KILLED proof of input_IBOK327fsf.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxWeightedTrs (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedTrs (11) CompletionProof [UPPER BOUND(ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CompletionProof [UPPER BOUND(ID), 0 ms] (20) CpxTypedWeightedCompleteTrs (21) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (24) CdtProblem (25) CdtLeafRemovalProof [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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 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), 2 ms] (44) CdtProblem (45) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 9 ms] (48) CdtProblem (49) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) 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: -(x, 0) -> x -(0, s(y)) -> 0 -(s(x), s(y)) -> -(x, y) f(0) -> 0 f(s(x)) -> -(s(x), g(f(x))) g(0) -> s(0) g(s(x)) -> -(s(x), f(g(x))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: -(x, 0') -> x -(0', s(y)) -> 0' -(s(x), s(y)) -> -(x, y) f(0') -> 0' f(s(x)) -> -(s(x), g(f(x))) g(0') -> s(0') g(s(x)) -> -(s(x), f(g(x))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: -(x, 0) -> x -(0, s(y)) -> 0 -(s(x), s(y)) -> -(x, y) f(0) -> 0 f(s(x)) -> -(s(x), g(f(x))) g(0) -> s(0) g(s(x)) -> -(s(x), f(g(x))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: -(x, 0) -> x [1] -(0, s(y)) -> 0 [1] -(s(x), s(y)) -> -(x, y) [1] f(0) -> 0 [1] f(s(x)) -> -(s(x), g(f(x))) [1] g(0) -> s(0) [1] g(s(x)) -> -(s(x), f(g(x))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: - => minus ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(0, s(y)) -> 0 [1] minus(s(x), s(y)) -> minus(x, y) [1] f(0) -> 0 [1] f(s(x)) -> minus(s(x), g(f(x))) [1] g(0) -> s(0) [1] g(s(x)) -> minus(s(x), f(g(x))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(0, s(y)) -> 0 [1] minus(s(x), s(y)) -> minus(x, y) [1] f(0) -> 0 [1] f(s(x)) -> minus(s(x), g(f(x))) [1] g(0) -> s(0) [1] g(s(x)) -> minus(s(x), f(g(x))) [1] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s f :: 0:s -> 0:s g :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (11) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: f_1 g_1 minus_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(0, s(y)) -> 0 [1] minus(s(x), s(y)) -> minus(x, y) [1] f(0) -> 0 [1] f(s(x)) -> minus(s(x), g(f(x))) [1] g(0) -> s(0) [1] g(s(x)) -> minus(s(x), f(g(x))) [1] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s f :: 0:s -> 0:s g :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (13) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (14) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(0, s(y)) -> 0 [1] minus(s(x), s(y)) -> minus(x, y) [1] f(0) -> 0 [1] f(s(0)) -> minus(s(0), g(0)) [2] f(s(s(x'))) -> minus(s(s(x')), g(minus(s(x'), g(f(x'))))) [2] g(0) -> s(0) [1] g(s(0)) -> minus(s(0), f(s(0))) [2] g(s(s(x''))) -> minus(s(s(x'')), f(minus(s(x''), f(g(x''))))) [2] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s f :: 0:s -> 0:s g :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (15) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> minus(1 + 0, g(0)) :|: z = 1 + 0 f(z) -{ 2 }-> minus(1 + (1 + x'), g(minus(1 + x', g(f(x'))))) :|: x' >= 0, z = 1 + (1 + x') f(z) -{ 1 }-> 0 :|: z = 0 g(z) -{ 2 }-> minus(1 + 0, f(1 + 0)) :|: z = 1 + 0 g(z) -{ 2 }-> minus(1 + (1 + x''), f(minus(1 + x'', f(g(x''))))) :|: x'' >= 0, z = 1 + (1 + x'') g(z) -{ 1 }-> 1 + 0 :|: z = 0 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> minus(1 + 0, g(0)) :|: z = 1 + 0 f(z) -{ 2 }-> minus(1 + (1 + (z - 2)), g(minus(1 + (z - 2), g(f(z - 2))))) :|: z - 2 >= 0 f(z) -{ 1 }-> 0 :|: z = 0 g(z) -{ 2 }-> minus(1 + 0, f(1 + 0)) :|: z = 1 + 0 g(z) -{ 2 }-> minus(1 + (1 + (z - 2)), f(minus(1 + (z - 2), f(g(z - 2))))) :|: z - 2 >= 0 g(z) -{ 1 }-> 1 + 0 :|: z = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 ---------------------------------------- (19) 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 ---------------------------------------- (20) 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: minus(x, 0) -> x [1] minus(0, s(y)) -> 0 [1] minus(s(x), s(y)) -> minus(x, y) [1] f(0) -> 0 [1] f(s(x)) -> minus(s(x), g(f(x))) [1] g(0) -> s(0) [1] g(s(x)) -> minus(s(x), f(g(x))) [1] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s f :: 0:s -> 0:s g :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (21) 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 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> minus(1 + x, g(f(x))) :|: x >= 0, z = 1 + x f(z) -{ 1 }-> 0 :|: z = 0 g(z) -{ 1 }-> minus(1 + x, f(g(x))) :|: x >= 0, z = 1 + x g(z) -{ 1 }-> 1 + 0 :|: z = 0 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (23) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: -'(z0, 0) -> c -'(0, s(z0)) -> c1 -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(0) -> c3 F(s(z0)) -> c4(-'(s(z0), g(f(z0))), G(f(z0)), F(z0)) G(0) -> c5 G(s(z0)) -> c6(-'(s(z0), f(g(z0))), F(g(z0)), G(z0)) S tuples: -'(z0, 0) -> c -'(0, s(z0)) -> c1 -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(0) -> c3 F(s(z0)) -> c4(-'(s(z0), g(f(z0))), G(f(z0)), F(z0)) G(0) -> c5 G(s(z0)) -> c6(-'(s(z0), f(g(z0))), F(g(z0)), G(z0)) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: -'_2, F_1, G_1 Compound Symbols: c, c1, c2_1, c3, c4_3, c5, c6_3 ---------------------------------------- (25) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: -'(0, s(z0)) -> c1 F(0) -> c3 -'(z0, 0) -> c G(0) -> c5 ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(z0)) -> c4(-'(s(z0), g(f(z0))), G(f(z0)), F(z0)) G(s(z0)) -> c6(-'(s(z0), f(g(z0))), F(g(z0)), G(z0)) S tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(z0)) -> c4(-'(s(z0), g(f(z0))), G(f(z0)), F(z0)) G(s(z0)) -> c6(-'(s(z0), f(g(z0))), F(g(z0)), G(z0)) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: -'_2, F_1, G_1 Compound Symbols: c2_1, c4_3, c6_3 ---------------------------------------- (27) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c4(-'(s(z0), g(f(z0))), G(f(z0)), F(z0)) by F(s(0)) -> c4(-'(s(0), g(0)), G(f(0)), F(0)) F(s(s(z0))) -> c4(-'(s(s(z0)), g(-(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) G(s(z0)) -> c6(-'(s(z0), f(g(z0))), F(g(z0)), G(z0)) F(s(0)) -> c4(-'(s(0), g(0)), G(f(0)), F(0)) F(s(s(z0))) -> c4(-'(s(s(z0)), g(-(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) S tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) G(s(z0)) -> c6(-'(s(z0), f(g(z0))), F(g(z0)), G(z0)) F(s(0)) -> c4(-'(s(0), g(0)), G(f(0)), F(0)) F(s(s(z0))) -> c4(-'(s(s(z0)), g(-(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: -'_2, G_1, F_1 Compound Symbols: c2_1, c6_3, c4_3 ---------------------------------------- (29) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) G(s(z0)) -> c6(-'(s(z0), f(g(z0))), F(g(z0)), G(z0)) F(s(s(z0))) -> c4(-'(s(s(z0)), g(-(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) F(s(0)) -> c4(-'(s(0), g(0)), G(f(0))) S tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) G(s(z0)) -> c6(-'(s(z0), f(g(z0))), F(g(z0)), G(z0)) F(s(s(z0))) -> c4(-'(s(s(z0)), g(-(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) F(s(0)) -> c4(-'(s(0), g(0)), G(f(0))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: -'_2, G_1, F_1 Compound Symbols: c2_1, c6_3, c4_3, c4_2 ---------------------------------------- (31) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(z0)) -> c6(-'(s(z0), f(g(z0))), F(g(z0)), G(z0)) by G(s(0)) -> c6(-'(s(0), f(s(0))), F(g(0)), G(0)) G(s(s(z0))) -> c6(-'(s(s(z0)), f(-(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(s(z0))) -> c4(-'(s(s(z0)), g(-(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) F(s(0)) -> c4(-'(s(0), g(0)), G(f(0))) G(s(0)) -> c6(-'(s(0), f(s(0))), F(g(0)), G(0)) G(s(s(z0))) -> c6(-'(s(s(z0)), f(-(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) S tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(s(z0))) -> c4(-'(s(s(z0)), g(-(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) F(s(0)) -> c4(-'(s(0), g(0)), G(f(0))) G(s(0)) -> c6(-'(s(0), f(s(0))), F(g(0)), G(0)) G(s(s(z0))) -> c6(-'(s(s(z0)), f(-(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: -'_2, F_1, G_1 Compound Symbols: c2_1, c4_3, c4_2, c6_3 ---------------------------------------- (33) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(s(z0))) -> c4(-'(s(s(z0)), g(-(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) F(s(0)) -> c4(-'(s(0), g(0)), G(f(0))) G(s(s(z0))) -> c6(-'(s(s(z0)), f(-(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) G(s(0)) -> c6(-'(s(0), f(s(0))), F(g(0))) S tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(s(z0))) -> c4(-'(s(s(z0)), g(-(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) F(s(0)) -> c4(-'(s(0), g(0)), G(f(0))) G(s(s(z0))) -> c6(-'(s(s(z0)), f(-(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) G(s(0)) -> c6(-'(s(0), f(s(0))), F(g(0))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: -'_2, F_1, G_1 Compound Symbols: c2_1, c4_3, c4_2, c6_3, c6_2 ---------------------------------------- (35) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0))) -> c4(-'(s(s(z0)), g(-(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) by F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c4(G(f(s(x0)))) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(0)) -> c4(-'(s(0), g(0)), G(f(0))) G(s(s(z0))) -> c6(-'(s(s(z0)), f(-(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) G(s(0)) -> c6(-'(s(0), f(s(0))), F(g(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c4(G(f(s(x0)))) S tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(0)) -> c4(-'(s(0), g(0)), G(f(0))) G(s(s(z0))) -> c6(-'(s(s(z0)), f(-(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) G(s(0)) -> c6(-'(s(0), f(s(0))), F(g(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c4(G(f(s(x0)))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: -'_2, F_1, G_1 Compound Symbols: c2_1, c4_2, c6_3, c6_2, c4_3, c4_1 ---------------------------------------- (37) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0)) -> c4(-'(s(0), g(0)), G(f(0))) by F(s(0)) -> c4(-'(s(0), s(0)), G(f(0))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) G(s(s(z0))) -> c6(-'(s(s(z0)), f(-(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) G(s(0)) -> c6(-'(s(0), f(s(0))), F(g(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c4(G(f(s(x0)))) F(s(0)) -> c4(-'(s(0), s(0)), G(f(0))) S tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) G(s(s(z0))) -> c6(-'(s(s(z0)), f(-(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) G(s(0)) -> c6(-'(s(0), f(s(0))), F(g(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c4(G(f(s(x0)))) F(s(0)) -> c4(-'(s(0), s(0)), G(f(0))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: -'_2, G_1, F_1 Compound Symbols: c2_1, c6_3, c6_2, c4_3, c4_1, c4_2 ---------------------------------------- (39) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(s(z0))) -> c6(-'(s(s(z0)), f(-(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) by G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c6(F(g(s(x0)))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) G(s(0)) -> c6(-'(s(0), f(s(0))), F(g(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c4(G(f(s(x0)))) F(s(0)) -> c4(-'(s(0), s(0)), G(f(0))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c6(F(g(s(x0)))) S tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) G(s(0)) -> c6(-'(s(0), f(s(0))), F(g(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c4(G(f(s(x0)))) F(s(0)) -> c4(-'(s(0), s(0)), G(f(0))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c6(F(g(s(x0)))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: -'_2, G_1, F_1 Compound Symbols: c2_1, c6_2, c4_3, c4_1, c4_2, c6_3, c6_1 ---------------------------------------- (41) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(0)) -> c6(-'(s(0), f(s(0))), F(g(0))) by G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) G(s(0)) -> c6(F(g(0))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c4(G(f(s(x0)))) F(s(0)) -> c4(-'(s(0), s(0)), G(f(0))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) G(s(0)) -> c6(F(g(0))) S tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c4(G(f(s(x0)))) F(s(0)) -> c4(-'(s(0), s(0)), G(f(0))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) G(s(0)) -> c6(F(g(0))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: -'_2, F_1, G_1 Compound Symbols: c2_1, c4_3, c4_1, c4_2, c6_3, c6_1, c6_2 ---------------------------------------- (43) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0)) -> c4(-'(s(0), s(0)), G(f(0))) by F(s(0)) -> c4(-'(s(0), s(0)), G(0)) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c4(G(f(s(x0)))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) G(s(0)) -> c6(F(g(0))) F(s(0)) -> c4(-'(s(0), s(0)), G(0)) S tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c4(G(f(s(x0)))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) G(s(0)) -> c6(F(g(0))) F(s(0)) -> c4(-'(s(0), s(0)), G(0)) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: -'_2, F_1, G_1 Compound Symbols: c2_1, c4_3, c4_1, c6_3, c6_1, c6_2, c4_2 ---------------------------------------- (45) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c4(G(f(s(x0)))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) G(s(0)) -> c6(F(g(0))) F(s(0)) -> c4(-'(s(0), s(0))) S tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c4(G(f(s(x0)))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) G(s(0)) -> c6(F(g(0))) F(s(0)) -> c4(-'(s(0), s(0))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: -'_2, F_1, G_1 Compound Symbols: c2_1, c4_3, c4_1, c6_3, c6_1, c6_2 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(0)) -> c6(F(g(0))) by G(s(0)) -> c6(F(s(0))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c4(G(f(s(x0)))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) F(s(0)) -> c4(-'(s(0), s(0))) G(s(0)) -> c6(F(s(0))) S tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c4(G(f(s(x0)))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) F(s(0)) -> c4(-'(s(0), s(0))) G(s(0)) -> c6(F(s(0))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: -'_2, F_1, G_1 Compound Symbols: c2_1, c4_3, c4_1, c6_3, c6_1, c6_2 ---------------------------------------- (49) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(f(s(0))), F(s(0))) by F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0)))), F(s(0))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c4(G(f(s(x0)))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) F(s(0)) -> c4(-'(s(0), s(0))) G(s(0)) -> c6(F(s(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0)))), F(s(0))) S tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c4(G(f(s(x0)))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) F(s(0)) -> c4(-'(s(0), s(0))) G(s(0)) -> c6(F(s(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0)))), F(s(0))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: -'_2, F_1, G_1 Compound Symbols: c2_1, c4_3, c4_1, c6_3, c6_1, c6_2 ---------------------------------------- (51) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) by F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(s(x0))) -> c4(G(f(s(x0)))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) F(s(0)) -> c4(-'(s(0), s(0))) G(s(0)) -> c6(F(s(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) S tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) F(s(s(x0))) -> c4(G(f(s(x0)))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) F(s(0)) -> c4(-'(s(0), s(0))) G(s(0)) -> c6(F(s(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: -'_2, F_1, G_1 Compound Symbols: c2_1, c4_1, c6_3, c6_1, c6_2, c4_3 ---------------------------------------- (53) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(x0))) -> c4(G(f(s(x0)))) by F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) F(s(0)) -> c4(-'(s(0), s(0))) G(s(0)) -> c6(F(s(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) S tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) F(s(0)) -> c4(-'(s(0), s(0))) G(s(0)) -> c6(F(s(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: -'_2, G_1, F_1 Compound Symbols: c2_1, c6_3, c6_1, c6_2, c4_1, c4_3 ---------------------------------------- (55) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(g(s(0))), G(s(0))) by G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0)))), G(s(0))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) F(s(0)) -> c4(-'(s(0), s(0))) G(s(0)) -> c6(F(s(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0)))), G(s(0))) S tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) F(s(0)) -> c4(-'(s(0), s(0))) G(s(0)) -> c6(F(s(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0)))), G(s(0))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: -'_2, G_1, F_1 Compound Symbols: c2_1, c6_3, c6_1, c6_2, c4_1, c4_3 ---------------------------------------- (57) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) by G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) F(s(0)) -> c4(-'(s(0), s(0))) G(s(0)) -> c6(F(s(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) S tuples: -'(s(z0), s(z1)) -> c2(-'(z0, z1)) G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) F(s(0)) -> c4(-'(s(0), s(0))) G(s(0)) -> c6(F(s(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: -'_2, G_1, F_1 Compound Symbols: c2_1, c6_1, c6_2, c4_1, c4_3, c6_3 ---------------------------------------- (59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace -'(s(z0), s(z1)) -> c2(-'(z0, z1)) by -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) F(s(0)) -> c4(-'(s(0), s(0))) G(s(0)) -> c6(F(s(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) S tuples: G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) F(s(0)) -> c4(-'(s(0), s(0))) G(s(0)) -> c6(F(s(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: G_1, F_1, -'_2 Compound Symbols: c6_1, c6_2, c4_1, c4_3, c6_3, c2_1 ---------------------------------------- (61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: G(s(0)) -> c6(F(s(0))) F(s(0)) -> c4(-'(s(0), s(0))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) S tuples: G(s(s(x0))) -> c6(F(g(s(x0)))) G(s(0)) -> c6(-'(s(0), -(s(0), g(f(0)))), F(g(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: G_1, F_1, -'_2 Compound Symbols: c6_1, c6_2, c4_3, c4_1, c6_3, c2_1 ---------------------------------------- (63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: G(s(s(x0))) -> c6(F(g(s(x0)))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) G(s(0)) -> c6(F(g(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0))))) S tuples: G(s(s(x0))) -> c6(F(g(s(x0)))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) G(s(0)) -> c6(F(g(0))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0))))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: G_1, F_1, -'_2 Compound Symbols: c6_1, c4_3, c4_1, c6_3, c2_1, c4_2 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(0)) -> c6(F(g(0))) by G(s(0)) -> c6(F(s(0))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: G(s(s(x0))) -> c6(F(g(s(x0)))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0))))) G(s(0)) -> c6(F(s(0))) S tuples: G(s(s(x0))) -> c6(F(g(s(x0)))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0))))) G(s(0)) -> c6(F(s(0))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: G_1, F_1, -'_2 Compound Symbols: c6_1, c4_3, c4_1, c6_3, c2_1, c4_2 ---------------------------------------- (67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: G(s(0)) -> c6(F(s(0))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: G(s(s(x0))) -> c6(F(g(s(x0)))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0))))) S tuples: G(s(s(x0))) -> c6(F(g(s(x0)))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0))))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: G_1, F_1, -'_2 Compound Symbols: c6_1, c4_3, c4_1, c6_3, c2_1, c4_2 ---------------------------------------- (69) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: G(s(s(x0))) -> c6(F(g(s(x0)))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0))))) S tuples: G(s(s(x0))) -> c6(F(g(s(x0)))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0))))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: G_1, F_1, -'_2 Compound Symbols: c6_1, c4_3, c4_1, c6_3, c2_1, c4_2, c6_2 ---------------------------------------- (71) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace G(s(s(x0))) -> c6(F(g(s(x0)))) by G(s(s(z0))) -> c6(F(-(s(z0), f(g(z0))))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0))))) G(s(s(z0))) -> c6(F(-(s(z0), f(g(z0))))) S tuples: F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0))))) G(s(s(z0))) -> c6(F(-(s(z0), f(g(z0))))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: F_1, G_1, -'_2 Compound Symbols: c4_3, c4_1, c6_3, c2_1, c4_2, c6_2, c6_1 ---------------------------------------- (73) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) by F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(-(s(z0), g(f(z0)))))), F(s(s(z0)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0))))) G(s(s(z0))) -> c6(F(-(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(-(s(z0), g(f(z0)))))), F(s(s(z0)))) S tuples: F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0))))) G(s(s(z0))) -> c6(F(-(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(-(s(z0), g(f(z0)))))), F(s(s(z0)))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: F_1, G_1, -'_2 Compound Symbols: c4_1, c6_3, c2_1, c4_2, c6_2, c6_1, c4_3 ---------------------------------------- (75) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) by G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(-(s(z0), f(g(z0)))))), G(s(s(z0)))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0))))) G(s(s(z0))) -> c6(F(-(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(-(s(z0), g(f(z0)))))), F(s(s(z0)))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(-(s(z0), f(g(z0)))))), G(s(s(z0)))) S tuples: F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0))))) G(s(s(z0))) -> c6(F(-(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(-(s(z0), g(f(z0)))))), F(s(s(z0)))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(-(s(z0), f(g(z0)))))), G(s(s(z0)))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: F_1, -'_2, G_1 Compound Symbols: c4_1, c2_1, c4_2, c6_2, c6_1, c4_3, c6_3 ---------------------------------------- (77) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace -'(s(s(y0)), s(s(y1))) -> c2(-'(s(y0), s(y1))) by -'(s(s(s(y0))), s(s(s(y1)))) -> c2(-'(s(s(y0)), s(s(y1)))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0))))) G(s(s(z0))) -> c6(F(-(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(-(s(z0), g(f(z0)))))), F(s(s(z0)))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(-(s(z0), f(g(z0)))))), G(s(s(z0)))) -'(s(s(s(y0))), s(s(s(y1)))) -> c2(-'(s(s(y0)), s(s(y1)))) S tuples: F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) F(s(s(0))) -> c4(-'(s(s(0)), g(-(s(0), g(0)))), G(-(s(0), g(f(0))))) G(s(s(0))) -> c6(-'(s(s(0)), f(-(s(0), f(s(0))))), F(-(s(0), f(g(0))))) G(s(s(z0))) -> c6(F(-(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(-(s(z0), g(f(z0)))))), F(s(s(z0)))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(-(s(z0), f(g(z0)))))), G(s(s(z0)))) -'(s(s(s(y0))), s(s(s(y1)))) -> c2(-'(s(s(y0)), s(s(y1)))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: F_1, G_1, -'_2 Compound Symbols: c4_1, c4_2, c6_2, c6_1, c4_3, c6_3, c2_1 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(z0))) -> c6(F(-(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(-(s(z0), g(f(z0)))))), F(s(s(z0)))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(-(s(z0), f(g(z0)))))), G(s(s(z0)))) -'(s(s(s(y0))), s(s(s(y1)))) -> c2(-'(s(s(y0)), s(s(y1)))) F(s(s(0))) -> c4(G(-(s(0), g(f(0))))) G(s(s(0))) -> c6(F(-(s(0), f(g(0))))) S tuples: F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(z0))) -> c6(F(-(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(-(s(z0), g(f(z0)))))), F(s(s(z0)))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(-(s(z0), f(g(z0)))))), G(s(s(z0)))) -'(s(s(s(y0))), s(s(s(y1)))) -> c2(-'(s(s(y0)), s(s(y1)))) F(s(s(0))) -> c4(G(-(s(0), g(f(0))))) G(s(s(0))) -> c6(F(-(s(0), f(g(0))))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: F_1, G_1, -'_2 Compound Symbols: c4_1, c6_1, c4_3, c6_3, c2_1 ---------------------------------------- (81) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(0))) -> c4(G(-(s(0), g(f(0))))) by F(s(s(0))) -> c4(G(-(s(0), g(0)))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(z0))) -> c6(F(-(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(-(s(z0), g(f(z0)))))), F(s(s(z0)))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(-(s(z0), f(g(z0)))))), G(s(s(z0)))) -'(s(s(s(y0))), s(s(s(y1)))) -> c2(-'(s(s(y0)), s(s(y1)))) G(s(s(0))) -> c6(F(-(s(0), f(g(0))))) F(s(s(0))) -> c4(G(-(s(0), g(0)))) S tuples: F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(z0))) -> c6(F(-(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(-(s(z0), g(f(z0)))))), F(s(s(z0)))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(-(s(z0), f(g(z0)))))), G(s(s(z0)))) -'(s(s(s(y0))), s(s(s(y1)))) -> c2(-'(s(s(y0)), s(s(y1)))) G(s(s(0))) -> c6(F(-(s(0), f(g(0))))) F(s(s(0))) -> c4(G(-(s(0), g(0)))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: F_1, G_1, -'_2 Compound Symbols: c4_1, c6_1, c4_3, c6_3, c2_1 ---------------------------------------- (83) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace G(s(s(0))) -> c6(F(-(s(0), f(g(0))))) by G(s(s(0))) -> c6(F(-(s(0), f(s(0))))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(0, s(z0)) -> 0 -(s(z0), s(z1)) -> -(z0, z1) f(0) -> 0 f(s(z0)) -> -(s(z0), g(f(z0))) g(0) -> s(0) g(s(z0)) -> -(s(z0), f(g(z0))) Tuples: F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(z0))) -> c6(F(-(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(-(s(z0), g(f(z0)))))), F(s(s(z0)))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(-(s(z0), f(g(z0)))))), G(s(s(z0)))) -'(s(s(s(y0))), s(s(s(y1)))) -> c2(-'(s(s(y0)), s(s(y1)))) F(s(s(0))) -> c4(G(-(s(0), g(0)))) G(s(s(0))) -> c6(F(-(s(0), f(s(0))))) S tuples: F(s(s(z0))) -> c4(G(-(s(z0), g(f(z0))))) G(s(s(z0))) -> c6(F(-(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c4(-'(s(s(s(z0))), g(-(s(s(z0)), g(-(s(z0), g(f(z0))))))), G(-(s(s(z0)), g(-(s(z0), g(f(z0)))))), F(s(s(z0)))) G(s(s(s(z0)))) -> c6(-'(s(s(s(z0))), f(-(s(s(z0)), f(-(s(z0), f(g(z0))))))), F(-(s(s(z0)), f(-(s(z0), f(g(z0)))))), G(s(s(z0)))) -'(s(s(s(y0))), s(s(s(y1)))) -> c2(-'(s(s(y0)), s(s(y1)))) F(s(s(0))) -> c4(G(-(s(0), g(0)))) G(s(s(0))) -> c6(F(-(s(0), f(s(0))))) K tuples:none Defined Rule Symbols: -_2, f_1, g_1 Defined Pair Symbols: F_1, G_1, -'_2 Compound Symbols: c4_1, c6_1, c4_3, c6_3, c2_1