KILLED proof of input_smM7IAYlVy.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CompletionProof [UPPER BOUND(ID), 0 ms] (18) CpxTypedWeightedCompleteTrs (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (22) CdtProblem (23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CdtProblem (25) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (64) 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: rev(nil) -> nil rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y)) rev1(x, nil) -> x rev1(x, ++(y, z)) -> rev1(y, z) rev2(x, nil) -> nil rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z)))) 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: rev(nil) -> nil rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y)) rev1(x, nil) -> x rev1(x, ++(y, z)) -> rev1(y, z) rev2(x, nil) -> nil rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z)))) 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: rev(nil) -> nil rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y)) rev1(x, nil) -> x rev1(x, ++(y, z)) -> rev1(y, z) rev2(x, nil) -> nil rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z)))) 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: rev(nil) -> nil [1] rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y)) [1] rev1(x, nil) -> x [1] rev1(x, ++(y, z)) -> rev1(y, z) [1] rev2(x, nil) -> nil [1] rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z)))) [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: rev(nil) -> nil [1] rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y)) [1] rev1(x, nil) -> x [1] rev1(x, ++(y, z)) -> rev1(y, z) [1] rev2(x, nil) -> nil [1] rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z)))) [1] The TRS has the following type information: rev :: nil:++ -> nil:++ nil :: nil:++ ++ :: rev1 -> nil:++ -> nil:++ rev1 :: rev1 -> nil:++ -> rev1 rev2 :: rev1 -> nil:++ -> nil:++ 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: rev_1 rev2_2 rev1_2 Due to the following rules being added: none And the following fresh constants: const ---------------------------------------- (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: rev(nil) -> nil [1] rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y)) [1] rev1(x, nil) -> x [1] rev1(x, ++(y, z)) -> rev1(y, z) [1] rev2(x, nil) -> nil [1] rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z)))) [1] The TRS has the following type information: rev :: nil:++ -> nil:++ nil :: nil:++ ++ :: rev1 -> nil:++ -> nil:++ rev1 :: rev1 -> nil:++ -> rev1 rev2 :: rev1 -> nil:++ -> nil:++ const :: rev1 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: rev(nil) -> nil [1] rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y)) [1] rev1(x, nil) -> x [1] rev1(x, ++(y, z)) -> rev1(y, z) [1] rev2(x, nil) -> nil [1] rev2(x, ++(y, nil)) -> rev(++(x, rev(nil))) [2] rev2(x, ++(y, ++(y', z'))) -> rev(++(x, rev(rev(++(y, rev(rev2(y', z'))))))) [2] The TRS has the following type information: rev :: nil:++ -> nil:++ nil :: nil:++ ++ :: rev1 -> nil:++ -> nil:++ rev1 :: rev1 -> nil:++ -> rev1 rev2 :: rev1 -> nil:++ -> nil:++ const :: rev1 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: nil => 0 const => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: rev(z'') -{ 1 }-> 0 :|: z'' = 0 rev(z'') -{ 1 }-> 1 + rev1(x, y) + rev2(x, y) :|: z'' = 1 + x + y, x >= 0, y >= 0 rev1(z'', z1) -{ 1 }-> x :|: z1 = 0, x >= 0, z'' = x rev1(z'', z1) -{ 1 }-> rev1(y, z) :|: z >= 0, z1 = 1 + y + z, x >= 0, y >= 0, z'' = x rev2(z'', z1) -{ 2 }-> rev(1 + x + rev(rev(1 + y + rev(rev2(y', z'))))) :|: z1 = 1 + y + (1 + y' + z'), x >= 0, y >= 0, z' >= 0, z'' = x, y' >= 0 rev2(z'', z1) -{ 2 }-> rev(1 + x + rev(0)) :|: x >= 0, y >= 0, z'' = x, z1 = 1 + y + 0 rev2(z'', z1) -{ 1 }-> 0 :|: z1 = 0, x >= 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: rev(z'') -{ 1 }-> 0 :|: z'' = 0 rev(z'') -{ 1 }-> 1 + rev1(x, y) + rev2(x, y) :|: z'' = 1 + x + y, x >= 0, y >= 0 rev1(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0 rev1(z'', z1) -{ 1 }-> rev1(y, z) :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 rev2(z'', z1) -{ 2 }-> rev(1 + z'' + rev(rev(1 + y + rev(rev2(y', z'))))) :|: z1 = 1 + y + (1 + y' + z'), z'' >= 0, y >= 0, z' >= 0, y' >= 0 rev2(z'', z1) -{ 2 }-> rev(1 + z'' + rev(0)) :|: z'' >= 0, z1 - 1 >= 0 rev2(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0 ---------------------------------------- (17) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: const ---------------------------------------- (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: rev(nil) -> nil [1] rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y)) [1] rev1(x, nil) -> x [1] rev1(x, ++(y, z)) -> rev1(y, z) [1] rev2(x, nil) -> nil [1] rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z)))) [1] The TRS has the following type information: rev :: nil:++ -> nil:++ nil :: nil:++ ++ :: rev1 -> nil:++ -> nil:++ rev1 :: rev1 -> nil:++ -> rev1 rev2 :: rev1 -> nil:++ -> nil:++ const :: rev1 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: nil => 0 const => 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: rev(z') -{ 1 }-> 0 :|: z' = 0 rev(z') -{ 1 }-> 1 + rev1(x, y) + rev2(x, y) :|: z' = 1 + x + y, x >= 0, y >= 0 rev1(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0 rev1(z', z'') -{ 1 }-> rev1(y, z) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z rev2(z', z'') -{ 1 }-> rev(1 + x + rev(rev2(y, z))) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z rev2(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = x, x >= 0 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: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV(nil) -> c REV(++(z0, z1)) -> c1(REV1(z0, z1)) REV(++(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, nil) -> c3 REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(z0, nil) -> c5 REV2(z0, ++(z1, z2)) -> c6(REV(++(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2)) S tuples: REV(nil) -> c REV(++(z0, z1)) -> c1(REV1(z0, z1)) REV(++(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, nil) -> c3 REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(z0, nil) -> c5 REV2(z0, ++(z1, z2)) -> c6(REV(++(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2)) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV_1, REV1_2, REV2_2 Compound Symbols: c, c1_1, c2_1, c3, c4_1, c5, c6_3 ---------------------------------------- (23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: REV1(z0, nil) -> c3 REV(nil) -> c REV2(z0, nil) -> c5 ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV(++(z0, z1)) -> c1(REV1(z0, z1)) REV(++(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(z0, ++(z1, z2)) -> c6(REV(++(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2)) S tuples: REV(++(z0, z1)) -> c1(REV1(z0, z1)) REV(++(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(z0, ++(z1, z2)) -> c6(REV(++(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2)) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV_1, REV1_2, REV2_2 Compound Symbols: c1_1, c2_1, c4_1, c6_3 ---------------------------------------- (25) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV2(z0, ++(z1, z2)) -> c6(REV(++(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2)) by REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(rev2(z0, nil)))), REV(nil), REV2(z0, nil)) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev2(z0, ++(z1, z2))))), REV(rev(++(z0, rev(rev2(z1, z2))))), REV2(z0, ++(z1, z2))) REV2(x0, ++(x1, x2)) -> c6(REV(++(x0, rev(rev2(x1, x2))))) ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV(++(z0, z1)) -> c1(REV1(z0, z1)) REV(++(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(rev2(z0, nil)))), REV(nil), REV2(z0, nil)) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev2(z0, ++(z1, z2))))), REV(rev(++(z0, rev(rev2(z1, z2))))), REV2(z0, ++(z1, z2))) REV2(x0, ++(x1, x2)) -> c6(REV(++(x0, rev(rev2(x1, x2))))) S tuples: REV(++(z0, z1)) -> c1(REV1(z0, z1)) REV(++(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(rev2(z0, nil)))), REV(nil), REV2(z0, nil)) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev2(z0, ++(z1, z2))))), REV(rev(++(z0, rev(rev2(z1, z2))))), REV2(z0, ++(z1, z2))) REV2(x0, ++(x1, x2)) -> c6(REV(++(x0, rev(rev2(x1, x2))))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV_1, REV1_2, REV2_2 Compound Symbols: c1_1, c2_1, c4_1, c6_3, c6_1 ---------------------------------------- (27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV(++(z0, z1)) -> c1(REV1(z0, z1)) REV(++(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev2(z0, ++(z1, z2))))), REV(rev(++(z0, rev(rev2(z1, z2))))), REV2(z0, ++(z1, z2))) REV2(x0, ++(x1, x2)) -> c6(REV(++(x0, rev(rev2(x1, x2))))) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(rev2(z0, nil))))) S tuples: REV(++(z0, z1)) -> c1(REV1(z0, z1)) REV(++(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev2(z0, ++(z1, z2))))), REV(rev(++(z0, rev(rev2(z1, z2))))), REV2(z0, ++(z1, z2))) REV2(x0, ++(x1, x2)) -> c6(REV(++(x0, rev(rev2(x1, x2))))) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(rev2(z0, nil))))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV_1, REV1_2, REV2_2 Compound Symbols: c1_1, c2_1, c4_1, c6_3, c6_1 ---------------------------------------- (29) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev2(z0, ++(z1, z2))))), REV(rev(++(z0, rev(rev2(z1, z2))))), REV2(z0, ++(z1, z2))) by REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2))))))), REV(rev(++(z0, rev(rev2(z1, z2))))), REV2(z0, ++(z1, z2))) REV2(x0, ++(x1, ++(x2, x3))) -> c6(REV(rev(++(x1, rev(rev2(x2, x3)))))) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV(++(z0, z1)) -> c1(REV1(z0, z1)) REV(++(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(x0, ++(x1, x2)) -> c6(REV(++(x0, rev(rev2(x1, x2))))) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(rev2(z0, nil))))) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2))))))), REV(rev(++(z0, rev(rev2(z1, z2))))), REV2(z0, ++(z1, z2))) REV2(x0, ++(x1, ++(x2, x3))) -> c6(REV(rev(++(x1, rev(rev2(x2, x3)))))) S tuples: REV(++(z0, z1)) -> c1(REV1(z0, z1)) REV(++(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(x0, ++(x1, x2)) -> c6(REV(++(x0, rev(rev2(x1, x2))))) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(rev2(z0, nil))))) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2))))))), REV(rev(++(z0, rev(rev2(z1, z2))))), REV2(z0, ++(z1, z2))) REV2(x0, ++(x1, ++(x2, x3))) -> c6(REV(rev(++(x1, rev(rev2(x2, x3)))))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV_1, REV1_2, REV2_2 Compound Symbols: c1_1, c2_1, c4_1, c6_1, c6_3 ---------------------------------------- (31) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(rev2(z0, nil))))) by REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(nil)))) ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV(++(z0, z1)) -> c1(REV1(z0, z1)) REV(++(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(x0, ++(x1, x2)) -> c6(REV(++(x0, rev(rev2(x1, x2))))) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2))))))), REV(rev(++(z0, rev(rev2(z1, z2))))), REV2(z0, ++(z1, z2))) REV2(x0, ++(x1, ++(x2, x3))) -> c6(REV(rev(++(x1, rev(rev2(x2, x3)))))) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(nil)))) S tuples: REV(++(z0, z1)) -> c1(REV1(z0, z1)) REV(++(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(x0, ++(x1, x2)) -> c6(REV(++(x0, rev(rev2(x1, x2))))) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2))))))), REV(rev(++(z0, rev(rev2(z1, z2))))), REV2(z0, ++(z1, z2))) REV2(x0, ++(x1, ++(x2, x3))) -> c6(REV(rev(++(x1, rev(rev2(x2, x3)))))) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(nil)))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV_1, REV1_2, REV2_2 Compound Symbols: c1_1, c2_1, c4_1, c6_1, c6_3 ---------------------------------------- (33) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2))))))), REV(rev(++(z0, rev(rev2(z1, z2))))), REV2(z0, ++(z1, z2))) by REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV(++(z0, z1)) -> c1(REV1(z0, z1)) REV(++(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(x0, ++(x1, x2)) -> c6(REV(++(x0, rev(rev2(x1, x2))))) REV2(x0, ++(x1, ++(x2, x3))) -> c6(REV(rev(++(x1, rev(rev2(x2, x3)))))) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(nil)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) S tuples: REV(++(z0, z1)) -> c1(REV1(z0, z1)) REV(++(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(x0, ++(x1, x2)) -> c6(REV(++(x0, rev(rev2(x1, x2))))) REV2(x0, ++(x1, ++(x2, x3))) -> c6(REV(rev(++(x1, rev(rev2(x2, x3)))))) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(nil)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV_1, REV1_2, REV2_2 Compound Symbols: c1_1, c2_1, c4_1, c6_1, c6_3 ---------------------------------------- (35) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REV(++(z0, z1)) -> c1(REV1(z0, z1)) by REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV(++(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(x0, ++(x1, x2)) -> c6(REV(++(x0, rev(rev2(x1, x2))))) REV2(x0, ++(x1, ++(x2, x3))) -> c6(REV(rev(++(x1, rev(rev2(x2, x3)))))) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(nil)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) S tuples: REV(++(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(x0, ++(x1, x2)) -> c6(REV(++(x0, rev(rev2(x1, x2))))) REV2(x0, ++(x1, ++(x2, x3))) -> c6(REV(rev(++(x1, rev(rev2(x2, x3)))))) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(nil)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV_1, REV1_2, REV2_2 Compound Symbols: c2_1, c4_1, c6_1, c6_3, c1_1 ---------------------------------------- (37) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV2(x0, ++(x1, ++(x2, x3))) -> c6(REV(rev(++(x1, rev(rev2(x2, x3)))))) by REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV(++(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(x0, ++(x1, x2)) -> c6(REV(++(x0, rev(rev2(x1, x2))))) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(nil)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) S tuples: REV(++(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(x0, ++(x1, x2)) -> c6(REV(++(x0, rev(rev2(x1, x2))))) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(nil)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV_1, REV1_2, REV2_2 Compound Symbols: c2_1, c4_1, c6_1, c6_3, c1_1 ---------------------------------------- (39) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REV(++(z0, z1)) -> c2(REV2(z0, z1)) by REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, nil))) -> c2(REV2(z0, ++(y1, nil))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(x0, ++(x1, x2)) -> c6(REV(++(x0, rev(rev2(x1, x2))))) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(nil)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, nil))) -> c2(REV2(z0, ++(y1, nil))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) S tuples: REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(x0, ++(x1, x2)) -> c6(REV(++(x0, rev(rev2(x1, x2))))) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(nil)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, nil))) -> c2(REV2(z0, ++(y1, nil))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV1_2, REV2_2, REV_1 Compound Symbols: c4_1, c6_1, c6_3, c1_1, c2_1 ---------------------------------------- (41) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV2(x0, ++(x1, x2)) -> c6(REV(++(x0, rev(rev2(x1, x2))))) by REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(nil)))) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2)))))))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(nil)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, nil))) -> c2(REV2(z0, ++(y1, nil))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2)))))))) S tuples: REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(nil)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, nil))) -> c2(REV2(z0, ++(y1, nil))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2)))))))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV1_2, REV2_2, REV_1 Compound Symbols: c4_1, c6_1, c6_3, c1_1, c2_1 ---------------------------------------- (43) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(nil)))) by REV2(x0, ++(x1, nil)) -> c6(REV(++(x0, nil))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(nil)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, nil))) -> c2(REV2(z0, ++(y1, nil))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2)))))))) REV2(x0, ++(x1, nil)) -> c6(REV(++(x0, nil))) S tuples: REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, nil))) -> c2(REV2(z0, ++(y1, nil))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2)))))))) REV2(x0, ++(x1, nil)) -> c6(REV(++(x0, nil))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV1_2, REV2_2, REV_1 Compound Symbols: c4_1, c6_1, c6_3, c1_1, c2_1 ---------------------------------------- (45) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV2(x0, ++(x1, nil)) -> c6(REV(++(x0, nil))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, nil))) -> c2(REV2(z0, ++(y1, nil))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(nil)))) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2)))))))) S tuples: REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, nil))) -> c2(REV2(z0, ++(y1, nil))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2)))))))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV1_2, REV2_2, REV_1 Compound Symbols: c4_1, c6_3, c1_1, c6_1, c2_1 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV2(x0, ++(z0, nil)) -> c6(REV(++(x0, rev(nil)))) by REV2(x0, ++(x1, nil)) -> c6(REV(++(x0, nil))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, nil))) -> c2(REV2(z0, ++(y1, nil))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2)))))))) REV2(x0, ++(x1, nil)) -> c6(REV(++(x0, nil))) S tuples: REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, nil))) -> c2(REV2(z0, ++(y1, nil))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2)))))))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV1_2, REV2_2, REV_1 Compound Symbols: c4_1, c6_3, c1_1, c6_1, c2_1 ---------------------------------------- (49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: REV2(x0, ++(x1, nil)) -> c6(REV(++(x0, nil))) REV(++(z0, ++(y1, nil))) -> c2(REV2(z0, ++(y1, nil))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2)))))))) S tuples: REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2)))))))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV1_2, REV2_2, REV_1 Compound Symbols: c4_1, c6_3, c1_1, c6_1, c2_1 ---------------------------------------- (51) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(rev(++(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) by REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2)))))))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) S tuples: REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2)))))))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV1_2, REV_1, REV2_2 Compound Symbols: c4_1, c1_1, c6_1, c2_1, c6_3 ---------------------------------------- (53) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV2(x0, ++(z0, ++(z1, z2))) -> c6(REV(++(x0, rev(rev(++(z0, rev(rev2(z1, z2)))))))) by REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))))) S tuples: REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV1_2, REV_1, REV2_2 Compound Symbols: c4_1, c1_1, c6_1, c2_1, c6_3 ---------------------------------------- (55) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))))) by REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, ++(rev1(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))), rev2(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, ++(rev1(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))), rev2(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))))) S tuples: REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, ++(rev1(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))), rev2(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV1_2, REV_1, REV2_2 Compound Symbols: c4_1, c1_1, c6_1, c2_1, c6_3 ---------------------------------------- (57) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REV1(z0, ++(z1, z2)) -> c4(REV1(z1, z2)) by REV1(z0, ++(z1, ++(y1, y2))) -> c4(REV1(z1, ++(y1, y2))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, ++(rev1(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))), rev2(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))))) REV1(z0, ++(z1, ++(y1, y2))) -> c4(REV1(z1, ++(y1, y2))) S tuples: REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, ++(rev1(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))), rev2(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))))) REV1(z0, ++(z1, ++(y1, y2))) -> c4(REV1(z1, ++(y1, y2))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV_1, REV2_2, REV1_2 Compound Symbols: c1_1, c6_1, c2_1, c6_3, c4_1 ---------------------------------------- (59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REV(++(z0, ++(y1, y2))) -> c1(REV1(z0, ++(y1, y2))) by REV(++(z0, ++(z1, ++(y2, y3)))) -> c1(REV1(z0, ++(z1, ++(y2, y3)))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, ++(rev1(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))), rev2(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))))) REV1(z0, ++(z1, ++(y1, y2))) -> c4(REV1(z1, ++(y1, y2))) REV(++(z0, ++(z1, ++(y2, y3)))) -> c1(REV1(z0, ++(z1, ++(y2, y3)))) S tuples: REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, ++(rev1(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))), rev2(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))))) REV1(z0, ++(z1, ++(y1, y2))) -> c4(REV1(z1, ++(y1, y2))) REV(++(z0, ++(z1, ++(y2, y3)))) -> c1(REV1(z0, ++(z1, ++(y2, y3)))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV2_2, REV_1, REV1_2 Compound Symbols: c6_1, c2_1, c6_3, c4_1, c1_1 ---------------------------------------- (61) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REV(++(z0, ++(y1, y2))) -> c2(REV2(z0, ++(y1, y2))) by REV(++(z0, ++(z1, ++(y2, y3)))) -> c2(REV2(z0, ++(z1, ++(y2, y3)))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, ++(rev1(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))), rev2(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))))) REV1(z0, ++(z1, ++(y1, y2))) -> c4(REV1(z1, ++(y1, y2))) REV(++(z0, ++(z1, ++(y2, y3)))) -> c1(REV1(z0, ++(z1, ++(y2, y3)))) S tuples: REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, ++(rev1(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))), rev2(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))))) REV1(z0, ++(z1, ++(y1, y2))) -> c4(REV1(z1, ++(y1, y2))) REV(++(z0, ++(z1, ++(y2, y3)))) -> c1(REV1(z0, ++(z1, ++(y2, y3)))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV2_2, REV_1, REV1_2 Compound Symbols: c6_1, c2_1, c6_3, c4_1, c1_1 ---------------------------------------- (63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REV1(z0, ++(z1, ++(y1, y2))) -> c4(REV1(z1, ++(y1, y2))) by REV1(z0, ++(z1, ++(z2, ++(y2, y3)))) -> c4(REV1(z1, ++(z2, ++(y2, y3)))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(++(z0, z1)) -> ++(rev1(z0, z1), rev2(z0, z1)) rev1(z0, nil) -> z0 rev1(z0, ++(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, ++(z1, z2)) -> rev(++(z0, rev(rev2(z1, z2)))) Tuples: REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, ++(rev1(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))), rev2(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))))) REV(++(z0, ++(z1, ++(y2, y3)))) -> c1(REV1(z0, ++(z1, ++(y2, y3)))) REV1(z0, ++(z1, ++(z2, ++(y2, y3)))) -> c4(REV1(z1, ++(z2, ++(y2, y3)))) S tuples: REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV(++(z0, ++(y1, ++(y2, y3)))) -> c2(REV2(z0, ++(y1, ++(y2, y3)))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, rev(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(++(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, ++(z2, z3))) REV2(z0, ++(z1, ++(z2, z3))) -> c6(REV(++(z0, ++(rev1(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))), rev2(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))))) REV(++(z0, ++(z1, ++(y2, y3)))) -> c1(REV1(z0, ++(z1, ++(y2, y3)))) REV1(z0, ++(z1, ++(z2, ++(y2, y3)))) -> c4(REV1(z1, ++(z2, ++(y2, y3)))) K tuples:none Defined Rule Symbols: rev_1, rev1_2, rev2_2 Defined Pair Symbols: REV2_2, REV_1, REV1_2 Compound Symbols: c6_1, c2_1, c6_3, c1_1, c4_1