S(S(z1)))))), g(z0, S(S(S(S(S(S(S(z1))))))))))), F(S(S(S(z0))), S(z1))) G(S(S(0)), S(z0)) -> c4(H(f(S(S(0)), S(z0)), h(h(g(0, S(S(S(z0)))), f(S(S(S(0))), S(S(z0)))), 0)), F(S(S(0)), S(z0))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1)) F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0)) G(S(S(S(z0))), S(z1)) -> c4(H(f(S(S(S(z0))), S(z1)), h(f(S(S(z0)), S(S(S(z1)))), h(h(g(z0, S(S(S(S(S(z1)))))), f(S(S(S(z0))), S(S(S(S(z1)))))), g(z0, S(S(S(S(S(S(S(z1))))))))))), F(S(S(S(z0))), S(z1))) G(S(S(0)), S(z0)) -> c4(H(f(S(S(0)), S(z0)), h(h(g(0, S(S(S(z0)))), f(S(S(S(0))), S(S(z0)))), 0)), F(S(S(0)), S(z0))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) by G(S(S(z0)), S(z1)) -> c4(H(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), f(S(S(S(S(S(z0))))), S(z1)))), g(z0, S(S(S(S(S(z1)))))))), F(S(S(z0)), S(z1))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1)) F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0)) F(S(S(z0)), S(0)) -> c1(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0)) G(S(S(S(z0))), S(z1)) -> c4(H(f(S(S(S(z0))), S(z1)), h(f(S(S(z0)), S(S(S(z1)))), h(h(g(z0, S(S(S(S(S(z1)))))), f(S(S(S(z0))), S(S(S(S(z1)))))), g(z0, S(S(S(S(S(S(S(z1))))))))))), F(S(S(S(z0))), S(z1))) G(S(S(0)), S(z0)) -> c4(H(f(S(S(0)), S(z0)), h(h(g(0, S(S(S(z0)))), f(S(S(S(0))), S(S(z0)))), 0)), F(S(S(0)), S(z0))) G(S(S(z0)), S(z1)) -> c4(H(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), f(S(S(S(S(S(z0))))), S(z1)))), g(z0, S(S(S(S(S(z1)))))))), F(S(S(z0)), S(z1))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1)) F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0)) G(S(S(S(z0))), S(z1)) -> c4(H(f(S(S(S(z0))), S(z1)), h(f(S(S(z0)), S(S(S(z1)))), h(h(g(z0, S(S(S(S(S(z1)))))), f(S(S(S(z0))), S(S(S(S(z1)))))), g(z0, S(S(S(S(S(S(S(z1))))))))))), F(S(S(S(z0))), S(z1))) G(S(S(0)), S(z0)) -> c4(H(f(S(S(0)), S(z0)), h(h(g(0, S(S(S(z0)))), f(S(S(S(0))), S(S(z0)))), 0)), F(S(S(0)), S(z0))) G(S(S(z0)), S(z1)) -> c4(H(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), f(S(S(S(S(S(z0))))), S(z1)))), g(z0, S(S(S(S(S(z1)))))))), F(S(S(z0)), S(z1))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2 ---------------------------------------- (163) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) by G(S(S(z0)), S(z1)) -> c4(H(h(g(S(z0), S(z1)), f(S(S(S(S(z0)))), z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), F(S(S(z0)), S(z1))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) g(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) f(S(z0), 0) -> 0 Tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1)) F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0)) F(S(S(z0)), S(0)) -> c1(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0)) G(S(S(S(z0))), S(z1)) -> c4(H(f(S(S(S(z0))), S(z1)), h(f(S(S(z0)), S(S(S(z1)))), h(h(g(z0, S(S(S(S(S(z1)))))), f(S(S(S(z0))), S(S(S(S(z1)))))), g(z0, S(S(S(S(S(S(S(z1))))))))))), F(S(S(S(z0))), S(z1))) G(S(S(0)), S(z0)) -> c4(H(f(S(S(0)), S(z0)), h(h(g(0, S(S(S(z0)))), f(S(S(S(0))), S(S(z0)))), 0)), F(S(S(0)), S(z0))) G(S(S(z0)), S(z1)) -> c4(H(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), f(S(S(S(S(S(z0))))), S(z1)))), g(z0, S(S(S(S(S(z1)))))))), F(S(S(z0)), S(z1))) G(S(S(z0)), S(z1)) -> c4(H(h(g(S(z0), S(z1)), f(S(S(S(S(z0)))), z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), F(S(S(z0)), S(z1))) S tuples: F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1)) G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1))))) F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1)))) F(S(x0), S(0)) -> c(G(x0, S(0))) F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1))) F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1)) F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0)) G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1))) G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) G(S(0), S(x0)) -> c4(F(S(0), S(x0))) G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1)))) G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0))) G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1))) G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1))) G(S(x0), S(x1)) -> c4(F(S(x0), S(x1))) G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1))))) G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1))))) G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1))))) G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1))))) G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1)))))) G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0))))) G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1))))) F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1))))) F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0)))) F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1)))) F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1)))) F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1))) F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0))) F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1)))) F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0))) F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1))) F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1))) F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0))) F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1)) F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0)) G(S(S(S(z0))), S(z1)) -> c4(H(f(S(S(S(z0))), S(z1)), h(f(S(S(z0)), S(S(S(z1)))), h(h(g(z0, S(S(S(S(S(z1)))))), f(S(S(S(z0))), S(S(S(S(z1)))))), g(z0, S(S(S(S(S(S(S(z1))))))))))), F(S(S(S(z0))), S(z1))) G(S(S(0)), S(z0)) -> c4(H(f(S(S(0)), S(z0)), h(h(g(0, S(S(S(z0)))), f(S(S(S(0))), S(S(z0)))), 0)), F(S(S(0)), S(z0))) G(S(S(z0)), S(z1)) -> c4(H(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), f(S(S(S(S(S(z0))))), S(z1)))), g(z0, S(S(S(S(S(z1)))))))), F(S(S(z0)), S(z1))) G(S(S(z0)), S(z1)) -> c4(H(h(g(S(z0), S(z1)), f(S(S(S(S(z0)))), z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), F(S(S(z0)), S(z1))) K tuples: H(0, S(z0)) -> c8(H(0, z0)) H(S(z0), z1) -> c10(H(z0, z1)) G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0)) G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1)))))) Defined Rule Symbols: g_2, h_2, f_2 Defined Pair Symbols: H_2, F_2, G_2 Compound Symbols: c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2