KILLED proof of input_ufjWoZGlaN.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 3 ms] (20) CpxRNTS (21) CpxTrsToCdtProof [UPPER BOUND(ID), 9 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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 5 ms] (30) CdtProblem (31) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 4 ms] (42) CdtProblem (43) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtForwardInstantiationProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 10 ms] (70) CdtProblem (71) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (76) 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: a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y)) mark(f(X1, X2)) -> a__f(mark(X1), X2) mark(g(X)) -> g(mark(X)) a__f(X1, X2) -> f(X1, X2) 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: a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y)) mark(f(X1, X2)) -> a__f(mark(X1), X2) mark(g(X)) -> g(mark(X)) a__f(X1, X2) -> f(X1, X2) 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: a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y)) mark(f(X1, X2)) -> a__f(mark(X1), X2) mark(g(X)) -> g(mark(X)) a__f(X1, X2) -> f(X1, X2) 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: a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y)) [1] mark(f(X1, X2)) -> a__f(mark(X1), X2) [1] mark(g(X)) -> g(mark(X)) [1] a__f(X1, X2) -> f(X1, X2) [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: a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y)) [1] mark(f(X1, X2)) -> a__f(mark(X1), X2) [1] mark(g(X)) -> g(mark(X)) [1] a__f(X1, X2) -> f(X1, X2) [1] The TRS has the following type information: a__f :: g:f -> g:f -> g:f g :: g:f -> g:f mark :: g:f -> g:f f :: g:f -> g:f -> g:f 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: mark(v0) -> null_mark [0] And the following fresh constants: null_mark ---------------------------------------- (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: a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y)) [1] mark(f(X1, X2)) -> a__f(mark(X1), X2) [1] mark(g(X)) -> g(mark(X)) [1] a__f(X1, X2) -> f(X1, X2) [1] mark(v0) -> null_mark [0] The TRS has the following type information: a__f :: g:f:null_mark -> g:f:null_mark -> g:f:null_mark g :: g:f:null_mark -> g:f:null_mark mark :: g:f:null_mark -> g:f:null_mark f :: g:f:null_mark -> g:f:null_mark -> g:f:null_mark null_mark :: g:f:null_mark 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: null_mark => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: a__f(z, z') -{ 1 }-> a__f(mark(X), 1 + (1 + X) + Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0 a__f(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 mark(z) -{ 1 }-> a__f(mark(X1), X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 mark(z) -{ 1 }-> 1 + mark(X) :|: z = 1 + X, X >= 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: mark_1 a__f_2 Due to the following rules being added: mark(v0) -> const [0] And the following fresh constants: const ---------------------------------------- (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: a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y)) [1] mark(f(X1, X2)) -> a__f(mark(X1), X2) [1] mark(g(X)) -> g(mark(X)) [1] a__f(X1, X2) -> f(X1, X2) [1] mark(v0) -> const [0] The TRS has the following type information: a__f :: g:f:const -> g:f:const -> g:f:const g :: g:f:const -> g:f:const mark :: g:f:const -> g:f:const f :: g:f:const -> g:f:const -> g:f:const const :: g:f:const 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: a__f(g(f(X1', X2')), Y) -> a__f(a__f(mark(X1'), X2'), f(g(f(X1', X2')), Y)) [2] a__f(g(g(X')), Y) -> a__f(g(mark(X')), f(g(g(X')), Y)) [2] a__f(g(X), Y) -> a__f(const, f(g(X), Y)) [1] mark(f(f(X1'', X2''), X2)) -> a__f(a__f(mark(X1''), X2''), X2) [2] mark(f(g(X''), X2)) -> a__f(g(mark(X'')), X2) [2] mark(f(X1, X2)) -> a__f(const, X2) [1] mark(g(X)) -> g(mark(X)) [1] a__f(X1, X2) -> f(X1, X2) [1] mark(v0) -> const [0] The TRS has the following type information: a__f :: g:f:const -> g:f:const -> g:f:const g :: g:f:const -> g:f:const mark :: g:f:const -> g:f:const f :: g:f:const -> g:f:const -> g:f:const const :: g:f:const 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: const => 0 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: a__f(z, z') -{ 2 }-> a__f(a__f(mark(X1'), X2'), 1 + (1 + (1 + X1' + X2')) + Y) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, z' = Y, X1' >= 0, Y >= 0 a__f(z, z') -{ 1 }-> a__f(0, 1 + (1 + X) + Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0 a__f(z, z') -{ 2 }-> a__f(1 + mark(X'), 1 + (1 + (1 + X')) + Y) :|: z' = Y, Y >= 0, X' >= 0, z = 1 + (1 + X') a__f(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 mark(z) -{ 2 }-> a__f(a__f(mark(X1''), X2''), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0 mark(z) -{ 1 }-> a__f(0, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 2 }-> a__f(1 + mark(X''), X2) :|: z = 1 + (1 + X'') + X2, X'' >= 0, X2 >= 0 mark(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 mark(z) -{ 1 }-> 1 + mark(X) :|: z = 1 + X, X >= 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: a__f(z, z') -{ 2 }-> a__f(a__f(mark(X1'), X2'), 1 + (1 + (1 + X1' + X2')) + z') :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0, z' >= 0 a__f(z, z') -{ 1 }-> a__f(0, 1 + (1 + (z - 1)) + z') :|: z' >= 0, z - 1 >= 0 a__f(z, z') -{ 2 }-> a__f(1 + mark(z - 2), 1 + (1 + (1 + (z - 2))) + z') :|: z' >= 0, z - 2 >= 0 a__f(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 mark(z) -{ 2 }-> a__f(a__f(mark(X1''), X2''), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0 mark(z) -{ 1 }-> a__f(0, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 2 }-> a__f(1 + mark(X''), X2) :|: z = 1 + (1 + X'') + X2, X'' >= 0, X2 >= 0 mark(z) -{ 0 }-> 0 :|: z >= 0 mark(z) -{ 1 }-> 1 + mark(z - 1) :|: z - 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: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(z0), z1) -> c(A__F(mark(z0), f(g(z0), z1)), MARK(z0)) A__F(z0, z1) -> c1 MARK(f(z0, z1)) -> c2(A__F(mark(z0), z1), MARK(z0)) MARK(g(z0)) -> c3(MARK(z0)) S tuples: A__F(g(z0), z1) -> c(A__F(mark(z0), f(g(z0), z1)), MARK(z0)) A__F(z0, z1) -> c1 MARK(f(z0, z1)) -> c2(A__F(mark(z0), z1), MARK(z0)) MARK(g(z0)) -> c3(MARK(z0)) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c1, c2_2, c3_1 ---------------------------------------- (23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__F(z0, z1) -> c1 ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(z0), z1) -> c(A__F(mark(z0), f(g(z0), z1)), MARK(z0)) MARK(f(z0, z1)) -> c2(A__F(mark(z0), z1), MARK(z0)) MARK(g(z0)) -> c3(MARK(z0)) S tuples: A__F(g(z0), z1) -> c(A__F(mark(z0), f(g(z0), z1)), MARK(z0)) MARK(f(z0, z1)) -> c2(A__F(mark(z0), z1), MARK(z0)) MARK(g(z0)) -> c3(MARK(z0)) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c2_2, c3_1 ---------------------------------------- (25) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__F(g(z0), z1) -> c(A__F(mark(z0), f(g(z0), z1)), MARK(z0)) by A__F(g(f(z0, z1)), x1) -> c(A__F(a__f(mark(z0), z1), f(g(f(z0, z1)), x1)), MARK(f(z0, z1))) A__F(g(g(z0)), x1) -> c(A__F(g(mark(z0)), f(g(g(z0)), x1)), MARK(g(z0))) ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: MARK(f(z0, z1)) -> c2(A__F(mark(z0), z1), MARK(z0)) MARK(g(z0)) -> c3(MARK(z0)) A__F(g(f(z0, z1)), x1) -> c(A__F(a__f(mark(z0), z1), f(g(f(z0, z1)), x1)), MARK(f(z0, z1))) A__F(g(g(z0)), x1) -> c(A__F(g(mark(z0)), f(g(g(z0)), x1)), MARK(g(z0))) S tuples: MARK(f(z0, z1)) -> c2(A__F(mark(z0), z1), MARK(z0)) MARK(g(z0)) -> c3(MARK(z0)) A__F(g(f(z0, z1)), x1) -> c(A__F(a__f(mark(z0), z1), f(g(f(z0, z1)), x1)), MARK(f(z0, z1))) A__F(g(g(z0)), x1) -> c(A__F(g(mark(z0)), f(g(g(z0)), x1)), MARK(g(z0))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: MARK_1, A__F_2 Compound Symbols: c2_2, c3_1, c_2 ---------------------------------------- (27) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(f(z0, z1)) -> c2(A__F(mark(z0), z1), MARK(z0)) by MARK(f(f(z0, z1), x1)) -> c2(A__F(a__f(mark(z0), z1), x1), MARK(f(z0, z1))) MARK(f(g(z0), x1)) -> c2(A__F(g(mark(z0)), x1), MARK(g(z0))) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: MARK(g(z0)) -> c3(MARK(z0)) A__F(g(f(z0, z1)), x1) -> c(A__F(a__f(mark(z0), z1), f(g(f(z0, z1)), x1)), MARK(f(z0, z1))) A__F(g(g(z0)), x1) -> c(A__F(g(mark(z0)), f(g(g(z0)), x1)), MARK(g(z0))) MARK(f(f(z0, z1), x1)) -> c2(A__F(a__f(mark(z0), z1), x1), MARK(f(z0, z1))) MARK(f(g(z0), x1)) -> c2(A__F(g(mark(z0)), x1), MARK(g(z0))) S tuples: MARK(g(z0)) -> c3(MARK(z0)) A__F(g(f(z0, z1)), x1) -> c(A__F(a__f(mark(z0), z1), f(g(f(z0, z1)), x1)), MARK(f(z0, z1))) A__F(g(g(z0)), x1) -> c(A__F(g(mark(z0)), f(g(g(z0)), x1)), MARK(g(z0))) MARK(f(f(z0, z1), x1)) -> c2(A__F(a__f(mark(z0), z1), x1), MARK(f(z0, z1))) MARK(f(g(z0), x1)) -> c2(A__F(g(mark(z0)), x1), MARK(g(z0))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: MARK_1, A__F_2 Compound Symbols: c3_1, c_2, c2_2 ---------------------------------------- (29) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__F(g(f(z0, z1)), x1) -> c(A__F(a__f(mark(z0), z1), f(g(f(z0, z1)), x1)), MARK(f(z0, z1))) by A__F(g(f(x0, z1)), x2) -> c(A__F(f(mark(x0), z1), f(g(f(x0, z1)), x2)), MARK(f(x0, z1))) A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: MARK(g(z0)) -> c3(MARK(z0)) A__F(g(g(z0)), x1) -> c(A__F(g(mark(z0)), f(g(g(z0)), x1)), MARK(g(z0))) MARK(f(f(z0, z1), x1)) -> c2(A__F(a__f(mark(z0), z1), x1), MARK(f(z0, z1))) MARK(f(g(z0), x1)) -> c2(A__F(g(mark(z0)), x1), MARK(g(z0))) A__F(g(f(x0, z1)), x2) -> c(A__F(f(mark(x0), z1), f(g(f(x0, z1)), x2)), MARK(f(x0, z1))) A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) S tuples: MARK(g(z0)) -> c3(MARK(z0)) A__F(g(g(z0)), x1) -> c(A__F(g(mark(z0)), f(g(g(z0)), x1)), MARK(g(z0))) MARK(f(f(z0, z1), x1)) -> c2(A__F(a__f(mark(z0), z1), x1), MARK(f(z0, z1))) MARK(f(g(z0), x1)) -> c2(A__F(g(mark(z0)), x1), MARK(g(z0))) A__F(g(f(x0, z1)), x2) -> c(A__F(f(mark(x0), z1), f(g(f(x0, z1)), x2)), MARK(f(x0, z1))) A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: MARK_1, A__F_2 Compound Symbols: c3_1, c_2, c2_2 ---------------------------------------- (31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: MARK(g(z0)) -> c3(MARK(z0)) A__F(g(g(z0)), x1) -> c(A__F(g(mark(z0)), f(g(g(z0)), x1)), MARK(g(z0))) MARK(f(f(z0, z1), x1)) -> c2(A__F(a__f(mark(z0), z1), x1), MARK(f(z0, z1))) MARK(f(g(z0), x1)) -> c2(A__F(g(mark(z0)), x1), MARK(g(z0))) A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(f(x0, z1)), x2) -> c(MARK(f(x0, z1))) S tuples: MARK(g(z0)) -> c3(MARK(z0)) A__F(g(g(z0)), x1) -> c(A__F(g(mark(z0)), f(g(g(z0)), x1)), MARK(g(z0))) MARK(f(f(z0, z1), x1)) -> c2(A__F(a__f(mark(z0), z1), x1), MARK(f(z0, z1))) MARK(f(g(z0), x1)) -> c2(A__F(g(mark(z0)), x1), MARK(g(z0))) A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(f(x0, z1)), x2) -> c(MARK(f(x0, z1))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: MARK_1, A__F_2 Compound Symbols: c3_1, c_2, c2_2, c_1 ---------------------------------------- (33) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__F(g(g(z0)), x1) -> c(A__F(g(mark(z0)), f(g(g(z0)), x1)), MARK(g(z0))) by A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) A__F(g(g(x0)), x1) -> c(MARK(g(x0))) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: MARK(g(z0)) -> c3(MARK(z0)) MARK(f(f(z0, z1), x1)) -> c2(A__F(a__f(mark(z0), z1), x1), MARK(f(z0, z1))) MARK(f(g(z0), x1)) -> c2(A__F(g(mark(z0)), x1), MARK(g(z0))) A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(f(x0, z1)), x2) -> c(MARK(f(x0, z1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) A__F(g(g(x0)), x1) -> c(MARK(g(x0))) S tuples: MARK(g(z0)) -> c3(MARK(z0)) MARK(f(f(z0, z1), x1)) -> c2(A__F(a__f(mark(z0), z1), x1), MARK(f(z0, z1))) MARK(f(g(z0), x1)) -> c2(A__F(g(mark(z0)), x1), MARK(g(z0))) A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(f(x0, z1)), x2) -> c(MARK(f(x0, z1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) A__F(g(g(x0)), x1) -> c(MARK(g(x0))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: MARK_1, A__F_2 Compound Symbols: c3_1, c2_2, c_2, c_1 ---------------------------------------- (35) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(f(f(z0, z1), x1)) -> c2(A__F(a__f(mark(z0), z1), x1), MARK(f(z0, z1))) by MARK(f(f(x0, z1), x2)) -> c2(A__F(f(mark(x0), z1), x2), MARK(f(x0, z1))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: MARK(g(z0)) -> c3(MARK(z0)) MARK(f(g(z0), x1)) -> c2(A__F(g(mark(z0)), x1), MARK(g(z0))) A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(f(x0, z1)), x2) -> c(MARK(f(x0, z1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) A__F(g(g(x0)), x1) -> c(MARK(g(x0))) MARK(f(f(x0, z1), x2)) -> c2(A__F(f(mark(x0), z1), x2), MARK(f(x0, z1))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) S tuples: MARK(g(z0)) -> c3(MARK(z0)) MARK(f(g(z0), x1)) -> c2(A__F(g(mark(z0)), x1), MARK(g(z0))) A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(f(x0, z1)), x2) -> c(MARK(f(x0, z1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) A__F(g(g(x0)), x1) -> c(MARK(g(x0))) MARK(f(f(x0, z1), x2)) -> c2(A__F(f(mark(x0), z1), x2), MARK(f(x0, z1))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: MARK_1, A__F_2 Compound Symbols: c3_1, c2_2, c_2, c_1 ---------------------------------------- (37) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: MARK(g(z0)) -> c3(MARK(z0)) MARK(f(g(z0), x1)) -> c2(A__F(g(mark(z0)), x1), MARK(g(z0))) A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(f(x0, z1)), x2) -> c(MARK(f(x0, z1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) A__F(g(g(x0)), x1) -> c(MARK(g(x0))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(f(x0, z1), x2)) -> c2(MARK(f(x0, z1))) S tuples: MARK(g(z0)) -> c3(MARK(z0)) MARK(f(g(z0), x1)) -> c2(A__F(g(mark(z0)), x1), MARK(g(z0))) A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(f(x0, z1)), x2) -> c(MARK(f(x0, z1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) A__F(g(g(x0)), x1) -> c(MARK(g(x0))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(f(x0, z1), x2)) -> c2(MARK(f(x0, z1))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: MARK_1, A__F_2 Compound Symbols: c3_1, c2_2, c_2, c_1, c2_1 ---------------------------------------- (39) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(f(g(z0), x1)) -> c2(A__F(g(mark(z0)), x1), MARK(g(z0))) by MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(f(g(x0), x1)) -> c2(MARK(g(x0))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: MARK(g(z0)) -> c3(MARK(z0)) A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(f(x0, z1)), x2) -> c(MARK(f(x0, z1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) A__F(g(g(x0)), x1) -> c(MARK(g(x0))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(f(x0, z1), x2)) -> c2(MARK(f(x0, z1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(f(g(x0), x1)) -> c2(MARK(g(x0))) S tuples: MARK(g(z0)) -> c3(MARK(z0)) A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(f(x0, z1)), x2) -> c(MARK(f(x0, z1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) A__F(g(g(x0)), x1) -> c(MARK(g(x0))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(f(x0, z1), x2)) -> c2(MARK(f(x0, z1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(f(g(x0), x1)) -> c2(MARK(g(x0))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: MARK_1, A__F_2 Compound Symbols: c3_1, c_2, c_1, c2_2, c2_1 ---------------------------------------- (41) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(g(z0)) -> c3(MARK(z0)) by MARK(g(g(y0))) -> c3(MARK(g(y0))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(f(y0, y1), y2))) -> c3(MARK(f(f(y0, y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) MARK(g(f(g(y0), y1))) -> c3(MARK(f(g(y0), y1))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(f(x0, z1)), x2) -> c(MARK(f(x0, z1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) A__F(g(g(x0)), x1) -> c(MARK(g(x0))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(f(x0, z1), x2)) -> c2(MARK(f(x0, z1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(f(g(x0), x1)) -> c2(MARK(g(x0))) MARK(g(g(y0))) -> c3(MARK(g(y0))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(f(y0, y1), y2))) -> c3(MARK(f(f(y0, y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) MARK(g(f(g(y0), y1))) -> c3(MARK(f(g(y0), y1))) S tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(f(x0, z1)), x2) -> c(MARK(f(x0, z1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) A__F(g(g(x0)), x1) -> c(MARK(g(x0))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(f(x0, z1), x2)) -> c2(MARK(f(x0, z1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(f(g(x0), x1)) -> c2(MARK(g(x0))) MARK(g(g(y0))) -> c3(MARK(g(y0))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(f(y0, y1), y2))) -> c3(MARK(f(f(y0, y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) MARK(g(f(g(y0), y1))) -> c3(MARK(f(g(y0), y1))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c_1, c2_2, c2_1, c3_1 ---------------------------------------- (43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace A__F(g(f(x0, z1)), x2) -> c(MARK(f(x0, z1))) by A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(f(y0, y1), z1)), z2) -> c(MARK(f(f(y0, y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(f(g(y0), z1)), z2) -> c(MARK(f(g(y0), z1))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) A__F(g(g(x0)), x1) -> c(MARK(g(x0))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(f(x0, z1), x2)) -> c2(MARK(f(x0, z1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(f(g(x0), x1)) -> c2(MARK(g(x0))) MARK(g(g(y0))) -> c3(MARK(g(y0))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(f(y0, y1), y2))) -> c3(MARK(f(f(y0, y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) MARK(g(f(g(y0), y1))) -> c3(MARK(f(g(y0), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(f(y0, y1), z1)), z2) -> c(MARK(f(f(y0, y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(f(g(y0), z1)), z2) -> c(MARK(f(g(y0), z1))) S tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) A__F(g(g(x0)), x1) -> c(MARK(g(x0))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(f(x0, z1), x2)) -> c2(MARK(f(x0, z1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(f(g(x0), x1)) -> c2(MARK(g(x0))) MARK(g(g(y0))) -> c3(MARK(g(y0))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(f(y0, y1), y2))) -> c3(MARK(f(f(y0, y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) MARK(g(f(g(y0), y1))) -> c3(MARK(f(g(y0), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(f(y0, y1), z1)), z2) -> c(MARK(f(f(y0, y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(f(g(y0), z1)), z2) -> c(MARK(f(g(y0), z1))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c_1, c2_2, c2_1, c3_1 ---------------------------------------- (45) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace A__F(g(g(x0)), x1) -> c(MARK(g(x0))) by A__F(g(g(g(y0))), z1) -> c(MARK(g(g(y0)))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(f(x0, z1), x2)) -> c2(MARK(f(x0, z1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(f(g(x0), x1)) -> c2(MARK(g(x0))) MARK(g(g(y0))) -> c3(MARK(g(y0))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(f(y0, y1), y2))) -> c3(MARK(f(f(y0, y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) MARK(g(f(g(y0), y1))) -> c3(MARK(f(g(y0), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(f(y0, y1), z1)), z2) -> c(MARK(f(f(y0, y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(f(g(y0), z1)), z2) -> c(MARK(f(g(y0), z1))) A__F(g(g(g(y0))), z1) -> c(MARK(g(g(y0)))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) S tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(f(x0, z1), x2)) -> c2(MARK(f(x0, z1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(f(g(x0), x1)) -> c2(MARK(g(x0))) MARK(g(g(y0))) -> c3(MARK(g(y0))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(f(y0, y1), y2))) -> c3(MARK(f(f(y0, y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) MARK(g(f(g(y0), y1))) -> c3(MARK(f(g(y0), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(f(y0, y1), z1)), z2) -> c(MARK(f(f(y0, y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(f(g(y0), z1)), z2) -> c(MARK(f(g(y0), z1))) A__F(g(g(g(y0))), z1) -> c(MARK(g(g(y0)))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c2_2, c2_1, c3_1, c_1 ---------------------------------------- (47) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(f(f(x0, z1), x2)) -> c2(MARK(f(x0, z1))) by MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(f(g(x0), x1)) -> c2(MARK(g(x0))) MARK(g(g(y0))) -> c3(MARK(g(y0))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(f(y0, y1), y2))) -> c3(MARK(f(f(y0, y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) MARK(g(f(g(y0), y1))) -> c3(MARK(f(g(y0), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(f(y0, y1), z1)), z2) -> c(MARK(f(f(y0, y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(f(g(y0), z1)), z2) -> c(MARK(f(g(y0), z1))) A__F(g(g(g(y0))), z1) -> c(MARK(g(g(y0)))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) S tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(f(g(x0), x1)) -> c2(MARK(g(x0))) MARK(g(g(y0))) -> c3(MARK(g(y0))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(f(y0, y1), y2))) -> c3(MARK(f(f(y0, y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) MARK(g(f(g(y0), y1))) -> c3(MARK(f(g(y0), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(f(y0, y1), z1)), z2) -> c(MARK(f(f(y0, y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(f(g(y0), z1)), z2) -> c(MARK(f(g(y0), z1))) A__F(g(g(g(y0))), z1) -> c(MARK(g(g(y0)))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c2_2, c2_1, c3_1, c_1 ---------------------------------------- (49) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(f(g(x0), x1)) -> c2(MARK(g(x0))) by MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(g(y0))) -> c3(MARK(g(y0))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(f(y0, y1), y2))) -> c3(MARK(f(f(y0, y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) MARK(g(f(g(y0), y1))) -> c3(MARK(f(g(y0), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(f(y0, y1), z1)), z2) -> c(MARK(f(f(y0, y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(f(g(y0), z1)), z2) -> c(MARK(f(g(y0), z1))) A__F(g(g(g(y0))), z1) -> c(MARK(g(g(y0)))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) S tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(g(y0))) -> c3(MARK(g(y0))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(f(y0, y1), y2))) -> c3(MARK(f(f(y0, y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) MARK(g(f(g(y0), y1))) -> c3(MARK(f(g(y0), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(f(y0, y1), z1)), z2) -> c(MARK(f(f(y0, y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(f(g(y0), z1)), z2) -> c(MARK(f(g(y0), z1))) A__F(g(g(g(y0))), z1) -> c(MARK(g(g(y0)))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c2_2, c3_1, c_1, c2_1 ---------------------------------------- (51) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(g(g(y0))) -> c3(MARK(g(y0))) by MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(f(y0, y1), y2))) -> c3(MARK(f(f(y0, y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) MARK(g(f(g(y0), y1))) -> c3(MARK(f(g(y0), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(f(y0, y1), z1)), z2) -> c(MARK(f(f(y0, y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(f(g(y0), z1)), z2) -> c(MARK(f(g(y0), z1))) A__F(g(g(g(y0))), z1) -> c(MARK(g(g(y0)))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) S tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(f(y0, y1), y2))) -> c3(MARK(f(f(y0, y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) MARK(g(f(g(y0), y1))) -> c3(MARK(f(g(y0), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(f(y0, y1), z1)), z2) -> c(MARK(f(f(y0, y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(f(g(y0), z1)), z2) -> c(MARK(f(g(y0), z1))) A__F(g(g(g(y0))), z1) -> c(MARK(g(g(y0)))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c2_2, c3_1, c_1, c2_1 ---------------------------------------- (53) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(g(f(f(y0, y1), y2))) -> c3(MARK(f(f(y0, y1), y2))) by MARK(g(f(f(f(y0, y1), z1), z2))) -> c3(MARK(f(f(f(y0, y1), z1), z2))) MARK(g(f(f(g(y0), z1), z2))) -> c3(MARK(f(f(g(y0), z1), z2))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) MARK(g(f(g(y0), y1))) -> c3(MARK(f(g(y0), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(f(y0, y1), z1)), z2) -> c(MARK(f(f(y0, y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(f(g(y0), z1)), z2) -> c(MARK(f(g(y0), z1))) A__F(g(g(g(y0))), z1) -> c(MARK(g(g(y0)))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) S tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) MARK(g(f(g(y0), y1))) -> c3(MARK(f(g(y0), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(f(y0, y1), z1)), z2) -> c(MARK(f(f(y0, y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(f(g(y0), z1)), z2) -> c(MARK(f(g(y0), z1))) A__F(g(g(g(y0))), z1) -> c(MARK(g(g(y0)))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c2_2, c3_1, c_1, c2_1 ---------------------------------------- (55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(g(f(g(y0), y1))) -> c3(MARK(f(g(y0), y1))) by MARK(g(f(g(f(y0, y1)), z1))) -> c3(MARK(f(g(f(y0, y1)), z1))) MARK(g(f(g(g(y0)), z1))) -> c3(MARK(f(g(g(y0)), z1))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(f(y0, y1), z1)), z2) -> c(MARK(f(f(y0, y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(f(g(y0), z1)), z2) -> c(MARK(f(g(y0), z1))) A__F(g(g(g(y0))), z1) -> c(MARK(g(g(y0)))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) S tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(f(y0, y1), z1)), z2) -> c(MARK(f(f(y0, y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(f(g(y0), z1)), z2) -> c(MARK(f(g(y0), z1))) A__F(g(g(g(y0))), z1) -> c(MARK(g(g(y0)))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c2_2, c3_1, c_1, c2_1 ---------------------------------------- (57) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace A__F(g(f(f(y0, y1), z1)), z2) -> c(MARK(f(f(y0, y1), z1))) by A__F(g(f(f(f(y0, y1), z1), z2)), z3) -> c(MARK(f(f(f(y0, y1), z1), z2))) A__F(g(f(f(g(y0), z1), z2)), z3) -> c(MARK(f(f(g(y0), z1), z2))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(f(g(y0), z1)), z2) -> c(MARK(f(g(y0), z1))) A__F(g(g(g(y0))), z1) -> c(MARK(g(g(y0)))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) S tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(f(g(y0), z1)), z2) -> c(MARK(f(g(y0), z1))) A__F(g(g(g(y0))), z1) -> c(MARK(g(g(y0)))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c2_2, c3_1, c_1, c2_1 ---------------------------------------- (59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace A__F(g(f(g(y0), z1)), z2) -> c(MARK(f(g(y0), z1))) by A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2) -> c(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) A__F(g(f(g(f(f(g(y0), y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(g(y0), y1), y2)), z1))) A__F(g(f(g(f(f(y0, y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(y0, y1), y2)), z1))) A__F(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2) -> c(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) A__F(g(f(g(f(g(g(y0)), y1)), z1)), z2) -> c(MARK(f(g(f(g(g(y0)), y1)), z1))) A__F(g(f(g(f(g(y0), y1)), z1)), z2) -> c(MARK(f(g(f(g(y0), y1)), z1))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(g(g(y0))), z1) -> c(MARK(g(g(y0)))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) A__F(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2) -> c(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) A__F(g(f(g(f(f(g(y0), y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(g(y0), y1), y2)), z1))) A__F(g(f(g(f(f(y0, y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(y0, y1), y2)), z1))) A__F(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2) -> c(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) A__F(g(f(g(f(g(g(y0)), y1)), z1)), z2) -> c(MARK(f(g(f(g(g(y0)), y1)), z1))) A__F(g(f(g(f(g(y0), y1)), z1)), z2) -> c(MARK(f(g(f(g(y0), y1)), z1))) S tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(g(g(y0))), z1) -> c(MARK(g(g(y0)))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) A__F(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2) -> c(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) A__F(g(f(g(f(f(g(y0), y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(g(y0), y1), y2)), z1))) A__F(g(f(g(f(f(y0, y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(y0, y1), y2)), z1))) A__F(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2) -> c(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) A__F(g(f(g(f(g(g(y0)), y1)), z1)), z2) -> c(MARK(f(g(f(g(g(y0)), y1)), z1))) A__F(g(f(g(f(g(y0), y1)), z1)), z2) -> c(MARK(f(g(f(g(y0), y1)), z1))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c2_2, c3_1, c_1, c2_1 ---------------------------------------- (61) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace A__F(g(g(g(y0))), z1) -> c(MARK(g(g(y0)))) by A__F(g(g(g(g(y0)))), z1) -> c(MARK(g(g(g(y0))))) A__F(g(g(g(f(f(f(y0, y1), y2), y3)))), z1) -> c(MARK(g(g(f(f(f(y0, y1), y2), y3))))) A__F(g(g(g(f(f(g(y0), y1), y2)))), z1) -> c(MARK(g(g(f(f(g(y0), y1), y2))))) A__F(g(g(g(f(f(y0, y1), y2)))), z1) -> c(MARK(g(g(f(f(y0, y1), y2))))) A__F(g(g(g(f(g(f(y0, y1)), y2)))), z1) -> c(MARK(g(g(f(g(f(y0, y1)), y2))))) A__F(g(g(g(f(g(g(y0)), y1)))), z1) -> c(MARK(g(g(f(g(g(y0)), y1))))) A__F(g(g(g(f(g(y0), y1)))), z1) -> c(MARK(g(g(f(g(y0), y1))))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) A__F(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2) -> c(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) A__F(g(f(g(f(f(g(y0), y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(g(y0), y1), y2)), z1))) A__F(g(f(g(f(f(y0, y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(y0, y1), y2)), z1))) A__F(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2) -> c(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) A__F(g(f(g(f(g(g(y0)), y1)), z1)), z2) -> c(MARK(f(g(f(g(g(y0)), y1)), z1))) A__F(g(f(g(f(g(y0), y1)), z1)), z2) -> c(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(g(g(g(y0)))), z1) -> c(MARK(g(g(g(y0))))) A__F(g(g(g(f(f(f(y0, y1), y2), y3)))), z1) -> c(MARK(g(g(f(f(f(y0, y1), y2), y3))))) A__F(g(g(g(f(f(g(y0), y1), y2)))), z1) -> c(MARK(g(g(f(f(g(y0), y1), y2))))) A__F(g(g(g(f(f(y0, y1), y2)))), z1) -> c(MARK(g(g(f(f(y0, y1), y2))))) A__F(g(g(g(f(g(f(y0, y1)), y2)))), z1) -> c(MARK(g(g(f(g(f(y0, y1)), y2))))) A__F(g(g(g(f(g(g(y0)), y1)))), z1) -> c(MARK(g(g(f(g(g(y0)), y1))))) A__F(g(g(g(f(g(y0), y1)))), z1) -> c(MARK(g(g(f(g(y0), y1))))) S tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) A__F(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2) -> c(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) A__F(g(f(g(f(f(g(y0), y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(g(y0), y1), y2)), z1))) A__F(g(f(g(f(f(y0, y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(y0, y1), y2)), z1))) A__F(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2) -> c(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) A__F(g(f(g(f(g(g(y0)), y1)), z1)), z2) -> c(MARK(f(g(f(g(g(y0)), y1)), z1))) A__F(g(f(g(f(g(y0), y1)), z1)), z2) -> c(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(g(g(g(y0)))), z1) -> c(MARK(g(g(g(y0))))) A__F(g(g(g(f(f(f(y0, y1), y2), y3)))), z1) -> c(MARK(g(g(f(f(f(y0, y1), y2), y3))))) A__F(g(g(g(f(f(g(y0), y1), y2)))), z1) -> c(MARK(g(g(f(f(g(y0), y1), y2))))) A__F(g(g(g(f(f(y0, y1), y2)))), z1) -> c(MARK(g(g(f(f(y0, y1), y2))))) A__F(g(g(g(f(g(f(y0, y1)), y2)))), z1) -> c(MARK(g(g(f(g(f(y0, y1)), y2))))) A__F(g(g(g(f(g(g(y0)), y1)))), z1) -> c(MARK(g(g(f(g(g(y0)), y1))))) A__F(g(g(g(f(g(y0), y1)))), z1) -> c(MARK(g(g(f(g(y0), y1))))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c2_2, c3_1, c_1, c2_1 ---------------------------------------- (63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace A__F(g(g(f(f(y0, y1), y2))), z1) -> c(MARK(g(f(f(y0, y1), y2)))) by A__F(g(g(f(f(f(y0, y1), z1), z2))), z3) -> c(MARK(g(f(f(f(y0, y1), z1), z2)))) A__F(g(g(f(f(g(y0), z1), z2))), z3) -> c(MARK(g(f(f(g(y0), z1), z2)))) A__F(g(g(f(f(f(f(y0, y1), y2), z1), z2))), z3) -> c(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) A__F(g(g(f(f(f(g(y0), y1), z1), z2))), z3) -> c(MARK(g(f(f(f(g(y0), y1), z1), z2)))) A__F(g(g(f(f(g(f(y0, y1)), z1), z2))), z3) -> c(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) A__F(g(g(f(f(g(g(y0)), z1), z2))), z3) -> c(MARK(g(f(f(g(g(y0)), z1), z2)))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) A__F(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2) -> c(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) A__F(g(f(g(f(f(g(y0), y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(g(y0), y1), y2)), z1))) A__F(g(f(g(f(f(y0, y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(y0, y1), y2)), z1))) A__F(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2) -> c(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) A__F(g(f(g(f(g(g(y0)), y1)), z1)), z2) -> c(MARK(f(g(f(g(g(y0)), y1)), z1))) A__F(g(f(g(f(g(y0), y1)), z1)), z2) -> c(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(g(g(g(y0)))), z1) -> c(MARK(g(g(g(y0))))) A__F(g(g(g(f(f(f(y0, y1), y2), y3)))), z1) -> c(MARK(g(g(f(f(f(y0, y1), y2), y3))))) A__F(g(g(g(f(f(g(y0), y1), y2)))), z1) -> c(MARK(g(g(f(f(g(y0), y1), y2))))) A__F(g(g(g(f(f(y0, y1), y2)))), z1) -> c(MARK(g(g(f(f(y0, y1), y2))))) A__F(g(g(g(f(g(f(y0, y1)), y2)))), z1) -> c(MARK(g(g(f(g(f(y0, y1)), y2))))) A__F(g(g(g(f(g(g(y0)), y1)))), z1) -> c(MARK(g(g(f(g(g(y0)), y1))))) A__F(g(g(g(f(g(y0), y1)))), z1) -> c(MARK(g(g(f(g(y0), y1))))) A__F(g(g(f(f(f(f(y0, y1), y2), z1), z2))), z3) -> c(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) A__F(g(g(f(f(f(g(y0), y1), z1), z2))), z3) -> c(MARK(g(f(f(f(g(y0), y1), z1), z2)))) A__F(g(g(f(f(g(f(y0, y1)), z1), z2))), z3) -> c(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) A__F(g(g(f(f(g(g(y0)), z1), z2))), z3) -> c(MARK(g(f(f(g(g(y0)), z1), z2)))) S tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) A__F(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2) -> c(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) A__F(g(f(g(f(f(g(y0), y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(g(y0), y1), y2)), z1))) A__F(g(f(g(f(f(y0, y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(y0, y1), y2)), z1))) A__F(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2) -> c(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) A__F(g(f(g(f(g(g(y0)), y1)), z1)), z2) -> c(MARK(f(g(f(g(g(y0)), y1)), z1))) A__F(g(f(g(f(g(y0), y1)), z1)), z2) -> c(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(g(g(g(y0)))), z1) -> c(MARK(g(g(g(y0))))) A__F(g(g(g(f(f(f(y0, y1), y2), y3)))), z1) -> c(MARK(g(g(f(f(f(y0, y1), y2), y3))))) A__F(g(g(g(f(f(g(y0), y1), y2)))), z1) -> c(MARK(g(g(f(f(g(y0), y1), y2))))) A__F(g(g(g(f(f(y0, y1), y2)))), z1) -> c(MARK(g(g(f(f(y0, y1), y2))))) A__F(g(g(g(f(g(f(y0, y1)), y2)))), z1) -> c(MARK(g(g(f(g(f(y0, y1)), y2))))) A__F(g(g(g(f(g(g(y0)), y1)))), z1) -> c(MARK(g(g(f(g(g(y0)), y1))))) A__F(g(g(g(f(g(y0), y1)))), z1) -> c(MARK(g(g(f(g(y0), y1))))) A__F(g(g(f(f(f(f(y0, y1), y2), z1), z2))), z3) -> c(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) A__F(g(g(f(f(f(g(y0), y1), z1), z2))), z3) -> c(MARK(g(f(f(f(g(y0), y1), z1), z2)))) A__F(g(g(f(f(g(f(y0, y1)), z1), z2))), z3) -> c(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) A__F(g(g(f(f(g(g(y0)), z1), z2))), z3) -> c(MARK(g(f(f(g(g(y0)), z1), z2)))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c2_2, c3_1, c_1, c2_1 ---------------------------------------- (65) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace A__F(g(g(f(g(y0), y1))), z1) -> c(MARK(g(f(g(y0), y1)))) by A__F(g(g(f(g(f(y0, y1)), z1))), z2) -> c(MARK(g(f(g(f(y0, y1)), z1)))) A__F(g(g(f(g(g(y0)), z1))), z2) -> c(MARK(g(f(g(g(y0)), z1)))) A__F(g(g(f(g(f(f(f(y0, y1), y2), y3)), z1))), z2) -> c(MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1)))) A__F(g(g(f(g(f(f(g(y0), y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(g(y0), y1), y2)), z1)))) A__F(g(g(f(g(f(f(y0, y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(y0, y1), y2)), z1)))) A__F(g(g(f(g(f(g(f(y0, y1)), y2)), z1))), z2) -> c(MARK(g(f(g(f(g(f(y0, y1)), y2)), z1)))) A__F(g(g(f(g(f(g(g(y0)), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(g(y0)), y1)), z1)))) A__F(g(g(f(g(f(g(y0), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(y0), y1)), z1)))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) A__F(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2) -> c(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) A__F(g(f(g(f(f(g(y0), y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(g(y0), y1), y2)), z1))) A__F(g(f(g(f(f(y0, y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(y0, y1), y2)), z1))) A__F(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2) -> c(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) A__F(g(f(g(f(g(g(y0)), y1)), z1)), z2) -> c(MARK(f(g(f(g(g(y0)), y1)), z1))) A__F(g(f(g(f(g(y0), y1)), z1)), z2) -> c(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(g(g(g(y0)))), z1) -> c(MARK(g(g(g(y0))))) A__F(g(g(g(f(f(f(y0, y1), y2), y3)))), z1) -> c(MARK(g(g(f(f(f(y0, y1), y2), y3))))) A__F(g(g(g(f(f(g(y0), y1), y2)))), z1) -> c(MARK(g(g(f(f(g(y0), y1), y2))))) A__F(g(g(g(f(f(y0, y1), y2)))), z1) -> c(MARK(g(g(f(f(y0, y1), y2))))) A__F(g(g(g(f(g(f(y0, y1)), y2)))), z1) -> c(MARK(g(g(f(g(f(y0, y1)), y2))))) A__F(g(g(g(f(g(g(y0)), y1)))), z1) -> c(MARK(g(g(f(g(g(y0)), y1))))) A__F(g(g(g(f(g(y0), y1)))), z1) -> c(MARK(g(g(f(g(y0), y1))))) A__F(g(g(f(f(f(f(y0, y1), y2), z1), z2))), z3) -> c(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) A__F(g(g(f(f(f(g(y0), y1), z1), z2))), z3) -> c(MARK(g(f(f(f(g(y0), y1), z1), z2)))) A__F(g(g(f(f(g(f(y0, y1)), z1), z2))), z3) -> c(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) A__F(g(g(f(f(g(g(y0)), z1), z2))), z3) -> c(MARK(g(f(f(g(g(y0)), z1), z2)))) A__F(g(g(f(g(f(f(f(y0, y1), y2), y3)), z1))), z2) -> c(MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1)))) A__F(g(g(f(g(f(f(g(y0), y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(g(y0), y1), y2)), z1)))) A__F(g(g(f(g(f(f(y0, y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(y0, y1), y2)), z1)))) A__F(g(g(f(g(f(g(f(y0, y1)), y2)), z1))), z2) -> c(MARK(g(f(g(f(g(f(y0, y1)), y2)), z1)))) A__F(g(g(f(g(f(g(g(y0)), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(g(y0)), y1)), z1)))) A__F(g(g(f(g(f(g(y0), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(y0), y1)), z1)))) S tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) A__F(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2) -> c(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) A__F(g(f(g(f(f(g(y0), y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(g(y0), y1), y2)), z1))) A__F(g(f(g(f(f(y0, y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(y0, y1), y2)), z1))) A__F(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2) -> c(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) A__F(g(f(g(f(g(g(y0)), y1)), z1)), z2) -> c(MARK(f(g(f(g(g(y0)), y1)), z1))) A__F(g(f(g(f(g(y0), y1)), z1)), z2) -> c(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(g(g(g(y0)))), z1) -> c(MARK(g(g(g(y0))))) A__F(g(g(g(f(f(f(y0, y1), y2), y3)))), z1) -> c(MARK(g(g(f(f(f(y0, y1), y2), y3))))) A__F(g(g(g(f(f(g(y0), y1), y2)))), z1) -> c(MARK(g(g(f(f(g(y0), y1), y2))))) A__F(g(g(g(f(f(y0, y1), y2)))), z1) -> c(MARK(g(g(f(f(y0, y1), y2))))) A__F(g(g(g(f(g(f(y0, y1)), y2)))), z1) -> c(MARK(g(g(f(g(f(y0, y1)), y2))))) A__F(g(g(g(f(g(g(y0)), y1)))), z1) -> c(MARK(g(g(f(g(g(y0)), y1))))) A__F(g(g(g(f(g(y0), y1)))), z1) -> c(MARK(g(g(f(g(y0), y1))))) A__F(g(g(f(f(f(f(y0, y1), y2), z1), z2))), z3) -> c(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) A__F(g(g(f(f(f(g(y0), y1), z1), z2))), z3) -> c(MARK(g(f(f(f(g(y0), y1), z1), z2)))) A__F(g(g(f(f(g(f(y0, y1)), z1), z2))), z3) -> c(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) A__F(g(g(f(f(g(g(y0)), z1), z2))), z3) -> c(MARK(g(f(f(g(g(y0)), z1), z2)))) A__F(g(g(f(g(f(f(f(y0, y1), y2), y3)), z1))), z2) -> c(MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1)))) A__F(g(g(f(g(f(f(g(y0), y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(g(y0), y1), y2)), z1)))) A__F(g(g(f(g(f(f(y0, y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(y0, y1), y2)), z1)))) A__F(g(g(f(g(f(g(f(y0, y1)), y2)), z1))), z2) -> c(MARK(g(f(g(f(g(f(y0, y1)), y2)), z1)))) A__F(g(g(f(g(f(g(g(y0)), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(g(y0)), y1)), z1)))) A__F(g(g(f(g(f(g(y0), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(y0), y1)), z1)))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c2_2, c3_1, c_1, c2_1 ---------------------------------------- (67) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(f(f(f(y0, y1), z1), z2)) -> c2(MARK(f(f(y0, y1), z1))) by MARK(f(f(f(f(y0, y1), z1), z2), z3)) -> c2(MARK(f(f(f(y0, y1), z1), z2))) MARK(f(f(f(g(y0), z1), z2), z3)) -> c2(MARK(f(f(g(y0), z1), z2))) MARK(f(f(f(f(f(y0, y1), y2), z1), z2), z3)) -> c2(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(f(f(f(f(g(y0), y1), z1), z2), z3)) -> c2(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(f(f(f(g(f(y0, y1)), z1), z2), z3)) -> c2(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(f(f(f(g(g(y0)), z1), z2), z3)) -> c2(MARK(f(f(g(g(y0)), z1), z2))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) A__F(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2) -> c(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) A__F(g(f(g(f(f(g(y0), y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(g(y0), y1), y2)), z1))) A__F(g(f(g(f(f(y0, y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(y0, y1), y2)), z1))) A__F(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2) -> c(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) A__F(g(f(g(f(g(g(y0)), y1)), z1)), z2) -> c(MARK(f(g(f(g(g(y0)), y1)), z1))) A__F(g(f(g(f(g(y0), y1)), z1)), z2) -> c(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(g(g(g(y0)))), z1) -> c(MARK(g(g(g(y0))))) A__F(g(g(g(f(f(f(y0, y1), y2), y3)))), z1) -> c(MARK(g(g(f(f(f(y0, y1), y2), y3))))) A__F(g(g(g(f(f(g(y0), y1), y2)))), z1) -> c(MARK(g(g(f(f(g(y0), y1), y2))))) A__F(g(g(g(f(f(y0, y1), y2)))), z1) -> c(MARK(g(g(f(f(y0, y1), y2))))) A__F(g(g(g(f(g(f(y0, y1)), y2)))), z1) -> c(MARK(g(g(f(g(f(y0, y1)), y2))))) A__F(g(g(g(f(g(g(y0)), y1)))), z1) -> c(MARK(g(g(f(g(g(y0)), y1))))) A__F(g(g(g(f(g(y0), y1)))), z1) -> c(MARK(g(g(f(g(y0), y1))))) A__F(g(g(f(f(f(f(y0, y1), y2), z1), z2))), z3) -> c(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) A__F(g(g(f(f(f(g(y0), y1), z1), z2))), z3) -> c(MARK(g(f(f(f(g(y0), y1), z1), z2)))) A__F(g(g(f(f(g(f(y0, y1)), z1), z2))), z3) -> c(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) A__F(g(g(f(f(g(g(y0)), z1), z2))), z3) -> c(MARK(g(f(f(g(g(y0)), z1), z2)))) A__F(g(g(f(g(f(f(f(y0, y1), y2), y3)), z1))), z2) -> c(MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1)))) A__F(g(g(f(g(f(f(g(y0), y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(g(y0), y1), y2)), z1)))) A__F(g(g(f(g(f(f(y0, y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(y0, y1), y2)), z1)))) A__F(g(g(f(g(f(g(f(y0, y1)), y2)), z1))), z2) -> c(MARK(g(f(g(f(g(f(y0, y1)), y2)), z1)))) A__F(g(g(f(g(f(g(g(y0)), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(g(y0)), y1)), z1)))) A__F(g(g(f(g(f(g(y0), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(y0), y1)), z1)))) MARK(f(f(f(f(f(y0, y1), y2), z1), z2), z3)) -> c2(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(f(f(f(f(g(y0), y1), z1), z2), z3)) -> c2(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(f(f(f(g(f(y0, y1)), z1), z2), z3)) -> c2(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(f(f(f(g(g(y0)), z1), z2), z3)) -> c2(MARK(f(f(g(g(y0)), z1), z2))) S tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) A__F(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2) -> c(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) A__F(g(f(g(f(f(g(y0), y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(g(y0), y1), y2)), z1))) A__F(g(f(g(f(f(y0, y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(y0, y1), y2)), z1))) A__F(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2) -> c(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) A__F(g(f(g(f(g(g(y0)), y1)), z1)), z2) -> c(MARK(f(g(f(g(g(y0)), y1)), z1))) A__F(g(f(g(f(g(y0), y1)), z1)), z2) -> c(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(g(g(g(y0)))), z1) -> c(MARK(g(g(g(y0))))) A__F(g(g(g(f(f(f(y0, y1), y2), y3)))), z1) -> c(MARK(g(g(f(f(f(y0, y1), y2), y3))))) A__F(g(g(g(f(f(g(y0), y1), y2)))), z1) -> c(MARK(g(g(f(f(g(y0), y1), y2))))) A__F(g(g(g(f(f(y0, y1), y2)))), z1) -> c(MARK(g(g(f(f(y0, y1), y2))))) A__F(g(g(g(f(g(f(y0, y1)), y2)))), z1) -> c(MARK(g(g(f(g(f(y0, y1)), y2))))) A__F(g(g(g(f(g(g(y0)), y1)))), z1) -> c(MARK(g(g(f(g(g(y0)), y1))))) A__F(g(g(g(f(g(y0), y1)))), z1) -> c(MARK(g(g(f(g(y0), y1))))) A__F(g(g(f(f(f(f(y0, y1), y2), z1), z2))), z3) -> c(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) A__F(g(g(f(f(f(g(y0), y1), z1), z2))), z3) -> c(MARK(g(f(f(f(g(y0), y1), z1), z2)))) A__F(g(g(f(f(g(f(y0, y1)), z1), z2))), z3) -> c(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) A__F(g(g(f(f(g(g(y0)), z1), z2))), z3) -> c(MARK(g(f(f(g(g(y0)), z1), z2)))) A__F(g(g(f(g(f(f(f(y0, y1), y2), y3)), z1))), z2) -> c(MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1)))) A__F(g(g(f(g(f(f(g(y0), y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(g(y0), y1), y2)), z1)))) A__F(g(g(f(g(f(f(y0, y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(y0, y1), y2)), z1)))) A__F(g(g(f(g(f(g(f(y0, y1)), y2)), z1))), z2) -> c(MARK(g(f(g(f(g(f(y0, y1)), y2)), z1)))) A__F(g(g(f(g(f(g(g(y0)), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(g(y0)), y1)), z1)))) A__F(g(g(f(g(f(g(y0), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(y0), y1)), z1)))) MARK(f(f(f(f(f(y0, y1), y2), z1), z2), z3)) -> c2(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(f(f(f(f(g(y0), y1), z1), z2), z3)) -> c2(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(f(f(f(g(f(y0, y1)), z1), z2), z3)) -> c2(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(f(f(f(g(g(y0)), z1), z2), z3)) -> c2(MARK(f(f(g(g(y0)), z1), z2))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c2_2, c3_1, c_1, c2_1 ---------------------------------------- (69) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(f(f(g(y0), z1), z2)) -> c2(MARK(f(g(y0), z1))) by MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(f(g(f(f(f(y0, y1), y2), y3)), z1), z2)) -> c2(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(f(f(g(f(f(g(y0), y1), y2)), z1), z2)) -> c2(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(f(f(g(f(f(y0, y1), y2)), z1), z2)) -> c2(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(f(f(g(f(g(f(y0, y1)), y2)), z1), z2)) -> c2(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(f(f(g(f(g(g(y0)), y1)), z1), z2)) -> c2(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(f(f(g(f(g(y0), y1)), z1), z2)) -> c2(MARK(f(g(f(g(y0), y1)), z1))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) A__F(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2) -> c(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) A__F(g(f(g(f(f(g(y0), y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(g(y0), y1), y2)), z1))) A__F(g(f(g(f(f(y0, y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(y0, y1), y2)), z1))) A__F(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2) -> c(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) A__F(g(f(g(f(g(g(y0)), y1)), z1)), z2) -> c(MARK(f(g(f(g(g(y0)), y1)), z1))) A__F(g(f(g(f(g(y0), y1)), z1)), z2) -> c(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(g(g(g(y0)))), z1) -> c(MARK(g(g(g(y0))))) A__F(g(g(g(f(f(f(y0, y1), y2), y3)))), z1) -> c(MARK(g(g(f(f(f(y0, y1), y2), y3))))) A__F(g(g(g(f(f(g(y0), y1), y2)))), z1) -> c(MARK(g(g(f(f(g(y0), y1), y2))))) A__F(g(g(g(f(f(y0, y1), y2)))), z1) -> c(MARK(g(g(f(f(y0, y1), y2))))) A__F(g(g(g(f(g(f(y0, y1)), y2)))), z1) -> c(MARK(g(g(f(g(f(y0, y1)), y2))))) A__F(g(g(g(f(g(g(y0)), y1)))), z1) -> c(MARK(g(g(f(g(g(y0)), y1))))) A__F(g(g(g(f(g(y0), y1)))), z1) -> c(MARK(g(g(f(g(y0), y1))))) A__F(g(g(f(f(f(f(y0, y1), y2), z1), z2))), z3) -> c(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) A__F(g(g(f(f(f(g(y0), y1), z1), z2))), z3) -> c(MARK(g(f(f(f(g(y0), y1), z1), z2)))) A__F(g(g(f(f(g(f(y0, y1)), z1), z2))), z3) -> c(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) A__F(g(g(f(f(g(g(y0)), z1), z2))), z3) -> c(MARK(g(f(f(g(g(y0)), z1), z2)))) A__F(g(g(f(g(f(f(f(y0, y1), y2), y3)), z1))), z2) -> c(MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1)))) A__F(g(g(f(g(f(f(g(y0), y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(g(y0), y1), y2)), z1)))) A__F(g(g(f(g(f(f(y0, y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(y0, y1), y2)), z1)))) A__F(g(g(f(g(f(g(f(y0, y1)), y2)), z1))), z2) -> c(MARK(g(f(g(f(g(f(y0, y1)), y2)), z1)))) A__F(g(g(f(g(f(g(g(y0)), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(g(y0)), y1)), z1)))) A__F(g(g(f(g(f(g(y0), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(y0), y1)), z1)))) MARK(f(f(f(f(f(y0, y1), y2), z1), z2), z3)) -> c2(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(f(f(f(f(g(y0), y1), z1), z2), z3)) -> c2(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(f(f(f(g(f(y0, y1)), z1), z2), z3)) -> c2(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(f(f(f(g(g(y0)), z1), z2), z3)) -> c2(MARK(f(f(g(g(y0)), z1), z2))) MARK(f(f(g(f(f(f(y0, y1), y2), y3)), z1), z2)) -> c2(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(f(f(g(f(f(g(y0), y1), y2)), z1), z2)) -> c2(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(f(f(g(f(f(y0, y1), y2)), z1), z2)) -> c2(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(f(f(g(f(g(f(y0, y1)), y2)), z1), z2)) -> c2(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(f(f(g(f(g(g(y0)), y1)), z1), z2)) -> c2(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(f(f(g(f(g(y0), y1)), z1), z2)) -> c2(MARK(f(g(f(g(y0), y1)), z1))) S tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) A__F(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2) -> c(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) A__F(g(f(g(f(f(g(y0), y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(g(y0), y1), y2)), z1))) A__F(g(f(g(f(f(y0, y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(y0, y1), y2)), z1))) A__F(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2) -> c(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) A__F(g(f(g(f(g(g(y0)), y1)), z1)), z2) -> c(MARK(f(g(f(g(g(y0)), y1)), z1))) A__F(g(f(g(f(g(y0), y1)), z1)), z2) -> c(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(g(g(g(y0)))), z1) -> c(MARK(g(g(g(y0))))) A__F(g(g(g(f(f(f(y0, y1), y2), y3)))), z1) -> c(MARK(g(g(f(f(f(y0, y1), y2), y3))))) A__F(g(g(g(f(f(g(y0), y1), y2)))), z1) -> c(MARK(g(g(f(f(g(y0), y1), y2))))) A__F(g(g(g(f(f(y0, y1), y2)))), z1) -> c(MARK(g(g(f(f(y0, y1), y2))))) A__F(g(g(g(f(g(f(y0, y1)), y2)))), z1) -> c(MARK(g(g(f(g(f(y0, y1)), y2))))) A__F(g(g(g(f(g(g(y0)), y1)))), z1) -> c(MARK(g(g(f(g(g(y0)), y1))))) A__F(g(g(g(f(g(y0), y1)))), z1) -> c(MARK(g(g(f(g(y0), y1))))) A__F(g(g(f(f(f(f(y0, y1), y2), z1), z2))), z3) -> c(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) A__F(g(g(f(f(f(g(y0), y1), z1), z2))), z3) -> c(MARK(g(f(f(f(g(y0), y1), z1), z2)))) A__F(g(g(f(f(g(f(y0, y1)), z1), z2))), z3) -> c(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) A__F(g(g(f(f(g(g(y0)), z1), z2))), z3) -> c(MARK(g(f(f(g(g(y0)), z1), z2)))) A__F(g(g(f(g(f(f(f(y0, y1), y2), y3)), z1))), z2) -> c(MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1)))) A__F(g(g(f(g(f(f(g(y0), y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(g(y0), y1), y2)), z1)))) A__F(g(g(f(g(f(f(y0, y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(y0, y1), y2)), z1)))) A__F(g(g(f(g(f(g(f(y0, y1)), y2)), z1))), z2) -> c(MARK(g(f(g(f(g(f(y0, y1)), y2)), z1)))) A__F(g(g(f(g(f(g(g(y0)), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(g(y0)), y1)), z1)))) A__F(g(g(f(g(f(g(y0), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(y0), y1)), z1)))) MARK(f(f(f(f(f(y0, y1), y2), z1), z2), z3)) -> c2(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(f(f(f(f(g(y0), y1), z1), z2), z3)) -> c2(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(f(f(f(g(f(y0, y1)), z1), z2), z3)) -> c2(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(f(f(f(g(g(y0)), z1), z2), z3)) -> c2(MARK(f(f(g(g(y0)), z1), z2))) MARK(f(f(g(f(f(f(y0, y1), y2), y3)), z1), z2)) -> c2(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(f(f(g(f(f(g(y0), y1), y2)), z1), z2)) -> c2(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(f(f(g(f(f(y0, y1), y2)), z1), z2)) -> c2(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(f(f(g(f(g(f(y0, y1)), y2)), z1), z2)) -> c2(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(f(f(g(f(g(g(y0)), y1)), z1), z2)) -> c2(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(f(f(g(f(g(y0), y1)), z1), z2)) -> c2(MARK(f(g(f(g(y0), y1)), z1))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c2_2, c3_1, c_1, c2_1 ---------------------------------------- (71) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(f(g(g(y0)), z1)) -> c2(MARK(g(g(y0)))) by MARK(f(g(g(g(y0))), z1)) -> c2(MARK(g(g(g(y0))))) MARK(f(g(g(f(f(f(y0, y1), y2), y3))), z1)) -> c2(MARK(g(g(f(f(f(y0, y1), y2), y3))))) MARK(f(g(g(f(f(g(y0), y1), y2))), z1)) -> c2(MARK(g(g(f(f(g(y0), y1), y2))))) MARK(f(g(g(f(f(y0, y1), y2))), z1)) -> c2(MARK(g(g(f(f(y0, y1), y2))))) MARK(f(g(g(f(g(f(y0, y1)), y2))), z1)) -> c2(MARK(g(g(f(g(f(y0, y1)), y2))))) MARK(f(g(g(f(g(g(y0)), y1))), z1)) -> c2(MARK(g(g(f(g(g(y0)), y1))))) MARK(f(g(g(f(g(y0), y1))), z1)) -> c2(MARK(g(g(f(g(y0), y1))))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) A__F(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2) -> c(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) A__F(g(f(g(f(f(g(y0), y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(g(y0), y1), y2)), z1))) A__F(g(f(g(f(f(y0, y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(y0, y1), y2)), z1))) A__F(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2) -> c(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) A__F(g(f(g(f(g(g(y0)), y1)), z1)), z2) -> c(MARK(f(g(f(g(g(y0)), y1)), z1))) A__F(g(f(g(f(g(y0), y1)), z1)), z2) -> c(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(g(g(g(y0)))), z1) -> c(MARK(g(g(g(y0))))) A__F(g(g(g(f(f(f(y0, y1), y2), y3)))), z1) -> c(MARK(g(g(f(f(f(y0, y1), y2), y3))))) A__F(g(g(g(f(f(g(y0), y1), y2)))), z1) -> c(MARK(g(g(f(f(g(y0), y1), y2))))) A__F(g(g(g(f(f(y0, y1), y2)))), z1) -> c(MARK(g(g(f(f(y0, y1), y2))))) A__F(g(g(g(f(g(f(y0, y1)), y2)))), z1) -> c(MARK(g(g(f(g(f(y0, y1)), y2))))) A__F(g(g(g(f(g(g(y0)), y1)))), z1) -> c(MARK(g(g(f(g(g(y0)), y1))))) A__F(g(g(g(f(g(y0), y1)))), z1) -> c(MARK(g(g(f(g(y0), y1))))) A__F(g(g(f(f(f(f(y0, y1), y2), z1), z2))), z3) -> c(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) A__F(g(g(f(f(f(g(y0), y1), z1), z2))), z3) -> c(MARK(g(f(f(f(g(y0), y1), z1), z2)))) A__F(g(g(f(f(g(f(y0, y1)), z1), z2))), z3) -> c(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) A__F(g(g(f(f(g(g(y0)), z1), z2))), z3) -> c(MARK(g(f(f(g(g(y0)), z1), z2)))) A__F(g(g(f(g(f(f(f(y0, y1), y2), y3)), z1))), z2) -> c(MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1)))) A__F(g(g(f(g(f(f(g(y0), y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(g(y0), y1), y2)), z1)))) A__F(g(g(f(g(f(f(y0, y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(y0, y1), y2)), z1)))) A__F(g(g(f(g(f(g(f(y0, y1)), y2)), z1))), z2) -> c(MARK(g(f(g(f(g(f(y0, y1)), y2)), z1)))) A__F(g(g(f(g(f(g(g(y0)), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(g(y0)), y1)), z1)))) A__F(g(g(f(g(f(g(y0), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(y0), y1)), z1)))) MARK(f(f(f(f(f(y0, y1), y2), z1), z2), z3)) -> c2(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(f(f(f(f(g(y0), y1), z1), z2), z3)) -> c2(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(f(f(f(g(f(y0, y1)), z1), z2), z3)) -> c2(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(f(f(f(g(g(y0)), z1), z2), z3)) -> c2(MARK(f(f(g(g(y0)), z1), z2))) MARK(f(f(g(f(f(f(y0, y1), y2), y3)), z1), z2)) -> c2(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(f(f(g(f(f(g(y0), y1), y2)), z1), z2)) -> c2(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(f(f(g(f(f(y0, y1), y2)), z1), z2)) -> c2(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(f(f(g(f(g(f(y0, y1)), y2)), z1), z2)) -> c2(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(f(f(g(f(g(g(y0)), y1)), z1), z2)) -> c2(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(f(f(g(f(g(y0), y1)), z1), z2)) -> c2(MARK(f(g(f(g(y0), y1)), z1))) MARK(f(g(g(g(y0))), z1)) -> c2(MARK(g(g(g(y0))))) MARK(f(g(g(f(f(f(y0, y1), y2), y3))), z1)) -> c2(MARK(g(g(f(f(f(y0, y1), y2), y3))))) MARK(f(g(g(f(f(g(y0), y1), y2))), z1)) -> c2(MARK(g(g(f(f(g(y0), y1), y2))))) MARK(f(g(g(f(f(y0, y1), y2))), z1)) -> c2(MARK(g(g(f(f(y0, y1), y2))))) MARK(f(g(g(f(g(f(y0, y1)), y2))), z1)) -> c2(MARK(g(g(f(g(f(y0, y1)), y2))))) MARK(f(g(g(f(g(g(y0)), y1))), z1)) -> c2(MARK(g(g(f(g(g(y0)), y1))))) MARK(f(g(g(f(g(y0), y1))), z1)) -> c2(MARK(g(g(f(g(y0), y1))))) S tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) A__F(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2) -> c(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) A__F(g(f(g(f(f(g(y0), y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(g(y0), y1), y2)), z1))) A__F(g(f(g(f(f(y0, y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(y0, y1), y2)), z1))) A__F(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2) -> c(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) A__F(g(f(g(f(g(g(y0)), y1)), z1)), z2) -> c(MARK(f(g(f(g(g(y0)), y1)), z1))) A__F(g(f(g(f(g(y0), y1)), z1)), z2) -> c(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(g(g(g(y0)))), z1) -> c(MARK(g(g(g(y0))))) A__F(g(g(g(f(f(f(y0, y1), y2), y3)))), z1) -> c(MARK(g(g(f(f(f(y0, y1), y2), y3))))) A__F(g(g(g(f(f(g(y0), y1), y2)))), z1) -> c(MARK(g(g(f(f(g(y0), y1), y2))))) A__F(g(g(g(f(f(y0, y1), y2)))), z1) -> c(MARK(g(g(f(f(y0, y1), y2))))) A__F(g(g(g(f(g(f(y0, y1)), y2)))), z1) -> c(MARK(g(g(f(g(f(y0, y1)), y2))))) A__F(g(g(g(f(g(g(y0)), y1)))), z1) -> c(MARK(g(g(f(g(g(y0)), y1))))) A__F(g(g(g(f(g(y0), y1)))), z1) -> c(MARK(g(g(f(g(y0), y1))))) A__F(g(g(f(f(f(f(y0, y1), y2), z1), z2))), z3) -> c(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) A__F(g(g(f(f(f(g(y0), y1), z1), z2))), z3) -> c(MARK(g(f(f(f(g(y0), y1), z1), z2)))) A__F(g(g(f(f(g(f(y0, y1)), z1), z2))), z3) -> c(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) A__F(g(g(f(f(g(g(y0)), z1), z2))), z3) -> c(MARK(g(f(f(g(g(y0)), z1), z2)))) A__F(g(g(f(g(f(f(f(y0, y1), y2), y3)), z1))), z2) -> c(MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1)))) A__F(g(g(f(g(f(f(g(y0), y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(g(y0), y1), y2)), z1)))) A__F(g(g(f(g(f(f(y0, y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(y0, y1), y2)), z1)))) A__F(g(g(f(g(f(g(f(y0, y1)), y2)), z1))), z2) -> c(MARK(g(f(g(f(g(f(y0, y1)), y2)), z1)))) A__F(g(g(f(g(f(g(g(y0)), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(g(y0)), y1)), z1)))) A__F(g(g(f(g(f(g(y0), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(y0), y1)), z1)))) MARK(f(f(f(f(f(y0, y1), y2), z1), z2), z3)) -> c2(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(f(f(f(f(g(y0), y1), z1), z2), z3)) -> c2(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(f(f(f(g(f(y0, y1)), z1), z2), z3)) -> c2(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(f(f(f(g(g(y0)), z1), z2), z3)) -> c2(MARK(f(f(g(g(y0)), z1), z2))) MARK(f(f(g(f(f(f(y0, y1), y2), y3)), z1), z2)) -> c2(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(f(f(g(f(f(g(y0), y1), y2)), z1), z2)) -> c2(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(f(f(g(f(f(y0, y1), y2)), z1), z2)) -> c2(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(f(f(g(f(g(f(y0, y1)), y2)), z1), z2)) -> c2(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(f(f(g(f(g(g(y0)), y1)), z1), z2)) -> c2(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(f(f(g(f(g(y0), y1)), z1), z2)) -> c2(MARK(f(g(f(g(y0), y1)), z1))) MARK(f(g(g(g(y0))), z1)) -> c2(MARK(g(g(g(y0))))) MARK(f(g(g(f(f(f(y0, y1), y2), y3))), z1)) -> c2(MARK(g(g(f(f(f(y0, y1), y2), y3))))) MARK(f(g(g(f(f(g(y0), y1), y2))), z1)) -> c2(MARK(g(g(f(f(g(y0), y1), y2))))) MARK(f(g(g(f(f(y0, y1), y2))), z1)) -> c2(MARK(g(g(f(f(y0, y1), y2))))) MARK(f(g(g(f(g(f(y0, y1)), y2))), z1)) -> c2(MARK(g(g(f(g(f(y0, y1)), y2))))) MARK(f(g(g(f(g(g(y0)), y1))), z1)) -> c2(MARK(g(g(f(g(g(y0)), y1))))) MARK(f(g(g(f(g(y0), y1))), z1)) -> c2(MARK(g(g(f(g(y0), y1))))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c2_2, c3_1, c_1, c2_1 ---------------------------------------- (73) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(f(g(f(f(y0, y1), y2)), z1)) -> c2(MARK(g(f(f(y0, y1), y2)))) by MARK(f(g(f(f(f(y0, y1), z1), z2)), z3)) -> c2(MARK(g(f(f(f(y0, y1), z1), z2)))) MARK(f(g(f(f(g(y0), z1), z2)), z3)) -> c2(MARK(g(f(f(g(y0), z1), z2)))) MARK(f(g(f(f(f(f(y0, y1), y2), z1), z2)), z3)) -> c2(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) MARK(f(g(f(f(f(g(y0), y1), z1), z2)), z3)) -> c2(MARK(g(f(f(f(g(y0), y1), z1), z2)))) MARK(f(g(f(f(g(f(y0, y1)), z1), z2)), z3)) -> c2(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) MARK(f(g(f(f(g(g(y0)), z1), z2)), z3)) -> c2(MARK(g(f(f(g(g(y0)), z1), z2)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) A__F(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2) -> c(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) A__F(g(f(g(f(f(g(y0), y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(g(y0), y1), y2)), z1))) A__F(g(f(g(f(f(y0, y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(y0, y1), y2)), z1))) A__F(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2) -> c(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) A__F(g(f(g(f(g(g(y0)), y1)), z1)), z2) -> c(MARK(f(g(f(g(g(y0)), y1)), z1))) A__F(g(f(g(f(g(y0), y1)), z1)), z2) -> c(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(g(g(g(y0)))), z1) -> c(MARK(g(g(g(y0))))) A__F(g(g(g(f(f(f(y0, y1), y2), y3)))), z1) -> c(MARK(g(g(f(f(f(y0, y1), y2), y3))))) A__F(g(g(g(f(f(g(y0), y1), y2)))), z1) -> c(MARK(g(g(f(f(g(y0), y1), y2))))) A__F(g(g(g(f(f(y0, y1), y2)))), z1) -> c(MARK(g(g(f(f(y0, y1), y2))))) A__F(g(g(g(f(g(f(y0, y1)), y2)))), z1) -> c(MARK(g(g(f(g(f(y0, y1)), y2))))) A__F(g(g(g(f(g(g(y0)), y1)))), z1) -> c(MARK(g(g(f(g(g(y0)), y1))))) A__F(g(g(g(f(g(y0), y1)))), z1) -> c(MARK(g(g(f(g(y0), y1))))) A__F(g(g(f(f(f(f(y0, y1), y2), z1), z2))), z3) -> c(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) A__F(g(g(f(f(f(g(y0), y1), z1), z2))), z3) -> c(MARK(g(f(f(f(g(y0), y1), z1), z2)))) A__F(g(g(f(f(g(f(y0, y1)), z1), z2))), z3) -> c(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) A__F(g(g(f(f(g(g(y0)), z1), z2))), z3) -> c(MARK(g(f(f(g(g(y0)), z1), z2)))) A__F(g(g(f(g(f(f(f(y0, y1), y2), y3)), z1))), z2) -> c(MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1)))) A__F(g(g(f(g(f(f(g(y0), y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(g(y0), y1), y2)), z1)))) A__F(g(g(f(g(f(f(y0, y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(y0, y1), y2)), z1)))) A__F(g(g(f(g(f(g(f(y0, y1)), y2)), z1))), z2) -> c(MARK(g(f(g(f(g(f(y0, y1)), y2)), z1)))) A__F(g(g(f(g(f(g(g(y0)), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(g(y0)), y1)), z1)))) A__F(g(g(f(g(f(g(y0), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(y0), y1)), z1)))) MARK(f(f(f(f(f(y0, y1), y2), z1), z2), z3)) -> c2(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(f(f(f(f(g(y0), y1), z1), z2), z3)) -> c2(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(f(f(f(g(f(y0, y1)), z1), z2), z3)) -> c2(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(f(f(f(g(g(y0)), z1), z2), z3)) -> c2(MARK(f(f(g(g(y0)), z1), z2))) MARK(f(f(g(f(f(f(y0, y1), y2), y3)), z1), z2)) -> c2(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(f(f(g(f(f(g(y0), y1), y2)), z1), z2)) -> c2(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(f(f(g(f(f(y0, y1), y2)), z1), z2)) -> c2(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(f(f(g(f(g(f(y0, y1)), y2)), z1), z2)) -> c2(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(f(f(g(f(g(g(y0)), y1)), z1), z2)) -> c2(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(f(f(g(f(g(y0), y1)), z1), z2)) -> c2(MARK(f(g(f(g(y0), y1)), z1))) MARK(f(g(g(g(y0))), z1)) -> c2(MARK(g(g(g(y0))))) MARK(f(g(g(f(f(f(y0, y1), y2), y3))), z1)) -> c2(MARK(g(g(f(f(f(y0, y1), y2), y3))))) MARK(f(g(g(f(f(g(y0), y1), y2))), z1)) -> c2(MARK(g(g(f(f(g(y0), y1), y2))))) MARK(f(g(g(f(f(y0, y1), y2))), z1)) -> c2(MARK(g(g(f(f(y0, y1), y2))))) MARK(f(g(g(f(g(f(y0, y1)), y2))), z1)) -> c2(MARK(g(g(f(g(f(y0, y1)), y2))))) MARK(f(g(g(f(g(g(y0)), y1))), z1)) -> c2(MARK(g(g(f(g(g(y0)), y1))))) MARK(f(g(g(f(g(y0), y1))), z1)) -> c2(MARK(g(g(f(g(y0), y1))))) MARK(f(g(f(f(f(f(y0, y1), y2), z1), z2)), z3)) -> c2(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) MARK(f(g(f(f(f(g(y0), y1), z1), z2)), z3)) -> c2(MARK(g(f(f(f(g(y0), y1), z1), z2)))) MARK(f(g(f(f(g(f(y0, y1)), z1), z2)), z3)) -> c2(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) MARK(f(g(f(f(g(g(y0)), z1), z2)), z3)) -> c2(MARK(g(f(f(g(g(y0)), z1), z2)))) S tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) A__F(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2) -> c(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) A__F(g(f(g(f(f(g(y0), y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(g(y0), y1), y2)), z1))) A__F(g(f(g(f(f(y0, y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(y0, y1), y2)), z1))) A__F(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2) -> c(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) A__F(g(f(g(f(g(g(y0)), y1)), z1)), z2) -> c(MARK(f(g(f(g(g(y0)), y1)), z1))) A__F(g(f(g(f(g(y0), y1)), z1)), z2) -> c(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(g(g(g(y0)))), z1) -> c(MARK(g(g(g(y0))))) A__F(g(g(g(f(f(f(y0, y1), y2), y3)))), z1) -> c(MARK(g(g(f(f(f(y0, y1), y2), y3))))) A__F(g(g(g(f(f(g(y0), y1), y2)))), z1) -> c(MARK(g(g(f(f(g(y0), y1), y2))))) A__F(g(g(g(f(f(y0, y1), y2)))), z1) -> c(MARK(g(g(f(f(y0, y1), y2))))) A__F(g(g(g(f(g(f(y0, y1)), y2)))), z1) -> c(MARK(g(g(f(g(f(y0, y1)), y2))))) A__F(g(g(g(f(g(g(y0)), y1)))), z1) -> c(MARK(g(g(f(g(g(y0)), y1))))) A__F(g(g(g(f(g(y0), y1)))), z1) -> c(MARK(g(g(f(g(y0), y1))))) A__F(g(g(f(f(f(f(y0, y1), y2), z1), z2))), z3) -> c(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) A__F(g(g(f(f(f(g(y0), y1), z1), z2))), z3) -> c(MARK(g(f(f(f(g(y0), y1), z1), z2)))) A__F(g(g(f(f(g(f(y0, y1)), z1), z2))), z3) -> c(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) A__F(g(g(f(f(g(g(y0)), z1), z2))), z3) -> c(MARK(g(f(f(g(g(y0)), z1), z2)))) A__F(g(g(f(g(f(f(f(y0, y1), y2), y3)), z1))), z2) -> c(MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1)))) A__F(g(g(f(g(f(f(g(y0), y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(g(y0), y1), y2)), z1)))) A__F(g(g(f(g(f(f(y0, y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(y0, y1), y2)), z1)))) A__F(g(g(f(g(f(g(f(y0, y1)), y2)), z1))), z2) -> c(MARK(g(f(g(f(g(f(y0, y1)), y2)), z1)))) A__F(g(g(f(g(f(g(g(y0)), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(g(y0)), y1)), z1)))) A__F(g(g(f(g(f(g(y0), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(y0), y1)), z1)))) MARK(f(f(f(f(f(y0, y1), y2), z1), z2), z3)) -> c2(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(f(f(f(f(g(y0), y1), z1), z2), z3)) -> c2(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(f(f(f(g(f(y0, y1)), z1), z2), z3)) -> c2(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(f(f(f(g(g(y0)), z1), z2), z3)) -> c2(MARK(f(f(g(g(y0)), z1), z2))) MARK(f(f(g(f(f(f(y0, y1), y2), y3)), z1), z2)) -> c2(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(f(f(g(f(f(g(y0), y1), y2)), z1), z2)) -> c2(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(f(f(g(f(f(y0, y1), y2)), z1), z2)) -> c2(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(f(f(g(f(g(f(y0, y1)), y2)), z1), z2)) -> c2(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(f(f(g(f(g(g(y0)), y1)), z1), z2)) -> c2(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(f(f(g(f(g(y0), y1)), z1), z2)) -> c2(MARK(f(g(f(g(y0), y1)), z1))) MARK(f(g(g(g(y0))), z1)) -> c2(MARK(g(g(g(y0))))) MARK(f(g(g(f(f(f(y0, y1), y2), y3))), z1)) -> c2(MARK(g(g(f(f(f(y0, y1), y2), y3))))) MARK(f(g(g(f(f(g(y0), y1), y2))), z1)) -> c2(MARK(g(g(f(f(g(y0), y1), y2))))) MARK(f(g(g(f(f(y0, y1), y2))), z1)) -> c2(MARK(g(g(f(f(y0, y1), y2))))) MARK(f(g(g(f(g(f(y0, y1)), y2))), z1)) -> c2(MARK(g(g(f(g(f(y0, y1)), y2))))) MARK(f(g(g(f(g(g(y0)), y1))), z1)) -> c2(MARK(g(g(f(g(g(y0)), y1))))) MARK(f(g(g(f(g(y0), y1))), z1)) -> c2(MARK(g(g(f(g(y0), y1))))) MARK(f(g(f(f(f(f(y0, y1), y2), z1), z2)), z3)) -> c2(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) MARK(f(g(f(f(f(g(y0), y1), z1), z2)), z3)) -> c2(MARK(g(f(f(f(g(y0), y1), z1), z2)))) MARK(f(g(f(f(g(f(y0, y1)), z1), z2)), z3)) -> c2(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) MARK(f(g(f(f(g(g(y0)), z1), z2)), z3)) -> c2(MARK(g(f(f(g(g(y0)), z1), z2)))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c2_2, c3_1, c_1, c2_1 ---------------------------------------- (75) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(f(g(f(g(y0), y1)), z1)) -> c2(MARK(g(f(g(y0), y1)))) by MARK(f(g(f(g(f(y0, y1)), z1)), z2)) -> c2(MARK(g(f(g(f(y0, y1)), z1)))) MARK(f(g(f(g(g(y0)), z1)), z2)) -> c2(MARK(g(f(g(g(y0)), z1)))) MARK(f(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2)) -> c2(MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1)))) MARK(f(g(f(g(f(f(g(y0), y1), y2)), z1)), z2)) -> c2(MARK(g(f(g(f(f(g(y0), y1), y2)), z1)))) MARK(f(g(f(g(f(f(y0, y1), y2)), z1)), z2)) -> c2(MARK(g(f(g(f(f(y0, y1), y2)), z1)))) MARK(f(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2)) -> c2(MARK(g(f(g(f(g(f(y0, y1)), y2)), z1)))) MARK(f(g(f(g(f(g(g(y0)), y1)), z1)), z2)) -> c2(MARK(g(f(g(f(g(g(y0)), y1)), z1)))) MARK(f(g(f(g(f(g(y0), y1)), z1)), z2)) -> c2(MARK(g(f(g(f(g(y0), y1)), z1)))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: a__f(g(z0), z1) -> a__f(mark(z0), f(g(z0), z1)) a__f(z0, z1) -> f(z0, z1) mark(f(z0, z1)) -> a__f(mark(z0), z1) mark(g(z0)) -> g(mark(z0)) Tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) A__F(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2) -> c(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) A__F(g(f(g(f(f(g(y0), y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(g(y0), y1), y2)), z1))) A__F(g(f(g(f(f(y0, y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(y0, y1), y2)), z1))) A__F(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2) -> c(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) A__F(g(f(g(f(g(g(y0)), y1)), z1)), z2) -> c(MARK(f(g(f(g(g(y0)), y1)), z1))) A__F(g(f(g(f(g(y0), y1)), z1)), z2) -> c(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(g(g(g(y0)))), z1) -> c(MARK(g(g(g(y0))))) A__F(g(g(g(f(f(f(y0, y1), y2), y3)))), z1) -> c(MARK(g(g(f(f(f(y0, y1), y2), y3))))) A__F(g(g(g(f(f(g(y0), y1), y2)))), z1) -> c(MARK(g(g(f(f(g(y0), y1), y2))))) A__F(g(g(g(f(f(y0, y1), y2)))), z1) -> c(MARK(g(g(f(f(y0, y1), y2))))) A__F(g(g(g(f(g(f(y0, y1)), y2)))), z1) -> c(MARK(g(g(f(g(f(y0, y1)), y2))))) A__F(g(g(g(f(g(g(y0)), y1)))), z1) -> c(MARK(g(g(f(g(g(y0)), y1))))) A__F(g(g(g(f(g(y0), y1)))), z1) -> c(MARK(g(g(f(g(y0), y1))))) A__F(g(g(f(f(f(f(y0, y1), y2), z1), z2))), z3) -> c(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) A__F(g(g(f(f(f(g(y0), y1), z1), z2))), z3) -> c(MARK(g(f(f(f(g(y0), y1), z1), z2)))) A__F(g(g(f(f(g(f(y0, y1)), z1), z2))), z3) -> c(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) A__F(g(g(f(f(g(g(y0)), z1), z2))), z3) -> c(MARK(g(f(f(g(g(y0)), z1), z2)))) A__F(g(g(f(g(f(f(f(y0, y1), y2), y3)), z1))), z2) -> c(MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1)))) A__F(g(g(f(g(f(f(g(y0), y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(g(y0), y1), y2)), z1)))) A__F(g(g(f(g(f(f(y0, y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(y0, y1), y2)), z1)))) A__F(g(g(f(g(f(g(f(y0, y1)), y2)), z1))), z2) -> c(MARK(g(f(g(f(g(f(y0, y1)), y2)), z1)))) A__F(g(g(f(g(f(g(g(y0)), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(g(y0)), y1)), z1)))) A__F(g(g(f(g(f(g(y0), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(y0), y1)), z1)))) MARK(f(f(f(f(f(y0, y1), y2), z1), z2), z3)) -> c2(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(f(f(f(f(g(y0), y1), z1), z2), z3)) -> c2(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(f(f(f(g(f(y0, y1)), z1), z2), z3)) -> c2(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(f(f(f(g(g(y0)), z1), z2), z3)) -> c2(MARK(f(f(g(g(y0)), z1), z2))) MARK(f(f(g(f(f(f(y0, y1), y2), y3)), z1), z2)) -> c2(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(f(f(g(f(f(g(y0), y1), y2)), z1), z2)) -> c2(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(f(f(g(f(f(y0, y1), y2)), z1), z2)) -> c2(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(f(f(g(f(g(f(y0, y1)), y2)), z1), z2)) -> c2(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(f(f(g(f(g(g(y0)), y1)), z1), z2)) -> c2(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(f(f(g(f(g(y0), y1)), z1), z2)) -> c2(MARK(f(g(f(g(y0), y1)), z1))) MARK(f(g(g(g(y0))), z1)) -> c2(MARK(g(g(g(y0))))) MARK(f(g(g(f(f(f(y0, y1), y2), y3))), z1)) -> c2(MARK(g(g(f(f(f(y0, y1), y2), y3))))) MARK(f(g(g(f(f(g(y0), y1), y2))), z1)) -> c2(MARK(g(g(f(f(g(y0), y1), y2))))) MARK(f(g(g(f(f(y0, y1), y2))), z1)) -> c2(MARK(g(g(f(f(y0, y1), y2))))) MARK(f(g(g(f(g(f(y0, y1)), y2))), z1)) -> c2(MARK(g(g(f(g(f(y0, y1)), y2))))) MARK(f(g(g(f(g(g(y0)), y1))), z1)) -> c2(MARK(g(g(f(g(g(y0)), y1))))) MARK(f(g(g(f(g(y0), y1))), z1)) -> c2(MARK(g(g(f(g(y0), y1))))) MARK(f(g(f(f(f(f(y0, y1), y2), z1), z2)), z3)) -> c2(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) MARK(f(g(f(f(f(g(y0), y1), z1), z2)), z3)) -> c2(MARK(g(f(f(f(g(y0), y1), z1), z2)))) MARK(f(g(f(f(g(f(y0, y1)), z1), z2)), z3)) -> c2(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) MARK(f(g(f(f(g(g(y0)), z1), z2)), z3)) -> c2(MARK(g(f(f(g(g(y0)), z1), z2)))) MARK(f(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2)) -> c2(MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1)))) MARK(f(g(f(g(f(f(g(y0), y1), y2)), z1)), z2)) -> c2(MARK(g(f(g(f(f(g(y0), y1), y2)), z1)))) MARK(f(g(f(g(f(f(y0, y1), y2)), z1)), z2)) -> c2(MARK(g(f(g(f(f(y0, y1), y2)), z1)))) MARK(f(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2)) -> c2(MARK(g(f(g(f(g(f(y0, y1)), y2)), z1)))) MARK(f(g(f(g(f(g(g(y0)), y1)), z1)), z2)) -> c2(MARK(g(f(g(f(g(g(y0)), y1)), z1)))) MARK(f(g(f(g(f(g(y0), y1)), z1)), z2)) -> c2(MARK(g(f(g(f(g(y0), y1)), z1)))) S tuples: A__F(g(f(f(z0, z1), x1)), x2) -> c(A__F(a__f(a__f(mark(z0), z1), x1), f(g(f(f(z0, z1), x1)), x2)), MARK(f(f(z0, z1), x1))) A__F(g(f(g(z0), x1)), x2) -> c(A__F(a__f(g(mark(z0)), x1), f(g(f(g(z0), x1)), x2)), MARK(f(g(z0), x1))) A__F(g(g(f(z0, z1))), x1) -> c(A__F(g(a__f(mark(z0), z1)), f(g(g(f(z0, z1))), x1)), MARK(g(f(z0, z1)))) A__F(g(g(g(z0))), x1) -> c(A__F(g(g(mark(z0))), f(g(g(g(z0))), x1)), MARK(g(g(z0)))) MARK(f(f(f(z0, z1), x1), x2)) -> c2(A__F(a__f(a__f(mark(z0), z1), x1), x2), MARK(f(f(z0, z1), x1))) MARK(f(f(g(z0), x1), x2)) -> c2(A__F(a__f(g(mark(z0)), x1), x2), MARK(f(g(z0), x1))) MARK(f(g(f(z0, z1)), x1)) -> c2(A__F(g(a__f(mark(z0), z1)), x1), MARK(g(f(z0, z1)))) MARK(f(g(g(z0)), x1)) -> c2(A__F(g(g(mark(z0))), x1), MARK(g(g(z0)))) MARK(g(f(f(f(y0, y1), y2), y3))) -> c3(MARK(f(f(f(y0, y1), y2), y3))) MARK(g(f(f(g(y0), y1), y2))) -> c3(MARK(f(f(g(y0), y1), y2))) MARK(g(f(g(f(y0, y1)), y2))) -> c3(MARK(f(g(f(y0, y1)), y2))) MARK(g(f(g(g(y0)), y1))) -> c3(MARK(f(g(g(y0)), y1))) A__F(g(f(f(f(y0, y1), y2), z1)), z2) -> c(MARK(f(f(f(y0, y1), y2), z1))) A__F(g(f(f(g(y0), y1), z1)), z2) -> c(MARK(f(f(g(y0), y1), z1))) A__F(g(f(g(f(y0, y1)), z1)), z2) -> c(MARK(f(g(f(y0, y1)), z1))) A__F(g(f(g(g(y0)), z1)), z2) -> c(MARK(f(g(g(y0)), z1))) A__F(g(g(f(f(f(y0, y1), y2), y3))), z1) -> c(MARK(g(f(f(f(y0, y1), y2), y3)))) A__F(g(g(f(f(g(y0), y1), y2))), z1) -> c(MARK(g(f(f(g(y0), y1), y2)))) A__F(g(g(f(g(f(y0, y1)), y2))), z1) -> c(MARK(g(f(g(f(y0, y1)), y2)))) A__F(g(g(f(g(g(y0)), y1))), z1) -> c(MARK(g(f(g(g(y0)), y1)))) MARK(f(f(f(f(y0, y1), y2), z1), z2)) -> c2(MARK(f(f(f(y0, y1), y2), z1))) MARK(f(f(f(g(y0), y1), z1), z2)) -> c2(MARK(f(f(g(y0), y1), z1))) MARK(f(f(g(f(y0, y1)), z1), z2)) -> c2(MARK(f(g(f(y0, y1)), z1))) MARK(f(f(g(g(y0)), z1), z2)) -> c2(MARK(f(g(g(y0)), z1))) MARK(f(g(f(f(f(y0, y1), y2), y3)), z1)) -> c2(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(f(g(f(f(g(y0), y1), y2)), z1)) -> c2(MARK(g(f(f(g(y0), y1), y2)))) MARK(f(g(f(g(f(y0, y1)), y2)), z1)) -> c2(MARK(g(f(g(f(y0, y1)), y2)))) MARK(f(g(f(g(g(y0)), y1)), z1)) -> c2(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(g(y0)))) -> c3(MARK(g(g(y0)))) MARK(g(g(f(f(f(y0, y1), y2), y3)))) -> c3(MARK(g(f(f(f(y0, y1), y2), y3)))) MARK(g(g(f(f(g(y0), y1), y2)))) -> c3(MARK(g(f(f(g(y0), y1), y2)))) MARK(g(g(f(f(y0, y1), y2)))) -> c3(MARK(g(f(f(y0, y1), y2)))) MARK(g(g(f(g(f(y0, y1)), y2)))) -> c3(MARK(g(f(g(f(y0, y1)), y2)))) MARK(g(g(f(g(g(y0)), y1)))) -> c3(MARK(g(f(g(g(y0)), y1)))) MARK(g(g(f(g(y0), y1)))) -> c3(MARK(g(f(g(y0), y1)))) MARK(g(f(f(f(f(y0, y1), y2), z1), z2))) -> c3(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(g(f(f(f(g(y0), y1), z1), z2))) -> c3(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(g(f(f(g(f(y0, y1)), z1), z2))) -> c3(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(g(f(f(g(g(y0)), z1), z2))) -> c3(MARK(f(f(g(g(y0)), z1), z2))) MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1))) -> c3(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(g(f(g(f(f(g(y0), y1), y2)), z1))) -> c3(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(g(f(g(f(f(y0, y1), y2)), z1))) -> c3(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(g(f(g(f(g(f(y0, y1)), y2)), z1))) -> c3(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(g(f(g(f(g(g(y0)), y1)), z1))) -> c3(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(g(f(g(f(g(y0), y1)), z1))) -> c3(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(f(f(f(f(y0, y1), y2), z1), z2)), z3) -> c(MARK(f(f(f(f(y0, y1), y2), z1), z2))) A__F(g(f(f(f(g(y0), y1), z1), z2)), z3) -> c(MARK(f(f(f(g(y0), y1), z1), z2))) A__F(g(f(f(g(f(y0, y1)), z1), z2)), z3) -> c(MARK(f(f(g(f(y0, y1)), z1), z2))) A__F(g(f(f(g(g(y0)), z1), z2)), z3) -> c(MARK(f(f(g(g(y0)), z1), z2))) A__F(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2) -> c(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) A__F(g(f(g(f(f(g(y0), y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(g(y0), y1), y2)), z1))) A__F(g(f(g(f(f(y0, y1), y2)), z1)), z2) -> c(MARK(f(g(f(f(y0, y1), y2)), z1))) A__F(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2) -> c(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) A__F(g(f(g(f(g(g(y0)), y1)), z1)), z2) -> c(MARK(f(g(f(g(g(y0)), y1)), z1))) A__F(g(f(g(f(g(y0), y1)), z1)), z2) -> c(MARK(f(g(f(g(y0), y1)), z1))) A__F(g(g(g(g(y0)))), z1) -> c(MARK(g(g(g(y0))))) A__F(g(g(g(f(f(f(y0, y1), y2), y3)))), z1) -> c(MARK(g(g(f(f(f(y0, y1), y2), y3))))) A__F(g(g(g(f(f(g(y0), y1), y2)))), z1) -> c(MARK(g(g(f(f(g(y0), y1), y2))))) A__F(g(g(g(f(f(y0, y1), y2)))), z1) -> c(MARK(g(g(f(f(y0, y1), y2))))) A__F(g(g(g(f(g(f(y0, y1)), y2)))), z1) -> c(MARK(g(g(f(g(f(y0, y1)), y2))))) A__F(g(g(g(f(g(g(y0)), y1)))), z1) -> c(MARK(g(g(f(g(g(y0)), y1))))) A__F(g(g(g(f(g(y0), y1)))), z1) -> c(MARK(g(g(f(g(y0), y1))))) A__F(g(g(f(f(f(f(y0, y1), y2), z1), z2))), z3) -> c(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) A__F(g(g(f(f(f(g(y0), y1), z1), z2))), z3) -> c(MARK(g(f(f(f(g(y0), y1), z1), z2)))) A__F(g(g(f(f(g(f(y0, y1)), z1), z2))), z3) -> c(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) A__F(g(g(f(f(g(g(y0)), z1), z2))), z3) -> c(MARK(g(f(f(g(g(y0)), z1), z2)))) A__F(g(g(f(g(f(f(f(y0, y1), y2), y3)), z1))), z2) -> c(MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1)))) A__F(g(g(f(g(f(f(g(y0), y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(g(y0), y1), y2)), z1)))) A__F(g(g(f(g(f(f(y0, y1), y2)), z1))), z2) -> c(MARK(g(f(g(f(f(y0, y1), y2)), z1)))) A__F(g(g(f(g(f(g(f(y0, y1)), y2)), z1))), z2) -> c(MARK(g(f(g(f(g(f(y0, y1)), y2)), z1)))) A__F(g(g(f(g(f(g(g(y0)), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(g(y0)), y1)), z1)))) A__F(g(g(f(g(f(g(y0), y1)), z1))), z2) -> c(MARK(g(f(g(f(g(y0), y1)), z1)))) MARK(f(f(f(f(f(y0, y1), y2), z1), z2), z3)) -> c2(MARK(f(f(f(f(y0, y1), y2), z1), z2))) MARK(f(f(f(f(g(y0), y1), z1), z2), z3)) -> c2(MARK(f(f(f(g(y0), y1), z1), z2))) MARK(f(f(f(g(f(y0, y1)), z1), z2), z3)) -> c2(MARK(f(f(g(f(y0, y1)), z1), z2))) MARK(f(f(f(g(g(y0)), z1), z2), z3)) -> c2(MARK(f(f(g(g(y0)), z1), z2))) MARK(f(f(g(f(f(f(y0, y1), y2), y3)), z1), z2)) -> c2(MARK(f(g(f(f(f(y0, y1), y2), y3)), z1))) MARK(f(f(g(f(f(g(y0), y1), y2)), z1), z2)) -> c2(MARK(f(g(f(f(g(y0), y1), y2)), z1))) MARK(f(f(g(f(f(y0, y1), y2)), z1), z2)) -> c2(MARK(f(g(f(f(y0, y1), y2)), z1))) MARK(f(f(g(f(g(f(y0, y1)), y2)), z1), z2)) -> c2(MARK(f(g(f(g(f(y0, y1)), y2)), z1))) MARK(f(f(g(f(g(g(y0)), y1)), z1), z2)) -> c2(MARK(f(g(f(g(g(y0)), y1)), z1))) MARK(f(f(g(f(g(y0), y1)), z1), z2)) -> c2(MARK(f(g(f(g(y0), y1)), z1))) MARK(f(g(g(g(y0))), z1)) -> c2(MARK(g(g(g(y0))))) MARK(f(g(g(f(f(f(y0, y1), y2), y3))), z1)) -> c2(MARK(g(g(f(f(f(y0, y1), y2), y3))))) MARK(f(g(g(f(f(g(y0), y1), y2))), z1)) -> c2(MARK(g(g(f(f(g(y0), y1), y2))))) MARK(f(g(g(f(f(y0, y1), y2))), z1)) -> c2(MARK(g(g(f(f(y0, y1), y2))))) MARK(f(g(g(f(g(f(y0, y1)), y2))), z1)) -> c2(MARK(g(g(f(g(f(y0, y1)), y2))))) MARK(f(g(g(f(g(g(y0)), y1))), z1)) -> c2(MARK(g(g(f(g(g(y0)), y1))))) MARK(f(g(g(f(g(y0), y1))), z1)) -> c2(MARK(g(g(f(g(y0), y1))))) MARK(f(g(f(f(f(f(y0, y1), y2), z1), z2)), z3)) -> c2(MARK(g(f(f(f(f(y0, y1), y2), z1), z2)))) MARK(f(g(f(f(f(g(y0), y1), z1), z2)), z3)) -> c2(MARK(g(f(f(f(g(y0), y1), z1), z2)))) MARK(f(g(f(f(g(f(y0, y1)), z1), z2)), z3)) -> c2(MARK(g(f(f(g(f(y0, y1)), z1), z2)))) MARK(f(g(f(f(g(g(y0)), z1), z2)), z3)) -> c2(MARK(g(f(f(g(g(y0)), z1), z2)))) MARK(f(g(f(g(f(f(f(y0, y1), y2), y3)), z1)), z2)) -> c2(MARK(g(f(g(f(f(f(y0, y1), y2), y3)), z1)))) MARK(f(g(f(g(f(f(g(y0), y1), y2)), z1)), z2)) -> c2(MARK(g(f(g(f(f(g(y0), y1), y2)), z1)))) MARK(f(g(f(g(f(f(y0, y1), y2)), z1)), z2)) -> c2(MARK(g(f(g(f(f(y0, y1), y2)), z1)))) MARK(f(g(f(g(f(g(f(y0, y1)), y2)), z1)), z2)) -> c2(MARK(g(f(g(f(g(f(y0, y1)), y2)), z1)))) MARK(f(g(f(g(f(g(g(y0)), y1)), z1)), z2)) -> c2(MARK(g(f(g(f(g(g(y0)), y1)), z1)))) MARK(f(g(f(g(f(g(y0), y1)), z1)), z2)) -> c2(MARK(g(f(g(f(g(y0), y1)), z1)))) K tuples:none Defined Rule Symbols: a__f_2, mark_1 Defined Pair Symbols: A__F_2, MARK_1 Compound Symbols: c_2, c2_2, c3_1, c_1, c2_1