MAYBE proof of input_NvBMtnYbl8.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CompletionProof [UPPER BOUND(ID), 0 ms] (18) CpxTypedWeightedCompleteTrs (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (22) CdtProblem (23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CdtProblem (25) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 1 ms] (32) CdtProblem (33) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 9 ms] (42) CdtProblem (43) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 13 ms] (44) CdtProblem (45) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(g(X), Y) -> f(X, n__f(g(X), activate(Y))) f(X1, X2) -> n__f(X1, X2) activate(n__f(X1, X2)) -> f(X1, X2) activate(X) -> X S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(g(X), Y) -> f(X, n__f(g(X), activate(Y))) f(X1, X2) -> n__f(X1, X2) activate(n__f(X1, X2)) -> f(X1, X2) activate(X) -> X S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(g(X), Y) -> f(X, n__f(g(X), activate(Y))) f(X1, X2) -> n__f(X1, X2) activate(n__f(X1, X2)) -> f(X1, X2) activate(X) -> X S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(g(X), Y) -> f(X, n__f(g(X), activate(Y))) [1] f(X1, X2) -> n__f(X1, X2) [1] activate(n__f(X1, X2)) -> f(X1, X2) [1] activate(X) -> X [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(g(X), Y) -> f(X, n__f(g(X), activate(Y))) [1] f(X1, X2) -> n__f(X1, X2) [1] activate(n__f(X1, X2)) -> f(X1, X2) [1] activate(X) -> X [1] The TRS has the following type information: f :: g -> n__f -> n__f g :: g -> g n__f :: g -> n__f -> n__f activate :: n__f -> n__f Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: activate_1 f_2 Due to the following rules being added: none And the following fresh constants: const, const1 ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(g(X), Y) -> f(X, n__f(g(X), activate(Y))) [1] f(X1, X2) -> n__f(X1, X2) [1] activate(n__f(X1, X2)) -> f(X1, X2) [1] activate(X) -> X [1] The TRS has the following type information: f :: g -> n__f -> n__f g :: g -> g n__f :: g -> n__f -> n__f activate :: n__f -> n__f const :: n__f const1 :: g Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(g(X), n__f(X1', X2')) -> f(X, n__f(g(X), f(X1', X2'))) [2] f(g(X), Y) -> f(X, n__f(g(X), Y)) [2] f(X1, X2) -> n__f(X1, X2) [1] activate(n__f(X1, X2)) -> f(X1, X2) [1] activate(X) -> X [1] The TRS has the following type information: f :: g -> n__f -> n__f g :: g -> g n__f :: g -> n__f -> n__f activate :: n__f -> n__f const :: n__f const1 :: g Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: const => 0 const1 => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> f(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 f(z, z') -{ 2 }-> f(X, 1 + (1 + X) + Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0 f(z, z') -{ 2 }-> f(X, 1 + (1 + X) + f(X1', X2')) :|: z = 1 + X, z' = 1 + X1' + X2', X2' >= 0, X1' >= 0, X >= 0 f(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> f(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 f(z, z') -{ 2 }-> f(z - 1, 1 + (1 + (z - 1)) + z') :|: z' >= 0, z - 1 >= 0 f(z, z') -{ 2 }-> f(z - 1, 1 + (1 + (z - 1)) + f(X1', X2')) :|: z' = 1 + X1' + X2', X2' >= 0, X1' >= 0, z - 1 >= 0 f(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 ---------------------------------------- (17) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: const, const1 ---------------------------------------- (18) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(g(X), Y) -> f(X, n__f(g(X), activate(Y))) [1] f(X1, X2) -> n__f(X1, X2) [1] activate(n__f(X1, X2)) -> f(X1, X2) [1] activate(X) -> X [1] The TRS has the following type information: f :: g -> n__f -> n__f g :: g -> g n__f :: g -> n__f -> n__f activate :: n__f -> n__f const :: n__f const1 :: g Rewrite Strategy: INNERMOST ---------------------------------------- (19) 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 const1 => 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> f(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 f(z, z') -{ 1 }-> f(X, 1 + (1 + X) + activate(Y)) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0 f(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (21) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0), z1) -> f(z0, n__f(g(z0), activate(z1))) f(z0, z1) -> n__f(z0, z1) activate(n__f(z0, z1)) -> f(z0, z1) activate(z0) -> z0 Tuples: F(g(z0), z1) -> c(F(z0, n__f(g(z0), activate(z1))), ACTIVATE(z1)) F(z0, z1) -> c1 ACTIVATE(n__f(z0, z1)) -> c2(F(z0, z1)) ACTIVATE(z0) -> c3 S tuples: F(g(z0), z1) -> c(F(z0, n__f(g(z0), activate(z1))), ACTIVATE(z1)) F(z0, z1) -> c1 ACTIVATE(n__f(z0, z1)) -> c2(F(z0, z1)) ACTIVATE(z0) -> c3 K tuples:none Defined Rule Symbols: f_2, activate_1 Defined Pair Symbols: F_2, ACTIVATE_1 Compound Symbols: c_2, c1, c2_1, c3 ---------------------------------------- (23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: F(z0, z1) -> c1 ACTIVATE(z0) -> c3 ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0), z1) -> f(z0, n__f(g(z0), activate(z1))) f(z0, z1) -> n__f(z0, z1) activate(n__f(z0, z1)) -> f(z0, z1) activate(z0) -> z0 Tuples: F(g(z0), z1) -> c(F(z0, n__f(g(z0), activate(z1))), ACTIVATE(z1)) ACTIVATE(n__f(z0, z1)) -> c2(F(z0, z1)) S tuples: F(g(z0), z1) -> c(F(z0, n__f(g(z0), activate(z1))), ACTIVATE(z1)) ACTIVATE(n__f(z0, z1)) -> c2(F(z0, z1)) K tuples:none Defined Rule Symbols: f_2, activate_1 Defined Pair Symbols: F_2, ACTIVATE_1 Compound Symbols: c_2, c2_1 ---------------------------------------- (25) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(g(z0), z1) -> c(F(z0, n__f(g(z0), activate(z1))), ACTIVATE(z1)) by F(g(x0), n__f(z0, z1)) -> c(F(x0, n__f(g(x0), f(z0, z1))), ACTIVATE(n__f(z0, z1))) F(g(x0), z0) -> c(F(x0, n__f(g(x0), z0)), ACTIVATE(z0)) ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0), z1) -> f(z0, n__f(g(z0), activate(z1))) f(z0, z1) -> n__f(z0, z1) activate(n__f(z0, z1)) -> f(z0, z1) activate(z0) -> z0 Tuples: ACTIVATE(n__f(z0, z1)) -> c2(F(z0, z1)) F(g(x0), n__f(z0, z1)) -> c(F(x0, n__f(g(x0), f(z0, z1))), ACTIVATE(n__f(z0, z1))) F(g(x0), z0) -> c(F(x0, n__f(g(x0), z0)), ACTIVATE(z0)) S tuples: ACTIVATE(n__f(z0, z1)) -> c2(F(z0, z1)) F(g(x0), n__f(z0, z1)) -> c(F(x0, n__f(g(x0), f(z0, z1))), ACTIVATE(n__f(z0, z1))) F(g(x0), z0) -> c(F(x0, n__f(g(x0), z0)), ACTIVATE(z0)) K tuples:none Defined Rule Symbols: f_2, activate_1 Defined Pair Symbols: ACTIVATE_1, F_2 Compound Symbols: c2_1, c_2 ---------------------------------------- (27) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(g(x0), n__f(z0, z1)) -> c(F(x0, n__f(g(x0), f(z0, z1))), ACTIVATE(n__f(z0, z1))) by F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) F(g(x0), n__f(z0, z1)) -> c(F(x0, n__f(g(x0), n__f(z0, z1))), ACTIVATE(n__f(z0, z1))) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0), z1) -> f(z0, n__f(g(z0), activate(z1))) f(z0, z1) -> n__f(z0, z1) activate(n__f(z0, z1)) -> f(z0, z1) activate(z0) -> z0 Tuples: ACTIVATE(n__f(z0, z1)) -> c2(F(z0, z1)) F(g(x0), z0) -> c(F(x0, n__f(g(x0), z0)), ACTIVATE(z0)) F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) F(g(x0), n__f(z0, z1)) -> c(F(x0, n__f(g(x0), n__f(z0, z1))), ACTIVATE(n__f(z0, z1))) S tuples: ACTIVATE(n__f(z0, z1)) -> c2(F(z0, z1)) F(g(x0), z0) -> c(F(x0, n__f(g(x0), z0)), ACTIVATE(z0)) F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) F(g(x0), n__f(z0, z1)) -> c(F(x0, n__f(g(x0), n__f(z0, z1))), ACTIVATE(n__f(z0, z1))) K tuples:none Defined Rule Symbols: f_2, activate_1 Defined Pair Symbols: ACTIVATE_1, F_2 Compound Symbols: c2_1, c_2 ---------------------------------------- (29) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__f(z0, z1)) -> c2(F(z0, z1)) by ACTIVATE(n__f(g(y0), z1)) -> c2(F(g(y0), z1)) ACTIVATE(n__f(g(y0), n__f(g(y1), y2))) -> c2(F(g(y0), n__f(g(y1), y2))) ACTIVATE(n__f(g(y0), n__f(y1, y2))) -> c2(F(g(y0), n__f(y1, y2))) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0), z1) -> f(z0, n__f(g(z0), activate(z1))) f(z0, z1) -> n__f(z0, z1) activate(n__f(z0, z1)) -> f(z0, z1) activate(z0) -> z0 Tuples: F(g(x0), z0) -> c(F(x0, n__f(g(x0), z0)), ACTIVATE(z0)) F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) F(g(x0), n__f(z0, z1)) -> c(F(x0, n__f(g(x0), n__f(z0, z1))), ACTIVATE(n__f(z0, z1))) ACTIVATE(n__f(g(y0), z1)) -> c2(F(g(y0), z1)) ACTIVATE(n__f(g(y0), n__f(g(y1), y2))) -> c2(F(g(y0), n__f(g(y1), y2))) ACTIVATE(n__f(g(y0), n__f(y1, y2))) -> c2(F(g(y0), n__f(y1, y2))) S tuples: F(g(x0), z0) -> c(F(x0, n__f(g(x0), z0)), ACTIVATE(z0)) F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) F(g(x0), n__f(z0, z1)) -> c(F(x0, n__f(g(x0), n__f(z0, z1))), ACTIVATE(n__f(z0, z1))) ACTIVATE(n__f(g(y0), z1)) -> c2(F(g(y0), z1)) ACTIVATE(n__f(g(y0), n__f(g(y1), y2))) -> c2(F(g(y0), n__f(g(y1), y2))) ACTIVATE(n__f(g(y0), n__f(y1, y2))) -> c2(F(g(y0), n__f(y1, y2))) K tuples:none Defined Rule Symbols: f_2, activate_1 Defined Pair Symbols: F_2, ACTIVATE_1 Compound Symbols: c_2, c2_1 ---------------------------------------- (31) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(g(x0), z0) -> c(F(x0, n__f(g(x0), z0)), ACTIVATE(z0)) by F(g(g(y0)), z1) -> c(F(g(y0), n__f(g(g(y0)), z1)), ACTIVATE(z1)) F(g(z0), n__f(g(y0), y1)) -> c(F(z0, n__f(g(z0), n__f(g(y0), y1))), ACTIVATE(n__f(g(y0), y1))) F(g(z0), n__f(g(y0), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(g(y1), y2)))), ACTIVATE(n__f(g(y0), n__f(g(y1), y2)))) F(g(z0), n__f(g(y0), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(y1, y2)))), ACTIVATE(n__f(g(y0), n__f(y1, y2)))) ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0), z1) -> f(z0, n__f(g(z0), activate(z1))) f(z0, z1) -> n__f(z0, z1) activate(n__f(z0, z1)) -> f(z0, z1) activate(z0) -> z0 Tuples: F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) F(g(x0), n__f(z0, z1)) -> c(F(x0, n__f(g(x0), n__f(z0, z1))), ACTIVATE(n__f(z0, z1))) ACTIVATE(n__f(g(y0), z1)) -> c2(F(g(y0), z1)) ACTIVATE(n__f(g(y0), n__f(g(y1), y2))) -> c2(F(g(y0), n__f(g(y1), y2))) ACTIVATE(n__f(g(y0), n__f(y1, y2))) -> c2(F(g(y0), n__f(y1, y2))) F(g(g(y0)), z1) -> c(F(g(y0), n__f(g(g(y0)), z1)), ACTIVATE(z1)) F(g(z0), n__f(g(y0), y1)) -> c(F(z0, n__f(g(z0), n__f(g(y0), y1))), ACTIVATE(n__f(g(y0), y1))) F(g(z0), n__f(g(y0), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(g(y1), y2)))), ACTIVATE(n__f(g(y0), n__f(g(y1), y2)))) F(g(z0), n__f(g(y0), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(y1, y2)))), ACTIVATE(n__f(g(y0), n__f(y1, y2)))) S tuples: F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) F(g(x0), n__f(z0, z1)) -> c(F(x0, n__f(g(x0), n__f(z0, z1))), ACTIVATE(n__f(z0, z1))) ACTIVATE(n__f(g(y0), z1)) -> c2(F(g(y0), z1)) ACTIVATE(n__f(g(y0), n__f(g(y1), y2))) -> c2(F(g(y0), n__f(g(y1), y2))) ACTIVATE(n__f(g(y0), n__f(y1, y2))) -> c2(F(g(y0), n__f(y1, y2))) F(g(g(y0)), z1) -> c(F(g(y0), n__f(g(g(y0)), z1)), ACTIVATE(z1)) F(g(z0), n__f(g(y0), y1)) -> c(F(z0, n__f(g(z0), n__f(g(y0), y1))), ACTIVATE(n__f(g(y0), y1))) F(g(z0), n__f(g(y0), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(g(y1), y2)))), ACTIVATE(n__f(g(y0), n__f(g(y1), y2)))) F(g(z0), n__f(g(y0), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(y1, y2)))), ACTIVATE(n__f(g(y0), n__f(y1, y2)))) K tuples:none Defined Rule Symbols: f_2, activate_1 Defined Pair Symbols: F_2, ACTIVATE_1 Compound Symbols: c_2, c2_1 ---------------------------------------- (33) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(g(x0), n__f(z0, z1)) -> c(F(x0, n__f(g(x0), n__f(z0, z1))), ACTIVATE(n__f(z0, z1))) by F(g(g(y0)), n__f(z1, z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(z0), n__f(g(y0), z2)) -> c(F(z0, n__f(g(z0), n__f(g(y0), z2))), ACTIVATE(n__f(g(y0), z2))) F(g(z0), n__f(g(y0), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(g(y1), y2)))), ACTIVATE(n__f(g(y0), n__f(g(y1), y2)))) F(g(z0), n__f(g(y0), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(y1, y2)))), ACTIVATE(n__f(g(y0), n__f(y1, y2)))) F(g(g(g(y0))), n__f(z1, z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(y0)), n__f(g(y2), z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(y2), z2))), ACTIVATE(n__f(g(y2), z2))) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0), z1) -> f(z0, n__f(g(z0), activate(z1))) f(z0, z1) -> n__f(z0, z1) activate(n__f(z0, z1)) -> f(z0, z1) activate(z0) -> z0 Tuples: F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) ACTIVATE(n__f(g(y0), z1)) -> c2(F(g(y0), z1)) ACTIVATE(n__f(g(y0), n__f(g(y1), y2))) -> c2(F(g(y0), n__f(g(y1), y2))) ACTIVATE(n__f(g(y0), n__f(y1, y2))) -> c2(F(g(y0), n__f(y1, y2))) F(g(g(y0)), z1) -> c(F(g(y0), n__f(g(g(y0)), z1)), ACTIVATE(z1)) F(g(z0), n__f(g(y0), y1)) -> c(F(z0, n__f(g(z0), n__f(g(y0), y1))), ACTIVATE(n__f(g(y0), y1))) F(g(z0), n__f(g(y0), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(g(y1), y2)))), ACTIVATE(n__f(g(y0), n__f(g(y1), y2)))) F(g(z0), n__f(g(y0), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(y1, y2)))), ACTIVATE(n__f(g(y0), n__f(y1, y2)))) F(g(g(y0)), n__f(z1, z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(g(y0))), n__f(z1, z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(y0)), n__f(g(y2), z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(y2), z2))), ACTIVATE(n__f(g(y2), z2))) S tuples: F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) ACTIVATE(n__f(g(y0), z1)) -> c2(F(g(y0), z1)) ACTIVATE(n__f(g(y0), n__f(g(y1), y2))) -> c2(F(g(y0), n__f(g(y1), y2))) ACTIVATE(n__f(g(y0), n__f(y1, y2))) -> c2(F(g(y0), n__f(y1, y2))) F(g(g(y0)), z1) -> c(F(g(y0), n__f(g(g(y0)), z1)), ACTIVATE(z1)) F(g(z0), n__f(g(y0), y1)) -> c(F(z0, n__f(g(z0), n__f(g(y0), y1))), ACTIVATE(n__f(g(y0), y1))) F(g(z0), n__f(g(y0), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(g(y1), y2)))), ACTIVATE(n__f(g(y0), n__f(g(y1), y2)))) F(g(z0), n__f(g(y0), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(y1, y2)))), ACTIVATE(n__f(g(y0), n__f(y1, y2)))) F(g(g(y0)), n__f(z1, z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(g(y0))), n__f(z1, z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(y0)), n__f(g(y2), z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(y2), z2))), ACTIVATE(n__f(g(y2), z2))) K tuples:none Defined Rule Symbols: f_2, activate_1 Defined Pair Symbols: F_2, ACTIVATE_1 Compound Symbols: c_2, c2_1 ---------------------------------------- (35) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__f(g(y0), z1)) -> c2(F(g(y0), z1)) by ACTIVATE(n__f(g(z0), n__f(g(y1), y2))) -> c2(F(g(z0), n__f(g(y1), y2))) ACTIVATE(n__f(g(g(y0)), z1)) -> c2(F(g(g(y0)), z1)) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(g(y2), y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(g(y2), y3)))) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(y2, y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(y2, y3)))) ACTIVATE(n__f(g(g(y0)), n__f(y1, y2))) -> c2(F(g(g(y0)), n__f(y1, y2))) ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2))) -> c2(F(g(g(g(y0))), n__f(y1, y2))) ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2))) -> c2(F(g(g(y0)), n__f(g(y1), y2))) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0), z1) -> f(z0, n__f(g(z0), activate(z1))) f(z0, z1) -> n__f(z0, z1) activate(n__f(z0, z1)) -> f(z0, z1) activate(z0) -> z0 Tuples: F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) ACTIVATE(n__f(g(y0), n__f(g(y1), y2))) -> c2(F(g(y0), n__f(g(y1), y2))) ACTIVATE(n__f(g(y0), n__f(y1, y2))) -> c2(F(g(y0), n__f(y1, y2))) F(g(g(y0)), z1) -> c(F(g(y0), n__f(g(g(y0)), z1)), ACTIVATE(z1)) F(g(z0), n__f(g(y0), y1)) -> c(F(z0, n__f(g(z0), n__f(g(y0), y1))), ACTIVATE(n__f(g(y0), y1))) F(g(z0), n__f(g(y0), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(g(y1), y2)))), ACTIVATE(n__f(g(y0), n__f(g(y1), y2)))) F(g(z0), n__f(g(y0), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(y1, y2)))), ACTIVATE(n__f(g(y0), n__f(y1, y2)))) F(g(g(y0)), n__f(z1, z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(g(y0))), n__f(z1, z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(y0)), n__f(g(y2), z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(y2), z2))), ACTIVATE(n__f(g(y2), z2))) ACTIVATE(n__f(g(g(y0)), z1)) -> c2(F(g(g(y0)), z1)) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(g(y2), y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(g(y2), y3)))) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(y2, y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(y2, y3)))) ACTIVATE(n__f(g(g(y0)), n__f(y1, y2))) -> c2(F(g(g(y0)), n__f(y1, y2))) ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2))) -> c2(F(g(g(g(y0))), n__f(y1, y2))) ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2))) -> c2(F(g(g(y0)), n__f(g(y1), y2))) S tuples: F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) ACTIVATE(n__f(g(y0), n__f(g(y1), y2))) -> c2(F(g(y0), n__f(g(y1), y2))) ACTIVATE(n__f(g(y0), n__f(y1, y2))) -> c2(F(g(y0), n__f(y1, y2))) F(g(g(y0)), z1) -> c(F(g(y0), n__f(g(g(y0)), z1)), ACTIVATE(z1)) F(g(z0), n__f(g(y0), y1)) -> c(F(z0, n__f(g(z0), n__f(g(y0), y1))), ACTIVATE(n__f(g(y0), y1))) F(g(z0), n__f(g(y0), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(g(y1), y2)))), ACTIVATE(n__f(g(y0), n__f(g(y1), y2)))) F(g(z0), n__f(g(y0), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(y1, y2)))), ACTIVATE(n__f(g(y0), n__f(y1, y2)))) F(g(g(y0)), n__f(z1, z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(g(y0))), n__f(z1, z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(y0)), n__f(g(y2), z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(y2), z2))), ACTIVATE(n__f(g(y2), z2))) ACTIVATE(n__f(g(g(y0)), z1)) -> c2(F(g(g(y0)), z1)) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(g(y2), y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(g(y2), y3)))) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(y2, y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(y2, y3)))) ACTIVATE(n__f(g(g(y0)), n__f(y1, y2))) -> c2(F(g(g(y0)), n__f(y1, y2))) ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2))) -> c2(F(g(g(g(y0))), n__f(y1, y2))) ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2))) -> c2(F(g(g(y0)), n__f(g(y1), y2))) K tuples:none Defined Rule Symbols: f_2, activate_1 Defined Pair Symbols: F_2, ACTIVATE_1 Compound Symbols: c_2, c2_1 ---------------------------------------- (37) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__f(g(y0), n__f(y1, y2))) -> c2(F(g(y0), n__f(y1, y2))) by ACTIVATE(n__f(g(z0), n__f(g(y1), z2))) -> c2(F(g(z0), n__f(g(y1), z2))) ACTIVATE(n__f(g(g(y0)), n__f(z1, z2))) -> c2(F(g(g(y0)), n__f(z1, z2))) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(g(y2), y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(g(y2), y3)))) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(y2, y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(y2, y3)))) ACTIVATE(n__f(g(g(g(y0))), n__f(z1, z2))) -> c2(F(g(g(g(y0))), n__f(z1, z2))) ACTIVATE(n__f(g(g(y0)), n__f(g(y1), z2))) -> c2(F(g(g(y0)), n__f(g(y1), z2))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0), z1) -> f(z0, n__f(g(z0), activate(z1))) f(z0, z1) -> n__f(z0, z1) activate(n__f(z0, z1)) -> f(z0, z1) activate(z0) -> z0 Tuples: F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) ACTIVATE(n__f(g(y0), n__f(g(y1), y2))) -> c2(F(g(y0), n__f(g(y1), y2))) F(g(g(y0)), z1) -> c(F(g(y0), n__f(g(g(y0)), z1)), ACTIVATE(z1)) F(g(z0), n__f(g(y0), y1)) -> c(F(z0, n__f(g(z0), n__f(g(y0), y1))), ACTIVATE(n__f(g(y0), y1))) F(g(z0), n__f(g(y0), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(g(y1), y2)))), ACTIVATE(n__f(g(y0), n__f(g(y1), y2)))) F(g(z0), n__f(g(y0), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(y1, y2)))), ACTIVATE(n__f(g(y0), n__f(y1, y2)))) F(g(g(y0)), n__f(z1, z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(g(y0))), n__f(z1, z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(y0)), n__f(g(y2), z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(y2), z2))), ACTIVATE(n__f(g(y2), z2))) ACTIVATE(n__f(g(g(y0)), z1)) -> c2(F(g(g(y0)), z1)) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(g(y2), y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(g(y2), y3)))) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(y2, y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(y2, y3)))) ACTIVATE(n__f(g(g(y0)), n__f(y1, y2))) -> c2(F(g(g(y0)), n__f(y1, y2))) ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2))) -> c2(F(g(g(g(y0))), n__f(y1, y2))) ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2))) -> c2(F(g(g(y0)), n__f(g(y1), y2))) S tuples: F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) ACTIVATE(n__f(g(y0), n__f(g(y1), y2))) -> c2(F(g(y0), n__f(g(y1), y2))) F(g(g(y0)), z1) -> c(F(g(y0), n__f(g(g(y0)), z1)), ACTIVATE(z1)) F(g(z0), n__f(g(y0), y1)) -> c(F(z0, n__f(g(z0), n__f(g(y0), y1))), ACTIVATE(n__f(g(y0), y1))) F(g(z0), n__f(g(y0), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(g(y1), y2)))), ACTIVATE(n__f(g(y0), n__f(g(y1), y2)))) F(g(z0), n__f(g(y0), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(y1, y2)))), ACTIVATE(n__f(g(y0), n__f(y1, y2)))) F(g(g(y0)), n__f(z1, z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(g(y0))), n__f(z1, z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(y0)), n__f(g(y2), z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(y2), z2))), ACTIVATE(n__f(g(y2), z2))) ACTIVATE(n__f(g(g(y0)), z1)) -> c2(F(g(g(y0)), z1)) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(g(y2), y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(g(y2), y3)))) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(y2, y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(y2, y3)))) ACTIVATE(n__f(g(g(y0)), n__f(y1, y2))) -> c2(F(g(g(y0)), n__f(y1, y2))) ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2))) -> c2(F(g(g(g(y0))), n__f(y1, y2))) ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2))) -> c2(F(g(g(y0)), n__f(g(y1), y2))) K tuples:none Defined Rule Symbols: f_2, activate_1 Defined Pair Symbols: F_2, ACTIVATE_1 Compound Symbols: c_2, c2_1 ---------------------------------------- (39) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(g(z0), n__f(g(y0), y1)) -> c(F(z0, n__f(g(z0), n__f(g(y0), y1))), ACTIVATE(n__f(g(y0), y1))) by F(g(g(y0)), n__f(g(z1), z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(z0), n__f(g(z1), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), y2)))), ACTIVATE(n__f(g(z1), n__f(g(y1), y2)))) F(g(g(g(y0))), n__f(g(z1), z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(g(g(g(y0)))), n__f(g(z1), z2)) -> c(F(g(g(g(y0))), n__f(g(g(g(g(y0)))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(z0), n__f(g(g(y0)), z2)) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), z2))), ACTIVATE(n__f(g(g(y0)), z2))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(y2, y3))))) F(g(z0), n__f(g(g(y0)), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(y1, y2)))), ACTIVATE(n__f(g(g(y0)), n__f(y1, y2)))) F(g(z0), n__f(g(g(g(y0))), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(g(y0))), n__f(y1, y2)))), ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2)))) F(g(z0), n__f(g(g(y0)), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(g(y1), y2)))), ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2)))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0), z1) -> f(z0, n__f(g(z0), activate(z1))) f(z0, z1) -> n__f(z0, z1) activate(n__f(z0, z1)) -> f(z0, z1) activate(z0) -> z0 Tuples: F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) ACTIVATE(n__f(g(y0), n__f(g(y1), y2))) -> c2(F(g(y0), n__f(g(y1), y2))) F(g(g(y0)), z1) -> c(F(g(y0), n__f(g(g(y0)), z1)), ACTIVATE(z1)) F(g(z0), n__f(g(y0), y1)) -> c(F(z0, n__f(g(z0), n__f(g(y0), y1))), ACTIVATE(n__f(g(y0), y1))) F(g(z0), n__f(g(y0), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(g(y1), y2)))), ACTIVATE(n__f(g(y0), n__f(g(y1), y2)))) F(g(z0), n__f(g(y0), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(y1, y2)))), ACTIVATE(n__f(g(y0), n__f(y1, y2)))) F(g(g(y0)), n__f(z1, z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(g(y0))), n__f(z1, z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(y0)), n__f(g(y2), z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(y2), z2))), ACTIVATE(n__f(g(y2), z2))) ACTIVATE(n__f(g(g(y0)), z1)) -> c2(F(g(g(y0)), z1)) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(g(y2), y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(g(y2), y3)))) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(y2, y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(y2, y3)))) ACTIVATE(n__f(g(g(y0)), n__f(y1, y2))) -> c2(F(g(g(y0)), n__f(y1, y2))) ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2))) -> c2(F(g(g(g(y0))), n__f(y1, y2))) ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2))) -> c2(F(g(g(y0)), n__f(g(y1), y2))) F(g(g(g(y0))), n__f(g(z1), z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(g(g(g(y0)))), n__f(g(z1), z2)) -> c(F(g(g(g(y0))), n__f(g(g(g(g(y0)))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(z0), n__f(g(g(y0)), z2)) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), z2))), ACTIVATE(n__f(g(g(y0)), z2))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(y2, y3))))) F(g(z0), n__f(g(g(y0)), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(y1, y2)))), ACTIVATE(n__f(g(g(y0)), n__f(y1, y2)))) F(g(z0), n__f(g(g(g(y0))), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(g(y0))), n__f(y1, y2)))), ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2)))) F(g(z0), n__f(g(g(y0)), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(g(y1), y2)))), ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2)))) S tuples: F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) ACTIVATE(n__f(g(y0), n__f(g(y1), y2))) -> c2(F(g(y0), n__f(g(y1), y2))) F(g(g(y0)), z1) -> c(F(g(y0), n__f(g(g(y0)), z1)), ACTIVATE(z1)) F(g(z0), n__f(g(y0), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(g(y1), y2)))), ACTIVATE(n__f(g(y0), n__f(g(y1), y2)))) F(g(z0), n__f(g(y0), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(y1, y2)))), ACTIVATE(n__f(g(y0), n__f(y1, y2)))) F(g(g(y0)), n__f(z1, z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(g(y0))), n__f(z1, z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(y0)), n__f(g(y2), z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(y2), z2))), ACTIVATE(n__f(g(y2), z2))) ACTIVATE(n__f(g(g(y0)), z1)) -> c2(F(g(g(y0)), z1)) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(g(y2), y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(g(y2), y3)))) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(y2, y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(y2, y3)))) ACTIVATE(n__f(g(g(y0)), n__f(y1, y2))) -> c2(F(g(g(y0)), n__f(y1, y2))) ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2))) -> c2(F(g(g(g(y0))), n__f(y1, y2))) ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2))) -> c2(F(g(g(y0)), n__f(g(y1), y2))) F(g(g(g(y0))), n__f(g(z1), z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(g(g(g(y0)))), n__f(g(z1), z2)) -> c(F(g(g(g(y0))), n__f(g(g(g(g(y0)))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(z0), n__f(g(g(y0)), z2)) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), z2))), ACTIVATE(n__f(g(g(y0)), z2))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(y2, y3))))) F(g(z0), n__f(g(g(y0)), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(y1, y2)))), ACTIVATE(n__f(g(g(y0)), n__f(y1, y2)))) F(g(z0), n__f(g(g(g(y0))), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(g(y0))), n__f(y1, y2)))), ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2)))) F(g(z0), n__f(g(g(y0)), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(g(y1), y2)))), ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2)))) K tuples:none Defined Rule Symbols: f_2, activate_1 Defined Pair Symbols: F_2, ACTIVATE_1 Compound Symbols: c_2, c2_1 ---------------------------------------- (41) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(g(z0), n__f(g(y0), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(y1, y2)))), ACTIVATE(n__f(g(y0), n__f(y1, y2)))) by F(g(g(y0)), n__f(g(z1), n__f(z2, z3))) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(z0), n__f(g(z1), n__f(g(y1), z3))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), z3)))), ACTIVATE(n__f(g(z1), n__f(g(y1), z3)))) F(g(g(g(y0))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(g(y0)))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(y0))), n__f(g(g(g(g(y0)))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(z0), n__f(g(g(y0)), n__f(z2, z3))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(z2, z3)))), ACTIVATE(n__f(g(g(y0)), n__f(z2, z3)))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(y2, y3))))) F(g(z0), n__f(g(g(g(y0))), n__f(z2, z3))) -> c(F(z0, n__f(g(z0), n__f(g(g(g(y0))), n__f(z2, z3)))), ACTIVATE(n__f(g(g(g(y0))), n__f(z2, z3)))) F(g(z0), n__f(g(g(y0)), n__f(g(y1), z3))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(g(y1), z3)))), ACTIVATE(n__f(g(g(y0)), n__f(g(y1), z3)))) F(g(g(g(g(g(y0))))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(g(y0)))), n__f(g(g(g(g(g(y0))))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(y0)), n__f(g(z1), n__f(g(y3), z3))) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), n__f(g(y3), z3)))), ACTIVATE(n__f(g(z1), n__f(g(y3), z3)))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0), z1) -> f(z0, n__f(g(z0), activate(z1))) f(z0, z1) -> n__f(z0, z1) activate(n__f(z0, z1)) -> f(z0, z1) activate(z0) -> z0 Tuples: F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) ACTIVATE(n__f(g(y0), n__f(g(y1), y2))) -> c2(F(g(y0), n__f(g(y1), y2))) F(g(g(y0)), z1) -> c(F(g(y0), n__f(g(g(y0)), z1)), ACTIVATE(z1)) F(g(z0), n__f(g(y0), y1)) -> c(F(z0, n__f(g(z0), n__f(g(y0), y1))), ACTIVATE(n__f(g(y0), y1))) F(g(z0), n__f(g(y0), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(g(y1), y2)))), ACTIVATE(n__f(g(y0), n__f(g(y1), y2)))) F(g(z0), n__f(g(y0), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(y1, y2)))), ACTIVATE(n__f(g(y0), n__f(y1, y2)))) F(g(g(y0)), n__f(z1, z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(g(y0))), n__f(z1, z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(y0)), n__f(g(y2), z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(y2), z2))), ACTIVATE(n__f(g(y2), z2))) ACTIVATE(n__f(g(g(y0)), z1)) -> c2(F(g(g(y0)), z1)) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(g(y2), y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(g(y2), y3)))) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(y2, y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(y2, y3)))) ACTIVATE(n__f(g(g(y0)), n__f(y1, y2))) -> c2(F(g(g(y0)), n__f(y1, y2))) ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2))) -> c2(F(g(g(g(y0))), n__f(y1, y2))) ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2))) -> c2(F(g(g(y0)), n__f(g(y1), y2))) F(g(g(g(y0))), n__f(g(z1), z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(g(g(g(y0)))), n__f(g(z1), z2)) -> c(F(g(g(g(y0))), n__f(g(g(g(g(y0)))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(z0), n__f(g(g(y0)), z2)) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), z2))), ACTIVATE(n__f(g(g(y0)), z2))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(y2, y3))))) F(g(z0), n__f(g(g(y0)), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(y1, y2)))), ACTIVATE(n__f(g(g(y0)), n__f(y1, y2)))) F(g(z0), n__f(g(g(g(y0))), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(g(y0))), n__f(y1, y2)))), ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2)))) F(g(z0), n__f(g(g(y0)), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(g(y1), y2)))), ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2)))) F(g(g(y0)), n__f(g(z1), n__f(z2, z3))) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(y0))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(g(y0)))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(y0))), n__f(g(g(g(g(y0)))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(g(g(y0))))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(g(y0)))), n__f(g(g(g(g(g(y0))))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(y0)), n__f(g(z1), n__f(g(y3), z3))) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), n__f(g(y3), z3)))), ACTIVATE(n__f(g(z1), n__f(g(y3), z3)))) S tuples: F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) ACTIVATE(n__f(g(y0), n__f(g(y1), y2))) -> c2(F(g(y0), n__f(g(y1), y2))) F(g(g(y0)), z1) -> c(F(g(y0), n__f(g(g(y0)), z1)), ACTIVATE(z1)) F(g(z0), n__f(g(y0), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(g(y1), y2)))), ACTIVATE(n__f(g(y0), n__f(g(y1), y2)))) F(g(g(y0)), n__f(z1, z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(g(y0))), n__f(z1, z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(y0)), n__f(g(y2), z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(y2), z2))), ACTIVATE(n__f(g(y2), z2))) ACTIVATE(n__f(g(g(y0)), z1)) -> c2(F(g(g(y0)), z1)) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(g(y2), y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(g(y2), y3)))) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(y2, y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(y2, y3)))) ACTIVATE(n__f(g(g(y0)), n__f(y1, y2))) -> c2(F(g(g(y0)), n__f(y1, y2))) ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2))) -> c2(F(g(g(g(y0))), n__f(y1, y2))) ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2))) -> c2(F(g(g(y0)), n__f(g(y1), y2))) F(g(g(g(y0))), n__f(g(z1), z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(g(g(g(y0)))), n__f(g(z1), z2)) -> c(F(g(g(g(y0))), n__f(g(g(g(g(y0)))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(z0), n__f(g(g(y0)), z2)) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), z2))), ACTIVATE(n__f(g(g(y0)), z2))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(y2, y3))))) F(g(z0), n__f(g(g(y0)), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(y1, y2)))), ACTIVATE(n__f(g(g(y0)), n__f(y1, y2)))) F(g(z0), n__f(g(g(g(y0))), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(g(y0))), n__f(y1, y2)))), ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2)))) F(g(z0), n__f(g(g(y0)), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(g(y1), y2)))), ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2)))) F(g(g(y0)), n__f(g(z1), n__f(z2, z3))) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(y0))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(g(y0)))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(y0))), n__f(g(g(g(g(y0)))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(g(g(y0))))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(g(y0)))), n__f(g(g(g(g(g(y0))))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(y0)), n__f(g(z1), n__f(g(y3), z3))) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), n__f(g(y3), z3)))), ACTIVATE(n__f(g(z1), n__f(g(y3), z3)))) K tuples:none Defined Rule Symbols: f_2, activate_1 Defined Pair Symbols: F_2, ACTIVATE_1 Compound Symbols: c_2, c2_1 ---------------------------------------- (43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(g(z0), n__f(g(y0), z2)) -> c(F(z0, n__f(g(z0), n__f(g(y0), z2))), ACTIVATE(n__f(g(y0), z2))) by F(g(g(y0)), n__f(g(z1), z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(z0), n__f(g(z1), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), y2)))), ACTIVATE(n__f(g(z1), n__f(g(y1), y2)))) F(g(g(g(y0))), n__f(g(z1), z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(g(g(g(y0)))), n__f(g(z1), z2)) -> c(F(g(g(g(y0))), n__f(g(g(g(g(y0)))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(z0), n__f(g(g(y0)), z2)) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), z2))), ACTIVATE(n__f(g(g(y0)), z2))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(y2, y3))))) F(g(z0), n__f(g(g(y0)), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(y1, y2)))), ACTIVATE(n__f(g(g(y0)), n__f(y1, y2)))) F(g(z0), n__f(g(g(g(y0))), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(g(y0))), n__f(y1, y2)))), ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2)))) F(g(z0), n__f(g(g(y0)), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(g(y1), y2)))), ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2)))) F(g(g(g(g(g(y0))))), n__f(g(z1), z2)) -> c(F(g(g(g(g(y0)))), n__f(g(g(g(g(g(y0))))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(g(y0)), n__f(g(z1), n__f(g(y3), y4))) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), n__f(g(y3), y4)))), ACTIVATE(n__f(g(z1), n__f(g(y3), y4)))) F(g(g(y0)), n__f(g(z1), n__f(y3, y4))) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), n__f(y3, y4)))), ACTIVATE(n__f(g(z1), n__f(y3, y4)))) F(g(g(g(g(g(g(y0)))))), n__f(g(z1), z2)) -> c(F(g(g(g(g(g(y0))))), n__f(g(g(g(g(g(g(y0)))))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0), z1) -> f(z0, n__f(g(z0), activate(z1))) f(z0, z1) -> n__f(z0, z1) activate(n__f(z0, z1)) -> f(z0, z1) activate(z0) -> z0 Tuples: F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) ACTIVATE(n__f(g(y0), n__f(g(y1), y2))) -> c2(F(g(y0), n__f(g(y1), y2))) F(g(g(y0)), z1) -> c(F(g(y0), n__f(g(g(y0)), z1)), ACTIVATE(z1)) F(g(z0), n__f(g(y0), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(g(y1), y2)))), ACTIVATE(n__f(g(y0), n__f(g(y1), y2)))) F(g(z0), n__f(g(y0), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(y1, y2)))), ACTIVATE(n__f(g(y0), n__f(y1, y2)))) F(g(g(y0)), n__f(z1, z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(g(y0))), n__f(z1, z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(y0)), n__f(g(y2), z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(y2), z2))), ACTIVATE(n__f(g(y2), z2))) ACTIVATE(n__f(g(g(y0)), z1)) -> c2(F(g(g(y0)), z1)) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(g(y2), y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(g(y2), y3)))) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(y2, y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(y2, y3)))) ACTIVATE(n__f(g(g(y0)), n__f(y1, y2))) -> c2(F(g(g(y0)), n__f(y1, y2))) ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2))) -> c2(F(g(g(g(y0))), n__f(y1, y2))) ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2))) -> c2(F(g(g(y0)), n__f(g(y1), y2))) F(g(g(g(y0))), n__f(g(z1), z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(g(g(g(y0)))), n__f(g(z1), z2)) -> c(F(g(g(g(y0))), n__f(g(g(g(g(y0)))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(z0), n__f(g(g(y0)), z2)) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), z2))), ACTIVATE(n__f(g(g(y0)), z2))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(y2, y3))))) F(g(z0), n__f(g(g(y0)), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(y1, y2)))), ACTIVATE(n__f(g(g(y0)), n__f(y1, y2)))) F(g(z0), n__f(g(g(g(y0))), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(g(y0))), n__f(y1, y2)))), ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2)))) F(g(z0), n__f(g(g(y0)), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(g(y1), y2)))), ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2)))) F(g(g(y0)), n__f(g(z1), n__f(z2, z3))) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(y0))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(g(y0)))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(y0))), n__f(g(g(g(g(y0)))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(g(g(y0))))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(g(y0)))), n__f(g(g(g(g(g(y0))))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(y0)), n__f(g(z1), n__f(g(y3), z3))) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), n__f(g(y3), z3)))), ACTIVATE(n__f(g(z1), n__f(g(y3), z3)))) F(g(g(g(g(g(y0))))), n__f(g(z1), z2)) -> c(F(g(g(g(g(y0)))), n__f(g(g(g(g(g(y0))))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(g(g(g(g(g(y0)))))), n__f(g(z1), z2)) -> c(F(g(g(g(g(g(y0))))), n__f(g(g(g(g(g(g(y0)))))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) S tuples: F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) ACTIVATE(n__f(g(y0), n__f(g(y1), y2))) -> c2(F(g(y0), n__f(g(y1), y2))) F(g(g(y0)), z1) -> c(F(g(y0), n__f(g(g(y0)), z1)), ACTIVATE(z1)) F(g(z0), n__f(g(y0), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(g(y1), y2)))), ACTIVATE(n__f(g(y0), n__f(g(y1), y2)))) F(g(g(y0)), n__f(z1, z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(g(y0))), n__f(z1, z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(y0)), n__f(g(y2), z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(y2), z2))), ACTIVATE(n__f(g(y2), z2))) ACTIVATE(n__f(g(g(y0)), z1)) -> c2(F(g(g(y0)), z1)) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(g(y2), y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(g(y2), y3)))) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(y2, y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(y2, y3)))) ACTIVATE(n__f(g(g(y0)), n__f(y1, y2))) -> c2(F(g(g(y0)), n__f(y1, y2))) ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2))) -> c2(F(g(g(g(y0))), n__f(y1, y2))) ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2))) -> c2(F(g(g(y0)), n__f(g(y1), y2))) F(g(g(g(y0))), n__f(g(z1), z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(g(g(g(y0)))), n__f(g(z1), z2)) -> c(F(g(g(g(y0))), n__f(g(g(g(g(y0)))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(z0), n__f(g(g(y0)), z2)) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), z2))), ACTIVATE(n__f(g(g(y0)), z2))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(y2, y3))))) F(g(z0), n__f(g(g(y0)), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(y1, y2)))), ACTIVATE(n__f(g(g(y0)), n__f(y1, y2)))) F(g(z0), n__f(g(g(g(y0))), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(g(y0))), n__f(y1, y2)))), ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2)))) F(g(z0), n__f(g(g(y0)), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(g(y1), y2)))), ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2)))) F(g(g(y0)), n__f(g(z1), n__f(z2, z3))) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(y0))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(g(y0)))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(y0))), n__f(g(g(g(g(y0)))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(g(g(y0))))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(g(y0)))), n__f(g(g(g(g(g(y0))))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(y0)), n__f(g(z1), n__f(g(y3), z3))) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), n__f(g(y3), z3)))), ACTIVATE(n__f(g(z1), n__f(g(y3), z3)))) K tuples:none Defined Rule Symbols: f_2, activate_1 Defined Pair Symbols: F_2, ACTIVATE_1 Compound Symbols: c_2, c2_1 ---------------------------------------- (45) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(g(z0), n__f(g(y0), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(y1, y2)))), ACTIVATE(n__f(g(y0), n__f(y1, y2)))) by F(g(g(y0)), n__f(g(z1), n__f(z2, z3))) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(z0), n__f(g(z1), n__f(g(y1), z3))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), z3)))), ACTIVATE(n__f(g(z1), n__f(g(y1), z3)))) F(g(g(g(y0))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(g(y0)))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(y0))), n__f(g(g(g(g(y0)))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(z0), n__f(g(g(y0)), n__f(z2, z3))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(z2, z3)))), ACTIVATE(n__f(g(g(y0)), n__f(z2, z3)))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(y2, y3))))) F(g(z0), n__f(g(g(g(y0))), n__f(z2, z3))) -> c(F(z0, n__f(g(z0), n__f(g(g(g(y0))), n__f(z2, z3)))), ACTIVATE(n__f(g(g(g(y0))), n__f(z2, z3)))) F(g(z0), n__f(g(g(y0)), n__f(g(y1), z3))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(g(y1), z3)))), ACTIVATE(n__f(g(g(y0)), n__f(g(y1), z3)))) F(g(g(g(g(g(y0))))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(g(y0)))), n__f(g(g(g(g(g(y0))))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(y0)), n__f(g(z1), n__f(g(y3), z3))) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), n__f(g(y3), z3)))), ACTIVATE(n__f(g(z1), n__f(g(y3), z3)))) F(g(g(g(g(g(g(y0)))))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(g(g(y0))))), n__f(g(g(g(g(g(g(y0)))))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(g(g(g(g(y0))))))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(g(g(g(y0)))))), n__f(g(g(g(g(g(g(g(y0))))))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0), z1) -> f(z0, n__f(g(z0), activate(z1))) f(z0, z1) -> n__f(z0, z1) activate(n__f(z0, z1)) -> f(z0, z1) activate(z0) -> z0 Tuples: F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) ACTIVATE(n__f(g(y0), n__f(g(y1), y2))) -> c2(F(g(y0), n__f(g(y1), y2))) F(g(g(y0)), z1) -> c(F(g(y0), n__f(g(g(y0)), z1)), ACTIVATE(z1)) F(g(z0), n__f(g(y0), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(g(y1), y2)))), ACTIVATE(n__f(g(y0), n__f(g(y1), y2)))) F(g(g(y0)), n__f(z1, z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(g(y0))), n__f(z1, z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(y0)), n__f(g(y2), z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(y2), z2))), ACTIVATE(n__f(g(y2), z2))) ACTIVATE(n__f(g(g(y0)), z1)) -> c2(F(g(g(y0)), z1)) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(g(y2), y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(g(y2), y3)))) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(y2, y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(y2, y3)))) ACTIVATE(n__f(g(g(y0)), n__f(y1, y2))) -> c2(F(g(g(y0)), n__f(y1, y2))) ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2))) -> c2(F(g(g(g(y0))), n__f(y1, y2))) ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2))) -> c2(F(g(g(y0)), n__f(g(y1), y2))) F(g(g(g(y0))), n__f(g(z1), z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(g(g(g(y0)))), n__f(g(z1), z2)) -> c(F(g(g(g(y0))), n__f(g(g(g(g(y0)))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(z0), n__f(g(g(y0)), z2)) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), z2))), ACTIVATE(n__f(g(g(y0)), z2))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(y2, y3))))) F(g(z0), n__f(g(g(y0)), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(y1, y2)))), ACTIVATE(n__f(g(g(y0)), n__f(y1, y2)))) F(g(z0), n__f(g(g(g(y0))), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(g(y0))), n__f(y1, y2)))), ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2)))) F(g(z0), n__f(g(g(y0)), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(g(y1), y2)))), ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2)))) F(g(g(y0)), n__f(g(z1), n__f(z2, z3))) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(y0))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(g(y0)))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(y0))), n__f(g(g(g(g(y0)))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(g(g(y0))))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(g(y0)))), n__f(g(g(g(g(g(y0))))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(y0)), n__f(g(z1), n__f(g(y3), z3))) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), n__f(g(y3), z3)))), ACTIVATE(n__f(g(z1), n__f(g(y3), z3)))) F(g(g(g(g(g(y0))))), n__f(g(z1), z2)) -> c(F(g(g(g(g(y0)))), n__f(g(g(g(g(g(y0))))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(g(g(g(g(g(y0)))))), n__f(g(z1), z2)) -> c(F(g(g(g(g(g(y0))))), n__f(g(g(g(g(g(g(y0)))))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(g(g(g(g(g(y0)))))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(g(g(y0))))), n__f(g(g(g(g(g(g(y0)))))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(g(g(g(g(y0))))))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(g(g(g(y0)))))), n__f(g(g(g(g(g(g(g(y0))))))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) S tuples: F(g(x0), n__f(g(z0), z1)) -> c(F(x0, n__f(g(x0), f(z0, n__f(g(z0), activate(z1))))), ACTIVATE(n__f(g(z0), z1))) ACTIVATE(n__f(g(y0), n__f(g(y1), y2))) -> c2(F(g(y0), n__f(g(y1), y2))) F(g(g(y0)), z1) -> c(F(g(y0), n__f(g(g(y0)), z1)), ACTIVATE(z1)) F(g(z0), n__f(g(y0), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(y0), n__f(g(y1), y2)))), ACTIVATE(n__f(g(y0), n__f(g(y1), y2)))) F(g(g(y0)), n__f(z1, z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(g(y0))), n__f(z1, z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(z1, z2))), ACTIVATE(n__f(z1, z2))) F(g(g(y0)), n__f(g(y2), z2)) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(y2), z2))), ACTIVATE(n__f(g(y2), z2))) ACTIVATE(n__f(g(g(y0)), z1)) -> c2(F(g(g(y0)), z1)) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(g(y2), y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(g(y2), y3)))) ACTIVATE(n__f(g(z0), n__f(g(y1), n__f(y2, y3)))) -> c2(F(g(z0), n__f(g(y1), n__f(y2, y3)))) ACTIVATE(n__f(g(g(y0)), n__f(y1, y2))) -> c2(F(g(g(y0)), n__f(y1, y2))) ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2))) -> c2(F(g(g(g(y0))), n__f(y1, y2))) ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2))) -> c2(F(g(g(y0)), n__f(g(y1), y2))) F(g(g(g(y0))), n__f(g(z1), z2)) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(g(g(g(y0)))), n__f(g(z1), z2)) -> c(F(g(g(g(y0))), n__f(g(g(g(g(y0)))), n__f(g(z1), z2))), ACTIVATE(n__f(g(z1), z2))) F(g(z0), n__f(g(g(y0)), z2)) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), z2))), ACTIVATE(n__f(g(g(y0)), z2))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(g(y2), y3))))) F(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3)))) -> c(F(z0, n__f(g(z0), n__f(g(z1), n__f(g(y1), n__f(y2, y3))))), ACTIVATE(n__f(g(z1), n__f(g(y1), n__f(y2, y3))))) F(g(z0), n__f(g(g(y0)), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(y1, y2)))), ACTIVATE(n__f(g(g(y0)), n__f(y1, y2)))) F(g(z0), n__f(g(g(g(y0))), n__f(y1, y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(g(y0))), n__f(y1, y2)))), ACTIVATE(n__f(g(g(g(y0))), n__f(y1, y2)))) F(g(z0), n__f(g(g(y0)), n__f(g(y1), y2))) -> c(F(z0, n__f(g(z0), n__f(g(g(y0)), n__f(g(y1), y2)))), ACTIVATE(n__f(g(g(y0)), n__f(g(y1), y2)))) F(g(g(y0)), n__f(g(z1), n__f(z2, z3))) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(y0))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(y0)), n__f(g(g(g(y0))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(g(y0)))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(y0))), n__f(g(g(g(g(y0)))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(g(g(g(y0))))), n__f(g(z1), n__f(z2, z3))) -> c(F(g(g(g(g(y0)))), n__f(g(g(g(g(g(y0))))), n__f(g(z1), n__f(z2, z3)))), ACTIVATE(n__f(g(z1), n__f(z2, z3)))) F(g(g(y0)), n__f(g(z1), n__f(g(y3), z3))) -> c(F(g(y0), n__f(g(g(y0)), n__f(g(z1), n__f(g(y3), z3)))), ACTIVATE(n__f(g(z1), n__f(g(y3), z3)))) K tuples:none Defined Rule Symbols: f_2, activate_1 Defined Pair Symbols: F_2, ACTIVATE_1 Compound Symbols: c_2, c2_1