MAYBE proof of input_gfq2UyOWZv.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CdtProblem (9) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (10) CpxWeightedTrs (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedTrs (13) CompletionProof [UPPER BOUND(ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) CompletionProof [UPPER BOUND(ID), 0 ms] (18) CpxTypedWeightedCompleteTrs (19) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxTypedWeightedCompleteTrs (21) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CpxRNTS (25) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 276 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 66 ms] (32) 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) -> g(h(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) -> g(h(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) -> g(h(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) -> g(h(f(z0))) Tuples: F(z0) -> c(F(z0)) S tuples: F(z0) -> c(F(z0)) K tuples:none Defined Rule Symbols: f_1 Defined Pair Symbols: F_1 Compound Symbols: c_1 ---------------------------------------- (7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: f(z0) -> g(h(f(z0))) ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(z0) -> c(F(z0)) S tuples: F(z0) -> c(F(z0)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: F_1 Compound Symbols: c_1 ---------------------------------------- (9) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (10) 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) -> g(h(f(X))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (11) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(X) -> g(h(f(X))) [1] The TRS has the following type information: f :: a -> g g :: h -> g h :: g -> h Rewrite Strategy: INNERMOST ---------------------------------------- (13) 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, const1, const2 ---------------------------------------- (14) 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) -> g(h(f(X))) [1] The TRS has the following type information: f :: a -> g g :: h -> g h :: g -> h const :: g const1 :: a const2 :: h Rewrite Strategy: INNERMOST ---------------------------------------- (15) 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 const2 => 0 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> 1 + (1 + f(X)) :|: X >= 0, z = X Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (17) 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: f_1 (c) The following functions are completely defined: none Due to the following rules being added: none And the following fresh constants: const, const1, const2 ---------------------------------------- (18) 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) -> g(h(f(X))) [1] The TRS has the following type information: f :: a -> g g :: h -> g h :: g -> h const :: g const1 :: a const2 :: h Rewrite Strategy: INNERMOST ---------------------------------------- (19) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (20) 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) -> g(h(f(X))) [1] The TRS has the following type information: f :: a -> g g :: h -> g h :: g -> h const :: g const1 :: a const2 :: h 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 const1 => 0 const2 => 0 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> 1 + (1 + f(X)) :|: X >= 0, z = X ---------------------------------------- (23) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> 1 + (1 + f(z)) :|: z >= 0 ---------------------------------------- (25) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { f } ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> 1 + (1 + f(z)) :|: z >= 0 Function symbols to be analyzed: {f} ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> 1 + (1 + f(z)) :|: z >= 0 Function symbols to be analyzed: {f} ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> 1 + (1 + f(z)) :|: z >= 0 Function symbols to be analyzed: {f} Previous analysis results are: f: runtime: ?, size: O(1) [0] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> 1 + (1 + f(z)) :|: z >= 0 Function symbols to be analyzed: {f} Previous analysis results are: f: runtime: INF, size: O(1) [0]