MAYBE proof of input_av4BJ6PzKk.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 2875 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 1390 ms] (24) CpxRNTS (25) CompletionProof [UPPER BOUND(ID), 0 ms] (26) CpxTypedWeightedCompleteTrs (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (30) CdtProblem (31) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 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), 7 ms] (42) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(s(x), y) -> f(x, s(s(x))) f(x, s(s(y))) -> f(y, 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(s(x), y) -> f(x, s(s(x))) f(x, s(s(y))) -> f(y, 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(s(x), y) -> f(x, s(s(x))) f(x, s(s(y))) -> f(y, 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(s(x), y) -> f(x, s(s(x))) [1] f(x, s(s(y))) -> f(y, 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(s(x), y) -> f(x, s(s(x))) [1] f(x, s(s(y))) -> f(y, x) [1] The TRS has the following type information: f :: s -> s -> f s :: s -> s 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: f_2 (c) The following functions are completely defined: none 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(s(x), y) -> f(x, s(s(x))) [1] f(x, s(s(y))) -> f(y, x) [1] The TRS has the following type information: f :: s -> s -> f s :: s -> s const :: f const1 :: s Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(s(x), y) -> f(x, s(s(x))) [1] f(x, s(s(y))) -> f(y, x) [1] The TRS has the following type information: f :: s -> s -> f s :: s -> s const :: f const1 :: s 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: f(z, z') -{ 1 }-> f(x, 1 + (1 + x)) :|: x >= 0, y >= 0, z = 1 + x, z' = y f(z, z') -{ 1 }-> f(y, x) :|: z' = 1 + (1 + y), x >= 0, y >= 0, z = x ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> f(z - 1, 1 + (1 + (z - 1))) :|: z - 1 >= 0, z' >= 0 f(z, z') -{ 1 }-> f(z' - 2, z) :|: z >= 0, z' - 2 >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { f } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> f(z - 1, 1 + (1 + (z - 1))) :|: z - 1 >= 0, z' >= 0 f(z, z') -{ 1 }-> f(z' - 2, z) :|: z >= 0, z' - 2 >= 0 Function symbols to be analyzed: {f} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> f(z - 1, 1 + (1 + (z - 1))) :|: z - 1 >= 0, z' >= 0 f(z, z') -{ 1 }-> f(z' - 2, z) :|: z >= 0, z' - 2 >= 0 Function symbols to be analyzed: {f} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> f(z - 1, 1 + (1 + (z - 1))) :|: z - 1 >= 0, z' >= 0 f(z, z') -{ 1 }-> f(z' - 2, z) :|: z >= 0, z' - 2 >= 0 Function symbols to be analyzed: {f} Previous analysis results are: f: runtime: ?, size: O(1) [0] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> f(z - 1, 1 + (1 + (z - 1))) :|: z - 1 >= 0, z' >= 0 f(z, z') -{ 1 }-> f(z' - 2, z) :|: z >= 0, z' - 2 >= 0 Function symbols to be analyzed: {f} Previous analysis results are: f: runtime: INF, size: O(1) [0] ---------------------------------------- (25) 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: f(v0, v1) -> null_f [0] And the following fresh constants: null_f, const ---------------------------------------- (26) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(s(x), y) -> f(x, s(s(x))) [1] f(x, s(s(y))) -> f(y, x) [1] f(v0, v1) -> null_f [0] The TRS has the following type information: f :: s -> s -> null_f s :: s -> s null_f :: null_f const :: s Rewrite Strategy: INNERMOST ---------------------------------------- (27) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: null_f => 0 const => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> f(x, 1 + (1 + x)) :|: x >= 0, y >= 0, z = 1 + x, z' = y f(z, z') -{ 1 }-> f(y, x) :|: z' = 1 + (1 + y), x >= 0, y >= 0, z = x f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (29) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: f(s(z0), z1) -> f(z0, s(s(z0))) f(z0, s(s(z1))) -> f(z1, z0) Tuples: F(s(z0), z1) -> c(F(z0, s(s(z0)))) F(z0, s(s(z1))) -> c1(F(z1, z0)) S tuples: F(s(z0), z1) -> c(F(z0, s(s(z0)))) F(z0, s(s(z1))) -> c1(F(z1, z0)) K tuples:none Defined Rule Symbols: f_2 Defined Pair Symbols: F_2 Compound Symbols: c_1, c1_1 ---------------------------------------- (31) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: f(s(z0), z1) -> f(z0, s(s(z0))) f(z0, s(s(z1))) -> f(z1, z0) ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(s(z0), z1) -> c(F(z0, s(s(z0)))) F(z0, s(s(z1))) -> c1(F(z1, z0)) S tuples: F(s(z0), z1) -> c(F(z0, s(s(z0)))) F(z0, s(s(z1))) -> c1(F(z1, z0)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: F_2 Compound Symbols: c_1, c1_1 ---------------------------------------- (33) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(z0, s(s(z1))) -> c1(F(z1, z0)) by F(z0, s(s(s(y0)))) -> c1(F(s(y0), z0)) F(s(s(y1)), s(s(z1))) -> c1(F(z1, s(s(y1)))) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(s(z0), z1) -> c(F(z0, s(s(z0)))) F(z0, s(s(s(y0)))) -> c1(F(s(y0), z0)) F(s(s(y1)), s(s(z1))) -> c1(F(z1, s(s(y1)))) S tuples: F(s(z0), z1) -> c(F(z0, s(s(z0)))) F(z0, s(s(s(y0)))) -> c1(F(s(y0), z0)) F(s(s(y1)), s(s(z1))) -> c1(F(z1, s(s(y1)))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: F_2 Compound Symbols: c_1, c1_1 ---------------------------------------- (35) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(z0), z1) -> c(F(z0, s(s(z0)))) by F(s(s(y0)), z1) -> c(F(s(y0), s(s(s(y0))))) F(s(s(s(y0))), z1) -> c(F(s(s(y0)), s(s(s(s(y0)))))) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(z0, s(s(s(y0)))) -> c1(F(s(y0), z0)) F(s(s(y1)), s(s(z1))) -> c1(F(z1, s(s(y1)))) F(s(s(y0)), z1) -> c(F(s(y0), s(s(s(y0))))) F(s(s(s(y0))), z1) -> c(F(s(s(y0)), s(s(s(s(y0)))))) S tuples: F(z0, s(s(s(y0)))) -> c1(F(s(y0), z0)) F(s(s(y1)), s(s(z1))) -> c1(F(z1, s(s(y1)))) F(s(s(y0)), z1) -> c(F(s(y0), s(s(s(y0))))) F(s(s(s(y0))), z1) -> c(F(s(s(y0)), s(s(s(s(y0)))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: F_2 Compound Symbols: c1_1, c_1 ---------------------------------------- (37) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(z0, s(s(s(y0)))) -> c1(F(s(y0), z0)) by F(s(s(s(y1))), s(s(s(z1)))) -> c1(F(s(z1), s(s(s(y1))))) F(s(s(y1)), s(s(s(s(y0))))) -> c1(F(s(s(y0)), s(s(y1)))) F(z0, s(s(s(s(y0))))) -> c1(F(s(s(y0)), z0)) F(z0, s(s(s(s(s(y0)))))) -> c1(F(s(s(s(y0))), z0)) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(s(s(y1)), s(s(z1))) -> c1(F(z1, s(s(y1)))) F(s(s(y0)), z1) -> c(F(s(y0), s(s(s(y0))))) F(s(s(s(y0))), z1) -> c(F(s(s(y0)), s(s(s(s(y0)))))) F(s(s(s(y1))), s(s(s(z1)))) -> c1(F(s(z1), s(s(s(y1))))) F(s(s(y1)), s(s(s(s(y0))))) -> c1(F(s(s(y0)), s(s(y1)))) F(z0, s(s(s(s(y0))))) -> c1(F(s(s(y0)), z0)) F(z0, s(s(s(s(s(y0)))))) -> c1(F(s(s(s(y0))), z0)) S tuples: F(s(s(y1)), s(s(z1))) -> c1(F(z1, s(s(y1)))) F(s(s(y0)), z1) -> c(F(s(y0), s(s(s(y0))))) F(s(s(s(y0))), z1) -> c(F(s(s(y0)), s(s(s(s(y0)))))) F(s(s(s(y1))), s(s(s(z1)))) -> c1(F(s(z1), s(s(s(y1))))) F(s(s(y1)), s(s(s(s(y0))))) -> c1(F(s(s(y0)), s(s(y1)))) F(z0, s(s(s(s(y0))))) -> c1(F(s(s(y0)), z0)) F(z0, s(s(s(s(s(y0)))))) -> c1(F(s(s(s(y0))), z0)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: F_2 Compound Symbols: c1_1, c_1 ---------------------------------------- (39) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(y1)), s(s(z1))) -> c1(F(z1, s(s(y1)))) by F(s(s(z0)), s(s(s(s(y0))))) -> c1(F(s(s(y0)), s(s(z0)))) F(s(s(z0)), s(s(s(s(s(y0)))))) -> c1(F(s(s(s(y0))), s(s(z0)))) F(s(s(s(y1))), s(s(s(s(s(y0)))))) -> c1(F(s(s(s(y0))), s(s(s(y1))))) F(s(s(s(s(y1)))), s(s(s(s(y0))))) -> c1(F(s(s(y0)), s(s(s(s(y1)))))) F(s(s(s(s(y1)))), s(s(z1))) -> c1(F(z1, s(s(s(s(y1)))))) F(s(s(s(s(s(y1))))), s(s(z1))) -> c1(F(z1, s(s(s(s(s(y1))))))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(s(s(y0)), z1) -> c(F(s(y0), s(s(s(y0))))) F(s(s(s(y0))), z1) -> c(F(s(s(y0)), s(s(s(s(y0)))))) F(s(s(s(y1))), s(s(s(z1)))) -> c1(F(s(z1), s(s(s(y1))))) F(s(s(y1)), s(s(s(s(y0))))) -> c1(F(s(s(y0)), s(s(y1)))) F(z0, s(s(s(s(y0))))) -> c1(F(s(s(y0)), z0)) F(z0, s(s(s(s(s(y0)))))) -> c1(F(s(s(s(y0))), z0)) F(s(s(z0)), s(s(s(s(s(y0)))))) -> c1(F(s(s(s(y0))), s(s(z0)))) F(s(s(s(y1))), s(s(s(s(s(y0)))))) -> c1(F(s(s(s(y0))), s(s(s(y1))))) F(s(s(s(s(y1)))), s(s(s(s(y0))))) -> c1(F(s(s(y0)), s(s(s(s(y1)))))) F(s(s(s(s(y1)))), s(s(z1))) -> c1(F(z1, s(s(s(s(y1)))))) F(s(s(s(s(s(y1))))), s(s(z1))) -> c1(F(z1, s(s(s(s(s(y1))))))) S tuples: F(s(s(y0)), z1) -> c(F(s(y0), s(s(s(y0))))) F(s(s(s(y0))), z1) -> c(F(s(s(y0)), s(s(s(s(y0)))))) F(s(s(s(y1))), s(s(s(z1)))) -> c1(F(s(z1), s(s(s(y1))))) F(s(s(y1)), s(s(s(s(y0))))) -> c1(F(s(s(y0)), s(s(y1)))) F(z0, s(s(s(s(y0))))) -> c1(F(s(s(y0)), z0)) F(z0, s(s(s(s(s(y0)))))) -> c1(F(s(s(s(y0))), z0)) F(s(s(z0)), s(s(s(s(s(y0)))))) -> c1(F(s(s(s(y0))), s(s(z0)))) F(s(s(s(y1))), s(s(s(s(s(y0)))))) -> c1(F(s(s(s(y0))), s(s(s(y1))))) F(s(s(s(s(y1)))), s(s(s(s(y0))))) -> c1(F(s(s(y0)), s(s(s(s(y1)))))) F(s(s(s(s(y1)))), s(s(z1))) -> c1(F(z1, s(s(s(s(y1)))))) F(s(s(s(s(s(y1))))), s(s(z1))) -> c1(F(z1, s(s(s(s(s(y1))))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: F_2 Compound Symbols: c_1, c1_1 ---------------------------------------- (41) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(y0)), z1) -> c(F(s(y0), s(s(s(y0))))) by F(s(s(s(y0))), z1) -> c(F(s(s(y0)), s(s(s(s(y0)))))) F(s(s(s(s(y0)))), z1) -> c(F(s(s(s(y0))), s(s(s(s(s(y0))))))) F(s(s(s(s(s(y0))))), z1) -> c(F(s(s(s(s(y0)))), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(s(s(y0)))))), z1) -> c(F(s(s(s(s(s(y0))))), s(s(s(s(s(s(s(y0))))))))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(s(s(s(y0))), z1) -> c(F(s(s(y0)), s(s(s(s(y0)))))) F(s(s(s(y1))), s(s(s(z1)))) -> c1(F(s(z1), s(s(s(y1))))) F(s(s(y1)), s(s(s(s(y0))))) -> c1(F(s(s(y0)), s(s(y1)))) F(z0, s(s(s(s(y0))))) -> c1(F(s(s(y0)), z0)) F(z0, s(s(s(s(s(y0)))))) -> c1(F(s(s(s(y0))), z0)) F(s(s(z0)), s(s(s(s(s(y0)))))) -> c1(F(s(s(s(y0))), s(s(z0)))) F(s(s(s(y1))), s(s(s(s(s(y0)))))) -> c1(F(s(s(s(y0))), s(s(s(y1))))) F(s(s(s(s(y1)))), s(s(s(s(y0))))) -> c1(F(s(s(y0)), s(s(s(s(y1)))))) F(s(s(s(s(y1)))), s(s(z1))) -> c1(F(z1, s(s(s(s(y1)))))) F(s(s(s(s(s(y1))))), s(s(z1))) -> c1(F(z1, s(s(s(s(s(y1))))))) F(s(s(s(s(y0)))), z1) -> c(F(s(s(s(y0))), s(s(s(s(s(y0))))))) F(s(s(s(s(s(y0))))), z1) -> c(F(s(s(s(s(y0)))), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(s(s(y0)))))), z1) -> c(F(s(s(s(s(s(y0))))), s(s(s(s(s(s(s(y0))))))))) S tuples: F(s(s(s(y0))), z1) -> c(F(s(s(y0)), s(s(s(s(y0)))))) F(s(s(s(y1))), s(s(s(z1)))) -> c1(F(s(z1), s(s(s(y1))))) F(s(s(y1)), s(s(s(s(y0))))) -> c1(F(s(s(y0)), s(s(y1)))) F(z0, s(s(s(s(y0))))) -> c1(F(s(s(y0)), z0)) F(z0, s(s(s(s(s(y0)))))) -> c1(F(s(s(s(y0))), z0)) F(s(s(z0)), s(s(s(s(s(y0)))))) -> c1(F(s(s(s(y0))), s(s(z0)))) F(s(s(s(y1))), s(s(s(s(s(y0)))))) -> c1(F(s(s(s(y0))), s(s(s(y1))))) F(s(s(s(s(y1)))), s(s(s(s(y0))))) -> c1(F(s(s(y0)), s(s(s(s(y1)))))) F(s(s(s(s(y1)))), s(s(z1))) -> c1(F(z1, s(s(s(s(y1)))))) F(s(s(s(s(s(y1))))), s(s(z1))) -> c1(F(z1, s(s(s(s(s(y1))))))) F(s(s(s(s(y0)))), z1) -> c(F(s(s(s(y0))), s(s(s(s(s(y0))))))) F(s(s(s(s(s(y0))))), z1) -> c(F(s(s(s(s(y0)))), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(s(s(y0)))))), z1) -> c(F(s(s(s(s(s(y0))))), s(s(s(s(s(s(s(y0))))))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: F_2 Compound Symbols: c_1, c1_1