MAYBE proof of input_sgzwH9cmXd.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (24) 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: a(b(x)) -> b(b(a(a(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: a(b(x)) -> b(b(a(a(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: a(b(x)) -> b(b(a(a(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: a(b(x)) -> b(b(a(a(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: a(b(x)) -> b(b(a(a(x)))) [1] The TRS has the following type information: a :: b -> b b :: b -> b 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: a_1 Due to the following rules being added: a(v0) -> const [0] 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: a(b(x)) -> b(b(a(a(x)))) [1] a(v0) -> const [0] The TRS has the following type information: a :: b:const -> b:const b :: b:const -> b:const const :: b:const 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: a(b(b(x'))) -> b(b(a(b(b(a(a(x'))))))) [2] a(b(x)) -> b(b(a(const))) [1] a(v0) -> const [0] The TRS has the following type information: a :: b:const -> b:const b :: b:const -> b:const const :: b:const 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 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a(z) -{ 1 }-> 1 + (1 + a(0)) :|: x >= 0, z = 1 + x a(z) -{ 2 }-> 1 + (1 + a(1 + (1 + a(a(x'))))) :|: x' >= 0, z = 1 + (1 + 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: a(z) -{ 0 }-> 0 :|: z >= 0 a(z) -{ 1 }-> 1 + (1 + a(0)) :|: z - 1 >= 0 a(z) -{ 2 }-> 1 + (1 + a(1 + (1 + a(a(z - 2))))) :|: z - 2 >= 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: a(v0) -> null_a [0] And the following fresh constants: null_a ---------------------------------------- (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: a(b(x)) -> b(b(a(a(x)))) [1] a(v0) -> null_a [0] The TRS has the following type information: a :: b:null_a -> b:null_a b :: b:null_a -> b:null_a null_a :: b:null_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: null_a => 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a(z) -{ 1 }-> 1 + (1 + a(a(x))) :|: x >= 0, z = 1 + x 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: a(b(z0)) -> b(b(a(a(z0)))) Tuples: A(b(z0)) -> c(A(a(z0)), A(z0)) S tuples: A(b(z0)) -> c(A(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: a_1 Defined Pair Symbols: A_1 Compound Symbols: c_2 ---------------------------------------- (23) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(b(z0)) -> c(A(a(z0)), A(z0)) by A(b(b(z0))) -> c(A(b(b(a(a(z0))))), A(b(z0))) ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: a(b(z0)) -> b(b(a(a(z0)))) Tuples: A(b(b(z0))) -> c(A(b(b(a(a(z0))))), A(b(z0))) S tuples: A(b(b(z0))) -> c(A(b(b(a(a(z0))))), A(b(z0))) K tuples:none Defined Rule Symbols: a_1 Defined Pair Symbols: A_1 Compound Symbols: c_2