KILLED proof of input_034lQ4nNXx.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (6) CdtProblem (7) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CdtProblem (9) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CdtProblem (11) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CdtProblem (13) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CdtProblem (15) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (16) CpxWeightedTrs (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxTypedWeightedTrs (19) CompletionProof [UPPER BOUND(ID), 0 ms] (20) CpxTypedWeightedCompleteTrs (21) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) CompletionProof [UPPER BOUND(ID), 0 ms] (24) CpxTypedWeightedCompleteTrs (25) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CpxTypedWeightedCompleteTrs (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CpxRNTS ---------------------------------------- (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: f(x) -> f(f(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: f(x) -> f(f(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: f(x) -> f(f(x)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: f(z0) -> f(f(z0)) Tuples: F(z0) -> c(F(f(z0)), F(z0)) S tuples: F(z0) -> c(F(f(z0)), F(z0)) K tuples:none Defined Rule Symbols: f_1 Defined Pair Symbols: F_1 Compound Symbols: c_2 ---------------------------------------- (7) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(z0) -> c(F(f(z0)), F(z0)) by F(z0) -> c(F(f(f(z0))), F(z0)) ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: f(z0) -> f(f(z0)) Tuples: F(z0) -> c(F(f(f(z0))), F(z0)) S tuples: F(z0) -> c(F(f(f(z0))), F(z0)) K tuples:none Defined Rule Symbols: f_1 Defined Pair Symbols: F_1 Compound Symbols: c_2 ---------------------------------------- (9) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(z0) -> c(F(f(f(z0))), F(z0)) by F(x0) -> c(F(f(f(f(x0)))), F(x0)) ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: f(z0) -> f(f(z0)) Tuples: F(x0) -> c(F(f(f(f(x0)))), F(x0)) S tuples: F(x0) -> c(F(f(f(f(x0)))), F(x0)) K tuples:none Defined Rule Symbols: f_1 Defined Pair Symbols: F_1 Compound Symbols: c_2 ---------------------------------------- (11) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(x0) -> c(F(f(f(f(x0)))), F(x0)) by F(z0) -> c(F(f(f(f(f(z0))))), F(z0)) ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: f(z0) -> f(f(z0)) Tuples: F(z0) -> c(F(f(f(f(f(z0))))), F(z0)) S tuples: F(z0) -> c(F(f(f(f(f(z0))))), F(z0)) K tuples:none Defined Rule Symbols: f_1 Defined Pair Symbols: F_1 Compound Symbols: c_2 ---------------------------------------- (13) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(z0) -> c(F(f(f(f(f(z0))))), F(z0)) by F(z0) -> c(F(f(f(f(f(f(z0)))))), F(z0)) ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules: f(z0) -> f(f(z0)) Tuples: F(z0) -> c(F(f(f(f(f(f(z0)))))), F(z0)) S tuples: F(z0) -> c(F(f(f(f(f(f(z0)))))), F(z0)) K tuples:none Defined Rule Symbols: f_1 Defined Pair Symbols: F_1 Compound Symbols: c_2 ---------------------------------------- (15) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (16) 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: f(x) -> f(f(x)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (17) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (18) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(x) -> f(f(x)) [1] The TRS has the following type information: f :: f -> f Rewrite Strategy: INNERMOST ---------------------------------------- (19) 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 ---------------------------------------- (20) 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: f(x) -> f(f(x)) [1] The TRS has the following type information: f :: f -> f const :: f Rewrite Strategy: INNERMOST ---------------------------------------- (21) 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 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> f(f(x)) :|: x >= 0, z = x Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (23) 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: f_1 Due to the following rules being added: none And the following fresh constants: const ---------------------------------------- (24) 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: f(x) -> f(f(x)) [1] The TRS has the following type information: f :: f -> f const :: f Rewrite Strategy: INNERMOST ---------------------------------------- (25) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (26) 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: f(x) -> f(f(f(x))) [2] The TRS has the following type information: f :: f -> f const :: f 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: const => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(f(f(x))) :|: x >= 0, z = x ---------------------------------------- (29) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(f(f(z))) :|: z >= 0