MAYBE proof of input_VmQDz8KrjM.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 858 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 137 ms] (24) CpxRNTS (25) CompletionProof [UPPER BOUND(ID), 0 ms] (26) CpxTypedWeightedCompleteTrs (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (30) CdtProblem (31) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (38) 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: 2nd(cons1(X, cons(Y, Z))) -> Y 2nd(cons(X, X1)) -> 2nd(cons1(X, X1)) from(X) -> cons(X, from(s(X))) 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: 2nd(cons1(X, cons(Y, Z))) -> Y 2nd(cons(X, X1)) -> 2nd(cons1(X, X1)) from(X) -> cons(X, from(s(X))) 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: 2nd(cons1(X, cons(Y, Z))) -> Y 2nd(cons(X, X1)) -> 2nd(cons1(X, X1)) from(X) -> cons(X, from(s(X))) 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: 2nd(cons1(X, cons(Y, Z))) -> Y [1] 2nd(cons(X, X1)) -> 2nd(cons1(X, X1)) [1] from(X) -> cons(X, from(s(X))) [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: 2nd(cons1(X, cons(Y, Z))) -> Y [1] 2nd(cons(X, X1)) -> 2nd(cons1(X, X1)) [1] from(X) -> cons(X, from(s(X))) [1] The TRS has the following type information: 2nd :: cons:cons1 -> s cons1 :: s -> cons:cons1 -> cons:cons1 cons :: s -> cons:cons1 -> cons:cons1 from :: s -> cons:cons1 s :: s -> 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: 2nd_1 from_1 (c) The following functions are completely defined: none Due to the following rules being added: none And the following fresh constants: const, const1 ---------------------------------------- (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: 2nd(cons1(X, cons(Y, Z))) -> Y [1] 2nd(cons(X, X1)) -> 2nd(cons1(X, X1)) [1] from(X) -> cons(X, from(s(X))) [1] The TRS has the following type information: 2nd :: cons:cons1 -> s cons1 :: s -> cons:cons1 -> cons:cons1 cons :: s -> cons:cons1 -> cons:cons1 from :: s -> cons:cons1 s :: s -> s const :: s const1 :: cons:cons1 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: 2nd(cons1(X, cons(Y, Z))) -> Y [1] 2nd(cons(X, X1)) -> 2nd(cons1(X, X1)) [1] from(X) -> cons(X, from(s(X))) [1] The TRS has the following type information: 2nd :: cons:cons1 -> s cons1 :: s -> cons:cons1 -> cons:cons1 cons :: s -> cons:cons1 -> cons:cons1 from :: s -> cons:cons1 s :: s -> s const :: s const1 :: cons:cons1 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: const => 0 const1 => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: 2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0 2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1 from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: 2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0 2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1 from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { from } { 2nd } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: 2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0 2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1 from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 Function symbols to be analyzed: {from}, {2nd} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: 2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0 2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1 from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 Function symbols to be analyzed: {from}, {2nd} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: from after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: 2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0 2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1 from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 Function symbols to be analyzed: {from}, {2nd} Previous analysis results are: from: runtime: ?, size: O(1) [0] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: from after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: 2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0 2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1 from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 Function symbols to be analyzed: {from}, {2nd} Previous analysis results are: from: runtime: INF, size: O(1) [0] ---------------------------------------- (25) 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: 2nd(v0) -> null_2nd [0] And the following fresh constants: null_2nd, const ---------------------------------------- (26) 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: 2nd(cons1(X, cons(Y, Z))) -> Y [1] 2nd(cons(X, X1)) -> 2nd(cons1(X, X1)) [1] from(X) -> cons(X, from(s(X))) [1] 2nd(v0) -> null_2nd [0] The TRS has the following type information: 2nd :: cons:cons1 -> s:null_2nd cons1 :: s:null_2nd -> cons:cons1 -> cons:cons1 cons :: s:null_2nd -> cons:cons1 -> cons:cons1 from :: s:null_2nd -> cons:cons1 s :: s:null_2nd -> s:null_2nd null_2nd :: s:null_2nd const :: cons:cons1 Rewrite Strategy: INNERMOST ---------------------------------------- (27) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: null_2nd => 0 const => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: 2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0 2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1 2nd(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (29) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: 2nd(cons1(z0, cons(z1, z2))) -> z1 2nd(cons(z0, z1)) -> 2nd(cons1(z0, z1)) from(z0) -> cons(z0, from(s(z0))) Tuples: 2ND(cons1(z0, cons(z1, z2))) -> c 2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, z1))) FROM(z0) -> c2(FROM(s(z0))) S tuples: 2ND(cons1(z0, cons(z1, z2))) -> c 2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, z1))) FROM(z0) -> c2(FROM(s(z0))) K tuples:none Defined Rule Symbols: 2nd_1, from_1 Defined Pair Symbols: 2ND_1, FROM_1 Compound Symbols: c, c1_1, c2_1 ---------------------------------------- (31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: 2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, z1))) 2ND(cons1(z0, cons(z1, z2))) -> c ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: 2nd(cons1(z0, cons(z1, z2))) -> z1 2nd(cons(z0, z1)) -> 2nd(cons1(z0, z1)) from(z0) -> cons(z0, from(s(z0))) Tuples: FROM(z0) -> c2(FROM(s(z0))) S tuples: FROM(z0) -> c2(FROM(s(z0))) K tuples:none Defined Rule Symbols: 2nd_1, from_1 Defined Pair Symbols: FROM_1 Compound Symbols: c2_1 ---------------------------------------- (33) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: 2nd(cons1(z0, cons(z1, z2))) -> z1 2nd(cons(z0, z1)) -> 2nd(cons1(z0, z1)) from(z0) -> cons(z0, from(s(z0))) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(z0) -> c2(FROM(s(z0))) S tuples: FROM(z0) -> c2(FROM(s(z0))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: FROM_1 Compound Symbols: c2_1 ---------------------------------------- (35) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FROM(z0) -> c2(FROM(s(z0))) by FROM(s(x0)) -> c2(FROM(s(s(x0)))) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(x0)) -> c2(FROM(s(s(x0)))) S tuples: FROM(s(x0)) -> c2(FROM(s(s(x0)))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: FROM_1 Compound Symbols: c2_1 ---------------------------------------- (37) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FROM(s(x0)) -> c2(FROM(s(s(x0)))) by FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) S tuples: FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: FROM_1 Compound Symbols: c2_1