KILLED proof of input_8nvDsNY9Kj.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (56) 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(cons(x, l)) -> cons(rev1(x, l), rev2(x, l)) rev1(0, nil) -> 0 rev1(s(x), nil) -> s(x) rev1(x, cons(y, l)) -> rev1(y, l) rev2(x, nil) -> nil rev2(x, cons(y, l)) -> rev(cons(x, 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: rev(nil) -> nil rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l)) rev1(0', nil) -> 0' rev1(s(x), nil) -> s(x) rev1(x, cons(y, l)) -> rev1(y, l) rev2(x, nil) -> nil rev2(x, cons(y, l)) -> rev(cons(x, 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: rev(nil) -> nil rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l)) rev1(0, nil) -> 0 rev1(s(x), nil) -> s(x) rev1(x, cons(y, l)) -> rev1(y, l) rev2(x, nil) -> nil rev2(x, cons(y, l)) -> rev(cons(x, 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: rev(nil) -> nil [1] rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l)) [1] rev1(0, nil) -> 0 [1] rev1(s(x), nil) -> s(x) [1] rev1(x, cons(y, l)) -> rev1(y, l) [1] rev2(x, nil) -> nil [1] rev2(x, cons(y, l)) -> rev(cons(x, 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: rev(nil) -> nil [1] rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l)) [1] rev1(0, nil) -> 0 [1] rev1(s(x), nil) -> s(x) [1] rev1(x, cons(y, l)) -> rev1(y, l) [1] rev2(x, nil) -> nil [1] rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l))) [1] The TRS has the following type information: rev :: nil:cons -> nil:cons nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons rev1 :: 0:s -> nil:cons -> 0:s rev2 :: 0:s -> nil:cons -> nil:cons 0 :: 0:s s :: a -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: rev2_2 rev_1 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(cons(x, l)) -> cons(rev1(x, l), rev2(x, l)) [1] rev1(0, nil) -> 0 [1] rev1(s(x), nil) -> s(x) [1] rev1(x, cons(y, l)) -> rev1(y, l) [1] rev2(x, nil) -> nil [1] rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l))) [1] The TRS has the following type information: rev :: nil:cons -> nil:cons nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons rev1 :: 0:s -> nil:cons -> 0:s rev2 :: 0:s -> nil:cons -> nil:cons 0 :: 0:s s :: a -> 0:s 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: rev(nil) -> nil [1] rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l)) [1] rev1(0, nil) -> 0 [1] rev1(s(x), nil) -> s(x) [1] rev1(x, cons(y, l)) -> rev1(y, l) [1] rev2(x, nil) -> nil [1] rev2(x, cons(y, nil)) -> rev(cons(x, nil)) [2] rev2(x, cons(y, cons(y', l'))) -> rev(cons(x, rev(cons(y, rev2(y', l'))))) [2] The TRS has the following type information: rev :: nil:cons -> nil:cons nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons rev1 :: 0:s -> nil:cons -> 0:s rev2 :: 0:s -> nil:cons -> nil:cons 0 :: 0:s s :: a -> 0:s 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: nil => 0 0 => 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) :|: x >= 0, l >= 0, z = 1 + x + l rev1(z, z') -{ 1 }-> rev1(y, l) :|: x >= 0, y >= 0, z' = 1 + y + l, l >= 0, z = x rev1(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 rev1(z, z') -{ 1 }-> 1 + x :|: x >= 0, z = 1 + x, z' = 0 rev2(z, z') -{ 2 }-> rev(1 + x + rev(1 + y + rev2(y', l'))) :|: x >= 0, y >= 0, l' >= 0, y' >= 0, z = x, z' = 1 + y + (1 + y' + l') rev2(z, z') -{ 2 }-> rev(1 + x + 0) :|: x >= 0, y >= 0, z' = 1 + y + 0, z = x 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) :|: x >= 0, l >= 0, z = 1 + x + l rev1(z, z') -{ 1 }-> rev1(y, l) :|: z >= 0, y >= 0, z' = 1 + y + l, l >= 0 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(1 + y + rev2(y', l'))) :|: z >= 0, y >= 0, l' >= 0, y' >= 0, z' = 1 + y + (1 + y' + l') rev2(z, z') -{ 2 }-> rev(1 + z + 0) :|: z >= 0, z' - 1 >= 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: rev(nil) -> nil [1] rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l)) [1] rev1(0, nil) -> 0 [1] rev1(s(x), nil) -> s(x) [1] rev1(x, cons(y, l)) -> rev1(y, l) [1] rev2(x, nil) -> nil [1] rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l))) [1] The TRS has the following type information: rev :: nil:cons -> nil:cons nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons rev1 :: 0:s -> nil:cons -> 0:s rev2 :: 0:s -> nil:cons -> nil:cons 0 :: 0:s s :: a -> 0:s 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: nil => 0 0 => 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) :|: x >= 0, l >= 0, z = 1 + x + l rev1(z, z') -{ 1 }-> rev1(y, l) :|: x >= 0, y >= 0, z' = 1 + y + l, l >= 0, z = x rev1(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 rev1(z, z') -{ 1 }-> 1 + x :|: x >= 0, z = 1 + x, z' = 0 rev2(z, z') -{ 1 }-> rev(1 + x + rev2(y, l)) :|: x >= 0, y >= 0, z' = 1 + y + l, l >= 0, z = x 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: rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2))) Tuples: REV(nil) -> c REV(cons(z0, z1)) -> c1(REV1(z0, z1)) REV(cons(z0, z1)) -> c2(REV2(z0, z1)) REV1(0, nil) -> c3 REV1(s(z0), nil) -> c4 REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) REV2(z0, nil) -> c6 REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2)) S tuples: REV(nil) -> c REV(cons(z0, z1)) -> c1(REV1(z0, z1)) REV(cons(z0, z1)) -> c2(REV2(z0, z1)) REV1(0, nil) -> c3 REV1(s(z0), nil) -> c4 REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) REV2(z0, nil) -> c6 REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, 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, c5_1, c6, c7_2 ---------------------------------------- (23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: REV2(z0, nil) -> c6 REV1(0, nil) -> c3 REV(nil) -> c REV1(s(z0), nil) -> c4 ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: rev(nil) -> nil rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2))) Tuples: REV(cons(z0, z1)) -> c1(REV1(z0, z1)) REV(cons(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2)) S tuples: REV(cons(z0, z1)) -> c1(REV1(z0, z1)) REV(cons(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, 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, c5_1, c7_2 ---------------------------------------- (25) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: rev(nil) -> nil ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2))) rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) Tuples: REV(cons(z0, z1)) -> c1(REV1(z0, z1)) REV(cons(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2)) S tuples: REV(cons(z0, z1)) -> c1(REV1(z0, z1)) REV(cons(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2)) K tuples:none Defined Rule Symbols: rev2_2, rev_1, rev1_2 Defined Pair Symbols: REV_1, REV1_2, REV2_2 Compound Symbols: c1_1, c2_1, c5_1, c7_2 ---------------------------------------- (27) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REV(cons(z0, z1)) -> c1(REV1(z0, z1)) by REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2))) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2))) rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) Tuples: REV(cons(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2)) REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2))) S tuples: REV(cons(z0, z1)) -> c2(REV2(z0, z1)) REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2)) REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2))) K tuples:none Defined Rule Symbols: rev2_2, rev_1, rev1_2 Defined Pair Symbols: REV_1, REV1_2, REV2_2 Compound Symbols: c2_1, c5_1, c7_2, c1_1 ---------------------------------------- (29) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REV(cons(z0, z1)) -> c2(REV2(z0, z1)) by REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2))) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2))) rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) Tuples: REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2)) REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2))) REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2))) S tuples: REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2)) REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2))) REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2))) K tuples:none Defined Rule Symbols: rev2_2, rev_1, rev1_2 Defined Pair Symbols: REV1_2, REV2_2, REV_1 Compound Symbols: c5_1, c7_2, c1_1, c2_1 ---------------------------------------- (31) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2)) by REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, nil)), REV2(z0, nil)) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(cons(z0, rev2(z1, z2))))), REV2(z0, cons(z1, z2))) ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2))) rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) Tuples: REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2))) REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2))) REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, nil)), REV2(z0, nil)) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(cons(z0, rev2(z1, z2))))), REV2(z0, cons(z1, z2))) S tuples: REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2))) REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2))) REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, nil)), REV2(z0, nil)) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(cons(z0, rev2(z1, z2))))), REV2(z0, cons(z1, z2))) K tuples:none Defined Rule Symbols: rev2_2, rev_1, rev1_2 Defined Pair Symbols: REV1_2, REV_1, REV2_2 Compound Symbols: c5_1, c1_1, c2_1, c7_2 ---------------------------------------- (33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, nil)), REV2(z0, nil)) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2))) rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) Tuples: REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2))) REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2))) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(cons(z0, rev2(z1, z2))))), REV2(z0, cons(z1, z2))) S tuples: REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2))) REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2))) REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(cons(z0, rev2(z1, z2))))), REV2(z0, cons(z1, z2))) K tuples:none Defined Rule Symbols: rev2_2, rev_1, rev1_2 Defined Pair Symbols: REV1_2, REV_1, REV2_2 Compound Symbols: c5_1, c1_1, c2_1, c7_2 ---------------------------------------- (35) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(cons(z0, rev2(z1, z2))))), REV2(z0, cons(z1, z2))) by REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil)))), REV2(x1, cons(z0, nil))) REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2)))) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2))) rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) Tuples: REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2))) REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2))) REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil)))), REV2(x1, cons(z0, nil))) REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2)))) S tuples: REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2))) REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2))) REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil)))), REV2(x1, cons(z0, nil))) REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2)))) K tuples:none Defined Rule Symbols: rev2_2, rev_1, rev1_2 Defined Pair Symbols: REV1_2, REV_1, REV2_2 Compound Symbols: c5_1, c1_1, c2_1, c7_2 ---------------------------------------- (37) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2))) rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) Tuples: REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2))) REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2))) REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2)))) REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil))))) S tuples: REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2))) REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2))) REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2)))) REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil))))) K tuples:none Defined Rule Symbols: rev2_2, rev_1, rev1_2 Defined Pair Symbols: REV1_2, REV_1, REV2_2 Compound Symbols: c5_1, c1_1, c2_1, c7_2, c7_1 ---------------------------------------- (39) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) by REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2))) rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) Tuples: REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2))) REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2))) REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2)))) REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil))))) REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2))) S tuples: REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2))) REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2))) REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2)))) REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil))))) REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2))) K tuples:none Defined Rule Symbols: rev2_2, rev_1, rev1_2 Defined Pair Symbols: REV_1, REV2_2, REV1_2 Compound Symbols: c1_1, c2_1, c7_2, c7_1, c5_1 ---------------------------------------- (41) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2))) by REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3)))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2))) rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) Tuples: REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2))) REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2)))) REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil))))) REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3)))) S tuples: REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2))) REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2)))) REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil))))) REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3)))) K tuples:none Defined Rule Symbols: rev2_2, rev_1, rev1_2 Defined Pair Symbols: REV_1, REV2_2, REV1_2 Compound Symbols: c2_1, c7_2, c7_1, c5_1, c1_1 ---------------------------------------- (43) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2)))) by REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, rev(cons(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4)))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2))) rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) Tuples: REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2))) REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil))))) REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3)))) REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, rev(cons(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4)))) S tuples: REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2))) REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil))))) REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3)))) REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, rev(cons(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4)))) K tuples:none Defined Rule Symbols: rev2_2, rev_1, rev1_2 Defined Pair Symbols: REV_1, REV2_2, REV1_2 Compound Symbols: c2_1, c7_2, c7_1, c5_1, c1_1 ---------------------------------------- (45) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2))) by REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil)))) REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4))))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2))) rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) Tuples: REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil))))) REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3)))) REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, rev(cons(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4)))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil)))) REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4))))) S tuples: REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil))))) REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3)))) REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, rev(cons(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4)))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil)))) REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4))))) K tuples:none Defined Rule Symbols: rev2_2, rev_1, rev1_2 Defined Pair Symbols: REV2_2, REV1_2, REV_1 Compound Symbols: c7_2, c7_1, c5_1, c1_1, c2_1 ---------------------------------------- (47) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil))))) by REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), rev2(z1, nil))))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2))) rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) Tuples: REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3)))) REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, rev(cons(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4)))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil)))) REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4))))) REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), rev2(z1, nil))))) S tuples: REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3)))) REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, rev(cons(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4)))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil)))) REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4))))) REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), rev2(z1, nil))))) K tuples:none Defined Rule Symbols: rev2_2, rev_1, rev1_2 Defined Pair Symbols: REV2_2, REV1_2, REV_1 Compound Symbols: c7_2, c5_1, c1_1, c2_1, c7_1 ---------------------------------------- (49) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, rev(cons(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4)))) by REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, cons(rev1(z2, rev2(z3, z4)), rev2(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4)))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2))) rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) Tuples: REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil)))) REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4))))) REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), rev2(z1, nil))))) REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, cons(rev1(z2, rev2(z3, z4)), rev2(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4)))) S tuples: REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil)))) REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4))))) REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), rev2(z1, nil))))) REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, cons(rev1(z2, rev2(z3, z4)), rev2(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4)))) K tuples:none Defined Rule Symbols: rev2_2, rev_1, rev1_2 Defined Pair Symbols: REV2_2, REV1_2, REV_1 Compound Symbols: c7_2, c5_1, c1_1, c2_1, c7_1 ---------------------------------------- (51) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), rev2(z1, nil))))) by REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), nil)))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2))) rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) Tuples: REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil)))) REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4))))) REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, cons(rev1(z2, rev2(z3, z4)), rev2(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4)))) REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), nil)))) S tuples: REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil)))) REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4))))) REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, cons(rev1(z2, rev2(z3, z4)), rev2(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4)))) REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), nil)))) K tuples:none Defined Rule Symbols: rev2_2, rev_1, rev1_2 Defined Pair Symbols: REV2_2, REV1_2, REV_1 Compound Symbols: c7_2, c5_1, c1_1, c2_1, c7_1 ---------------------------------------- (53) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), nil)))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2))) rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) Tuples: REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil)))) REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4))))) REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, cons(rev1(z2, rev2(z3, z4)), rev2(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4)))) S tuples: REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil)))) REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4))))) REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, cons(rev1(z2, rev2(z3, z4)), rev2(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4)))) K tuples:none Defined Rule Symbols: rev2_2, rev_1, rev1_2 Defined Pair Symbols: REV2_2, REV1_2, REV_1 Compound Symbols: c7_2, c5_1, c1_1, c2_1 ---------------------------------------- (55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2))) by REV1(z0, cons(z1, cons(z2, cons(y2, y3)))) -> c5(REV1(z1, cons(z2, cons(y2, y3)))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: rev2(z0, nil) -> nil rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2))) rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1)) rev1(0, nil) -> 0 rev1(s(z0), nil) -> s(z0) rev1(z0, cons(z1, z2)) -> rev1(z1, z2) Tuples: REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil)))) REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4))))) REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, cons(rev1(z2, rev2(z3, z4)), rev2(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4)))) REV1(z0, cons(z1, cons(z2, cons(y2, y3)))) -> c5(REV1(z1, cons(z2, cons(y2, y3)))) S tuples: REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3)))) REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil)))) REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4))))) REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, cons(rev1(z2, rev2(z3, z4)), rev2(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4)))) REV1(z0, cons(z1, cons(z2, cons(y2, y3)))) -> c5(REV1(z1, cons(z2, cons(y2, y3)))) K tuples:none Defined Rule Symbols: rev2_2, rev_1, rev1_2 Defined Pair Symbols: REV2_2, REV_1, REV1_2 Compound Symbols: c7_2, c1_1, c2_1, c5_1