KILLED proof of input_raszjBeHQh.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 7 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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtForwardInstantiationProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 1 ms] (54) CdtProblem (55) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 8 ms] (58) CdtProblem (59) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 31 ms] (62) 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: plus(s(s(x)), y) -> s(plus(x, s(y))) plus(x, s(s(y))) -> s(plus(s(x), y)) plus(s(0), y) -> s(y) plus(0, y) -> y ack(0, y) -> s(y) ack(s(x), 0) -> ack(x, s(0)) ack(s(x), s(y)) -> ack(x, plus(y, 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: plus(s(s(x)), y) -> s(plus(x, s(y))) plus(x, s(s(y))) -> s(plus(s(x), y)) plus(s(0'), y) -> s(y) plus(0', y) -> y ack(0', y) -> s(y) ack(s(x), 0') -> ack(x, s(0')) ack(s(x), s(y)) -> ack(x, plus(y, 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: plus(s(s(x)), y) -> s(plus(x, s(y))) plus(x, s(s(y))) -> s(plus(s(x), y)) plus(s(0), y) -> s(y) plus(0, y) -> y ack(0, y) -> s(y) ack(s(x), 0) -> ack(x, s(0)) ack(s(x), s(y)) -> ack(x, plus(y, 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: plus(s(s(x)), y) -> s(plus(x, s(y))) [1] plus(x, s(s(y))) -> s(plus(s(x), y)) [1] plus(s(0), y) -> s(y) [1] plus(0, y) -> y [1] ack(0, y) -> s(y) [1] ack(s(x), 0) -> ack(x, s(0)) [1] ack(s(x), s(y)) -> ack(x, plus(y, 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: plus(s(s(x)), y) -> s(plus(x, s(y))) [1] plus(x, s(s(y))) -> s(plus(s(x), y)) [1] plus(s(0), y) -> s(y) [1] plus(0, y) -> y [1] ack(0, y) -> s(y) [1] ack(s(x), 0) -> ack(x, s(0)) [1] ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y))) [1] The TRS has the following type information: plus :: s:0 -> s:0 -> s:0 s :: s:0 -> s:0 0 :: s:0 ack :: s:0 -> s:0 -> s:0 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: plus_2 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: plus(s(s(x)), y) -> s(plus(x, s(y))) [1] plus(x, s(s(y))) -> s(plus(s(x), y)) [1] plus(s(0), y) -> s(y) [1] plus(0, y) -> y [1] ack(0, y) -> s(y) [1] ack(s(x), 0) -> ack(x, s(0)) [1] ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y))) [1] The TRS has the following type information: plus :: s:0 -> s:0 -> s:0 s :: s:0 -> s:0 0 :: s:0 ack :: s:0 -> s:0 -> s:0 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: plus(s(s(x)), y) -> s(plus(x, s(y))) [1] plus(x, s(s(y))) -> s(plus(s(x), y)) [1] plus(s(0), y) -> s(y) [1] plus(0, y) -> y [1] ack(0, y) -> s(y) [1] ack(s(x), 0) -> ack(x, s(0)) [1] ack(s(x), s(0)) -> ack(x, plus(0, ack(x, s(0)))) [2] ack(s(x), s(s(y'))) -> ack(x, plus(s(y'), ack(x, plus(y', ack(s(x), y'))))) [2] The TRS has the following type information: plus :: s:0 -> s:0 -> s:0 s :: s:0 -> s:0 0 :: s:0 ack :: s:0 -> s:0 -> s:0 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, plus(0, ack(x, 1 + 0))) :|: x >= 0, z' = 1 + 0, z = 1 + x ack(z, z') -{ 2 }-> ack(x, plus(1 + y', ack(x, plus(y', ack(1 + x, y'))))) :|: z' = 1 + (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 plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y plus(z, z') -{ 1 }-> 1 + y :|: z = 1 + 0, y >= 0, z' = y plus(z, z') -{ 1 }-> 1 + plus(x, 1 + y) :|: x >= 0, y >= 0, z' = y, z = 1 + (1 + x) plus(z, z') -{ 1 }-> 1 + plus(1 + x, y) :|: z' = 1 + (1 + y), x >= 0, y >= 0, z = x ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: ack(z, z') -{ 2 }-> ack(z - 1, plus(0, ack(z - 1, 1 + 0))) :|: z - 1 >= 0, z' = 1 + 0 ack(z, z') -{ 2 }-> ack(z - 1, plus(1 + (z' - 2), ack(z - 1, plus(z' - 2, ack(1 + (z - 1), z' - 2))))) :|: z - 1 >= 0, z' - 2 >= 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 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + z' :|: z = 1 + 0, z' >= 0 plus(z, z') -{ 1 }-> 1 + plus(z - 2, 1 + z') :|: z - 2 >= 0, z' >= 0 plus(z, z') -{ 1 }-> 1 + plus(1 + z, z' - 2) :|: z >= 0, z' - 2 >= 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: plus(s(s(x)), y) -> s(plus(x, s(y))) [1] plus(x, s(s(y))) -> s(plus(s(x), y)) [1] plus(s(0), y) -> s(y) [1] plus(0, y) -> y [1] ack(0, y) -> s(y) [1] ack(s(x), 0) -> ack(x, s(0)) [1] ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y))) [1] The TRS has the following type information: plus :: s:0 -> s:0 -> s:0 s :: s:0 -> s:0 0 :: s:0 ack :: s:0 -> s:0 -> s:0 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, plus(y, 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 plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y plus(z, z') -{ 1 }-> 1 + y :|: z = 1 + 0, y >= 0, z' = y plus(z, z') -{ 1 }-> 1 + plus(x, 1 + y) :|: x >= 0, y >= 0, z' = y, z = 1 + (1 + x) plus(z, z') -{ 1 }-> 1 + plus(1 + x, y) :|: z' = 1 + (1 + y), x >= 0, y >= 0, z = x 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: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) PLUS(s(0), z0) -> c2 PLUS(0, z0) -> c3 ACK(0, z0) -> c4 ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c6(ACK(z0, plus(z1, ack(s(z0), z1))), PLUS(z1, ack(s(z0), z1)), ACK(s(z0), z1)) S tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) PLUS(s(0), z0) -> c2 PLUS(0, z0) -> c3 ACK(0, z0) -> c4 ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c6(ACK(z0, plus(z1, ack(s(z0), z1))), PLUS(z1, ack(s(z0), z1)), ACK(s(z0), z1)) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: PLUS_2, ACK_2 Compound Symbols: c_1, c1_1, c2, c3, c4, c5_1, c6_3 ---------------------------------------- (23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: PLUS(0, z0) -> c3 PLUS(s(0), z0) -> c2 ACK(0, z0) -> c4 ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c6(ACK(z0, plus(z1, ack(s(z0), z1))), PLUS(z1, ack(s(z0), z1)), ACK(s(z0), z1)) S tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c6(ACK(z0, plus(z1, ack(s(z0), z1))), PLUS(z1, ack(s(z0), z1)), ACK(s(z0), z1)) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: PLUS_2, ACK_2 Compound Symbols: c_1, c1_1, c5_1, c6_3 ---------------------------------------- (25) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(z0), s(z1)) -> c6(ACK(z0, plus(z1, ack(s(z0), z1))), PLUS(z1, ack(s(z0), z1)), ACK(s(z0), z1)) by ACK(s(x0), s(s(s(z0)))) -> c6(ACK(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))), PLUS(s(s(z0)), ack(s(x0), s(s(z0)))), ACK(s(x0), s(s(z0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, s(ack(s(x0), s(0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(x0), s(0)) -> c6(ACK(x0, ack(s(x0), 0)), PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(z0), s(0)) -> c6(ACK(z0, plus(0, ack(z0, s(0)))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c6(ACK(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))), PLUS(s(z1), ack(s(z0), s(z1))), ACK(s(z0), s(z1))) ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(x0), s(s(s(z0)))) -> c6(ACK(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))), PLUS(s(s(z0)), ack(s(x0), s(s(z0)))), ACK(s(x0), s(s(z0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, s(ack(s(x0), s(0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(x0), s(0)) -> c6(ACK(x0, ack(s(x0), 0)), PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(z0), s(0)) -> c6(ACK(z0, plus(0, ack(z0, s(0)))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c6(ACK(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))), PLUS(s(z1), ack(s(z0), s(z1))), ACK(s(z0), s(z1))) S tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(x0), s(s(s(z0)))) -> c6(ACK(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))), PLUS(s(s(z0)), ack(s(x0), s(s(z0)))), ACK(s(x0), s(s(z0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, s(ack(s(x0), s(0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(x0), s(0)) -> c6(ACK(x0, ack(s(x0), 0)), PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(z0), s(0)) -> c6(ACK(z0, plus(0, ack(z0, s(0)))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c6(ACK(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))), PLUS(s(z1), ack(s(z0), s(z1))), ACK(s(z0), s(z1))) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: PLUS_2, ACK_2 Compound Symbols: c_1, c1_1, c5_1, c6_3 ---------------------------------------- (27) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(x0), s(s(s(z0)))) -> c6(ACK(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))), PLUS(s(s(z0)), ack(s(x0), s(s(z0)))), ACK(s(x0), s(s(z0)))) by ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, s(ack(s(x0), s(0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(x0), s(0)) -> c6(ACK(x0, ack(s(x0), 0)), PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(z0), s(0)) -> c6(ACK(z0, plus(0, ack(z0, s(0)))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c6(ACK(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))), PLUS(s(z1), ack(s(z0), s(z1))), ACK(s(z0), s(z1))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) S tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, s(ack(s(x0), s(0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(x0), s(0)) -> c6(ACK(x0, ack(s(x0), 0)), PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(z0), s(0)) -> c6(ACK(z0, plus(0, ack(z0, s(0)))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c6(ACK(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))), PLUS(s(z1), ack(s(z0), s(z1))), ACK(s(z0), s(z1))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: PLUS_2, ACK_2 Compound Symbols: c_1, c1_1, c5_1, c6_3, c6_2 ---------------------------------------- (29) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(x0), s(s(0))) -> c6(ACK(x0, s(ack(s(x0), s(0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) by ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c6(ACK(x0, ack(s(x0), 0)), PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(z0), s(0)) -> c6(ACK(z0, plus(0, ack(z0, s(0)))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c6(ACK(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))), PLUS(s(z1), ack(s(z0), s(z1))), ACK(s(z0), s(z1))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) S tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c6(ACK(x0, ack(s(x0), 0)), PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(z0), s(0)) -> c6(ACK(z0, plus(0, ack(z0, s(0)))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c6(ACK(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))), PLUS(s(z1), ack(s(z0), s(z1))), ACK(s(z0), s(z1))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: PLUS_2, ACK_2 Compound Symbols: c_1, c1_1, c5_1, c6_3, c6_2, c6_1 ---------------------------------------- (31) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(x0), s(0)) -> c6(ACK(x0, ack(s(x0), 0)), PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) by ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(z0), s(0)) -> c6(ACK(z0, plus(0, ack(z0, s(0)))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c6(ACK(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))), PLUS(s(z1), ack(s(z0), s(z1))), ACK(s(z0), s(z1))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) S tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(z0), s(0)) -> c6(ACK(z0, plus(0, ack(z0, s(0)))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c6(ACK(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))), PLUS(s(z1), ack(s(z0), s(z1))), ACK(s(z0), s(z1))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: PLUS_2, ACK_2 Compound Symbols: c_1, c1_1, c5_1, c6_3, c6_2, c6_1 ---------------------------------------- (33) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(z0), s(0)) -> c6(ACK(z0, plus(0, ack(z0, s(0)))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) by ACK(s(x0), s(0)) -> c6(ACK(x0, ack(x0, s(0))), PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(0), s(0)) -> c6(ACK(0, plus(0, s(s(0)))), PLUS(0, ack(s(0), 0)), ACK(s(0), 0)) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(z0), s(s(z1))) -> c6(ACK(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))), PLUS(s(z1), ack(s(z0), s(z1))), ACK(s(z0), s(z1))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(0), s(0)) -> c6(ACK(0, plus(0, s(s(0)))), PLUS(0, ack(s(0), 0)), ACK(s(0), 0)) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) S tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(z0), s(s(z1))) -> c6(ACK(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))), PLUS(s(z1), ack(s(z0), s(z1))), ACK(s(z0), s(z1))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(0), s(0)) -> c6(ACK(0, plus(0, s(s(0)))), PLUS(0, ack(s(0), 0)), ACK(s(0), 0)) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: PLUS_2, ACK_2 Compound Symbols: c_1, c1_1, c5_1, c6_3, c6_2, c6_1 ---------------------------------------- (35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(z0), s(s(z1))) -> c6(ACK(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))), PLUS(s(z1), ack(s(z0), s(z1))), ACK(s(z0), s(z1))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0)), ACK(s(0), 0)) S tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(z0), s(s(z1))) -> c6(ACK(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))), PLUS(s(z1), ack(s(z0), s(z1))), ACK(s(z0), s(z1))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0)), ACK(s(0), 0)) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: PLUS_2, ACK_2 Compound Symbols: c_1, c1_1, c5_1, c6_3, c6_2, c6_1 ---------------------------------------- (37) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(z0), s(s(z1))) -> c6(ACK(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))), PLUS(s(z1), ack(s(z0), s(z1))), ACK(s(z0), s(z1))) by ACK(s(x0), s(s(s(z0)))) -> c6(ACK(x0, s(plus(z0, s(ack(x0, plus(s(z0), ack(s(x0), s(z0)))))))), PLUS(s(s(z0)), ack(s(x0), s(s(z0)))), ACK(s(x0), s(s(z0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, s(ack(x0, plus(0, ack(s(x0), 0))))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(0), s(s(x1))) -> c6(ACK(0, plus(s(x1), s(plus(x1, ack(s(0), x1))))), PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0)), ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c6(ACK(0, plus(s(x1), s(plus(x1, ack(s(0), x1))))), PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) S tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0)), ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c6(ACK(0, plus(s(x1), s(plus(x1, ack(s(0), x1))))), PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: PLUS_2, ACK_2 Compound Symbols: c_1, c1_1, c5_1, c6_3, c6_2, c6_1 ---------------------------------------- (39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0)), ACK(s(0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) S tuples: PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0)), ACK(s(0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: PLUS_2, ACK_2 Compound Symbols: c_1, c1_1, c5_1, c6_3, c6_2, c6_1 ---------------------------------------- (41) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(s(z0)), z1) -> c(PLUS(z0, s(z1))) by PLUS(s(s(s(s(y0)))), z1) -> c(PLUS(s(s(y0)), s(z1))) PLUS(s(s(z0)), s(y1)) -> c(PLUS(z0, s(s(y1)))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0)), ACK(s(0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(s(s(s(s(y0)))), z1) -> c(PLUS(s(s(y0)), s(z1))) PLUS(s(s(z0)), s(y1)) -> c(PLUS(z0, s(s(y1)))) S tuples: PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0)), ACK(s(0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(s(s(s(s(y0)))), z1) -> c(PLUS(s(s(y0)), s(z1))) PLUS(s(s(z0)), s(y1)) -> c(PLUS(z0, s(s(y1)))) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: PLUS_2, ACK_2 Compound Symbols: c1_1, c5_1, c6_3, c6_2, c6_1, c_1 ---------------------------------------- (43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(z0, s(s(z1))) -> c1(PLUS(s(z0), z1)) by PLUS(z0, s(s(s(s(y1))))) -> c1(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(s(z1))) -> c1(PLUS(s(s(s(s(y0)))), z1)) PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0)), ACK(s(0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(s(s(s(s(y0)))), z1) -> c(PLUS(s(s(y0)), s(z1))) PLUS(s(s(z0)), s(y1)) -> c(PLUS(z0, s(s(y1)))) PLUS(z0, s(s(s(s(y1))))) -> c1(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(s(z1))) -> c1(PLUS(s(s(s(s(y0)))), z1)) PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) S tuples: ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0)), ACK(s(0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(s(s(s(s(y0)))), z1) -> c(PLUS(s(s(y0)), s(z1))) PLUS(s(s(z0)), s(y1)) -> c(PLUS(z0, s(s(y1)))) PLUS(z0, s(s(s(s(y1))))) -> c1(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(s(z1))) -> c1(PLUS(s(s(s(s(y0)))), z1)) PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: ACK_2, PLUS_2 Compound Symbols: c5_1, c6_3, c6_2, c6_1, c_1, c1_1 ---------------------------------------- (45) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) by ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(s(z0), 0)), ACK(s(z0), 0)) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0)), ACK(s(0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(s(s(s(s(y0)))), z1) -> c(PLUS(s(s(y0)), s(z1))) PLUS(s(s(z0)), s(y1)) -> c(PLUS(z0, s(s(y1)))) PLUS(z0, s(s(s(s(y1))))) -> c1(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(s(z1))) -> c1(PLUS(s(s(s(s(y0)))), z1)) PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) S tuples: ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0)), ACK(s(0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(s(s(s(s(y0)))), z1) -> c(PLUS(s(s(y0)), s(z1))) PLUS(s(s(z0)), s(y1)) -> c(PLUS(z0, s(s(y1)))) PLUS(z0, s(s(s(s(y1))))) -> c1(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(s(z1))) -> c1(PLUS(s(s(s(s(y0)))), z1)) PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: ACK_2, PLUS_2 Compound Symbols: c5_1, c6_3, c6_2, c6_1, c_1, c1_1 ---------------------------------------- (47) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ACK(s(x0), s(0)) -> c6(ACK(x0, ack(x0, s(0))), PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) by ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0)), ACK(s(0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(s(s(s(s(y0)))), z1) -> c(PLUS(s(s(y0)), s(z1))) PLUS(s(s(z0)), s(y1)) -> c(PLUS(z0, s(s(y1)))) PLUS(z0, s(s(s(s(y1))))) -> c1(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(s(z1))) -> c1(PLUS(s(s(s(s(y0)))), z1)) PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) S tuples: ACK(s(z0), 0) -> c5(ACK(z0, s(0))) ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0)), ACK(s(0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(s(s(s(s(y0)))), z1) -> c(PLUS(s(s(y0)), s(z1))) PLUS(s(s(z0)), s(y1)) -> c(PLUS(z0, s(s(y1)))) PLUS(z0, s(s(s(s(y1))))) -> c1(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(s(z1))) -> c1(PLUS(s(s(s(s(y0)))), z1)) PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: ACK_2, PLUS_2 Compound Symbols: c5_1, c6_3, c6_2, c6_1, c_1, c1_1 ---------------------------------------- (49) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACK(s(z0), 0) -> c5(ACK(z0, s(0))) by ACK(s(s(s(y0))), 0) -> c5(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), 0) -> c5(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c5(ACK(s(0), s(0))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0)), ACK(s(0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(s(s(s(s(y0)))), z1) -> c(PLUS(s(s(y0)), s(z1))) PLUS(s(s(z0)), s(y1)) -> c(PLUS(z0, s(s(y1)))) PLUS(z0, s(s(s(s(y1))))) -> c1(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(s(z1))) -> c1(PLUS(s(s(s(s(y0)))), z1)) PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(s(s(y0))), 0) -> c5(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), 0) -> c5(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c5(ACK(s(0), s(0))) S tuples: ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0)), ACK(s(0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(s(s(s(s(y0)))), z1) -> c(PLUS(s(s(y0)), s(z1))) PLUS(s(s(z0)), s(y1)) -> c(PLUS(z0, s(s(y1)))) PLUS(z0, s(s(s(s(y1))))) -> c1(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(s(z1))) -> c1(PLUS(s(s(s(s(y0)))), z1)) PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(s(s(y0))), 0) -> c5(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), 0) -> c5(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c5(ACK(s(0), s(0))) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: ACK_2, PLUS_2 Compound Symbols: c6_3, c6_2, c6_1, c_1, c1_1, c5_1 ---------------------------------------- (51) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(s(s(s(s(y0)))), z1) -> c(PLUS(s(s(y0)), s(z1))) PLUS(s(s(z0)), s(y1)) -> c(PLUS(z0, s(s(y1)))) PLUS(z0, s(s(s(s(y1))))) -> c1(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(s(z1))) -> c1(PLUS(s(s(s(s(y0)))), z1)) PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(s(s(y0))), 0) -> c5(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), 0) -> c5(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c5(ACK(s(0), s(0))) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0))) S tuples: ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(s(s(s(s(y0)))), z1) -> c(PLUS(s(s(y0)), s(z1))) PLUS(s(s(z0)), s(y1)) -> c(PLUS(z0, s(s(y1)))) PLUS(z0, s(s(s(s(y1))))) -> c1(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(s(z1))) -> c1(PLUS(s(s(s(s(y0)))), z1)) PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(s(s(y0))), 0) -> c5(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), 0) -> c5(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c5(ACK(s(0), s(0))) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0))) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: ACK_2, PLUS_2 Compound Symbols: c6_3, c6_2, c6_1, c_1, c1_1, c5_1 ---------------------------------------- (53) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(s(z0)), s(y1)) -> c(PLUS(z0, s(s(y1)))) by PLUS(s(s(s(s(s(s(y0)))))), s(z1)) -> c(PLUS(s(s(s(s(y0)))), s(s(z1)))) PLUS(s(s(s(s(y0)))), s(z1)) -> c(PLUS(s(s(y0)), s(s(z1)))) PLUS(s(s(z0)), s(s(s(y1)))) -> c(PLUS(z0, s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(z1)) -> c(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(y1))) -> c(PLUS(s(y0), s(s(s(y1))))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(s(s(s(s(y0)))), z1) -> c(PLUS(s(s(y0)), s(z1))) PLUS(z0, s(s(s(s(y1))))) -> c1(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(s(z1))) -> c1(PLUS(s(s(s(s(y0)))), z1)) PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(s(s(y0))), 0) -> c5(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), 0) -> c5(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c5(ACK(s(0), s(0))) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0))) PLUS(s(s(s(s(s(s(y0)))))), s(z1)) -> c(PLUS(s(s(s(s(y0)))), s(s(z1)))) PLUS(s(s(s(s(y0)))), s(z1)) -> c(PLUS(s(s(y0)), s(s(z1)))) PLUS(s(s(z0)), s(s(s(y1)))) -> c(PLUS(z0, s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(z1)) -> c(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(y1))) -> c(PLUS(s(y0), s(s(s(y1))))) S tuples: ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(s(s(s(s(y0)))), z1) -> c(PLUS(s(s(y0)), s(z1))) PLUS(z0, s(s(s(s(y1))))) -> c1(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(s(z1))) -> c1(PLUS(s(s(s(s(y0)))), z1)) PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(s(s(y0))), 0) -> c5(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), 0) -> c5(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c5(ACK(s(0), s(0))) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0))) PLUS(s(s(s(s(s(s(y0)))))), s(z1)) -> c(PLUS(s(s(s(s(y0)))), s(s(z1)))) PLUS(s(s(s(s(y0)))), s(z1)) -> c(PLUS(s(s(y0)), s(s(z1)))) PLUS(s(s(z0)), s(s(s(y1)))) -> c(PLUS(z0, s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(z1)) -> c(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(y1))) -> c(PLUS(s(y0), s(s(s(y1))))) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: ACK_2, PLUS_2 Compound Symbols: c6_3, c6_2, c6_1, c_1, c1_1, c5_1 ---------------------------------------- (55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(s(s(s(y0)))), z1) -> c(PLUS(s(s(y0)), s(z1))) by PLUS(s(s(s(s(s(s(y0)))))), z1) -> c(PLUS(s(s(s(s(y0)))), s(z1))) PLUS(s(s(s(s(z0)))), s(s(s(y1)))) -> c(PLUS(s(s(z0)), s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(y1)) -> c(PLUS(s(s(s(y0))), s(s(y1)))) PLUS(s(s(s(s(z0)))), s(s(y1))) -> c(PLUS(s(s(z0)), s(s(s(y1))))) PLUS(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(PLUS(s(s(s(s(s(s(y0)))))), s(z1))) PLUS(s(s(s(s(s(s(s(y0))))))), z1) -> c(PLUS(s(s(s(s(s(y0))))), s(z1))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(z0, s(s(s(s(y1))))) -> c1(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(s(z1))) -> c1(PLUS(s(s(s(s(y0)))), z1)) PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(s(s(y0))), 0) -> c5(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), 0) -> c5(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c5(ACK(s(0), s(0))) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0))) PLUS(s(s(s(s(s(s(y0)))))), s(z1)) -> c(PLUS(s(s(s(s(y0)))), s(s(z1)))) PLUS(s(s(s(s(y0)))), s(z1)) -> c(PLUS(s(s(y0)), s(s(z1)))) PLUS(s(s(z0)), s(s(s(y1)))) -> c(PLUS(z0, s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(z1)) -> c(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(y1))) -> c(PLUS(s(y0), s(s(s(y1))))) PLUS(s(s(s(s(s(s(y0)))))), z1) -> c(PLUS(s(s(s(s(y0)))), s(z1))) PLUS(s(s(s(s(z0)))), s(s(s(y1)))) -> c(PLUS(s(s(z0)), s(s(s(s(y1)))))) PLUS(s(s(s(s(z0)))), s(s(y1))) -> c(PLUS(s(s(z0)), s(s(s(y1))))) PLUS(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(PLUS(s(s(s(s(s(s(y0)))))), s(z1))) PLUS(s(s(s(s(s(s(s(y0))))))), z1) -> c(PLUS(s(s(s(s(s(y0))))), s(z1))) S tuples: ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(z0, s(s(s(s(y1))))) -> c1(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(s(z1))) -> c1(PLUS(s(s(s(s(y0)))), z1)) PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(s(s(y0))), 0) -> c5(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), 0) -> c5(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c5(ACK(s(0), s(0))) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0))) PLUS(s(s(s(s(s(s(y0)))))), s(z1)) -> c(PLUS(s(s(s(s(y0)))), s(s(z1)))) PLUS(s(s(s(s(y0)))), s(z1)) -> c(PLUS(s(s(y0)), s(s(z1)))) PLUS(s(s(z0)), s(s(s(y1)))) -> c(PLUS(z0, s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(z1)) -> c(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(y1))) -> c(PLUS(s(y0), s(s(s(y1))))) PLUS(s(s(s(s(s(s(y0)))))), z1) -> c(PLUS(s(s(s(s(y0)))), s(z1))) PLUS(s(s(s(s(z0)))), s(s(s(y1)))) -> c(PLUS(s(s(z0)), s(s(s(s(y1)))))) PLUS(s(s(s(s(z0)))), s(s(y1))) -> c(PLUS(s(s(z0)), s(s(s(y1))))) PLUS(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(PLUS(s(s(s(s(s(s(y0)))))), s(z1))) PLUS(s(s(s(s(s(s(s(y0))))))), z1) -> c(PLUS(s(s(s(s(s(y0))))), s(z1))) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: ACK_2, PLUS_2 Compound Symbols: c6_3, c6_2, c6_1, c1_1, c5_1, c_1 ---------------------------------------- (57) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(z0, s(s(s(s(y1))))) -> c1(PLUS(s(z0), s(s(y1)))) by PLUS(z0, s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(z0), s(s(s(s(y1)))))) PLUS(s(s(y0)), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(z0, s(s(s(s(s(y1)))))) -> c1(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(s(s(s(y0))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(y0)))), s(s(z1)))) PLUS(s(y0), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(y0)), s(s(s(y1))))) PLUS(s(s(s(s(y0)))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(s(s(y0)))), s(s(s(y1))))) PLUS(s(s(s(s(s(s(s(y0))))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(s(s(y0)))))))), s(s(z1)))) PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), s(s(z1)))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(s(s(s(y0))), s(s(z1))) -> c1(PLUS(s(s(s(s(y0)))), z1)) PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(s(s(y0))), 0) -> c5(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), 0) -> c5(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c5(ACK(s(0), s(0))) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0))) PLUS(s(s(s(s(s(s(y0)))))), s(z1)) -> c(PLUS(s(s(s(s(y0)))), s(s(z1)))) PLUS(s(s(s(s(y0)))), s(z1)) -> c(PLUS(s(s(y0)), s(s(z1)))) PLUS(s(s(z0)), s(s(s(y1)))) -> c(PLUS(z0, s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(z1)) -> c(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(y1))) -> c(PLUS(s(y0), s(s(s(y1))))) PLUS(s(s(s(s(s(s(y0)))))), z1) -> c(PLUS(s(s(s(s(y0)))), s(z1))) PLUS(s(s(s(s(z0)))), s(s(s(y1)))) -> c(PLUS(s(s(z0)), s(s(s(s(y1)))))) PLUS(s(s(s(s(z0)))), s(s(y1))) -> c(PLUS(s(s(z0)), s(s(s(y1))))) PLUS(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(PLUS(s(s(s(s(s(s(y0)))))), s(z1))) PLUS(s(s(s(s(s(s(s(y0))))))), z1) -> c(PLUS(s(s(s(s(s(y0))))), s(z1))) PLUS(z0, s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(z0), s(s(s(s(y1)))))) PLUS(s(s(y0)), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(z0, s(s(s(s(s(y1)))))) -> c1(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(s(s(s(y0))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(y0)))), s(s(z1)))) PLUS(s(y0), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(y0)), s(s(s(y1))))) PLUS(s(s(s(s(y0)))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(s(s(y0)))), s(s(s(y1))))) PLUS(s(s(s(s(s(s(s(y0))))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(s(s(y0)))))))), s(s(z1)))) PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), s(s(z1)))) S tuples: ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(s(s(s(y0))), s(s(z1))) -> c1(PLUS(s(s(s(s(y0)))), z1)) PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(s(s(y0))), 0) -> c5(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), 0) -> c5(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c5(ACK(s(0), s(0))) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0))) PLUS(s(s(s(s(s(s(y0)))))), s(z1)) -> c(PLUS(s(s(s(s(y0)))), s(s(z1)))) PLUS(s(s(s(s(y0)))), s(z1)) -> c(PLUS(s(s(y0)), s(s(z1)))) PLUS(s(s(z0)), s(s(s(y1)))) -> c(PLUS(z0, s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(z1)) -> c(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(y1))) -> c(PLUS(s(y0), s(s(s(y1))))) PLUS(s(s(s(s(s(s(y0)))))), z1) -> c(PLUS(s(s(s(s(y0)))), s(z1))) PLUS(s(s(s(s(z0)))), s(s(s(y1)))) -> c(PLUS(s(s(z0)), s(s(s(s(y1)))))) PLUS(s(s(s(s(z0)))), s(s(y1))) -> c(PLUS(s(s(z0)), s(s(s(y1))))) PLUS(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(PLUS(s(s(s(s(s(s(y0)))))), s(z1))) PLUS(s(s(s(s(s(s(s(y0))))))), z1) -> c(PLUS(s(s(s(s(s(y0))))), s(z1))) PLUS(z0, s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(z0), s(s(s(s(y1)))))) PLUS(s(s(y0)), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(z0, s(s(s(s(s(y1)))))) -> c1(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(s(s(s(y0))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(y0)))), s(s(z1)))) PLUS(s(y0), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(y0)), s(s(s(y1))))) PLUS(s(s(s(s(y0)))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(s(s(y0)))), s(s(s(y1))))) PLUS(s(s(s(s(s(s(s(y0))))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(s(s(y0)))))))), s(s(z1)))) PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), s(s(z1)))) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: ACK_2, PLUS_2 Compound Symbols: c6_3, c6_2, c6_1, c1_1, c5_1, c_1 ---------------------------------------- (59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(s(s(y0))), s(s(z1))) -> c1(PLUS(s(s(s(s(y0)))), z1)) by PLUS(s(s(s(z0))), s(s(s(s(y1))))) -> c1(PLUS(s(s(s(s(z0)))), s(s(y1)))) PLUS(s(s(s(z0))), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(s(s(z0)))), s(s(s(y1))))) PLUS(s(s(s(s(s(y0))))), s(s(s(y1)))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(y1))) PLUS(s(s(s(z0))), s(s(s(y1)))) -> c1(PLUS(s(s(s(s(z0)))), s(y1))) PLUS(s(s(s(s(y0)))), s(s(s(y1)))) -> c1(PLUS(s(s(s(s(s(y0))))), s(y1))) PLUS(s(s(s(s(s(y0))))), s(s(z1))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), z1)) PLUS(s(s(s(s(s(s(s(y0))))))), s(s(z1))) -> c1(PLUS(s(s(s(s(s(s(s(s(y0)))))))), z1)) PLUS(s(s(s(s(s(s(y0)))))), s(s(z1))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), z1)) PLUS(s(s(s(z0))), s(s(s(s(s(s(s(s(y1))))))))) -> c1(PLUS(s(s(s(s(z0)))), s(s(s(s(s(s(y1)))))))) PLUS(s(s(s(z0))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(z0)))), s(s(s(s(y1)))))) PLUS(s(s(s(z0))), s(s(s(s(s(s(s(y1)))))))) -> c1(PLUS(s(s(s(s(z0)))), s(s(s(s(s(y1))))))) PLUS(s(s(s(s(y0)))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(s(s(y1)))))) PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(y1)))))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(s(s(y0))), 0) -> c5(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), 0) -> c5(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c5(ACK(s(0), s(0))) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0))) PLUS(s(s(s(s(s(s(y0)))))), s(z1)) -> c(PLUS(s(s(s(s(y0)))), s(s(z1)))) PLUS(s(s(s(s(y0)))), s(z1)) -> c(PLUS(s(s(y0)), s(s(z1)))) PLUS(s(s(z0)), s(s(s(y1)))) -> c(PLUS(z0, s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(z1)) -> c(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(y1))) -> c(PLUS(s(y0), s(s(s(y1))))) PLUS(s(s(s(s(s(s(y0)))))), z1) -> c(PLUS(s(s(s(s(y0)))), s(z1))) PLUS(s(s(s(s(z0)))), s(s(s(y1)))) -> c(PLUS(s(s(z0)), s(s(s(s(y1)))))) PLUS(s(s(s(s(z0)))), s(s(y1))) -> c(PLUS(s(s(z0)), s(s(s(y1))))) PLUS(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(PLUS(s(s(s(s(s(s(y0)))))), s(z1))) PLUS(s(s(s(s(s(s(s(y0))))))), z1) -> c(PLUS(s(s(s(s(s(y0))))), s(z1))) PLUS(z0, s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(z0), s(s(s(s(y1)))))) PLUS(s(s(y0)), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(z0, s(s(s(s(s(y1)))))) -> c1(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(s(s(s(y0))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(y0)))), s(s(z1)))) PLUS(s(y0), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(y0)), s(s(s(y1))))) PLUS(s(s(s(s(y0)))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(s(s(y0)))), s(s(s(y1))))) PLUS(s(s(s(s(s(s(s(y0))))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(s(s(y0)))))))), s(s(z1)))) PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), s(s(z1)))) PLUS(s(s(s(s(s(y0))))), s(s(s(y1)))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(y1))) PLUS(s(s(s(z0))), s(s(s(y1)))) -> c1(PLUS(s(s(s(s(z0)))), s(y1))) PLUS(s(s(s(s(y0)))), s(s(s(y1)))) -> c1(PLUS(s(s(s(s(s(y0))))), s(y1))) PLUS(s(s(s(s(s(y0))))), s(s(z1))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), z1)) PLUS(s(s(s(s(s(s(s(y0))))))), s(s(z1))) -> c1(PLUS(s(s(s(s(s(s(s(s(y0)))))))), z1)) PLUS(s(s(s(s(s(s(y0)))))), s(s(z1))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), z1)) PLUS(s(s(s(z0))), s(s(s(s(s(s(s(s(y1))))))))) -> c1(PLUS(s(s(s(s(z0)))), s(s(s(s(s(s(y1)))))))) PLUS(s(s(s(z0))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(z0)))), s(s(s(s(y1)))))) PLUS(s(s(s(z0))), s(s(s(s(s(s(s(y1)))))))) -> c1(PLUS(s(s(s(s(z0)))), s(s(s(s(s(y1))))))) PLUS(s(s(s(s(y0)))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(s(s(y1)))))) PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(y1)))))) S tuples: ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(s(s(y0))), 0) -> c5(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), 0) -> c5(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c5(ACK(s(0), s(0))) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0))) PLUS(s(s(s(s(s(s(y0)))))), s(z1)) -> c(PLUS(s(s(s(s(y0)))), s(s(z1)))) PLUS(s(s(s(s(y0)))), s(z1)) -> c(PLUS(s(s(y0)), s(s(z1)))) PLUS(s(s(z0)), s(s(s(y1)))) -> c(PLUS(z0, s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(z1)) -> c(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(y1))) -> c(PLUS(s(y0), s(s(s(y1))))) PLUS(s(s(s(s(s(s(y0)))))), z1) -> c(PLUS(s(s(s(s(y0)))), s(z1))) PLUS(s(s(s(s(z0)))), s(s(s(y1)))) -> c(PLUS(s(s(z0)), s(s(s(s(y1)))))) PLUS(s(s(s(s(z0)))), s(s(y1))) -> c(PLUS(s(s(z0)), s(s(s(y1))))) PLUS(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(PLUS(s(s(s(s(s(s(y0)))))), s(z1))) PLUS(s(s(s(s(s(s(s(y0))))))), z1) -> c(PLUS(s(s(s(s(s(y0))))), s(z1))) PLUS(z0, s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(z0), s(s(s(s(y1)))))) PLUS(s(s(y0)), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(z0, s(s(s(s(s(y1)))))) -> c1(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(s(s(s(y0))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(y0)))), s(s(z1)))) PLUS(s(y0), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(y0)), s(s(s(y1))))) PLUS(s(s(s(s(y0)))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(s(s(y0)))), s(s(s(y1))))) PLUS(s(s(s(s(s(s(s(y0))))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(s(s(y0)))))))), s(s(z1)))) PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), s(s(z1)))) PLUS(s(s(s(s(s(y0))))), s(s(s(y1)))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(y1))) PLUS(s(s(s(z0))), s(s(s(y1)))) -> c1(PLUS(s(s(s(s(z0)))), s(y1))) PLUS(s(s(s(s(y0)))), s(s(s(y1)))) -> c1(PLUS(s(s(s(s(s(y0))))), s(y1))) PLUS(s(s(s(s(s(y0))))), s(s(z1))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), z1)) PLUS(s(s(s(s(s(s(s(y0))))))), s(s(z1))) -> c1(PLUS(s(s(s(s(s(s(s(s(y0)))))))), z1)) PLUS(s(s(s(s(s(s(y0)))))), s(s(z1))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), z1)) PLUS(s(s(s(z0))), s(s(s(s(s(s(s(s(y1))))))))) -> c1(PLUS(s(s(s(s(z0)))), s(s(s(s(s(s(y1)))))))) PLUS(s(s(s(z0))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(z0)))), s(s(s(s(y1)))))) PLUS(s(s(s(z0))), s(s(s(s(s(s(s(y1)))))))) -> c1(PLUS(s(s(s(s(z0)))), s(s(s(s(s(y1))))))) PLUS(s(s(s(s(y0)))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(s(s(y1)))))) PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(y1)))))) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: ACK_2, PLUS_2 Compound Symbols: c6_3, c6_2, c6_1, c1_1, c5_1, c_1 ---------------------------------------- (61) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(y0), s(s(s(y1)))) -> c1(PLUS(s(s(y0)), s(y1))) by PLUS(s(z0), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(z0)), s(s(s(y1))))) PLUS(s(s(s(s(s(y0))))), s(s(s(z1)))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(z1))) PLUS(s(s(s(y0))), s(s(s(z1)))) -> c1(PLUS(s(s(s(s(y0)))), s(z1))) PLUS(s(s(s(s(y0)))), s(s(s(z1)))) -> c1(PLUS(s(s(s(s(s(y0))))), s(z1))) PLUS(s(s(y0)), s(s(s(s(y1))))) -> c1(PLUS(s(s(s(y0))), s(s(y1)))) PLUS(s(s(s(y0))), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(s(s(y0)))), s(s(s(y1))))) PLUS(s(s(s(y0))), s(s(s(s(y1))))) -> c1(PLUS(s(s(s(s(y0)))), s(s(y1)))) PLUS(s(s(s(s(s(s(s(y0))))))), s(s(s(z1)))) -> c1(PLUS(s(s(s(s(s(s(s(s(y0)))))))), s(z1))) PLUS(s(s(s(s(s(s(y0)))))), s(s(s(z1)))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), s(z1))) PLUS(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c1(PLUS(s(s(z0)), s(s(s(s(s(s(y1)))))))) PLUS(s(z0), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(z0)), s(s(s(s(y1)))))) PLUS(s(z0), s(s(s(s(s(s(s(y1)))))))) -> c1(PLUS(s(s(z0)), s(s(s(s(s(y1))))))) PLUS(s(s(s(s(y0)))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(s(s(y1)))))) PLUS(s(s(y0)), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(y0))), s(s(s(s(y1)))))) PLUS(s(s(s(y0))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(y0)))), s(s(s(s(y1)))))) PLUS(s(s(y0)), s(s(s(s(s(s(s(y1)))))))) -> c1(PLUS(s(s(s(y0))), s(s(s(s(s(y1))))))) PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(y1)))))) PLUS(s(s(s(s(y0)))), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(s(y1))))) PLUS(s(s(y0)), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(s(y0))), s(s(s(y1))))) PLUS(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(y1)))) PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(y1))))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), s(s(y1)))) PLUS(s(s(s(s(s(y0))))), s(s(s(s(y1))))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(s(y1)))) PLUS(s(s(y0)), s(s(s(s(s(s(s(s(s(s(y1))))))))))) -> c1(PLUS(s(s(s(y0))), s(s(s(s(s(s(s(s(y1)))))))))) PLUS(s(s(y0)), s(s(s(s(s(s(s(s(y1))))))))) -> c1(PLUS(s(s(s(y0))), s(s(s(s(s(s(y1)))))))) PLUS(s(s(y0)), s(s(s(s(s(s(s(s(s(y1)))))))))) -> c1(PLUS(s(s(s(y0))), s(s(s(s(s(s(s(y1))))))))) PLUS(s(s(s(y0))), s(s(s(s(s(s(s(s(y1))))))))) -> c1(PLUS(s(s(s(s(y0)))), s(s(s(s(s(s(y1)))))))) PLUS(s(s(s(s(s(y0))))), s(s(s(s(s(s(s(s(y1))))))))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(s(s(y1)))))))) PLUS(s(s(s(s(y0)))), s(s(s(s(s(s(s(s(y1))))))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(s(s(s(s(y1)))))))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: plus(s(s(z0)), z1) -> s(plus(z0, s(z1))) plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) plus(s(0), z0) -> s(z0) plus(0, z0) -> z0 ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, plus(z1, ack(s(z0), z1))) Tuples: ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(s(s(y0))), 0) -> c5(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), 0) -> c5(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c5(ACK(s(0), s(0))) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0))) PLUS(s(s(s(s(s(s(y0)))))), s(z1)) -> c(PLUS(s(s(s(s(y0)))), s(s(z1)))) PLUS(s(s(s(s(y0)))), s(z1)) -> c(PLUS(s(s(y0)), s(s(z1)))) PLUS(s(s(z0)), s(s(s(y1)))) -> c(PLUS(z0, s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(z1)) -> c(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(y1))) -> c(PLUS(s(y0), s(s(s(y1))))) PLUS(s(s(s(s(s(s(y0)))))), z1) -> c(PLUS(s(s(s(s(y0)))), s(z1))) PLUS(s(s(s(s(z0)))), s(s(s(y1)))) -> c(PLUS(s(s(z0)), s(s(s(s(y1)))))) PLUS(s(s(s(s(z0)))), s(s(y1))) -> c(PLUS(s(s(z0)), s(s(s(y1))))) PLUS(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(PLUS(s(s(s(s(s(s(y0)))))), s(z1))) PLUS(s(s(s(s(s(s(s(y0))))))), z1) -> c(PLUS(s(s(s(s(s(y0))))), s(z1))) PLUS(z0, s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(z0), s(s(s(s(y1)))))) PLUS(s(s(y0)), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(z0, s(s(s(s(s(y1)))))) -> c1(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(s(s(s(y0))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(y0)))), s(s(z1)))) PLUS(s(y0), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(y0)), s(s(s(y1))))) PLUS(s(s(s(s(y0)))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(s(s(y0)))), s(s(s(y1))))) PLUS(s(s(s(s(s(s(s(y0))))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(s(s(y0)))))))), s(s(z1)))) PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), s(s(z1)))) PLUS(s(s(s(s(s(y0))))), s(s(s(y1)))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(y1))) PLUS(s(s(s(z0))), s(s(s(y1)))) -> c1(PLUS(s(s(s(s(z0)))), s(y1))) PLUS(s(s(s(s(y0)))), s(s(s(y1)))) -> c1(PLUS(s(s(s(s(s(y0))))), s(y1))) PLUS(s(s(s(s(s(y0))))), s(s(z1))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), z1)) PLUS(s(s(s(s(s(s(s(y0))))))), s(s(z1))) -> c1(PLUS(s(s(s(s(s(s(s(s(y0)))))))), z1)) PLUS(s(s(s(s(s(s(y0)))))), s(s(z1))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), z1)) PLUS(s(s(s(z0))), s(s(s(s(s(s(s(s(y1))))))))) -> c1(PLUS(s(s(s(s(z0)))), s(s(s(s(s(s(y1)))))))) PLUS(s(s(s(z0))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(z0)))), s(s(s(s(y1)))))) PLUS(s(s(s(z0))), s(s(s(s(s(s(s(y1)))))))) -> c1(PLUS(s(s(s(s(z0)))), s(s(s(s(s(y1))))))) PLUS(s(s(s(s(y0)))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(s(s(y1)))))) PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(y1)))))) PLUS(s(s(s(s(s(s(s(y0))))))), s(s(s(z1)))) -> c1(PLUS(s(s(s(s(s(s(s(s(y0)))))))), s(z1))) PLUS(s(s(s(s(s(s(y0)))))), s(s(s(z1)))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), s(z1))) PLUS(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c1(PLUS(s(s(z0)), s(s(s(s(s(s(y1)))))))) PLUS(s(z0), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(z0)), s(s(s(s(y1)))))) PLUS(s(z0), s(s(s(s(s(s(s(y1)))))))) -> c1(PLUS(s(s(z0)), s(s(s(s(s(y1))))))) PLUS(s(s(y0)), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(y0))), s(s(s(s(y1)))))) PLUS(s(s(y0)), s(s(s(s(s(s(s(y1)))))))) -> c1(PLUS(s(s(s(y0))), s(s(s(s(s(y1))))))) PLUS(s(s(s(s(y0)))), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(s(y1))))) PLUS(s(s(y0)), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(s(y0))), s(s(s(y1))))) PLUS(s(s(y0)), s(s(s(s(s(s(s(s(s(s(y1))))))))))) -> c1(PLUS(s(s(s(y0))), s(s(s(s(s(s(s(s(y1)))))))))) PLUS(s(s(y0)), s(s(s(s(s(s(s(s(y1))))))))) -> c1(PLUS(s(s(s(y0))), s(s(s(s(s(s(y1)))))))) PLUS(s(s(y0)), s(s(s(s(s(s(s(s(s(y1)))))))))) -> c1(PLUS(s(s(s(y0))), s(s(s(s(s(s(s(y1))))))))) PLUS(s(s(s(s(s(y0))))), s(s(s(s(s(s(s(s(y1))))))))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(s(s(y1)))))))) PLUS(s(s(s(s(y0)))), s(s(s(s(s(s(s(s(y1))))))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(s(s(s(s(y1)))))))) S tuples: ACK(s(x0), s(s(s(s(s(z0)))))) -> c6(ACK(x0, s(s(plus(z0, s(s(ack(s(x0), s(s(s(s(z0))))))))))), PLUS(s(s(s(s(z0)))), ack(s(x0), s(s(s(s(z0)))))), ACK(s(x0), s(s(s(s(z0)))))) ACK(s(x0), s(s(s(s(0))))) -> c6(ACK(x0, s(s(s(ack(s(x0), s(s(s(0)))))))), PLUS(s(s(s(0))), ack(s(x0), s(s(s(0))))), ACK(s(x0), s(s(s(0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, s(s(ack(s(x0), s(s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(z0), s(s(s(x1)))) -> c6(ACK(z0, s(plus(x1, s(ack(z0, plus(s(x1), ack(s(z0), s(x1)))))))), PLUS(s(s(x1)), ack(s(z0), s(s(x1)))), ACK(s(z0), s(s(x1)))) ACK(s(x0), s(s(s(x1)))) -> c6(PLUS(s(s(x1)), ack(s(x0), s(s(x1)))), ACK(s(x0), s(s(x1)))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, s(ack(z0, plus(0, ack(s(z0), 0))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(x0), s(s(0))) -> c6(ACK(s(x0), s(0))) ACK(s(s(z0)), s(0)) -> c6(ACK(s(z0), plus(0, ack(z0, plus(0, ack(s(z0), 0))))), PLUS(0, ack(s(s(z0)), 0)), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c6(PLUS(0, ack(s(x0), 0)), ACK(s(x0), 0)) ACK(s(x0), s(s(s(s(z0))))) -> c6(ACK(x0, plus(s(s(s(z0))), ack(x0, s(plus(z0, s(ack(s(x0), s(s(z0))))))))), PLUS(s(s(s(z0))), ack(s(x0), s(s(s(z0))))), ACK(s(x0), s(s(s(z0))))) ACK(s(x0), s(s(s(0)))) -> c6(ACK(x0, plus(s(s(0)), ack(x0, s(ack(s(x0), s(0)))))), PLUS(s(s(0)), ack(s(x0), s(s(0)))), ACK(s(x0), s(s(0)))) ACK(s(x0), s(s(0))) -> c6(ACK(x0, plus(s(0), ack(x0, ack(s(x0), 0)))), PLUS(s(0), ack(s(x0), s(0))), ACK(s(x0), s(0))) ACK(s(z0), s(s(0))) -> c6(ACK(z0, plus(s(0), ack(z0, plus(0, ack(z0, s(0)))))), PLUS(s(0), ack(s(z0), s(0))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c6(ACK(z0, plus(s(s(z1)), ack(z0, plus(s(z1), ack(z0, plus(z1, ack(s(z0), z1))))))), PLUS(s(s(z1)), ack(s(z0), s(s(z1)))), ACK(s(z0), s(s(z1)))) ACK(s(x0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(x0), s(x1)))) ACK(s(0), s(s(x1))) -> c6(PLUS(s(x1), ack(s(0), s(x1))), ACK(s(0), s(x1))) ACK(s(z0), s(0)) -> c6(ACK(z0, ack(z0, s(0))), PLUS(0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(s(s(y0))), 0) -> c5(ACK(s(s(y0)), s(0))) ACK(s(s(y0)), 0) -> c5(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c5(ACK(s(0), s(0))) ACK(s(0), s(0)) -> c6(PLUS(0, ack(s(0), 0))) PLUS(s(s(s(s(s(s(y0)))))), s(z1)) -> c(PLUS(s(s(s(s(y0)))), s(s(z1)))) PLUS(s(s(s(s(y0)))), s(z1)) -> c(PLUS(s(s(y0)), s(s(z1)))) PLUS(s(s(z0)), s(s(s(y1)))) -> c(PLUS(z0, s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(z1)) -> c(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(y1))) -> c(PLUS(s(y0), s(s(s(y1))))) PLUS(s(s(s(s(s(s(y0)))))), z1) -> c(PLUS(s(s(s(s(y0)))), s(z1))) PLUS(s(s(s(s(z0)))), s(s(s(y1)))) -> c(PLUS(s(s(z0)), s(s(s(s(y1)))))) PLUS(s(s(s(s(z0)))), s(s(y1))) -> c(PLUS(s(s(z0)), s(s(s(y1))))) PLUS(s(s(s(s(s(s(s(s(y0)))))))), z1) -> c(PLUS(s(s(s(s(s(s(y0)))))), s(z1))) PLUS(s(s(s(s(s(s(s(y0))))))), z1) -> c(PLUS(s(s(s(s(s(y0))))), s(z1))) PLUS(z0, s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(z0), s(s(s(s(y1)))))) PLUS(s(s(y0)), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(z0, s(s(s(s(s(y1)))))) -> c1(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(s(s(s(y0))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(y0)))), s(s(z1)))) PLUS(s(y0), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(y0)), s(s(s(y1))))) PLUS(s(s(s(s(y0)))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(s(s(y0)))), s(s(s(y1))))) PLUS(s(s(s(s(s(s(s(y0))))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(s(s(y0)))))))), s(s(z1)))) PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(z1))))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), s(s(z1)))) PLUS(s(s(s(s(s(y0))))), s(s(s(y1)))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(y1))) PLUS(s(s(s(z0))), s(s(s(y1)))) -> c1(PLUS(s(s(s(s(z0)))), s(y1))) PLUS(s(s(s(s(y0)))), s(s(s(y1)))) -> c1(PLUS(s(s(s(s(s(y0))))), s(y1))) PLUS(s(s(s(s(s(y0))))), s(s(z1))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), z1)) PLUS(s(s(s(s(s(s(s(y0))))))), s(s(z1))) -> c1(PLUS(s(s(s(s(s(s(s(s(y0)))))))), z1)) PLUS(s(s(s(s(s(s(y0)))))), s(s(z1))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), z1)) PLUS(s(s(s(z0))), s(s(s(s(s(s(s(s(y1))))))))) -> c1(PLUS(s(s(s(s(z0)))), s(s(s(s(s(s(y1)))))))) PLUS(s(s(s(z0))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(z0)))), s(s(s(s(y1)))))) PLUS(s(s(s(z0))), s(s(s(s(s(s(s(y1)))))))) -> c1(PLUS(s(s(s(s(z0)))), s(s(s(s(s(y1))))))) PLUS(s(s(s(s(y0)))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(s(s(y1)))))) PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(y1)))))) PLUS(s(s(s(s(s(s(s(y0))))))), s(s(s(z1)))) -> c1(PLUS(s(s(s(s(s(s(s(s(y0)))))))), s(z1))) PLUS(s(s(s(s(s(s(y0)))))), s(s(s(z1)))) -> c1(PLUS(s(s(s(s(s(s(s(y0))))))), s(z1))) PLUS(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c1(PLUS(s(s(z0)), s(s(s(s(s(s(y1)))))))) PLUS(s(z0), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(z0)), s(s(s(s(y1)))))) PLUS(s(z0), s(s(s(s(s(s(s(y1)))))))) -> c1(PLUS(s(s(z0)), s(s(s(s(s(y1))))))) PLUS(s(s(y0)), s(s(s(s(s(s(y1))))))) -> c1(PLUS(s(s(s(y0))), s(s(s(s(y1)))))) PLUS(s(s(y0)), s(s(s(s(s(s(s(y1)))))))) -> c1(PLUS(s(s(s(y0))), s(s(s(s(s(y1))))))) PLUS(s(s(s(s(y0)))), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(s(y1))))) PLUS(s(s(y0)), s(s(s(s(s(y1)))))) -> c1(PLUS(s(s(s(y0))), s(s(s(y1))))) PLUS(s(s(y0)), s(s(s(s(s(s(s(s(s(s(y1))))))))))) -> c1(PLUS(s(s(s(y0))), s(s(s(s(s(s(s(s(y1)))))))))) PLUS(s(s(y0)), s(s(s(s(s(s(s(s(y1))))))))) -> c1(PLUS(s(s(s(y0))), s(s(s(s(s(s(y1)))))))) PLUS(s(s(y0)), s(s(s(s(s(s(s(s(s(y1)))))))))) -> c1(PLUS(s(s(s(y0))), s(s(s(s(s(s(s(y1))))))))) PLUS(s(s(s(s(s(y0))))), s(s(s(s(s(s(s(s(y1))))))))) -> c1(PLUS(s(s(s(s(s(s(y0)))))), s(s(s(s(s(s(y1)))))))) PLUS(s(s(s(s(y0)))), s(s(s(s(s(s(s(s(y1))))))))) -> c1(PLUS(s(s(s(s(s(y0))))), s(s(s(s(s(s(y1)))))))) K tuples:none Defined Rule Symbols: plus_2, ack_2 Defined Pair Symbols: ACK_2, PLUS_2 Compound Symbols: c6_3, c6_2, c6_1, c5_1, c_1, c1_1