KILLED proof of input_BnkykRsJF7.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 1 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) CdtForwardInstantiationProof [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), 0 ms] (30) CdtProblem (31) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtRewritingProof [BOTH BOUNDS(ID, ID), 3 ms] (60) CdtProblem (61) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 6 ms] (80) 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: dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y)) 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: dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y)) 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: dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y)) 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: dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y)) [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: dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y)) [1] The TRS has the following type information: dfib :: s -> dfib -> dfib 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: none (c) The following functions are completely defined: dfib_2 Due to the following rules being added: dfib(v0, v1) -> const [0] 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: dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y)) [1] dfib(v0, v1) -> const [0] The TRS has the following type information: dfib :: s -> const -> const s :: s -> s const :: const 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: dfib(s(s(s(s(x')))), y) -> dfib(s(s(s(x'))), dfib(s(x'), dfib(x', y))) [2] dfib(s(s(x)), y) -> dfib(s(x), const) [1] dfib(v0, v1) -> const [0] The TRS has the following type information: dfib :: s -> const -> const s :: s -> s const :: const 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: dfib(z, z') -{ 1 }-> dfib(1 + x, 0) :|: x >= 0, y >= 0, z' = y, z = 1 + (1 + x) dfib(z, z') -{ 2 }-> dfib(1 + (1 + (1 + x')), dfib(1 + x', dfib(x', y))) :|: z = 1 + (1 + (1 + (1 + x'))), x' >= 0, y >= 0, z' = y dfib(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 ---------------------------------------- (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: dfib(z, z') -{ 1 }-> dfib(1 + (z - 2), 0) :|: z - 2 >= 0, z' >= 0 dfib(z, z') -{ 2 }-> dfib(1 + (1 + (1 + (z - 4))), dfib(1 + (z - 4), dfib(z - 4, z'))) :|: z - 4 >= 0, z' >= 0 dfib(z, z') -{ 0 }-> 0 :|: 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: dfib(v0, v1) -> null_dfib [0] And the following fresh constants: null_dfib, const ---------------------------------------- (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: dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y)) [1] dfib(v0, v1) -> null_dfib [0] The TRS has the following type information: dfib :: s -> null_dfib -> null_dfib s :: s -> s null_dfib :: null_dfib const :: s 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: null_dfib => 0 const => 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: dfib(z, z') -{ 1 }-> dfib(1 + x, dfib(x, y)) :|: x >= 0, y >= 0, z' = y, z = 1 + (1 + x) dfib(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 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: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(z0)), z1) -> c(DFIB(s(z0), dfib(z0, z1)), DFIB(z0, z1)) S tuples: DFIB(s(s(z0)), z1) -> c(DFIB(s(z0), dfib(z0, z1)), DFIB(z0, z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (23) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DFIB(s(s(z0)), z1) -> c(DFIB(s(z0), dfib(z0, z1)), DFIB(z0, z1)) by DFIB(s(s(s(y0))), z1) -> c(DFIB(s(s(y0)), dfib(s(y0), z1)), DFIB(s(y0), z1)) DFIB(s(s(s(s(y0)))), z1) -> c(DFIB(s(s(s(y0))), dfib(s(s(y0)), z1)), DFIB(s(s(y0)), z1)) ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(y0))), z1) -> c(DFIB(s(s(y0)), dfib(s(y0), z1)), DFIB(s(y0), z1)) DFIB(s(s(s(s(y0)))), z1) -> c(DFIB(s(s(s(y0))), dfib(s(s(y0)), z1)), DFIB(s(s(y0)), z1)) S tuples: DFIB(s(s(s(y0))), z1) -> c(DFIB(s(s(y0)), dfib(s(y0), z1)), DFIB(s(y0), z1)) DFIB(s(s(s(s(y0)))), z1) -> c(DFIB(s(s(s(y0))), dfib(s(s(y0)), z1)), DFIB(s(s(y0)), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (25) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DFIB(s(s(s(y0))), z1) -> c(DFIB(s(s(y0)), dfib(s(y0), z1)), DFIB(s(y0), z1)) by DFIB(s(s(s(s(z0)))), z1) -> c(DFIB(s(s(s(z0))), dfib(s(z0), dfib(z0, z1))), DFIB(s(s(z0)), z1)) ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(y0)))), z1) -> c(DFIB(s(s(s(y0))), dfib(s(s(y0)), z1)), DFIB(s(s(y0)), z1)) DFIB(s(s(s(s(z0)))), z1) -> c(DFIB(s(s(s(z0))), dfib(s(z0), dfib(z0, z1))), DFIB(s(s(z0)), z1)) S tuples: DFIB(s(s(s(s(y0)))), z1) -> c(DFIB(s(s(s(y0))), dfib(s(s(y0)), z1)), DFIB(s(s(y0)), z1)) DFIB(s(s(s(s(z0)))), z1) -> c(DFIB(s(s(s(z0))), dfib(s(z0), dfib(z0, z1))), DFIB(s(s(z0)), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (27) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DFIB(s(s(s(s(y0)))), z1) -> c(DFIB(s(s(s(y0))), dfib(s(s(y0)), z1)), DFIB(s(s(y0)), z1)) by DFIB(s(s(s(s(z0)))), z1) -> c(DFIB(s(s(s(z0))), dfib(s(z0), dfib(z0, z1))), DFIB(s(s(z0)), z1)) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(z0)))), z1) -> c(DFIB(s(s(s(z0))), dfib(s(z0), dfib(z0, z1))), DFIB(s(s(z0)), z1)) S tuples: DFIB(s(s(s(s(z0)))), z1) -> c(DFIB(s(s(s(z0))), dfib(s(z0), dfib(z0, z1))), DFIB(s(s(z0)), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (29) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DFIB(s(s(s(s(z0)))), z1) -> c(DFIB(s(s(s(z0))), dfib(s(z0), dfib(z0, z1))), DFIB(s(s(z0)), z1)) by DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(s(s(z0))), dfib(s(z0), dfib(z0, z1)))), DFIB(s(s(s(s(z0)))), z1)) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(z0)))), z1) -> c(DFIB(s(s(s(z0))), dfib(s(z0), dfib(z0, z1))), DFIB(s(s(z0)), z1)) DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(s(s(z0))), dfib(s(z0), dfib(z0, z1)))), DFIB(s(s(s(s(z0)))), z1)) S tuples: DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(s(s(z0))), dfib(s(z0), dfib(z0, z1)))), DFIB(s(s(s(s(z0)))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (31) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DFIB(s(s(s(s(z0)))), z1) -> c(DFIB(s(s(s(z0))), dfib(s(z0), dfib(z0, z1))), DFIB(s(s(z0)), z1)) by DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(s(s(z0))), dfib(s(z0), dfib(z0, z1)))), DFIB(s(s(s(s(z0)))), z1)) ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(s(s(z0))), dfib(s(z0), dfib(z0, z1)))), DFIB(s(s(s(s(z0)))), z1)) S tuples: DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(s(s(z0))), dfib(s(z0), dfib(z0, z1)))), DFIB(s(s(s(s(z0)))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (33) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(s(s(z0))), dfib(s(z0), dfib(z0, z1)))), DFIB(s(s(s(s(z0)))), z1)) by DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(z0), dfib(z0, z1))))), DFIB(s(s(s(s(z0)))), z1)) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(s(s(z0))), dfib(s(z0), dfib(z0, z1)))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(z0), dfib(z0, z1))))), DFIB(s(s(s(s(z0)))), z1)) S tuples: DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(z0), dfib(z0, z1))))), DFIB(s(s(s(s(z0)))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (35) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(s(s(z0))), dfib(s(z0), dfib(z0, z1)))), DFIB(s(s(s(s(z0)))), z1)) by DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(z0), dfib(z0, z1))))), DFIB(s(s(s(s(z0)))), z1)) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(z0), dfib(z0, z1))))), DFIB(s(s(s(s(z0)))), z1)) S tuples: DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(z0), dfib(z0, z1))))), DFIB(s(s(s(s(z0)))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (37) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(z0), dfib(z0, z1))))), DFIB(s(s(s(s(z0)))), z1)) by DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(z0), dfib(z0, z1))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) S tuples: DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (39) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(z0), dfib(z0, z1))))), DFIB(s(s(s(s(z0)))), z1)) by DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) S tuples: DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (41) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) by DFIB(s(s(s(s(s(s(y0)))))), z1) -> c(DFIB(s(s(s(s(s(y0))))), dfib(s(s(y0)), dfib(s(y0), dfib(s(s(y0)), z1)))), DFIB(s(s(s(s(y0)))), z1)) DFIB(s(s(s(s(s(s(s(y0))))))), z1) -> c(DFIB(s(s(s(s(s(s(y0)))))), dfib(s(s(s(y0))), dfib(s(s(y0)), dfib(s(s(s(y0))), z1)))), DFIB(s(s(s(s(s(y0))))), z1)) DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), dfib(s(s(s(s(y0)))), z1)))), DFIB(s(s(s(s(s(s(y0)))))), z1)) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(y0)))))), z1) -> c(DFIB(s(s(s(s(s(y0))))), dfib(s(s(y0)), dfib(s(y0), dfib(s(s(y0)), z1)))), DFIB(s(s(s(s(y0)))), z1)) DFIB(s(s(s(s(s(s(s(y0))))))), z1) -> c(DFIB(s(s(s(s(s(s(y0)))))), dfib(s(s(s(y0))), dfib(s(s(y0)), dfib(s(s(s(y0))), z1)))), DFIB(s(s(s(s(s(y0))))), z1)) DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), dfib(s(s(s(s(y0)))), z1)))), DFIB(s(s(s(s(s(s(y0)))))), z1)) S tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(y0)))))), z1) -> c(DFIB(s(s(s(s(s(y0))))), dfib(s(s(y0)), dfib(s(y0), dfib(s(s(y0)), z1)))), DFIB(s(s(s(s(y0)))), z1)) DFIB(s(s(s(s(s(s(s(y0))))))), z1) -> c(DFIB(s(s(s(s(s(s(y0)))))), dfib(s(s(s(y0))), dfib(s(s(y0)), dfib(s(s(s(y0))), z1)))), DFIB(s(s(s(s(s(y0))))), z1)) DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), dfib(s(s(s(s(y0)))), z1)))), DFIB(s(s(s(s(s(s(y0)))))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (43) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(y0)))))), z1) -> c(DFIB(s(s(s(s(s(y0))))), dfib(s(s(y0)), dfib(s(y0), dfib(s(s(y0)), z1)))), DFIB(s(s(s(s(y0)))), z1)) by DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), z1))))), DFIB(s(s(s(s(z0)))), z1)) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(y0))))))), z1) -> c(DFIB(s(s(s(s(s(s(y0)))))), dfib(s(s(s(y0))), dfib(s(s(y0)), dfib(s(s(s(y0))), z1)))), DFIB(s(s(s(s(s(y0))))), z1)) DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), dfib(s(s(s(s(y0)))), z1)))), DFIB(s(s(s(s(s(s(y0)))))), z1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), z1))))), DFIB(s(s(s(s(z0)))), z1)) S tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(y0))))))), z1) -> c(DFIB(s(s(s(s(s(s(y0)))))), dfib(s(s(s(y0))), dfib(s(s(y0)), dfib(s(s(s(y0))), z1)))), DFIB(s(s(s(s(s(y0))))), z1)) DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), dfib(s(s(s(s(y0)))), z1)))), DFIB(s(s(s(s(s(s(y0)))))), z1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), z1))))), DFIB(s(s(s(s(z0)))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (45) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(s(y0))))))), z1) -> c(DFIB(s(s(s(s(s(s(y0)))))), dfib(s(s(s(y0))), dfib(s(s(y0)), dfib(s(s(s(y0))), z1)))), DFIB(s(s(s(s(s(y0))))), z1)) by DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1))))), DFIB(s(s(s(s(s(z0))))), z1)) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), dfib(s(s(s(s(y0)))), z1)))), DFIB(s(s(s(s(s(s(y0)))))), z1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), z1))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1))))), DFIB(s(s(s(s(s(z0))))), z1)) S tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), dfib(s(s(s(s(y0)))), z1)))), DFIB(s(s(s(s(s(s(y0)))))), z1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), z1))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1))))), DFIB(s(s(s(s(s(z0))))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (47) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), dfib(s(s(s(s(y0)))), z1)))), DFIB(s(s(s(s(s(s(y0)))))), z1)) by DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), z1))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) S tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), z1))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (49) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), z1))))), DFIB(s(s(s(s(z0)))), z1)) by DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) S tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (51) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1))))), DFIB(s(s(s(s(s(z0))))), z1)) by DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) S tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (53) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) by DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) S tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DFIB(s(s(s(s(s(z0))))), x1) -> c(DFIB(s(s(s(s(z0)))), dfib(s(z0), dfib(z0, dfib(s(z0), x1)))), DFIB(s(s(s(z0))), x1)) by DFIB(s(s(s(s(s(s(y0)))))), z1) -> c(DFIB(s(s(s(s(s(y0))))), dfib(s(s(y0)), dfib(s(y0), dfib(s(s(y0)), z1)))), DFIB(s(s(s(s(y0)))), z1)) DFIB(s(s(s(s(s(s(s(y0))))))), z1) -> c(DFIB(s(s(s(s(s(s(y0)))))), dfib(s(s(s(y0))), dfib(s(s(y0)), dfib(s(s(s(y0))), z1)))), DFIB(s(s(s(s(s(y0))))), z1)) DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), dfib(s(s(s(s(y0)))), z1)))), DFIB(s(s(s(s(s(s(y0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(s(s(y0))))), z1)))), DFIB(s(s(s(s(s(s(s(y0))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(s(s(y0)))))), z1)))), DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1)) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(y0)))))), z1) -> c(DFIB(s(s(s(s(s(y0))))), dfib(s(s(y0)), dfib(s(y0), dfib(s(s(y0)), z1)))), DFIB(s(s(s(s(y0)))), z1)) DFIB(s(s(s(s(s(s(s(y0))))))), z1) -> c(DFIB(s(s(s(s(s(s(y0)))))), dfib(s(s(s(y0))), dfib(s(s(y0)), dfib(s(s(s(y0))), z1)))), DFIB(s(s(s(s(s(y0))))), z1)) DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), dfib(s(s(s(s(y0)))), z1)))), DFIB(s(s(s(s(s(s(y0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(s(s(y0))))), z1)))), DFIB(s(s(s(s(s(s(s(y0))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(s(s(y0)))))), z1)))), DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1)) S tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (57) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(y0)))))), z1) -> c(DFIB(s(s(s(s(s(y0))))), dfib(s(s(y0)), dfib(s(y0), dfib(s(s(y0)), z1)))), DFIB(s(s(s(s(y0)))), z1)) by DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), z1))))), DFIB(s(s(s(s(z0)))), z1)) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(s(y0))))))), z1) -> c(DFIB(s(s(s(s(s(s(y0)))))), dfib(s(s(s(y0))), dfib(s(s(y0)), dfib(s(s(s(y0))), z1)))), DFIB(s(s(s(s(s(y0))))), z1)) DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), dfib(s(s(s(s(y0)))), z1)))), DFIB(s(s(s(s(s(s(y0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(s(s(y0))))), z1)))), DFIB(s(s(s(s(s(s(s(y0))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(s(s(y0)))))), z1)))), DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), z1))))), DFIB(s(s(s(s(z0)))), z1)) S tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (59) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(s(y0))))))), z1) -> c(DFIB(s(s(s(s(s(s(y0)))))), dfib(s(s(s(y0))), dfib(s(s(y0)), dfib(s(s(s(y0))), z1)))), DFIB(s(s(s(s(s(y0))))), z1)) by DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1))))), DFIB(s(s(s(s(s(z0))))), z1)) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), dfib(s(s(s(s(y0)))), z1)))), DFIB(s(s(s(s(s(s(y0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(s(s(y0))))), z1)))), DFIB(s(s(s(s(s(s(s(y0))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(s(s(y0)))))), z1)))), DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), z1))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1))))), DFIB(s(s(s(s(s(z0))))), z1)) S tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (61) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), dfib(s(s(s(s(y0)))), z1)))), DFIB(s(s(s(s(s(s(y0)))))), z1)) by DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(s(s(y0))))), z1)))), DFIB(s(s(s(s(s(s(s(y0))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(s(s(y0)))))), z1)))), DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), z1))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) S tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (63) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(s(s(y0))))), z1)))), DFIB(s(s(s(s(s(s(s(y0))))))), z1)) by DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(z0)))))))), dfib(s(s(s(s(z0)))), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), z1))))), DFIB(s(s(s(s(s(s(s(z0))))))), z1)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(s(s(y0)))))), z1)))), DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), z1))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(z0)))))))), dfib(s(s(s(s(z0)))), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), z1))))), DFIB(s(s(s(s(s(s(s(z0))))))), z1)) S tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (65) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(s(s(y0)))))), z1)))), DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1)) by DFIB(s(s(s(s(s(s(s(s(s(s(z0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), dfib(s(s(s(s(s(z0))))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), dfib(s(s(s(s(s(s(z0)))))), z1))))), DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1)) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), z1))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(z0)))))))), dfib(s(s(s(s(z0)))), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), z1))))), DFIB(s(s(s(s(s(s(s(z0))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(z0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), dfib(s(s(s(s(s(z0))))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), dfib(s(s(s(s(s(s(z0)))))), z1))))), DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1)) S tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (67) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), z1))))), DFIB(s(s(s(s(z0)))), z1)) by DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(z0)))))))), dfib(s(s(s(s(z0)))), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), z1))))), DFIB(s(s(s(s(s(s(s(z0))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(z0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), dfib(s(s(s(s(s(z0))))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), dfib(s(s(s(s(s(s(z0)))))), z1))))), DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1)) S tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (69) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1))))), DFIB(s(s(s(s(s(z0))))), z1)) by DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(z0)))))))), dfib(s(s(s(s(z0)))), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), z1))))), DFIB(s(s(s(s(s(s(s(z0))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(z0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), dfib(s(s(s(s(s(z0))))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), dfib(s(s(s(s(s(s(z0)))))), z1))))), DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1)) S tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (71) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) by DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(z0)))))))), dfib(s(s(s(s(z0)))), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), z1))))), DFIB(s(s(s(s(s(s(s(z0))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(z0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), dfib(s(s(s(s(s(z0))))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), dfib(s(s(s(s(s(s(z0)))))), z1))))), DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1)) S tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (73) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(z0)))))))), dfib(s(s(s(s(z0)))), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), z1))))), DFIB(s(s(s(s(s(s(s(z0))))))), z1)) by DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(z0)))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), z1)))))), DFIB(s(s(s(s(s(s(s(z0))))))), z1)) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(z0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), dfib(s(s(s(s(s(z0))))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), dfib(s(s(s(s(s(s(z0)))))), z1))))), DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(z0)))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), z1)))))), DFIB(s(s(s(s(s(s(s(z0))))))), z1)) S tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (75) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DFIB(s(s(s(s(s(s(s(s(s(s(z0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), dfib(s(s(s(s(s(z0))))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), dfib(s(s(s(s(s(s(z0)))))), z1))))), DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1)) by DFIB(s(s(s(s(s(s(s(s(s(s(z0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), dfib(s(s(s(s(z0)))), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), dfib(s(s(s(s(s(s(z0)))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1)) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(z0)))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), z1)))))), DFIB(s(s(s(s(s(s(s(z0))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(z0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), dfib(s(s(s(s(z0)))), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), dfib(s(s(s(s(s(s(z0)))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1)) S tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (77) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) by DFIB(s(s(s(s(s(s(s(y0))))))), z1) -> c(DFIB(s(s(s(s(s(s(y0)))))), dfib(s(s(y0)), dfib(s(y0), dfib(s(s(y0)), dfib(s(s(y0)), dfib(s(y0), z1)))))), DFIB(s(s(s(s(s(y0))))), z1)) DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(y0))), dfib(s(s(y0)), dfib(s(s(s(y0))), dfib(s(s(s(y0))), dfib(s(s(y0)), z1)))))), DFIB(s(s(s(s(s(s(y0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), dfib(s(s(s(s(y0)))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), z1)))))), DFIB(s(s(s(s(s(s(s(y0))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(y0)))), z1)))))), DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(s(y0))))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(y0))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(s(s(y0)))))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(s(s(y0))))))))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(y0)))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), z1)) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(z0)))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), z1)))))), DFIB(s(s(s(s(s(s(s(z0))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(z0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), dfib(s(s(s(s(z0)))), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), dfib(s(s(s(s(s(s(z0)))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1)) DFIB(s(s(s(s(s(s(s(y0))))))), z1) -> c(DFIB(s(s(s(s(s(s(y0)))))), dfib(s(s(y0)), dfib(s(y0), dfib(s(s(y0)), dfib(s(s(y0)), dfib(s(y0), z1)))))), DFIB(s(s(s(s(s(y0))))), z1)) DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(y0))), dfib(s(s(y0)), dfib(s(s(s(y0))), dfib(s(s(s(y0))), dfib(s(s(y0)), z1)))))), DFIB(s(s(s(s(s(s(y0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), dfib(s(s(s(s(y0)))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), z1)))))), DFIB(s(s(s(s(s(s(s(y0))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(y0)))), z1)))))), DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(s(y0))))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(y0))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(s(s(y0)))))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(s(s(y0))))))))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(y0)))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), z1)) S tuples: DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(s(y0))))))), z1) -> c(DFIB(s(s(s(s(s(s(y0)))))), dfib(s(s(y0)), dfib(s(y0), dfib(s(s(y0)), dfib(s(s(y0)), dfib(s(y0), z1)))))), DFIB(s(s(s(s(s(y0))))), z1)) DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(y0))), dfib(s(s(y0)), dfib(s(s(s(y0))), dfib(s(s(s(y0))), dfib(s(s(y0)), z1)))))), DFIB(s(s(s(s(s(s(y0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), dfib(s(s(s(s(y0)))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), z1)))))), DFIB(s(s(s(s(s(s(s(y0))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(y0)))), z1)))))), DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(s(y0))))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(y0))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(s(s(y0)))))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(s(s(y0))))))))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(y0)))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2 ---------------------------------------- (79) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) by DFIB(s(s(s(s(s(s(s(y0))))))), z1) -> c(DFIB(s(s(s(s(s(s(y0)))))), dfib(s(s(y0)), dfib(s(y0), dfib(s(s(y0)), dfib(s(s(y0)), dfib(s(y0), z1)))))), DFIB(s(s(s(s(s(y0))))), z1)) DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(y0))), dfib(s(s(y0)), dfib(s(s(s(y0))), dfib(s(s(s(y0))), dfib(s(s(y0)), z1)))))), DFIB(s(s(s(s(s(s(y0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), dfib(s(s(s(s(y0)))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), z1)))))), DFIB(s(s(s(s(s(s(s(y0))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(y0)))), z1)))))), DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(s(y0))))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(y0))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(s(s(y0)))))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(s(s(y0))))))))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(y0)))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(s(s(s(y0))))))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(s(s(s(y0)))))))))))), dfib(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(s(s(s(y0))))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(s(s(s(y0))))))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(s(s(s(s(y0)))))))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(s(s(s(s(y0))))))))))))), dfib(s(s(s(s(s(s(s(s(s(y0))))))))), dfib(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(s(s(s(s(s(y0))))))))), dfib(s(s(s(s(s(s(s(s(s(y0))))))))), dfib(s(s(s(s(s(s(s(s(y0)))))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(s(s(s(s(y0)))))))))))), z1)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: dfib(s(s(z0)), z1) -> dfib(s(z0), dfib(z0, z1)) Tuples: DFIB(s(s(s(s(s(s(z0)))))), z1) -> c(DFIB(s(s(s(s(s(z0))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(z0), dfib(z0, z1)))))), DFIB(s(s(s(s(z0)))), z1)) DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(z0)))))))), dfib(s(s(s(z0))), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), z1)))))), DFIB(s(s(s(s(s(s(s(z0))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(z0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(z0))))))))), dfib(s(s(s(s(z0)))), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), dfib(s(s(s(s(s(z0))))), dfib(s(s(s(s(s(s(z0)))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1)) DFIB(s(s(s(s(s(s(s(y0))))))), z1) -> c(DFIB(s(s(s(s(s(s(y0)))))), dfib(s(s(y0)), dfib(s(y0), dfib(s(s(y0)), dfib(s(s(y0)), dfib(s(y0), z1)))))), DFIB(s(s(s(s(s(y0))))), z1)) DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(y0))), dfib(s(s(y0)), dfib(s(s(s(y0))), dfib(s(s(s(y0))), dfib(s(s(y0)), z1)))))), DFIB(s(s(s(s(s(s(y0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), dfib(s(s(s(s(y0)))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), z1)))))), DFIB(s(s(s(s(s(s(s(y0))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(y0)))), z1)))))), DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(s(y0))))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(y0))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(s(s(y0)))))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(s(s(y0))))))))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(y0)))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(s(s(s(y0))))))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(s(s(s(y0)))))))))))), dfib(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(s(s(s(y0))))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(s(s(s(y0))))))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(s(s(s(s(y0)))))))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(s(s(s(s(y0))))))))))))), dfib(s(s(s(s(s(s(s(s(s(y0))))))))), dfib(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(s(s(s(s(s(y0))))))))), dfib(s(s(s(s(s(s(s(s(s(y0))))))))), dfib(s(s(s(s(s(s(s(s(y0)))))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(s(s(s(s(y0)))))))))))), z1)) S tuples: DFIB(s(s(s(s(s(s(s(z0))))))), z1) -> c(DFIB(s(s(s(s(s(s(z0)))))), dfib(s(z0), dfib(z0, dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), z1)))))), DFIB(s(s(s(s(s(z0))))), z1)) DFIB(s(s(s(s(s(s(s(s(z0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(z0))))))), dfib(s(s(z0)), dfib(s(z0), dfib(s(s(z0)), dfib(s(s(s(z0))), dfib(s(s(s(s(z0)))), z1)))))), DFIB(s(s(s(s(s(s(z0)))))), z1)) DFIB(s(s(s(s(s(s(s(y0))))))), z1) -> c(DFIB(s(s(s(s(s(s(y0)))))), dfib(s(s(y0)), dfib(s(y0), dfib(s(s(y0)), dfib(s(s(y0)), dfib(s(y0), z1)))))), DFIB(s(s(s(s(s(y0))))), z1)) DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(y0))), dfib(s(s(y0)), dfib(s(s(s(y0))), dfib(s(s(s(y0))), dfib(s(s(y0)), z1)))))), DFIB(s(s(s(s(s(s(y0)))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(y0)))))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), dfib(s(s(s(s(y0)))), dfib(s(s(s(s(y0)))), dfib(s(s(s(y0))), z1)))))), DFIB(s(s(s(s(s(s(s(y0))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(y0)))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(y0)))), z1)))))), DFIB(s(s(s(s(s(s(s(s(y0)))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(s(y0))))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(y0))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(y0))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(s(y0))))))))), z1)) DFIB(s(s(s(s(s(s(s(s(s(s(s(s(y0)))))))))))), z1) -> c(DFIB(s(s(s(s(s(s(s(s(s(s(s(y0))))))))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(y0)))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(s(y0))))))), dfib(s(s(s(s(s(s(y0)))))), z1)))))), DFIB(s(s(s(s(s(s(s(s(s(s(y0)))))))))), z1)) K tuples:none Defined Rule Symbols: dfib_2 Defined Pair Symbols: DFIB_2 Compound Symbols: c_2