KILLED proof of input_YbMzQg3e23.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CompletionProof [UPPER BOUND(ID), 0 ms] (18) CpxTypedWeightedCompleteTrs (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (22) CdtProblem (23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CdtProblem (25) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem ---------------------------------------- (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: ack(0, y) -> s(y) ack(s(x), 0) -> ack(x, s(0)) ack(s(x), s(y)) -> ack(x, ack(s(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: ack(0', y) -> s(y) ack(s(x), 0') -> ack(x, s(0')) ack(s(x), s(y)) -> ack(x, ack(s(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: ack(0, y) -> s(y) ack(s(x), 0) -> ack(x, s(0)) ack(s(x), s(y)) -> ack(x, ack(s(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: ack(0, y) -> s(y) [1] ack(s(x), 0) -> ack(x, s(0)) [1] ack(s(x), s(y)) -> ack(x, ack(s(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: ack(0, y) -> s(y) [1] ack(s(x), 0) -> ack(x, s(0)) [1] ack(s(x), s(y)) -> ack(x, ack(s(x), y)) [1] The TRS has the following type information: ack :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0: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: ack_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (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: ack(0, y) -> s(y) [1] ack(s(x), 0) -> ack(x, s(0)) [1] ack(s(x), s(y)) -> ack(x, ack(s(x), y)) [1] The TRS has the following type information: ack :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0: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: ack(0, y) -> s(y) [1] ack(s(x), 0) -> ack(x, s(0)) [1] ack(s(x), s(0)) -> ack(x, ack(x, s(0))) [2] ack(s(x), s(s(y'))) -> ack(x, ack(x, ack(s(x), y'))) [2] The TRS has the following type information: ack :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0: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: 0 => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: ack(z, z') -{ 2 }-> ack(x, ack(x, ack(1 + x, y'))) :|: z' = 1 + (1 + y'), x >= 0, y' >= 0, z = 1 + x ack(z, z') -{ 2 }-> ack(x, ack(x, 1 + 0)) :|: x >= 0, z' = 1 + 0, z = 1 + x ack(z, z') -{ 1 }-> ack(x, 1 + 0) :|: x >= 0, z = 1 + x, z' = 0 ack(z, z') -{ 1 }-> 1 + y :|: y >= 0, z = 0, z' = y ---------------------------------------- (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: ack(z, z') -{ 2 }-> ack(z - 1, ack(z - 1, ack(1 + (z - 1), z' - 2))) :|: z - 1 >= 0, z' - 2 >= 0 ack(z, z') -{ 2 }-> ack(z - 1, ack(z - 1, 1 + 0)) :|: z - 1 >= 0, z' = 1 + 0 ack(z, z') -{ 1 }-> ack(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 0 ack(z, z') -{ 1 }-> 1 + z' :|: z' >= 0, z = 0 ---------------------------------------- (17) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: none ---------------------------------------- (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: ack(0, y) -> s(y) [1] ack(s(x), 0) -> ack(x, s(0)) [1] ack(s(x), s(y)) -> ack(x, ack(s(x), y)) [1] The TRS has the following type information: ack :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0: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: 0 => 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: ack(z, z') -{ 1 }-> ack(x, ack(1 + x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x ack(z, z') -{ 1 }-> ack(x, 1 + 0) :|: x >= 0, z = 1 + x, z' = 0 ack(z, z') -{ 1 }-> 1 + y :|: y >= 0, z = 0, z' = y 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: ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: ACK(0, z0) -> c ACK(s(z0), 0) -> c1(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c2(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) S tuples: ACK(0, z0) -> c ACK(s(z0), 0) -> c1(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c2(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c, c1_1, c2_2 ---------------------------------------- (23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ACK(0, z0) -> c ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: ACK(s(z0), 0) -> c1(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c2(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) S tuples: ACK(s(z0), 0) -> c1(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c2(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c1_1, c2_2 ---------------------------------------- (25) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(z0), s(z1)) -> c2(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) by ACK(s(z0), s(0)) -> c2(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c2(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: ACK(s(z0), 0) -> c1(ACK(z0, s(0))) ACK(s(z0), s(0)) -> c2(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c2(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) S tuples: ACK(s(z0), 0) -> c1(ACK(z0, s(0))) ACK(s(z0), s(0)) -> c2(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c2(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c1_1, c2_2 ---------------------------------------- (27) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(z0), s(0)) -> c2(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) by ACK(s(0), s(0)) -> c2(ACK(0, s(s(0))), ACK(s(0), 0)) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c2(ACK(s(x0), 0)) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: ACK(s(z0), 0) -> c1(ACK(z0, s(0))) ACK(s(z0), s(s(z1))) -> c2(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) ACK(s(0), s(0)) -> c2(ACK(0, s(s(0))), ACK(s(0), 0)) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c2(ACK(s(x0), 0)) S tuples: ACK(s(z0), 0) -> c1(ACK(z0, s(0))) ACK(s(z0), s(s(z1))) -> c2(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) ACK(s(0), s(0)) -> c2(ACK(0, s(s(0))), ACK(s(0), 0)) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c2(ACK(s(x0), 0)) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c1_1, c2_2, c2_1 ---------------------------------------- (29) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: ACK(s(z0), 0) -> c1(ACK(z0, s(0))) ACK(s(z0), s(s(z1))) -> c2(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c2(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c2(ACK(s(0), 0)) S tuples: ACK(s(z0), 0) -> c1(ACK(z0, s(0))) ACK(s(z0), s(s(z1))) -> c2(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c2(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c2(ACK(s(0), 0)) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c1_1, c2_2, c2_1 ---------------------------------------- (31) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(z0), s(s(z1))) -> c2(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) by ACK(s(0), s(s(x1))) -> c2(ACK(0, s(ack(s(0), x1))), ACK(s(0), s(x1))) ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: ACK(s(z0), 0) -> c1(ACK(z0, s(0))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c2(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c2(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c2(ACK(0, s(ack(s(0), x1))), ACK(s(0), s(x1))) ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) S tuples: ACK(s(z0), 0) -> c1(ACK(z0, s(0))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c2(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c2(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c2(ACK(0, s(ack(s(0), x1))), ACK(s(0), s(x1))) ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c1_1, c2_2, c2_1 ---------------------------------------- (33) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: ACK(s(z0), 0) -> c1(ACK(z0, s(0))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c2(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c2(ACK(s(0), 0)) ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c2(ACK(s(0), s(x1))) S tuples: ACK(s(z0), 0) -> c1(ACK(z0, s(0))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c2(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c2(ACK(s(0), 0)) ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c2(ACK(s(0), s(x1))) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c1_1, c2_2, c2_1 ---------------------------------------- (35) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) by ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: ACK(s(z0), 0) -> c1(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c2(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c2(ACK(s(0), 0)) ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c2(ACK(s(0), s(x1))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) S tuples: ACK(s(z0), 0) -> c1(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c2(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c2(ACK(s(0), 0)) ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c2(ACK(s(0), s(x1))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c1_1, c2_1, c2_2 ---------------------------------------- (37) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACK(s(z0), 0) -> c1(ACK(z0, s(0))) by ACK(s(s(y0)), 0) -> c1(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c1(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: ACK(s(x0), s(0)) -> c2(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c2(ACK(s(0), 0)) ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c2(ACK(s(0), s(x1))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(y0)), 0) -> c1(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c1(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) S tuples: ACK(s(x0), s(0)) -> c2(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c2(ACK(s(0), 0)) ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c2(ACK(s(0), s(x1))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(y0)), 0) -> c1(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c1(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c2_1, c2_2, c1_1 ---------------------------------------- (39) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ACK(s(0), s(0)) -> c2(ACK(s(0), 0)) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: ACK(s(x0), s(0)) -> c2(ACK(s(x0), 0)) ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c2(ACK(s(0), s(x1))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(y0)), 0) -> c1(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c1(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) S tuples: ACK(s(x0), s(0)) -> c2(ACK(s(x0), 0)) ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c2(ACK(s(0), s(x1))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(y0)), 0) -> c1(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c1(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c2_1, c2_2, c1_1 ---------------------------------------- (41) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACK(s(x0), s(0)) -> c2(ACK(s(x0), 0)) by ACK(s(s(y0)), s(0)) -> c2(ACK(s(s(y0)), 0)) ACK(s(s(0)), s(0)) -> c2(ACK(s(s(0)), 0)) ACK(s(s(s(y0))), s(0)) -> c2(ACK(s(s(s(y0))), 0)) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c2(ACK(s(0), s(x1))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(y0)), 0) -> c1(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c1(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), s(0)) -> c2(ACK(s(s(y0)), 0)) ACK(s(s(0)), s(0)) -> c2(ACK(s(s(0)), 0)) ACK(s(s(s(y0))), s(0)) -> c2(ACK(s(s(s(y0))), 0)) S tuples: ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c2(ACK(s(0), s(x1))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(y0)), 0) -> c1(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c1(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), s(0)) -> c2(ACK(s(s(y0)), 0)) ACK(s(s(0)), s(0)) -> c2(ACK(s(s(0)), 0)) ACK(s(s(s(y0))), s(0)) -> c2(ACK(s(s(s(y0))), 0)) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c2_2, c2_1, c1_1 ---------------------------------------- (43) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ACK(s(s(0)), 0) -> c1(ACK(s(0), s(0))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c2(ACK(s(0), s(x1))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(y0)), 0) -> c1(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), s(0)) -> c2(ACK(s(s(y0)), 0)) ACK(s(s(0)), s(0)) -> c2(ACK(s(s(0)), 0)) ACK(s(s(s(y0))), s(0)) -> c2(ACK(s(s(s(y0))), 0)) S tuples: ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c2(ACK(s(0), s(x1))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(y0)), 0) -> c1(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), s(0)) -> c2(ACK(s(s(y0)), 0)) ACK(s(s(0)), s(0)) -> c2(ACK(s(s(0)), 0)) ACK(s(s(s(y0))), s(0)) -> c2(ACK(s(s(s(y0))), 0)) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c2_2, c2_1, c1_1 ---------------------------------------- (45) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACK(s(0), s(s(x1))) -> c2(ACK(s(0), s(x1))) by ACK(s(0), s(s(s(0)))) -> c2(ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(s(y1))))) -> c2(ACK(s(0), s(s(s(y1))))) ACK(s(0), s(s(s(y0)))) -> c2(ACK(s(0), s(s(y0)))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(y0)), 0) -> c1(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), s(0)) -> c2(ACK(s(s(y0)), 0)) ACK(s(s(0)), s(0)) -> c2(ACK(s(s(0)), 0)) ACK(s(s(s(y0))), s(0)) -> c2(ACK(s(s(s(y0))), 0)) ACK(s(0), s(s(s(0)))) -> c2(ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(s(y1))))) -> c2(ACK(s(0), s(s(s(y1))))) ACK(s(0), s(s(s(y0)))) -> c2(ACK(s(0), s(s(y0)))) S tuples: ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(y0)), 0) -> c1(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), s(0)) -> c2(ACK(s(s(y0)), 0)) ACK(s(s(0)), s(0)) -> c2(ACK(s(s(0)), 0)) ACK(s(s(s(y0))), s(0)) -> c2(ACK(s(s(s(y0))), 0)) ACK(s(0), s(s(s(0)))) -> c2(ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(s(y1))))) -> c2(ACK(s(0), s(s(s(y1))))) ACK(s(0), s(s(s(y0)))) -> c2(ACK(s(0), s(s(y0)))) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c2_2, c1_1, c2_1 ---------------------------------------- (47) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACK(s(s(y0)), 0) -> c1(ACK(s(y0), s(0))) by ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c1(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c1(ACK(s(s(s(y0))), s(0))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), s(0)) -> c2(ACK(s(s(y0)), 0)) ACK(s(s(0)), s(0)) -> c2(ACK(s(s(0)), 0)) ACK(s(s(s(y0))), s(0)) -> c2(ACK(s(s(s(y0))), 0)) ACK(s(0), s(s(s(0)))) -> c2(ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(s(y1))))) -> c2(ACK(s(0), s(s(s(y1))))) ACK(s(0), s(s(s(y0)))) -> c2(ACK(s(0), s(s(y0)))) ACK(s(s(s(0))), 0) -> c1(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c1(ACK(s(s(s(y0))), s(0))) S tuples: ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), s(0)) -> c2(ACK(s(s(y0)), 0)) ACK(s(s(0)), s(0)) -> c2(ACK(s(s(0)), 0)) ACK(s(s(s(y0))), s(0)) -> c2(ACK(s(s(s(y0))), 0)) ACK(s(0), s(s(s(0)))) -> c2(ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(s(y1))))) -> c2(ACK(s(0), s(s(s(y1))))) ACK(s(0), s(s(s(y0)))) -> c2(ACK(s(0), s(s(y0)))) ACK(s(s(s(0))), 0) -> c1(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c1(ACK(s(s(s(y0))), s(0))) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c2_2, c1_1, c2_1 ---------------------------------------- (49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ACK(s(s(0)), s(0)) -> c2(ACK(s(s(0)), 0)) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), s(0)) -> c2(ACK(s(s(y0)), 0)) ACK(s(s(s(y0))), s(0)) -> c2(ACK(s(s(s(y0))), 0)) ACK(s(0), s(s(s(0)))) -> c2(ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(s(y1))))) -> c2(ACK(s(0), s(s(s(y1))))) ACK(s(0), s(s(s(y0)))) -> c2(ACK(s(0), s(s(y0)))) ACK(s(s(s(0))), 0) -> c1(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c1(ACK(s(s(s(y0))), s(0))) S tuples: ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), s(0)) -> c2(ACK(s(s(y0)), 0)) ACK(s(s(s(y0))), s(0)) -> c2(ACK(s(s(s(y0))), 0)) ACK(s(0), s(s(s(0)))) -> c2(ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(s(y1))))) -> c2(ACK(s(0), s(s(s(y1))))) ACK(s(0), s(s(s(y0)))) -> c2(ACK(s(0), s(s(y0)))) ACK(s(s(s(0))), 0) -> c1(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c1(ACK(s(s(s(y0))), s(0))) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c2_2, c1_1, c2_1 ---------------------------------------- (51) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACK(s(s(y0)), s(0)) -> c2(ACK(s(s(y0)), 0)) by ACK(s(s(s(y0))), s(0)) -> c2(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c2(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c2(ACK(s(s(s(s(y0)))), 0)) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c2(ACK(s(s(s(y0))), 0)) ACK(s(0), s(s(s(0)))) -> c2(ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(s(y1))))) -> c2(ACK(s(0), s(s(s(y1))))) ACK(s(0), s(s(s(y0)))) -> c2(ACK(s(0), s(s(y0)))) ACK(s(s(s(0))), 0) -> c1(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c1(ACK(s(s(s(y0))), s(0))) ACK(s(s(s(0))), s(0)) -> c2(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c2(ACK(s(s(s(s(y0)))), 0)) S tuples: ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c2(ACK(s(s(s(y0))), 0)) ACK(s(0), s(s(s(0)))) -> c2(ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(s(y1))))) -> c2(ACK(s(0), s(s(s(y1))))) ACK(s(0), s(s(s(y0)))) -> c2(ACK(s(0), s(s(y0)))) ACK(s(s(s(0))), 0) -> c1(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c1(ACK(s(s(s(y0))), s(0))) ACK(s(s(s(0))), s(0)) -> c2(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c2(ACK(s(s(s(s(y0)))), 0)) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c2_2, c1_1, c2_1 ---------------------------------------- (53) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACK(s(0), s(s(s(y0)))) -> c2(ACK(s(0), s(s(y0)))) by ACK(s(0), s(s(s(0)))) -> c2(ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(s(y1))))) -> c2(ACK(s(0), s(s(s(y1))))) ACK(s(0), s(s(s(s(0))))) -> c2(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(y0)))))) -> c2(ACK(s(0), s(s(s(s(y0)))))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c2(ACK(s(s(s(y0))), 0)) ACK(s(0), s(s(s(0)))) -> c2(ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(s(y1))))) -> c2(ACK(s(0), s(s(s(y1))))) ACK(s(s(s(0))), 0) -> c1(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c1(ACK(s(s(s(y0))), s(0))) ACK(s(s(s(0))), s(0)) -> c2(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c2(ACK(s(s(s(s(y0)))), 0)) ACK(s(0), s(s(s(s(0))))) -> c2(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(y0)))))) -> c2(ACK(s(0), s(s(s(s(y0)))))) S tuples: ACK(s(z0), s(s(0))) -> c2(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c2(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(s(z0)), s(0)) -> c2(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(s(y0))), 0) -> c1(ACK(s(s(y0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c2(ACK(s(s(s(y0))), 0)) ACK(s(0), s(s(s(0)))) -> c2(ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(s(y1))))) -> c2(ACK(s(0), s(s(s(y1))))) ACK(s(s(s(0))), 0) -> c1(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c1(ACK(s(s(s(y0))), s(0))) ACK(s(s(s(0))), s(0)) -> c2(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c2(ACK(s(s(s(s(y0)))), 0)) ACK(s(0), s(s(s(s(0))))) -> c2(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(y0)))))) -> c2(ACK(s(0), s(s(s(s(y0)))))) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c2_2, c1_1, c2_1