MAYBE proof of input_sBim049yQJ.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxWeightedTrs (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedTrs (11) CompletionProof [UPPER BOUND(ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) CompletionProof [UPPER BOUND(ID), 0 ms] (16) CpxTypedWeightedCompleteTrs (17) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxTypedWeightedCompleteTrs (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 851 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 79 ms] (30) CpxRNTS (31) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (32) CdtProblem (33) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (42) 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: +(1, x) -> +(+(0, 1), x) +(0, x) -> 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: +'(1', x) -> +'(+'(0', 1'), x) +'(0', x) -> 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: +(1, x) -> +(+(0, 1), x) +(0, x) -> 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: +(1, x) -> +(+(0, 1), x) [1] +(0, x) -> x [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: + => plus ---------------------------------------- (8) 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: plus(1, x) -> plus(plus(0, 1), x) [1] plus(0, x) -> x [1] Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: plus(1, x) -> plus(plus(0, 1), x) [1] plus(0, x) -> x [1] The TRS has the following type information: plus :: 1:0 -> 1:0 -> 1:0 1 :: 1:0 0 :: 1:0 Rewrite Strategy: INNERMOST ---------------------------------------- (11) 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: none ---------------------------------------- (12) 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: plus(1, x) -> plus(plus(0, 1), x) [1] plus(0, x) -> x [1] The TRS has the following type information: plus :: 1:0 -> 1:0 -> 1:0 1 :: 1:0 0 :: 1:0 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: 1 => 1 0 => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0 plus(z, z') -{ 1 }-> plus(plus(0, 1), x) :|: z' = x, z = 1, x >= 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (15) 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: plus_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (16) 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: plus(1, x) -> plus(plus(0, 1), x) [1] plus(0, x) -> x [1] The TRS has the following type information: plus :: 1:0 -> 1:0 -> 1:0 1 :: 1:0 0 :: 1:0 Rewrite Strategy: INNERMOST ---------------------------------------- (17) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (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: plus(1, x) -> plus(1, x) [2] plus(0, x) -> x [1] The TRS has the following type information: plus :: 1:0 -> 1:0 -> 1:0 1 :: 1:0 0 :: 1:0 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: 1 => 1 0 => 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0 plus(z, z') -{ 2 }-> plus(1, x) :|: z' = x, z = 1, x >= 0 ---------------------------------------- (21) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> plus(1, z') :|: z = 1, z' >= 0 ---------------------------------------- (23) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { plus } ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> plus(1, z') :|: z = 1, z' >= 0 Function symbols to be analyzed: {plus} ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> plus(1, z') :|: z = 1, z' >= 0 Function symbols to be analyzed: {plus} ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> plus(1, z') :|: z = 1, z' >= 0 Function symbols to be analyzed: {plus} Previous analysis results are: plus: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> plus(1, z') :|: z = 1, z' >= 0 Function symbols to be analyzed: {plus} Previous analysis results are: plus: runtime: INF, size: O(n^1) [z'] ---------------------------------------- (31) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: +(1, z0) -> +(+(0, 1), z0) +(0, z0) -> z0 Tuples: +'(1, z0) -> c(+'(+(0, 1), z0), +'(0, 1)) +'(0, z0) -> c1 S tuples: +'(1, z0) -> c(+'(+(0, 1), z0), +'(0, 1)) +'(0, z0) -> c1 K tuples:none Defined Rule Symbols: +_2 Defined Pair Symbols: +'_2 Compound Symbols: c_2, c1 ---------------------------------------- (33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: +'(0, z0) -> c1 ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: +(1, z0) -> +(+(0, 1), z0) +(0, z0) -> z0 Tuples: +'(1, z0) -> c(+'(+(0, 1), z0), +'(0, 1)) S tuples: +'(1, z0) -> c(+'(+(0, 1), z0), +'(0, 1)) K tuples:none Defined Rule Symbols: +_2 Defined Pair Symbols: +'_2 Compound Symbols: c_2 ---------------------------------------- (35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: +(1, z0) -> +(+(0, 1), z0) +(0, z0) -> z0 Tuples: +'(1, z0) -> c(+'(+(0, 1), z0)) S tuples: +'(1, z0) -> c(+'(+(0, 1), z0)) K tuples:none Defined Rule Symbols: +_2 Defined Pair Symbols: +'_2 Compound Symbols: c_1 ---------------------------------------- (37) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: +(1, z0) -> +(+(0, 1), z0) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: +(0, z0) -> z0 Tuples: +'(1, z0) -> c(+'(+(0, 1), z0)) S tuples: +'(1, z0) -> c(+'(+(0, 1), z0)) K tuples:none Defined Rule Symbols: +_2 Defined Pair Symbols: +'_2 Compound Symbols: c_1 ---------------------------------------- (39) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace +'(1, z0) -> c(+'(+(0, 1), z0)) by +'(1, x0) -> c(+'(1, x0)) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: +(0, z0) -> z0 Tuples: +'(1, x0) -> c(+'(1, x0)) S tuples: +'(1, x0) -> c(+'(1, x0)) K tuples:none Defined Rule Symbols: +_2 Defined Pair Symbols: +'_2 Compound Symbols: c_1 ---------------------------------------- (41) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: +(0, z0) -> z0 ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: +'(1, x0) -> c(+'(1, x0)) S tuples: +'(1, x0) -> c(+'(1, x0)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: +'_2 Compound Symbols: c_1