KILLED proof of input_xmAs9GKB5J.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) CompletionProof [UPPER BOUND(ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxTypedWeightedCompleteTrs (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (22) CdtProblem (23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CdtProblem (25) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 4 ms] (38) CdtProblem (39) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 22 ms] (48) CdtProblem (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 1 ms] (66) CdtProblem (67) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 9 ms] (102) CdtProblem (103) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 9 ms] (104) CdtProblem (105) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 10 ms] (106) CdtProblem (107) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 7 ms] (116) CdtProblem (117) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 18 ms] (118) CdtProblem (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 14 ms] (122) CdtProblem (123) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 64 ms] (136) CdtProblem (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 6 ms] (140) CdtProblem (141) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 91 ms] (142) 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(x, 0, 0) -> s(x) f(0, y, 0) -> s(y) f(0, 0, z) -> s(z) f(s(0), y, z) -> f(0, s(y), s(z)) f(s(x), s(y), 0) -> f(x, y, s(0)) f(s(x), 0, s(z)) -> f(x, s(0), z) f(0, s(0), s(0)) -> s(s(0)) f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) f(0, s(s(y)), s(0)) -> f(0, y, s(0)) f(0, s(0), s(s(z))) -> f(0, s(0), z) f(0, s(s(y)), s(s(z))) -> f(0, y, f(0, s(s(y)), s(z))) 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(x, 0', 0') -> s(x) f(0', y, 0') -> s(y) f(0', 0', z) -> s(z) f(s(0'), y, z) -> f(0', s(y), s(z)) f(s(x), s(y), 0') -> f(x, y, s(0')) f(s(x), 0', s(z)) -> f(x, s(0'), z) f(0', s(0'), s(0')) -> s(s(0')) f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) f(0', s(s(y)), s(0')) -> f(0', y, s(0')) f(0', s(0'), s(s(z))) -> f(0', s(0'), z) f(0', s(s(y)), s(s(z))) -> f(0', y, f(0', s(s(y)), s(z))) 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(x, 0, 0) -> s(x) f(0, y, 0) -> s(y) f(0, 0, z) -> s(z) f(s(0), y, z) -> f(0, s(y), s(z)) f(s(x), s(y), 0) -> f(x, y, s(0)) f(s(x), 0, s(z)) -> f(x, s(0), z) f(0, s(0), s(0)) -> s(s(0)) f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) f(0, s(s(y)), s(0)) -> f(0, y, s(0)) f(0, s(0), s(s(z))) -> f(0, s(0), z) f(0, s(s(y)), s(s(z))) -> f(0, y, f(0, s(s(y)), s(z))) 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(x, 0, 0) -> s(x) [1] f(0, y, 0) -> s(y) [1] f(0, 0, z) -> s(z) [1] f(s(0), y, z) -> f(0, s(y), s(z)) [1] f(s(x), s(y), 0) -> f(x, y, s(0)) [1] f(s(x), 0, s(z)) -> f(x, s(0), z) [1] f(0, s(0), s(0)) -> s(s(0)) [1] f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) [1] f(0, s(s(y)), s(0)) -> f(0, y, s(0)) [1] f(0, s(0), s(s(z))) -> f(0, s(0), z) [1] f(0, s(s(y)), s(s(z))) -> f(0, y, f(0, s(s(y)), s(z))) [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(x, 0, 0) -> s(x) [1] f(0, y, 0) -> s(y) [1] f(0, 0, z) -> s(z) [1] f(s(0), y, z) -> f(0, s(y), s(z)) [1] f(s(x), s(y), 0) -> f(x, y, s(0)) [1] f(s(x), 0, s(z)) -> f(x, s(0), z) [1] f(0, s(0), s(0)) -> s(s(0)) [1] f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) [1] f(0, s(s(y)), s(0)) -> f(0, y, s(0)) [1] f(0, s(0), s(s(z))) -> f(0, s(0), z) [1] f(0, s(s(y)), s(s(z))) -> f(0, y, f(0, s(s(y)), s(z))) [1] The TRS has the following type information: f :: 0:s -> 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (9) 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 ---------------------------------------- (10) 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(x, 0, 0) -> s(x) [1] f(0, y, 0) -> s(y) [1] f(0, 0, z) -> s(z) [1] f(s(0), y, z) -> f(0, s(y), s(z)) [1] f(s(x), s(y), 0) -> f(x, y, s(0)) [1] f(s(x), 0, s(z)) -> f(x, s(0), z) [1] f(0, s(0), s(0)) -> s(s(0)) [1] f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) [1] f(0, s(s(y)), s(0)) -> f(0, y, s(0)) [1] f(0, s(0), s(s(z))) -> f(0, s(0), z) [1] f(0, s(s(y)), s(s(z))) -> f(0, y, f(0, s(s(y)), s(z))) [1] The TRS has the following type information: f :: 0:s -> 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (11) 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 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: f(z', z'', z1) -{ 1 }-> f(x, y, f(1 + x, 1 + y, z)) :|: z' = 1 + x, z >= 0, x >= 0, y >= 0, z'' = 1 + y, z1 = 1 + z f(z', z'', z1) -{ 1 }-> f(x, y, 1 + 0) :|: z' = 1 + x, z1 = 0, x >= 0, y >= 0, z'' = 1 + y f(z', z'', z1) -{ 1 }-> f(x, 1 + 0, z) :|: z'' = 0, z' = 1 + x, z >= 0, x >= 0, z1 = 1 + z f(z', z'', z1) -{ 1 }-> f(0, y, f(0, 1 + (1 + y), 1 + z)) :|: z >= 0, y >= 0, z'' = 1 + (1 + y), z1 = 1 + (1 + z), z' = 0 f(z', z'', z1) -{ 1 }-> f(0, y, 1 + 0) :|: z1 = 1 + 0, y >= 0, z'' = 1 + (1 + y), z' = 0 f(z', z'', z1) -{ 1 }-> f(0, 1 + y, 1 + z) :|: z1 = z, z >= 0, z'' = y, y >= 0, z' = 1 + 0 f(z', z'', z1) -{ 1 }-> f(0, 1 + 0, z) :|: z >= 0, z1 = 1 + (1 + z), z'' = 1 + 0, z' = 0 f(z', z'', z1) -{ 1 }-> 1 + x :|: z'' = 0, z1 = 0, z' = x, x >= 0 f(z', z'', z1) -{ 1 }-> 1 + y :|: z1 = 0, z'' = y, y >= 0, z' = 0 f(z', z'', z1) -{ 1 }-> 1 + z :|: z'' = 0, z1 = z, z >= 0, z' = 0 f(z', z'', z1) -{ 1 }-> 1 + (1 + 0) :|: z1 = 1 + 0, z'' = 1 + 0, z' = 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (13) 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: f_3 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (14) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(x, 0, 0) -> s(x) [1] f(0, y, 0) -> s(y) [1] f(0, 0, z) -> s(z) [1] f(s(0), y, z) -> f(0, s(y), s(z)) [1] f(s(x), s(y), 0) -> f(x, y, s(0)) [1] f(s(x), 0, s(z)) -> f(x, s(0), z) [1] f(0, s(0), s(0)) -> s(s(0)) [1] f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) [1] f(0, s(s(y)), s(0)) -> f(0, y, s(0)) [1] f(0, s(0), s(s(z))) -> f(0, s(0), z) [1] f(0, s(s(y)), s(s(z))) -> f(0, y, f(0, s(s(y)), s(z))) [1] The TRS has the following type information: f :: 0:s -> 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (15) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (16) 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(x, 0, 0) -> s(x) [1] f(0, y, 0) -> s(y) [1] f(0, 0, z) -> s(z) [1] f(s(0), y, z) -> f(0, s(y), s(z)) [1] f(s(x), s(y), 0) -> f(x, y, s(0)) [1] f(s(x), 0, s(z)) -> f(x, s(0), z) [1] f(0, s(0), s(0)) -> s(s(0)) [1] f(s(0), s(y), s(z)) -> f(0, y, f(0, s(s(y)), s(z))) [2] f(s(x), s(y), s(0)) -> f(x, y, f(x, y, s(0))) [2] f(s(x), s(y), s(s(z'))) -> f(x, y, f(x, y, f(s(x), s(y), z'))) [2] f(0, s(s(y)), s(0)) -> f(0, y, s(0)) [1] f(0, s(0), s(s(z))) -> f(0, s(0), z) [1] f(0, s(s(y)), s(s(0))) -> f(0, y, f(0, y, s(0))) [2] f(0, s(s(y)), s(s(s(z'')))) -> f(0, y, f(0, y, f(0, s(s(y)), s(z'')))) [2] The TRS has the following type information: f :: 0:s -> 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (17) 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 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z1, z2, z3) -{ 2 }-> f(x, y, f(x, y, f(1 + x, 1 + y, z'))) :|: z2 = 1 + y, x >= 0, y >= 0, z' >= 0, z1 = 1 + x, z3 = 1 + (1 + z') f(z1, z2, z3) -{ 2 }-> f(x, y, f(x, y, 1 + 0)) :|: z2 = 1 + y, x >= 0, y >= 0, z1 = 1 + x, z3 = 1 + 0 f(z1, z2, z3) -{ 1 }-> f(x, y, 1 + 0) :|: z2 = 1 + y, z3 = 0, x >= 0, y >= 0, z1 = 1 + x f(z1, z2, z3) -{ 1 }-> f(x, 1 + 0, z) :|: z >= 0, z2 = 0, x >= 0, z3 = 1 + z, z1 = 1 + x f(z1, z2, z3) -{ 2 }-> f(0, y, f(0, y, f(0, 1 + (1 + y), 1 + z''))) :|: z2 = 1 + (1 + y), z1 = 0, z'' >= 0, y >= 0, z3 = 1 + (1 + (1 + z'')) f(z1, z2, z3) -{ 2 }-> f(0, y, f(0, y, 1 + 0)) :|: z2 = 1 + (1 + y), z1 = 0, z3 = 1 + (1 + 0), y >= 0 f(z1, z2, z3) -{ 2 }-> f(0, y, f(0, 1 + (1 + y), 1 + z)) :|: z >= 0, z2 = 1 + y, z1 = 1 + 0, y >= 0, z3 = 1 + z f(z1, z2, z3) -{ 1 }-> f(0, y, 1 + 0) :|: z2 = 1 + (1 + y), z1 = 0, y >= 0, z3 = 1 + 0 f(z1, z2, z3) -{ 1 }-> f(0, 1 + y, 1 + z) :|: z2 = y, z >= 0, z1 = 1 + 0, z3 = z, y >= 0 f(z1, z2, z3) -{ 1 }-> f(0, 1 + 0, z) :|: z1 = 0, z2 = 1 + 0, z >= 0, z3 = 1 + (1 + z) f(z1, z2, z3) -{ 1 }-> 1 + x :|: z2 = 0, z3 = 0, x >= 0, z1 = x f(z1, z2, z3) -{ 1 }-> 1 + y :|: z1 = 0, z2 = y, z3 = 0, y >= 0 f(z1, z2, z3) -{ 1 }-> 1 + z :|: z1 = 0, z >= 0, z2 = 0, z3 = z f(z1, z2, z3) -{ 1 }-> 1 + (1 + 0) :|: z1 = 0, z2 = 1 + 0, z3 = 1 + 0 ---------------------------------------- (19) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z1, z2, z3) -{ 2 }-> f(0, z2 - 2, f(0, z2 - 2, f(0, 1 + (1 + (z2 - 2)), 1 + (z3 - 3)))) :|: z1 = 0, z3 - 3 >= 0, z2 - 2 >= 0 f(z1, z2, z3) -{ 2 }-> f(0, z2 - 2, f(0, z2 - 2, 1 + 0)) :|: z1 = 0, z3 = 1 + (1 + 0), z2 - 2 >= 0 f(z1, z2, z3) -{ 1 }-> f(0, z2 - 2, 1 + 0) :|: z1 = 0, z2 - 2 >= 0, z3 = 1 + 0 f(z1, z2, z3) -{ 2 }-> f(0, z2 - 1, f(0, 1 + (1 + (z2 - 1)), 1 + (z3 - 1))) :|: z3 - 1 >= 0, z1 = 1 + 0, z2 - 1 >= 0 f(z1, z2, z3) -{ 1 }-> f(0, 1 + z2, 1 + z3) :|: z3 >= 0, z1 = 1 + 0, z2 >= 0 f(z1, z2, z3) -{ 1 }-> f(0, 1 + 0, z3 - 2) :|: z1 = 0, z2 = 1 + 0, z3 - 2 >= 0 f(z1, z2, z3) -{ 2 }-> f(z1 - 1, z2 - 1, f(z1 - 1, z2 - 1, f(1 + (z1 - 1), 1 + (z2 - 1), z3 - 2))) :|: z1 - 1 >= 0, z2 - 1 >= 0, z3 - 2 >= 0 f(z1, z2, z3) -{ 2 }-> f(z1 - 1, z2 - 1, f(z1 - 1, z2 - 1, 1 + 0)) :|: z1 - 1 >= 0, z2 - 1 >= 0, z3 = 1 + 0 f(z1, z2, z3) -{ 1 }-> f(z1 - 1, z2 - 1, 1 + 0) :|: z3 = 0, z1 - 1 >= 0, z2 - 1 >= 0 f(z1, z2, z3) -{ 1 }-> f(z1 - 1, 1 + 0, z3 - 1) :|: z3 - 1 >= 0, z2 = 0, z1 - 1 >= 0 f(z1, z2, z3) -{ 1 }-> 1 + z1 :|: z2 = 0, z3 = 0, z1 >= 0 f(z1, z2, z3) -{ 1 }-> 1 + z2 :|: z1 = 0, z3 = 0, z2 >= 0 f(z1, z2, z3) -{ 1 }-> 1 + z3 :|: z1 = 0, z3 >= 0, z2 = 0 f(z1, z2, z3) -{ 1 }-> 1 + (1 + 0) :|: z1 = 0, z2 = 1 + 0, z3 = 1 + 0 ---------------------------------------- (21) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(z0, 0, 0) -> c F(0, z0, 0) -> c1 F(0, 0, z0) -> c2 F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(0), s(0)) -> c6 F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) S tuples: F(z0, 0, 0) -> c F(0, z0, 0) -> c1 F(0, 0, z0) -> c2 F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(0), s(0)) -> c6 F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c, c1, c2, c3_1, c4_1, c5_1, c6, c7_2, c8_1, c9_1, c10_2 ---------------------------------------- (23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: F(0, s(0), s(0)) -> c6 F(0, 0, z0) -> c2 F(0, z0, 0) -> c1 F(z0, 0, 0) -> c ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) S tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c3_1, c4_1, c5_1, c7_2, c8_1, c9_1, c10_2 ---------------------------------------- (25) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) by F(0, s(s(z0)), s(s(0))) -> c10(F(0, z0, f(0, z0, s(0))), F(0, s(s(z0)), s(0))) F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(0))) -> c10(F(0, z0, f(0, z0, s(0))), F(0, s(s(z0)), s(0))) F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) S tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(0))) -> c10(F(0, z0, f(0, z0, s(0))), F(0, s(s(z0)), s(0))) F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c3_1, c4_1, c5_1, c7_2, c8_1, c9_1, c10_2 ---------------------------------------- (27) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(0, s(s(z0)), s(s(0))) -> c10(F(0, z0, f(0, z0, s(0))), F(0, s(s(z0)), s(0))) by F(0, s(s(0)), s(s(0))) -> c10(F(0, 0, s(s(0))), F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c10(F(0, s(0), s(s(0))), F(0, s(s(s(0))), s(0))) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) F(0, s(s(0)), s(s(0))) -> c10(F(0, 0, s(s(0))), F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c10(F(0, s(0), s(s(0))), F(0, s(s(s(0))), s(0))) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) S tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) F(0, s(s(0)), s(s(0))) -> c10(F(0, 0, s(s(0))), F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c10(F(0, s(0), s(s(0))), F(0, s(s(s(0))), s(0))) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c3_1, c4_1, c5_1, c7_2, c8_1, c9_1, c10_2, c10_1 ---------------------------------------- (29) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) F(0, s(s(s(0))), s(s(0))) -> c10(F(0, s(0), s(s(0))), F(0, s(s(s(0))), s(0))) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) S tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) F(0, s(s(s(0))), s(s(0))) -> c10(F(0, s(0), s(s(0))), F(0, s(s(s(0))), s(0))) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c3_1, c4_1, c5_1, c7_2, c8_1, c9_1, c10_2, c10_1 ---------------------------------------- (31) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) S tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c3_1, c4_1, c5_1, c7_2, c8_1, c9_1, c10_2, c10_1, c_1 ---------------------------------------- (33) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) by F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, 0, s(f(0, s(s(0)), s(x1)))), F(0, s(s(0)), s(s(x1)))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, 0, s(f(0, s(s(0)), s(x1)))), F(0, s(s(0)), s(s(x1)))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) S tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, 0, s(f(0, s(s(0)), s(x1)))), F(0, s(s(0)), s(s(x1)))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c3_1, c4_1, c5_1, c7_2, c8_1, c9_1, c10_2, c10_1, c_1 ---------------------------------------- (35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) S tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c3_1, c4_1, c5_1, c7_2, c8_1, c9_1, c10_2, c10_1, c_1 ---------------------------------------- (37) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) by F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) S tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c4_1, c5_1, c7_2, c8_1, c9_1, c10_2, c10_1, c_1, c3_1 ---------------------------------------- (39) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) by F(s(0), s(x1), s(z1)) -> c7(F(0, x1, f(0, s(s(x1)), s(z1))), F(s(0), s(x1), z1)) F(s(z0), s(z1), s(0)) -> c7(F(z0, z1, f(z0, z1, s(0))), F(s(z0), s(z1), 0)) F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(x1), s(z1)) -> c7(F(0, x1, f(0, s(s(x1)), s(z1))), F(s(0), s(x1), z1)) F(s(z0), s(z1), s(0)) -> c7(F(z0, z1, f(z0, z1, s(0))), F(s(z0), s(z1), 0)) F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) S tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(x1), s(z1)) -> c7(F(0, x1, f(0, s(s(x1)), s(z1))), F(s(0), s(x1), z1)) F(s(z0), s(z1), s(0)) -> c7(F(z0, z1, f(z0, z1, s(0))), F(s(z0), s(z1), 0)) F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c4_1, c5_1, c8_1, c9_1, c10_2, c10_1, c_1, c3_1, c7_2 ---------------------------------------- (41) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(x1), s(z1)) -> c7(F(0, x1, f(0, s(s(x1)), s(z1))), F(s(0), s(x1), z1)) by F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(z0), s(z1), s(0)) -> c7(F(z0, z1, f(z0, z1, s(0))), F(s(z0), s(z1), 0)) F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) S tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(z0), s(z1), s(0)) -> c7(F(z0, z1, f(z0, z1, s(0))), F(s(z0), s(z1), 0)) F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c4_1, c5_1, c8_1, c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1 ---------------------------------------- (43) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1), s(0)) -> c7(F(z0, z1, f(z0, z1, s(0))), F(s(z0), s(z1), 0)) by F(s(0), s(0), s(0)) -> c7(F(0, 0, s(s(0))), F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(0), s(0), s(0)) -> c7(F(0, 0, s(s(0))), F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) S tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(0), s(0), s(0)) -> c7(F(0, 0, s(s(0))), F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c4_1, c5_1, c8_1, c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1 ---------------------------------------- (45) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) S tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c4_1, c5_1, c8_1, c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) by F(s(0), s(0), s(s(x2))) -> c7(F(0, 0, s(f(s(0), s(0), x2))), F(s(0), s(0), s(x2))) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(0), s(x1), s(s(z1))) -> c7(F(0, x1, f(0, x1, f(0, s(s(x1)), s(z1)))), F(s(0), s(x1), s(z1))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(0), s(0), s(s(x2))) -> c7(F(0, 0, s(f(s(0), s(0), x2))), F(s(0), s(0), s(x2))) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) S tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(0), s(0), s(s(x2))) -> c7(F(0, 0, s(f(s(0), s(0), x2))), F(s(0), s(0), s(x2))) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c4_1, c5_1, c8_1, c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) S tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c4_1, c5_1, c8_1, c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1 ---------------------------------------- (51) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) by F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) S tuples: F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c5_1, c8_1, c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1, c4_1 ---------------------------------------- (53) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) S tuples: F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c5_1, c8_1, c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1, c4_1, c1_1 ---------------------------------------- (55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) by F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) S tuples: F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c8_1, c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1 ---------------------------------------- (57) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) by F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) S tuples: F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1 ---------------------------------------- (59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) S tuples: F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1 ---------------------------------------- (61) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) by F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c10_1, c_1, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1 ---------------------------------------- (63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c10_1, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1 ---------------------------------------- (65) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) by F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c10_1, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1 ---------------------------------------- (67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c10_1, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1 ---------------------------------------- (69) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) by F(0, s(s(0)), s(s(s(s(0))))) -> c10(F(0, s(s(0)), s(s(s(0))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(0))))) -> c10(F(0, s(s(0)), s(s(s(0))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(0))))) -> c10(F(0, s(s(0)), s(s(s(0))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) by F(0, s(s(0)), s(s(s(0)))) -> c10(F(0, 0, s(f(0, 0, s(0)))), F(0, s(s(0)), s(s(0)))) F(0, s(s(0)), s(s(s(0)))) -> c10(F(0, 0, f(0, 0, s(s(0)))), F(0, s(s(0)), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0)))), F(0, s(s(s(0))), s(s(0)))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(0))))) -> c10(F(0, s(s(0)), s(s(s(0))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(0)), s(s(s(0)))) -> c10(F(0, 0, s(f(0, 0, s(0)))), F(0, s(s(0)), s(s(0)))) F(0, s(s(0)), s(s(s(0)))) -> c10(F(0, 0, f(0, 0, s(s(0)))), F(0, s(s(0)), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0)))), F(0, s(s(s(0))), s(s(0)))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(0))))) -> c10(F(0, s(s(0)), s(s(s(0))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(0)), s(s(s(0)))) -> c10(F(0, 0, s(f(0, 0, s(0)))), F(0, s(s(0)), s(s(0)))) F(0, s(s(0)), s(s(s(0)))) -> c10(F(0, 0, f(0, 0, s(s(0)))), F(0, s(s(0)), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0)))), F(0, s(s(s(0))), s(s(0)))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: F(0, s(s(0)), s(s(s(s(0))))) -> c10(F(0, s(s(0)), s(s(s(0))))) F(0, s(s(0)), s(s(s(0)))) -> c10(F(0, 0, s(f(0, 0, s(0)))), F(0, s(s(0)), s(s(0)))) F(0, s(s(0)), s(s(s(0)))) -> c10(F(0, 0, f(0, 0, s(s(0)))), F(0, s(s(0)), s(s(0)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0)))), F(0, s(s(s(0))), s(s(0)))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0)))), F(0, s(s(s(0))), s(s(0)))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (77) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) by F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (79) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (81) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) by F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (85) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) by F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (87) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) by F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (89) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) by F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (91) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) by F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (93) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) by F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (95) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) by F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), 0))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), 0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), 0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (97) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), 0))) by F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), s(s(0)))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), s(s(0)))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), s(s(0)))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (99) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (101) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) by F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z1), s(s(s(s(y1))))) -> c7(F(s(0), s(z1), s(s(s(y1))))) F(s(0), s(z1), s(s(s(y1)))) -> c7(F(s(0), s(z1), s(s(y1)))) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(z1), s(s(0))) -> c7(F(s(0), s(z1), s(0))) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(0)))) -> c7(F(s(0), s(z1), s(s(0)))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (103) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) by F(s(0), s(0), s(s(s(s(y1))))) -> c7(F(s(0), s(0), s(s(s(y1))))) F(s(0), s(0), s(s(s(y1)))) -> c7(F(s(0), s(0), s(s(y1)))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (105) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) by F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (107) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (109) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) by F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) by F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(s(0))), s(s(z1)), 0) -> c4(F(s(s(0)), s(z1), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(s(z1)), 0) -> c4(F(s(0), s(z1), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) by F(s(0), s(s(s(y0))), s(0)) -> c1(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (115) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) by F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) by F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y1)))) -> c5(F(s(0), s(0), s(s(y1)))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) by F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (121) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) by F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(z1)))) -> c5(F(s(0), s(0), s(s(z1)))) F(s(s(s(0))), 0, s(s(s(z1)))) -> c5(F(s(s(0)), s(0), s(s(z1)))) F(s(s(z0)), 0, s(s(s(0)))) -> c5(F(s(z0), s(0), s(s(0)))) F(s(s(z0)), 0, s(s(s(s(y2))))) -> c5(F(s(z0), s(0), s(s(s(y2))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (123) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) by F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (125) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (127) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) by F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c9_1, c10_1, c7_1, c8_1 ---------------------------------------- (129) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) by F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c10_1, c7_1, c8_1, c9_1 ---------------------------------------- (131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) by F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c10_1, c7_1, c8_1, c9_1 ---------------------------------------- (133) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) by F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c10_1, c7_1, c8_1, c9_1 ---------------------------------------- (135) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) by F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(0), s(z0), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z0), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(s(s(0))))) -> c7(F(s(0), s(z0), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(0), s(z0), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z0), s(s(s(s(s(y1))))))) F(s(0), s(z0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z0), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(0))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(0))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(z0), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z0), s(s(s(s(s(y1))))))) F(s(0), s(z0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z0), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(0))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(z0), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z0), s(s(s(s(s(y1))))))) F(s(0), s(z0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z0), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(0))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c10_1, c7_1, c8_1, c9_1 ---------------------------------------- (137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) by F(s(s(s(s(y0)))), s(0), s(0)) -> c7(F(s(s(s(s(y0)))), s(0), 0)) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(z0), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z0), s(s(s(s(s(y1))))))) F(s(0), s(z0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z0), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(0))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(0))) F(s(s(s(s(y0)))), s(0), s(0)) -> c7(F(s(s(s(s(y0)))), s(0), 0)) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(z0), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z0), s(s(s(s(s(y1))))))) F(s(0), s(z0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z0), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(0))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(0))) F(s(s(s(s(y0)))), s(0), s(0)) -> c7(F(s(s(s(s(y0)))), s(0), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c10_1, c7_1, c8_1, c9_1 ---------------------------------------- (139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) by F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(z1)), s(0)) -> c7(F(s(s(0)), s(s(z1)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(z1)), s(0)) -> c7(F(s(s(s(0))), s(s(z1)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(s(0))), s(s(s(s(s(y0))))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(s(y0))))), 0)) F(s(s(s(0))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(y0))), 0)) F(s(s(s(0))), s(s(s(0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(0))), 0)) F(s(s(s(0))), s(s(s(s(0)))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(0)))), 0)) F(s(s(s(s(0)))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(s(0)))), s(s(s(y0))), 0)) F(s(s(s(s(y0)))), s(s(s(0))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(0))), 0)) F(s(s(s(s(y0)))), s(s(s(s(y1)))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0)) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(z0), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z0), s(s(s(s(s(y1))))))) F(s(0), s(z0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z0), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(0))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(0))) F(s(s(s(s(y0)))), s(0), s(0)) -> c7(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(0))), s(s(s(s(s(y0))))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(s(y0))))), 0)) F(s(s(s(0))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(y0))), 0)) F(s(s(s(0))), s(s(s(0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(0))), 0)) F(s(s(s(0))), s(s(s(s(0)))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(0)))), 0)) F(s(s(s(s(0)))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(s(0)))), s(s(s(y0))), 0)) F(s(s(s(s(y0)))), s(s(s(0))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(0))), 0)) F(s(s(s(s(y0)))), s(s(s(s(y1)))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0)) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(z0), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z0), s(s(s(s(s(y1))))))) F(s(0), s(z0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z0), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(0))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(0))) F(s(s(s(s(y0)))), s(0), s(0)) -> c7(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(0))), s(s(s(s(s(y0))))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(s(y0))))), 0)) F(s(s(s(0))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(y0))), 0)) F(s(s(s(0))), s(s(s(0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(0))), 0)) F(s(s(s(0))), s(s(s(s(0)))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(0)))), 0)) F(s(s(s(s(0)))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(s(0)))), s(s(s(y0))), 0)) F(s(s(s(s(y0)))), s(s(s(0))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(0))), 0)) F(s(s(s(s(y0)))), s(s(s(s(y1)))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c10_1, c7_1, c8_1, c9_1 ---------------------------------------- (141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) by F(s(0), s(z1), s(s(s(s(y1))))) -> c7(F(s(0), s(z1), s(s(s(y1))))) F(s(0), s(z1), s(s(s(z2)))) -> c7(F(s(0), s(z1), s(s(z2)))) F(s(s(0)), s(z1), s(s(s(z2)))) -> c7(F(s(s(0)), s(z1), s(s(z2)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(0)))) -> c7(F(s(0), s(z1), s(s(0)))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(z1), s(s(s(0)))) -> c7(F(s(s(0)), s(z1), s(s(0)))) F(s(s(y0)), s(0), s(s(s(0)))) -> c7(F(s(s(y0)), s(0), s(s(0)))) F(s(s(y0)), s(s(y1)), s(s(s(0)))) -> c7(F(s(s(y0)), s(s(y1)), s(s(0)))) F(s(s(0)), s(z1), s(s(s(s(y1))))) -> c7(F(s(s(0)), s(z1), s(s(s(y1))))) F(s(z0), s(z1), s(s(s(s(0))))) -> c7(F(s(z0), s(z1), s(s(s(0))))) F(s(z0), s(z1), s(s(s(s(s(y2)))))) -> c7(F(s(z0), s(z1), s(s(s(s(y2)))))) F(s(0), s(z1), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z1), s(s(s(s(s(y1))))))) F(s(0), s(z1), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z1), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(s(0)))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(s(0)))) F(s(s(0)), s(s(y0)), s(s(s(0)))) -> c7(F(s(s(0)), s(s(y0)), s(s(0)))) F(s(s(0)), s(s(0)), s(s(s(0)))) -> c7(F(s(s(0)), s(s(0)), s(s(0)))) F(s(s(0)), s(s(s(0))), s(s(s(0)))) -> c7(F(s(s(0)), s(s(s(0))), s(s(0)))) F(s(s(s(0))), s(s(y0)), s(s(s(0)))) -> c7(F(s(s(s(0))), s(s(y0)), s(s(0)))) F(s(s(s(y0))), s(s(0)), s(s(s(0)))) -> c7(F(s(s(s(y0))), s(s(0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1))), s(s(s(0)))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(s(0)))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(z1), s(s(s(s(s(s(s(y1)))))))) -> c7(F(s(0), s(z1), s(s(s(s(s(s(y1)))))))) F(s(0), s(z1), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(z1), s(s(s(s(s(0))))))) F(s(0), s(s(s(y0))), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(s(0))))))) F(s(0), s(s(0)), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(s(0)), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(0)))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(s(y0)))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(s(y0)))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(s(0)))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(s(0)))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(z0), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z0), s(s(s(s(s(y1))))))) F(s(0), s(z0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z0), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(0))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(0))) F(s(s(s(s(y0)))), s(0), s(0)) -> c7(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(0))), s(s(s(s(s(y0))))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(s(y0))))), 0)) F(s(s(s(0))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(y0))), 0)) F(s(s(s(0))), s(s(s(0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(0))), 0)) F(s(s(s(0))), s(s(s(s(0)))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(0)))), 0)) F(s(s(s(s(0)))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(s(0)))), s(s(s(y0))), 0)) F(s(s(s(s(y0)))), s(s(s(0))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(0))), 0)) F(s(s(s(s(y0)))), s(s(s(s(y1)))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0)) F(s(s(0)), s(z1), s(s(s(0)))) -> c7(F(s(s(0)), s(z1), s(s(0)))) F(s(s(y0)), s(0), s(s(s(0)))) -> c7(F(s(s(y0)), s(0), s(s(0)))) F(s(s(y0)), s(s(y1)), s(s(s(0)))) -> c7(F(s(s(y0)), s(s(y1)), s(s(0)))) F(s(s(0)), s(z1), s(s(s(s(y1))))) -> c7(F(s(s(0)), s(z1), s(s(s(y1))))) F(s(z0), s(z1), s(s(s(s(0))))) -> c7(F(s(z0), s(z1), s(s(s(0))))) F(s(z0), s(z1), s(s(s(s(s(y2)))))) -> c7(F(s(z0), s(z1), s(s(s(s(y2)))))) F(s(s(0)), s(s(s(s(y0)))), s(s(s(0)))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(s(0)))) F(s(s(0)), s(s(y0)), s(s(s(0)))) -> c7(F(s(s(0)), s(s(y0)), s(s(0)))) F(s(s(0)), s(s(0)), s(s(s(0)))) -> c7(F(s(s(0)), s(s(0)), s(s(0)))) F(s(s(0)), s(s(s(0))), s(s(s(0)))) -> c7(F(s(s(0)), s(s(s(0))), s(s(0)))) F(s(s(s(0))), s(s(y0)), s(s(s(0)))) -> c7(F(s(s(s(0))), s(s(y0)), s(s(0)))) F(s(s(s(y0))), s(s(0)), s(s(s(0)))) -> c7(F(s(s(s(y0))), s(s(0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1))), s(s(s(0)))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(s(0)))) F(s(0), s(z1), s(s(s(s(s(s(s(y1)))))))) -> c7(F(s(0), s(z1), s(s(s(s(s(s(y1)))))))) F(s(0), s(z1), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(z1), s(s(s(s(s(0))))))) F(s(0), s(s(s(y0))), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(s(0))))))) F(s(0), s(s(0)), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(s(0)), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(0)))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(s(y0)))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(s(y0)))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(s(0)))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(s(0)))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(z0), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z0), s(s(s(s(s(y1))))))) F(s(0), s(z0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z0), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(0))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(0))) F(s(s(s(s(y0)))), s(0), s(0)) -> c7(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(0))), s(s(s(s(s(y0))))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(s(y0))))), 0)) F(s(s(s(0))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(y0))), 0)) F(s(s(s(0))), s(s(s(0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(0))), 0)) F(s(s(s(0))), s(s(s(s(0)))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(0)))), 0)) F(s(s(s(s(0)))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(s(0)))), s(s(s(y0))), 0)) F(s(s(s(s(y0)))), s(s(s(0))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(0))), 0)) F(s(s(s(s(y0)))), s(s(s(s(y1)))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0)) F(s(s(0)), s(z1), s(s(s(0)))) -> c7(F(s(s(0)), s(z1), s(s(0)))) F(s(s(y0)), s(0), s(s(s(0)))) -> c7(F(s(s(y0)), s(0), s(s(0)))) F(s(s(y0)), s(s(y1)), s(s(s(0)))) -> c7(F(s(s(y0)), s(s(y1)), s(s(0)))) F(s(s(0)), s(z1), s(s(s(s(y1))))) -> c7(F(s(s(0)), s(z1), s(s(s(y1))))) F(s(z0), s(z1), s(s(s(s(0))))) -> c7(F(s(z0), s(z1), s(s(s(0))))) F(s(z0), s(z1), s(s(s(s(s(y2)))))) -> c7(F(s(z0), s(z1), s(s(s(s(y2)))))) F(s(s(0)), s(s(s(s(y0)))), s(s(s(0)))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(s(0)))) F(s(s(0)), s(s(y0)), s(s(s(0)))) -> c7(F(s(s(0)), s(s(y0)), s(s(0)))) F(s(s(0)), s(s(0)), s(s(s(0)))) -> c7(F(s(s(0)), s(s(0)), s(s(0)))) F(s(s(0)), s(s(s(0))), s(s(s(0)))) -> c7(F(s(s(0)), s(s(s(0))), s(s(0)))) F(s(s(s(0))), s(s(y0)), s(s(s(0)))) -> c7(F(s(s(s(0))), s(s(y0)), s(s(0)))) F(s(s(s(y0))), s(s(0)), s(s(s(0)))) -> c7(F(s(s(s(y0))), s(s(0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1))), s(s(s(0)))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(s(0)))) F(s(0), s(z1), s(s(s(s(s(s(s(y1)))))))) -> c7(F(s(0), s(z1), s(s(s(s(s(s(y1)))))))) F(s(0), s(z1), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(z1), s(s(s(s(s(0))))))) F(s(0), s(s(s(y0))), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(s(0))))))) F(s(0), s(s(0)), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(s(0)), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(0)))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(s(y0)))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(s(y0)))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(s(0)))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(s(0)))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c10_1, c7_1, c8_1, c9_1