MAYBE proof of input_4DX32nrcGy.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 7 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), 1639 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 204 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) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (44) 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: *(X, +(Y, 1)) -> +(*(X, +(Y, *(1, 0))), X) *(X, 1) -> X *(X, 0) -> X *(X, 0) -> 0 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: *'(X, +'(Y, 1')) -> +'(*'(X, +'(Y, *'(1', 0'))), X) *'(X, 1') -> X *'(X, 0') -> X *'(X, 0') -> 0' 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: *(X, +(Y, 1)) -> +(*(X, +(Y, *(1, 0))), X) *(X, 1) -> X *(X, 0) -> X *(X, 0) -> 0 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: *(X, +(Y, 1)) -> +(*(X, +(Y, *(1, 0))), X) [1] *(X, 1) -> X [1] *(X, 0) -> X [1] *(X, 0) -> 0 [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: * => times ---------------------------------------- (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: times(X, +(Y, 1)) -> +(times(X, +(Y, times(1, 0))), X) [1] times(X, 1) -> X [1] times(X, 0) -> X [1] times(X, 0) -> 0 [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: times(X, +(Y, 1)) -> +(times(X, +(Y, times(1, 0))), X) [1] times(X, 1) -> X [1] times(X, 0) -> X [1] times(X, 0) -> 0 [1] The TRS has the following type information: times :: 1:+:0 -> 1:+:0 -> 1:+:0 + :: 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: times(v0, v1) -> null_times [0] And the following fresh constants: null_times ---------------------------------------- (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: times(X, +(Y, 1)) -> +(times(X, +(Y, times(1, 0))), X) [1] times(X, 1) -> X [1] times(X, 0) -> X [1] times(X, 0) -> 0 [1] times(v0, v1) -> null_times [0] The TRS has the following type information: times :: 1:+:0:null_times -> 1:+:0:null_times -> 1:+:0:null_times + :: 1:+:0:null_times -> 1:+:0:null_times -> 1:+:0:null_times 1 :: 1:+:0:null_times 0 :: 1:+:0:null_times null_times :: 1:+:0:null_times 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 null_times => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: times(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 1 times(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 0 times(z, z') -{ 1 }-> 0 :|: X >= 0, z = X, z' = 0 times(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 times(z, z') -{ 1 }-> 1 + times(X, 1 + Y + times(1, 0)) + X :|: Y >= 0, X >= 0, z = X, z' = 1 + Y + 1 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: times_2 Due to the following rules being added: times(v0, v1) -> null_times [0] And the following fresh constants: null_times ---------------------------------------- (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: times(X, +(Y, 1)) -> +(times(X, +(Y, times(1, 0))), X) [1] times(X, 1) -> X [1] times(X, 0) -> X [1] times(X, 0) -> 0 [1] times(v0, v1) -> null_times [0] The TRS has the following type information: times :: 1:+:0:null_times -> 1:+:0:null_times -> 1:+:0:null_times + :: 1:+:0:null_times -> 1:+:0:null_times -> 1:+:0:null_times 1 :: 1:+:0:null_times 0 :: 1:+:0:null_times null_times :: 1:+:0:null_times 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: times(X, +(Y, 1)) -> +(times(X, +(Y, 1)), X) [2] times(X, +(Y, 1)) -> +(times(X, +(Y, 0)), X) [2] times(X, +(Y, 1)) -> +(times(X, +(Y, null_times)), X) [1] times(X, 1) -> X [1] times(X, 0) -> X [1] times(X, 0) -> 0 [1] times(v0, v1) -> null_times [0] The TRS has the following type information: times :: 1:+:0:null_times -> 1:+:0:null_times -> 1:+:0:null_times + :: 1:+:0:null_times -> 1:+:0:null_times -> 1:+:0:null_times 1 :: 1:+:0:null_times 0 :: 1:+:0:null_times null_times :: 1:+:0:null_times 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 null_times => 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: times(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 1 times(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 0 times(z, z') -{ 1 }-> 0 :|: X >= 0, z = X, z' = 0 times(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 times(z, z') -{ 2 }-> 1 + times(X, 1 + Y + 1) + X :|: Y >= 0, X >= 0, z = X, z' = 1 + Y + 1 times(z, z') -{ 2 }-> 1 + times(X, 1 + Y + 0) + X :|: Y >= 0, X >= 0, z = X, z' = 1 + Y + 1 times(z, z') -{ 1 }-> 1 + times(X, 1 + Y + 0) + X :|: Y >= 0, X >= 0, z = X, z' = 1 + Y + 1 ---------------------------------------- (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: times(z, z') -{ 1 }-> z :|: z >= 0, z' = 1 times(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 times(z, z') -{ 2 }-> 1 + times(z, 1 + (z' - 2) + 1) + z :|: z' - 2 >= 0, z >= 0 times(z, z') -{ 2 }-> 1 + times(z, 1 + (z' - 2) + 0) + z :|: z' - 2 >= 0, z >= 0 times(z, z') -{ 1 }-> 1 + times(z, 1 + (z' - 2) + 0) + z :|: z' - 2 >= 0, z >= 0 ---------------------------------------- (23) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { times } ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: times(z, z') -{ 1 }-> z :|: z >= 0, z' = 1 times(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 times(z, z') -{ 2 }-> 1 + times(z, 1 + (z' - 2) + 1) + z :|: z' - 2 >= 0, z >= 0 times(z, z') -{ 2 }-> 1 + times(z, 1 + (z' - 2) + 0) + z :|: z' - 2 >= 0, z >= 0 times(z, z') -{ 1 }-> 1 + times(z, 1 + (z' - 2) + 0) + z :|: z' - 2 >= 0, z >= 0 Function symbols to be analyzed: {times} ---------------------------------------- (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: times(z, z') -{ 1 }-> z :|: z >= 0, z' = 1 times(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 times(z, z') -{ 2 }-> 1 + times(z, 1 + (z' - 2) + 1) + z :|: z' - 2 >= 0, z >= 0 times(z, z') -{ 2 }-> 1 + times(z, 1 + (z' - 2) + 0) + z :|: z' - 2 >= 0, z >= 0 times(z, z') -{ 1 }-> 1 + times(z, 1 + (z' - 2) + 0) + z :|: z' - 2 >= 0, z >= 0 Function symbols to be analyzed: {times} ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: times after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: times(z, z') -{ 1 }-> z :|: z >= 0, z' = 1 times(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 times(z, z') -{ 2 }-> 1 + times(z, 1 + (z' - 2) + 1) + z :|: z' - 2 >= 0, z >= 0 times(z, z') -{ 2 }-> 1 + times(z, 1 + (z' - 2) + 0) + z :|: z' - 2 >= 0, z >= 0 times(z, z') -{ 1 }-> 1 + times(z, 1 + (z' - 2) + 0) + z :|: z' - 2 >= 0, z >= 0 Function symbols to be analyzed: {times} Previous analysis results are: times: runtime: ?, size: INF ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: times after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: times(z, z') -{ 1 }-> z :|: z >= 0, z' = 1 times(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 times(z, z') -{ 2 }-> 1 + times(z, 1 + (z' - 2) + 1) + z :|: z' - 2 >= 0, z >= 0 times(z, z') -{ 2 }-> 1 + times(z, 1 + (z' - 2) + 0) + z :|: z' - 2 >= 0, z >= 0 times(z, z') -{ 1 }-> 1 + times(z, 1 + (z' - 2) + 0) + z :|: z' - 2 >= 0, z >= 0 Function symbols to be analyzed: {times} Previous analysis results are: times: runtime: INF, size: INF ---------------------------------------- (31) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, +(z1, 1)) -> +(*(z0, +(z1, *(1, 0))), z0) *(z0, 1) -> z0 *(z0, 0) -> z0 *(z0, 0) -> 0 Tuples: *'(z0, +(z1, 1)) -> c(*'(z0, +(z1, *(1, 0))), *'(1, 0)) *'(z0, 1) -> c1 *'(z0, 0) -> c2 *'(z0, 0) -> c3 S tuples: *'(z0, +(z1, 1)) -> c(*'(z0, +(z1, *(1, 0))), *'(1, 0)) *'(z0, 1) -> c1 *'(z0, 0) -> c2 *'(z0, 0) -> c3 K tuples:none Defined Rule Symbols: *_2 Defined Pair Symbols: *'_2 Compound Symbols: c_2, c1, c2, c3 ---------------------------------------- (33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: *'(z0, 0) -> c2 *'(z0, 1) -> c1 *'(z0, 0) -> c3 ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, +(z1, 1)) -> +(*(z0, +(z1, *(1, 0))), z0) *(z0, 1) -> z0 *(z0, 0) -> z0 *(z0, 0) -> 0 Tuples: *'(z0, +(z1, 1)) -> c(*'(z0, +(z1, *(1, 0))), *'(1, 0)) S tuples: *'(z0, +(z1, 1)) -> c(*'(z0, +(z1, *(1, 0))), *'(1, 0)) 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: *(z0, +(z1, 1)) -> +(*(z0, +(z1, *(1, 0))), z0) *(z0, 1) -> z0 *(z0, 0) -> z0 *(z0, 0) -> 0 Tuples: *'(z0, +(z1, 1)) -> c(*'(z0, +(z1, *(1, 0)))) S tuples: *'(z0, +(z1, 1)) -> c(*'(z0, +(z1, *(1, 0)))) 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: *(z0, +(z1, 1)) -> +(*(z0, +(z1, *(1, 0))), z0) *(z0, 1) -> z0 ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> z0 *(z0, 0) -> 0 Tuples: *'(z0, +(z1, 1)) -> c(*'(z0, +(z1, *(1, 0)))) S tuples: *'(z0, +(z1, 1)) -> c(*'(z0, +(z1, *(1, 0)))) 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 *'(z0, +(z1, 1)) -> c(*'(z0, +(z1, *(1, 0)))) by *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 1))) *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 0))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> z0 *(z0, 0) -> 0 Tuples: *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 1))) *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 0))) S tuples: *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 1))) *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 0))) K tuples:none Defined Rule Symbols: *_2 Defined Pair Symbols: *'_2 Compound Symbols: c_1 ---------------------------------------- (41) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 0))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> z0 *(z0, 0) -> 0 Tuples: *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 1))) S tuples: *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 1))) K tuples:none Defined Rule Symbols: *_2 Defined Pair Symbols: *'_2 Compound Symbols: c_1 ---------------------------------------- (43) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: *(z0, 0) -> z0 *(z0, 0) -> 0 ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 1))) S tuples: *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 1))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: *'_2 Compound Symbols: c_1