KILLED proof of input_JgtXAVHV5a.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 4 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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms] (52) CdtProblem (53) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRewritingProof [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: rev1(0, nil) -> 0 rev1(s(X), nil) -> s(X) rev1(X, cons(Y, L)) -> rev1(Y, L) rev(nil) -> nil rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L)) rev2(X, nil) -> nil rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L)))) 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: rev1(0', nil) -> 0' rev1(s(X), nil) -> s(X) rev1(X, cons(Y, L)) -> rev1(Y, L) rev(nil) -> nil rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L)) rev2(X, nil) -> nil rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L)))) 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: rev1(0, nil) -> 0 rev1(s(X), nil) -> s(X) rev1(X, cons(Y, L)) -> rev1(Y, L) rev(nil) -> nil rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L)) rev2(X, nil) -> nil rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L)))) 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: rev1(0, nil) -> 0 [1] rev1(s(X), nil) -> s(X) [1] rev1(X, cons(Y, L)) -> rev1(Y, L) [1] rev(nil) -> nil [1] rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L)) [1] rev2(X, nil) -> nil [1] rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L)))) [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: rev1(0, nil) -> 0 [1] rev1(s(X), nil) -> s(X) [1] rev1(X, cons(Y, L)) -> rev1(Y, L) [1] rev(nil) -> nil [1] rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L)) [1] rev2(X, nil) -> nil [1] rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L)))) [1] The TRS has the following type information: rev1 :: 0:s -> nil:cons -> 0:s 0 :: 0:s nil :: nil:cons s :: a -> 0:s cons :: 0:s -> nil:cons -> nil:cons rev :: nil:cons -> nil:cons rev2 :: 0:s -> nil:cons -> nil:cons 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: rev1(0, nil) -> 0 [1] rev1(s(X), nil) -> s(X) [1] rev1(X, cons(Y, L)) -> rev1(Y, L) [1] rev(nil) -> nil [1] rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L)) [1] rev2(X, nil) -> nil [1] rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L)))) [1] The TRS has the following type information: rev1 :: 0:s -> nil:cons -> 0:s 0 :: 0:s nil :: nil:cons s :: a -> 0:s cons :: 0:s -> nil:cons -> nil:cons rev :: nil:cons -> nil:cons rev2 :: 0:s -> nil:cons -> nil:cons const :: a 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: rev1(0, nil) -> 0 [1] rev1(s(X), nil) -> s(X) [1] rev1(X, cons(Y, L)) -> rev1(Y, L) [1] rev(nil) -> nil [1] rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L)) [1] rev2(X, nil) -> nil [1] rev2(X, cons(Y, nil)) -> rev(cons(X, rev(nil))) [2] rev2(X, cons(Y, cons(Y', L'))) -> rev(cons(X, rev(rev(cons(Y, rev(rev2(Y', L'))))))) [2] The TRS has the following type information: rev1 :: 0:s -> nil:cons -> 0:s 0 :: 0:s nil :: nil:cons s :: a -> 0:s cons :: 0:s -> nil:cons -> nil:cons rev :: nil:cons -> nil:cons rev2 :: 0:s -> nil:cons -> nil:cons const :: a 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 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, L) + rev2(X, L) :|: z = 1 + X + L, X >= 0, L >= 0 rev1(z, z') -{ 1 }-> rev1(Y, L) :|: Y >= 0, X >= 0, z = X, L >= 0, z' = 1 + Y + L rev1(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 rev1(z, z') -{ 1 }-> 1 + X :|: z = 1 + X, X >= 0, z' = 0 rev2(z, z') -{ 2 }-> rev(1 + X + rev(rev(1 + Y + rev(rev2(Y', L'))))) :|: L' >= 0, Y >= 0, Y' >= 0, X >= 0, z' = 1 + Y + (1 + Y' + L'), z = X rev2(z, z') -{ 2 }-> rev(1 + X + rev(0)) :|: Y >= 0, X >= 0, z = X, z' = 1 + Y + 0 rev2(z, z') -{ 1 }-> 0 :|: X >= 0, z = X, z' = 0 ---------------------------------------- (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, L) + rev2(X, L) :|: z = 1 + X + L, X >= 0, L >= 0 rev1(z, z') -{ 1 }-> rev1(Y, L) :|: Y >= 0, z >= 0, L >= 0, z' = 1 + Y + L rev1(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 rev1(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 rev2(z, z') -{ 2 }-> rev(1 + z + rev(rev(1 + Y + rev(rev2(Y', L'))))) :|: L' >= 0, Y >= 0, Y' >= 0, z >= 0, z' = 1 + Y + (1 + Y' + L') rev2(z, z') -{ 2 }-> rev(1 + z + rev(0)) :|: z' - 1 >= 0, z >= 0 rev2(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 ---------------------------------------- (17) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 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: rev1(0, nil) -> 0 [1] rev1(s(X), nil) -> s(X) [1] rev1(X, cons(Y, L)) -> rev1(Y, L) [1] rev(nil) -> nil [1] rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L)) [1] rev2(X, nil) -> nil [1] rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L)))) [1] The TRS has the following type information: rev1 :: 0:s -> nil:cons -> 0:s 0 :: 0:s nil :: nil:cons s :: a -> 0:s cons :: 0:s -> nil:cons -> nil:cons rev :: nil:cons -> nil:cons rev2 :: 0:s -> nil:cons -> nil:cons const :: a 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 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, L) + rev2(X, L) :|: z = 1 + X + L, X >= 0, L >= 0 rev1(z, z') -{ 1 }-> rev1(Y, L) :|: Y >= 0, X >= 0, z = X, L >= 0, z' = 1 + Y + L rev1(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 rev1(z, z') -{ 1 }-> 1 + X :|: z = 1 + X, X >= 0, z' = 0 rev2(z, z') -{ 1 }-> rev(1 + X + rev(rev2(Y, L))) :|: Y >= 0, X >= 0, z = X, L >= 0, z' = 1 + Y + L rev2(z, z') -{ 1 }-> 0 :|: X >= 0, z = X, z' = 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: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV1(0, nil) -> c REV1(s(z0), nil) -> c1 REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2)) REV(nil) -> c3 REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(z0, nil) -> c6 REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2)) S tuples: REV1(0, nil) -> c REV1(s(z0), nil) -> c1 REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2)) REV(nil) -> c3 REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(z0, nil) -> c6 REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2)) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV1_2, REV_1, REV2_2 Compound Symbols: c, c1, c2_1, c3, c4_1, c5_1, c6, c7_3 ---------------------------------------- (23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: REV1(s(z0), nil) -> c1 REV2(z0, nil) -> c6 REV1(0, nil) -> c REV(nil) -> c3 ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2)) REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2)) S tuples: REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2)) REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2)) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV1_2, REV_1, REV2_2 Compound Symbols: c2_1, c4_1, c5_1, c7_3 ---------------------------------------- (25) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2)) by REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(rev2(z0, nil)))), REV(nil), REV2(z0, nil)) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev2(z0, cons(z1, z2))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2))) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2)) REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(rev2(z0, nil)))), REV(nil), REV2(z0, nil)) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev2(z0, cons(z1, z2))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2))) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) S tuples: REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2)) REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(rev2(z0, nil)))), REV(nil), REV2(z0, nil)) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev2(z0, cons(z1, z2))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2))) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV1_2, REV_1, REV2_2 Compound Symbols: c2_1, c4_1, c5_1, c7_3, c7_1 ---------------------------------------- (27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2)) REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev2(z0, cons(z1, z2))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2))) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(rev2(z0, nil))))) S tuples: REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2)) REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev2(z0, cons(z1, z2))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2))) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(rev2(z0, nil))))) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV1_2, REV_1, REV2_2 Compound Symbols: c2_1, c4_1, c5_1, c7_3, c7_1 ---------------------------------------- (29) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev2(z0, cons(z1, z2))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2))) by REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2))) REV2(x0, cons(x1, cons(x2, x3))) -> c7(REV(rev(cons(x1, rev(rev2(x2, x3)))))) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2)) REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(rev2(z0, nil))))) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2))) REV2(x0, cons(x1, cons(x2, x3))) -> c7(REV(rev(cons(x1, rev(rev2(x2, x3)))))) S tuples: REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2)) REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(rev2(z0, nil))))) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2))) REV2(x0, cons(x1, cons(x2, x3))) -> c7(REV(rev(cons(x1, rev(rev2(x2, x3)))))) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV1_2, REV_1, REV2_2 Compound Symbols: c2_1, c4_1, c5_1, c7_1, c7_3 ---------------------------------------- (31) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(rev2(z0, nil))))) by REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil)))) ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2)) REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2))) REV2(x0, cons(x1, cons(x2, x3))) -> c7(REV(rev(cons(x1, rev(rev2(x2, x3)))))) REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil)))) S tuples: REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2)) REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2))) REV2(x0, cons(x1, cons(x2, x3))) -> c7(REV(rev(cons(x1, rev(rev2(x2, x3)))))) REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil)))) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV1_2, REV_1, REV2_2 Compound Symbols: c2_1, c4_1, c5_1, c7_1, c7_3 ---------------------------------------- (33) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2)) by REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2))) REV2(x0, cons(x1, cons(x2, x3))) -> c7(REV(rev(cons(x1, rev(rev2(x2, x3)))))) REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil)))) REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) S tuples: REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2))) REV2(x0, cons(x1, cons(x2, x3))) -> c7(REV(rev(cons(x1, rev(rev2(x2, x3)))))) REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil)))) REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV_1, REV2_2, REV1_2 Compound Symbols: c4_1, c5_1, c7_1, c7_3, c2_1 ---------------------------------------- (35) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2))) by REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(rev(cons(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV2(x0, cons(x1, cons(x2, x3))) -> c7(REV(rev(cons(x1, rev(rev2(x2, x3)))))) REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil)))) REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(rev(cons(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) S tuples: REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV2(x0, cons(x1, cons(x2, x3))) -> c7(REV(rev(cons(x1, rev(rev2(x2, x3)))))) REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil)))) REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(rev(cons(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV_1, REV2_2, REV1_2 Compound Symbols: c4_1, c5_1, c7_1, c2_1, c7_3 ---------------------------------------- (37) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV2(x0, cons(x1, cons(x2, x3))) -> c7(REV(rev(cons(x1, rev(rev2(x2, x3)))))) by REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil)))) REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(rev(cons(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) S tuples: REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil)))) REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(rev(cons(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV_1, REV2_2, REV1_2 Compound Symbols: c4_1, c5_1, c7_1, c2_1, c7_3 ---------------------------------------- (39) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil)))) by REV2(z0, cons(z1, nil)) -> c7(REV(cons(z0, nil))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(rev(cons(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, nil)) -> c7(REV(cons(z0, nil))) S tuples: REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(rev(cons(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, nil)) -> c7(REV(cons(z0, nil))) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV_1, REV2_2, REV1_2 Compound Symbols: c4_1, c5_1, c7_1, c2_1, c7_3 ---------------------------------------- (41) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(rev(cons(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) by REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, nil)) -> c7(REV(cons(z0, nil))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) S tuples: REV(cons(z0, z1)) -> c4(REV1(z0, z1)) REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, nil)) -> c7(REV(cons(z0, nil))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV_1, REV2_2, REV1_2 Compound Symbols: c4_1, c5_1, c7_1, c2_1, c7_3 ---------------------------------------- (43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REV(cons(z0, z1)) -> c4(REV1(z0, z1)) by REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, nil)) -> c7(REV(cons(z0, nil))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) S tuples: REV(cons(z0, z1)) -> c5(REV2(z0, z1)) REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, nil)) -> c7(REV(cons(z0, nil))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV_1, REV2_2, REV1_2 Compound Symbols: c5_1, c7_1, c2_1, c7_3, c4_1 ---------------------------------------- (45) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REV(cons(z0, z1)) -> c5(REV2(z0, z1)) by REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, nil)) -> c7(REV(cons(z0, nil))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil))) S tuples: REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, nil)) -> c7(REV(cons(z0, nil))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil))) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV2_2, REV1_2, REV_1 Compound Symbols: c7_1, c2_1, c7_3, c4_1, c5_1 ---------------------------------------- (47) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV2(z0, cons(z1, nil)) -> c7(REV(cons(z0, nil))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil))) S tuples: REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil))) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV2_2, REV1_2, REV_1 Compound Symbols: c7_1, c2_1, c7_3, c4_1, c5_1 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2))))) by REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil)))) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2)))))))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil))) REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil)))) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2)))))))) S tuples: REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil))) REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil)))) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2)))))))) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV1_2, REV2_2, REV_1 Compound Symbols: c2_1, c7_1, c7_3, c4_1, c5_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil)))) by REV2(x0, cons(x1, nil)) -> c7(REV(cons(x0, nil))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil))) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2)))))))) REV2(x0, cons(x1, nil)) -> c7(REV(cons(x0, nil))) S tuples: REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil))) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2)))))))) REV2(x0, cons(x1, nil)) -> c7(REV(cons(x0, nil))) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV1_2, REV2_2, REV_1 Compound Symbols: c2_1, c7_1, c7_3, c4_1, c5_1 ---------------------------------------- (53) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil))) REV2(x0, cons(x1, nil)) -> c7(REV(cons(x0, nil))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2)))))))) S tuples: REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2)))))))) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV1_2, REV2_2, REV_1 Compound Symbols: c2_1, c7_1, c7_3, c4_1, c5_1 ---------------------------------------- (55) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2)))))))) by REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))))) S tuples: REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))))) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV1_2, REV2_2, REV_1 Compound Symbols: c2_1, c7_1, c7_3, c4_1, c5_1 ---------------------------------------- (57) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))))) by REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, cons(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)))))))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, cons(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, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, cons(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: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV1_2, REV2_2, REV_1 Compound Symbols: c2_1, c7_1, c7_3, c4_1, c5_1 ---------------------------------------- (59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) by REV1(z0, cons(z1, cons(z2, cons(y2, y3)))) -> c2(REV1(z1, cons(z2, cons(y2, y3)))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, cons(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, cons(z1, cons(z2, cons(y2, y3)))) -> c2(REV1(z1, cons(z2, cons(y2, y3)))) S tuples: REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, cons(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, cons(z1, cons(z2, cons(y2, y3)))) -> c2(REV1(z1, cons(z2, cons(y2, y3)))) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV2_2, REV_1, REV1_2 Compound Symbols: c7_1, c7_3, c4_1, c5_1, c2_1 ---------------------------------------- (61) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) by REV(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c4(REV1(z0, cons(z1, cons(z2, cons(y3, y4))))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, cons(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, cons(z1, cons(z2, cons(y2, y3)))) -> c2(REV1(z1, cons(z2, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c4(REV1(z0, cons(z1, cons(z2, cons(y3, y4))))) S tuples: REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, cons(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, cons(z1, cons(z2, cons(y2, y3)))) -> c2(REV1(z1, cons(z2, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c4(REV1(z0, cons(z1, cons(z2, cons(y3, y4))))) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV2_2, REV_1, REV1_2 Compound Symbols: c7_1, c7_3, c5_1, c2_1, c4_1 ---------------------------------------- (63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) by REV(cons(z0, cons(z1, cons(y2, y3)))) -> c5(REV2(z0, cons(z1, cons(y2, y3)))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2)))) Tuples: REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, cons(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, cons(z1, cons(z2, cons(y2, y3)))) -> c2(REV1(z1, cons(z2, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c4(REV1(z0, cons(z1, cons(z2, cons(y3, y4))))) S tuples: REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3))) REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3)))) REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, cons(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, cons(z1, cons(z2, cons(y2, y3)))) -> c2(REV1(z1, cons(z2, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c4(REV1(z0, cons(z1, cons(z2, cons(y3, y4))))) K tuples:none Defined Rule Symbols: rev1_2, rev_1, rev2_2 Defined Pair Symbols: REV2_2, REV_1, REV1_2 Compound Symbols: c7_1, c7_3, c5_1, c2_1, c4_1