MAYBE proof of input_y9B4STMns3.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CdtProblem (9) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CdtProblem (11) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CdtProblem (13) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CdtProblem (15) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CdtProblem (17) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CdtProblem (19) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (20) CpxWeightedTrs (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxTypedWeightedTrs (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 (31) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 124 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 55 ms] (38) CpxRNTS (39) CompletionProof [UPPER BOUND(ID), 0 ms] (40) CpxTypedWeightedCompleteTrs (41) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (42) 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(c) c -> b 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(c) c -> b 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(c) c -> b 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(c) c -> b Tuples: F(z0) -> c1(F(c), C) C -> c2 S tuples: F(z0) -> c1(F(c), C) C -> c2 K tuples:none Defined Rule Symbols: f_1, c Defined Pair Symbols: F_1, C Compound Symbols: c1_2, c2 ---------------------------------------- (7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: C -> c2 ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: f(z0) -> f(c) c -> b Tuples: F(z0) -> c1(F(c), C) S tuples: F(z0) -> c1(F(c), C) K tuples:none Defined Rule Symbols: f_1, c Defined Pair Symbols: F_1 Compound Symbols: c1_2 ---------------------------------------- (9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: f(z0) -> f(c) c -> b Tuples: F(z0) -> c1(F(c)) S tuples: F(z0) -> c1(F(c)) K tuples:none Defined Rule Symbols: f_1, c Defined Pair Symbols: F_1 Compound Symbols: c1_1 ---------------------------------------- (11) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: f(z0) -> f(c) ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: c -> b Tuples: F(z0) -> c1(F(c)) S tuples: F(z0) -> c1(F(c)) K tuples:none Defined Rule Symbols: c Defined Pair Symbols: F_1 Compound Symbols: c1_1 ---------------------------------------- (13) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(z0) -> c1(F(c)) by F(x0) -> c1(F(b)) ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules: c -> b Tuples: F(x0) -> c1(F(b)) S tuples: F(x0) -> c1(F(b)) K tuples:none Defined Rule Symbols: c Defined Pair Symbols: F_1 Compound Symbols: c1_1 ---------------------------------------- (15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: c -> b ---------------------------------------- (16) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(x0) -> c1(F(b)) S tuples: F(x0) -> c1(F(b)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: F_1 Compound Symbols: c1_1 ---------------------------------------- (17) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace F(x0) -> c1(F(b)) by F(b) -> c1(F(b)) ---------------------------------------- (18) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(b) -> c1(F(b)) S tuples: F(b) -> c1(F(b)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: F_1 Compound Symbols: c1_1 ---------------------------------------- (19) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (20) 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(c) [1] c -> b [1] Rewrite Strategy: INNERMOST ---------------------------------------- (21) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (22) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(X) -> f(c) [1] c -> b [1] The TRS has the following type information: f :: b -> f c :: b b :: b Rewrite Strategy: INNERMOST ---------------------------------------- (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: f_1 (c) The following functions are completely defined: c 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(c) [1] c -> b [1] The TRS has the following type information: f :: b -> f c :: b b :: b 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(b) [2] c -> b [1] The TRS has the following type information: f :: b -> f c :: b b :: b 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: b => 0 const => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: c -{ 1 }-> 0 :|: f(z) -{ 2 }-> f(0) :|: 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: c -{ 1 }-> 0 :|: f(z) -{ 2 }-> f(0) :|: z >= 0 ---------------------------------------- (31) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { f } { c } ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: c -{ 1 }-> 0 :|: f(z) -{ 2 }-> f(0) :|: z >= 0 Function symbols to be analyzed: {f}, {c} ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: c -{ 1 }-> 0 :|: f(z) -{ 2 }-> f(0) :|: z >= 0 Function symbols to be analyzed: {f}, {c} ---------------------------------------- (35) 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 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: c -{ 1 }-> 0 :|: f(z) -{ 2 }-> f(0) :|: z >= 0 Function symbols to be analyzed: {f}, {c} Previous analysis results are: f: runtime: ?, size: O(1) [0] ---------------------------------------- (37) 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: ? ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: c -{ 1 }-> 0 :|: f(z) -{ 2 }-> f(0) :|: z >= 0 Function symbols to be analyzed: {f}, {c} Previous analysis results are: f: runtime: INF, size: O(1) [0] ---------------------------------------- (39) 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 ---------------------------------------- (40) 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(c) [1] c -> b [1] The TRS has the following type information: f :: b -> f c :: b b :: b const :: f Rewrite Strategy: INNERMOST ---------------------------------------- (41) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: b => 0 const => 0 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: c -{ 1 }-> 0 :|: f(z) -{ 1 }-> f(c) :|: X >= 0, z = X Only complete derivations are relevant for the runtime complexity.