KILLED proof of input_lauZLYpchv.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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 [ComplexityIfPolyImplication, 0 ms] (24) CdtProblem (25) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtForwardInstantiationProof [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: ack(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil)) ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs)) ack(Nil, n) -> Cons(Cons(Nil, Nil), n) goal(m, n) -> ack(m, n) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: ack(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil)) ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs)) ack(Nil, n) -> Cons(Cons(Nil, Nil), n) goal(m, n) -> ack(m, n) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: ack(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil)) ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs)) ack(Nil, n) -> Cons(Cons(Nil, Nil), n) goal(m, n) -> ack(m, n) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: ack(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil)) [1] ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs)) [1] ack(Nil, n) -> Cons(Cons(Nil, Nil), n) [1] goal(m, n) -> ack(m, n) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: ack(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil)) [1] ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs)) [1] ack(Nil, n) -> Cons(Cons(Nil, Nil), n) [1] goal(m, n) -> ack(m, n) [1] The TRS has the following type information: ack :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: goal_2 (c) The following functions are completely defined: ack_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: ack(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil)) [1] ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs)) [1] ack(Nil, n) -> Cons(Cons(Nil, Nil), n) [1] goal(m, n) -> ack(m, n) [1] The TRS has the following type information: ack :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: ack(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil)) [1] ack(Cons(x', xs'), Cons(x, Nil)) -> ack(xs', ack(xs', Cons(Nil, Nil))) [2] ack(Cons(x', xs'), Cons(x, Cons(x1, xs1))) -> ack(xs', ack(xs', ack(Cons(x', xs'), xs1))) [2] ack(Nil, n) -> Cons(Cons(Nil, Nil), n) [1] goal(m, n) -> ack(m, n) [1] The TRS has the following type information: ack :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil 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 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: ack(z, z') -{ 1 }-> ack(xs, 1 + 0 + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 ack(z, z') -{ 2 }-> ack(xs', ack(xs', ack(1 + x' + xs', xs1))) :|: x1 >= 0, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x + (1 + x1 + xs1), xs1 >= 0, z = 1 + x' + xs' ack(z, z') -{ 2 }-> ack(xs', ack(xs', 1 + 0 + 0)) :|: x' >= 0, xs' >= 0, x >= 0, z' = 1 + x + 0, z = 1 + x' + xs' ack(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + n :|: n >= 0, z' = n, z = 0 goal(z, z') -{ 1 }-> ack(m, n) :|: z = m, n >= 0, z' = n, m >= 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: ack(z, z') -{ 1 }-> ack(xs, 1 + 0 + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 ack(z, z') -{ 2 }-> ack(xs', ack(xs', ack(1 + x' + xs', xs1))) :|: x1 >= 0, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x + (1 + x1 + xs1), xs1 >= 0, z = 1 + x' + xs' ack(z, z') -{ 2 }-> ack(xs', ack(xs', 1 + 0 + 0)) :|: x' >= 0, xs' >= 0, z' - 1 >= 0, z = 1 + x' + xs' ack(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + z' :|: z' >= 0, z = 0 goal(z, z') -{ 1 }-> ack(z, z') :|: z' >= 0, z >= 0 ---------------------------------------- (17) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: none ---------------------------------------- (18) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: ack(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil)) [1] ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs)) [1] ack(Nil, n) -> Cons(Cons(Nil, Nil), n) [1] goal(m, n) -> ack(m, n) [1] The TRS has the following type information: ack :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil 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 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: ack(z, z') -{ 1 }-> ack(xs, 1 + 0 + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 ack(z, z') -{ 1 }-> ack(xs', ack(1 + x' + xs', xs)) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' ack(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + n :|: n >= 0, z' = n, z = 0 goal(z, z') -{ 1 }-> ack(m, n) :|: z = m, n >= 0, z' = n, m >= 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: ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) goal(z0, z1) -> ack(z0, z1) Tuples: ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) ACK(Cons(z0, z1), Cons(z2, z3)) -> c1(ACK(z1, ack(Cons(z0, z1), z3)), ACK(Cons(z0, z1), z3)) ACK(Nil, z0) -> c2 GOAL(z0, z1) -> c3(ACK(z0, z1)) S tuples: ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) ACK(Cons(z0, z1), Cons(z2, z3)) -> c1(ACK(z1, ack(Cons(z0, z1), z3)), ACK(Cons(z0, z1), z3)) ACK(Nil, z0) -> c2 GOAL(z0, z1) -> c3(ACK(z0, z1)) K tuples:none Defined Rule Symbols: ack_2, goal_2 Defined Pair Symbols: ACK_2, GOAL_2 Compound Symbols: c_1, c1_2, c2, c3_1 ---------------------------------------- (23) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: GOAL(z0, z1) -> c3(ACK(z0, z1)) Removed 1 trailing nodes: ACK(Nil, z0) -> c2 ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) goal(z0, z1) -> ack(z0, z1) Tuples: ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) ACK(Cons(z0, z1), Cons(z2, z3)) -> c1(ACK(z1, ack(Cons(z0, z1), z3)), ACK(Cons(z0, z1), z3)) S tuples: ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) ACK(Cons(z0, z1), Cons(z2, z3)) -> c1(ACK(z1, ack(Cons(z0, z1), z3)), ACK(Cons(z0, z1), z3)) K tuples:none Defined Rule Symbols: ack_2, goal_2 Defined Pair Symbols: ACK_2 Compound Symbols: c_1, c1_2 ---------------------------------------- (25) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: goal(z0, z1) -> ack(z0, z1) ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) Tuples: ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) ACK(Cons(z0, z1), Cons(z2, z3)) -> c1(ACK(z1, ack(Cons(z0, z1), z3)), ACK(Cons(z0, z1), z3)) S tuples: ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) ACK(Cons(z0, z1), Cons(z2, z3)) -> c1(ACK(z1, ack(Cons(z0, z1), z3)), ACK(Cons(z0, z1), z3)) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c_1, c1_2 ---------------------------------------- (27) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(Cons(z0, z1), Cons(z2, z3)) -> c1(ACK(z1, ack(Cons(z0, z1), z3)), ACK(Cons(z0, z1), z3)) by ACK(Cons(z0, z1), Cons(x2, Nil)) -> c1(ACK(z1, ack(z1, Cons(Nil, Nil))), ACK(Cons(z0, z1), Nil)) ACK(Cons(z0, z1), Cons(x2, Cons(z2, z3))) -> c1(ACK(z1, ack(z1, ack(Cons(z0, z1), z3))), ACK(Cons(z0, z1), Cons(z2, z3))) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) Tuples: ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) ACK(Cons(z0, z1), Cons(x2, Nil)) -> c1(ACK(z1, ack(z1, Cons(Nil, Nil))), ACK(Cons(z0, z1), Nil)) ACK(Cons(z0, z1), Cons(x2, Cons(z2, z3))) -> c1(ACK(z1, ack(z1, ack(Cons(z0, z1), z3))), ACK(Cons(z0, z1), Cons(z2, z3))) S tuples: ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) ACK(Cons(z0, z1), Cons(x2, Nil)) -> c1(ACK(z1, ack(z1, Cons(Nil, Nil))), ACK(Cons(z0, z1), Nil)) ACK(Cons(z0, z1), Cons(x2, Cons(z2, z3))) -> c1(ACK(z1, ack(z1, ack(Cons(z0, z1), z3))), ACK(Cons(z0, z1), Cons(z2, z3))) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c_1, c1_2 ---------------------------------------- (29) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(Cons(z0, z1), Cons(x2, Nil)) -> c1(ACK(z1, ack(z1, Cons(Nil, Nil))), ACK(Cons(z0, z1), Nil)) by ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil)) ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Nil, Cons(Cons(Nil, Nil), Cons(Nil, Nil))), ACK(Cons(x0, Nil), Nil)) ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil)) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) Tuples: ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(z2, z3))) -> c1(ACK(z1, ack(z1, ack(Cons(z0, z1), z3))), ACK(Cons(z0, z1), Cons(z2, z3))) ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil)) ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Nil, Cons(Cons(Nil, Nil), Cons(Nil, Nil))), ACK(Cons(x0, Nil), Nil)) ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil)) S tuples: ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(z2, z3))) -> c1(ACK(z1, ack(z1, ack(Cons(z0, z1), z3))), ACK(Cons(z0, z1), Cons(z2, z3))) ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil)) ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Nil, Cons(Cons(Nil, Nil), Cons(Nil, Nil))), ACK(Cons(x0, Nil), Nil)) ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil)) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c_1, c1_2, c1_1 ---------------------------------------- (31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) Tuples: ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(z2, z3))) -> c1(ACK(z1, ack(z1, ack(Cons(z0, z1), z3))), ACK(Cons(z0, z1), Cons(z2, z3))) ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil)) ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil)) ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil)) S tuples: ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(z2, z3))) -> c1(ACK(z1, ack(z1, ack(Cons(z0, z1), z3))), ACK(Cons(z0, z1), Cons(z2, z3))) ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil)) ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil)) ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil)) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c_1, c1_2, c1_1 ---------------------------------------- (33) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(Cons(z0, z1), Cons(x2, Cons(z2, z3))) -> c1(ACK(z1, ack(z1, ack(Cons(z0, z1), z3))), ACK(Cons(z0, z1), Cons(z2, z3))) by ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Nil, Cons(Cons(Nil, Nil), ack(Cons(x0, Nil), x4))), ACK(Cons(x0, Nil), Cons(x3, x4))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) Tuples: ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil)) ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil)) ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil)) ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Nil, Cons(Cons(Nil, Nil), ack(Cons(x0, Nil), x4))), ACK(Cons(x0, Nil), Cons(x3, x4))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) S tuples: ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil)) ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil)) ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil)) ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Nil, Cons(Cons(Nil, Nil), ack(Cons(x0, Nil), x4))), ACK(Cons(x0, Nil), Cons(x3, x4))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c_1, c1_2, c1_1 ---------------------------------------- (35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) Tuples: ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil)) ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil)) ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil)) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4))) S tuples: ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil)) ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil)) ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil)) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4))) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c_1, c1_2, c1_1 ---------------------------------------- (37) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil)) by ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) Tuples: ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil)) ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil)) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) S tuples: ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil)) ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil)) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c_1, c1_1, c1_2 ---------------------------------------- (39) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) by ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y0, Nil)), Nil) -> c(ACK(Cons(y0, Nil), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) Tuples: ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil)) ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil)) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y0, Nil)), Nil) -> c(ACK(Cons(y0, Nil), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil))) S tuples: ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil)) ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil)) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y0, Nil)), Nil) -> c(ACK(Cons(y0, Nil), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c1_1, c1_2, c_1 ---------------------------------------- (41) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil)) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) Tuples: ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil)) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y0, Nil)), Nil) -> c(ACK(Cons(y0, Nil), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil))) S tuples: ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil)) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y0, Nil)), Nil) -> c(ACK(Cons(y0, Nil), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c1_1, c1_2, c_1 ---------------------------------------- (43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil)) by ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil)) ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil)) ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil)) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) Tuples: ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y0, Nil)), Nil) -> c(ACK(Cons(y0, Nil), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil)) ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil)) ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil)) S tuples: ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y0, Nil)), Nil) -> c(ACK(Cons(y0, Nil), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil)) ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil)) ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil)) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c1_2, c1_1, c_1 ---------------------------------------- (45) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ACK(Cons(z0, Cons(y0, Nil)), Nil) -> c(ACK(Cons(y0, Nil), Cons(Nil, Nil))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) Tuples: ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil)) ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil)) ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil)) S tuples: ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil)) ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil)) ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil)) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c1_2, c1_1, c_1 ---------------------------------------- (47) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4))) by ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil)))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5))))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3)))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) Tuples: ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil)) ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil)) ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil)) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil)))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5))))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3)))) S tuples: ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil)) ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil)) ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil)) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil)))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5))))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3)))) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c1_2, c_1, c1_1 ---------------------------------------- (49) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil))) by ACK(Cons(z0, Cons(z1, Cons(y1, y2))), Nil) -> c(ACK(Cons(z1, Cons(y1, y2)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(z1, Cons(y1, Nil))), Nil) -> c(ACK(Cons(z1, Cons(y1, Nil)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Nil) -> c(ACK(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(Nil, Nil))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) Tuples: ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil)) ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil)) ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil)) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil)))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5))))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3)))) ACK(Cons(z0, Cons(z1, Cons(y1, Nil))), Nil) -> c(ACK(Cons(z1, Cons(y1, Nil)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Nil) -> c(ACK(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(Nil, Nil))) S tuples: ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil)) ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil)) ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil)) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil)))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5))))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3)))) ACK(Cons(z0, Cons(z1, Cons(y1, Nil))), Nil) -> c(ACK(Cons(z1, Cons(y1, Nil)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Nil) -> c(ACK(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c1_2, c_1, c1_1 ---------------------------------------- (51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil)) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) Tuples: ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil)) ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil)) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil)))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5))))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3)))) ACK(Cons(z0, Cons(z1, Cons(y1, Nil))), Nil) -> c(ACK(Cons(z1, Cons(y1, Nil)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Nil) -> c(ACK(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(Nil, Nil))) S tuples: ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil)) ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil)) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil)))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5))))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3)))) ACK(Cons(z0, Cons(z1, Cons(y1, Nil))), Nil) -> c(ACK(Cons(z1, Cons(y1, Nil)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Nil) -> c(ACK(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c1_2, c_1, c1_1 ---------------------------------------- (53) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil)) by ACK(Cons(z0, Cons(z1, Cons(y2, y3))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, y3))), Nil)) ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Nil)) ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Nil)) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) Tuples: ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil)) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil)))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5))))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3)))) ACK(Cons(z0, Cons(z1, Cons(y1, Nil))), Nil) -> c(ACK(Cons(z1, Cons(y1, Nil)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Nil) -> c(ACK(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(Nil, Nil))) ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Nil)) ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Nil)) S tuples: ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil)) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil)))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5))))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3)))) ACK(Cons(z0, Cons(z1, Cons(y1, Nil))), Nil) -> c(ACK(Cons(z1, Cons(y1, Nil)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Nil) -> c(ACK(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(Nil, Nil))) ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Nil)) ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Nil)) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c1_2, c_1, c1_1 ---------------------------------------- (55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3)))) by ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(z3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(z3, Nil)))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(z3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(z3, Cons(y4, y5))))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(z3, Cons(y3, Nil))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(z3, Cons(y3, Nil))))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(z3, Cons(y3, Cons(y4, y5)))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(z3, Cons(y3, Cons(y4, y5)))))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) Tuples: ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil)) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil)))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5))))) ACK(Cons(z0, Cons(z1, Cons(y1, Nil))), Nil) -> c(ACK(Cons(z1, Cons(y1, Nil)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Nil) -> c(ACK(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(Nil, Nil))) ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Nil)) ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Nil)) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(z3, Cons(y3, Nil))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(z3, Cons(y3, Nil))))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(z3, Cons(y3, Cons(y4, y5)))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(z3, Cons(y3, Cons(y4, y5)))))) S tuples: ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil))) ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil)) ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil)) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil)))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5))))) ACK(Cons(z0, Cons(z1, Cons(y1, Nil))), Nil) -> c(ACK(Cons(z1, Cons(y1, Nil)), Cons(Nil, Nil))) ACK(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Nil) -> c(ACK(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(Nil, Nil))) ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Nil)) ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Nil)) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(z3, Cons(y3, Nil))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(z3, Cons(y3, Nil))))) ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(z3, Cons(y3, Cons(y4, y5)))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(z3, Cons(y3, Cons(y4, y5)))))) K tuples:none Defined Rule Symbols: ack_2 Defined Pair Symbols: ACK_2 Compound Symbols: c1_2, c_1, c1_1