MAYBE proof of input_nroLuiqqel.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 103 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 55 ms] (24) CpxRNTS (25) CompletionProof [UPPER BOUND(ID), 0 ms] (26) CpxTypedWeightedCompleteTrs (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (30) CdtProblem (31) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (34) 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: f(S(x)) -> x f(0) -> 0 g(x) -> g(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(S(x)) -> x f(0') -> 0' g(x) -> g(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(S(x)) -> x f(0) -> 0 g(x) -> g(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: f(S(x)) -> x [1] f(0) -> 0 [1] g(x) -> g(x) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(S(x)) -> x [1] f(0) -> 0 [1] g(x) -> g(x) [1] The TRS has the following type information: f :: S:0 -> S:0 S :: S:0 -> S:0 0 :: S:0 g :: a -> g Rewrite Strategy: INNERMOST ---------------------------------------- (9) 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 g_1 (c) The following functions are completely defined: none Due to the following rules being added: none And the following fresh constants: const, const1 ---------------------------------------- (10) 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(S(x)) -> x [1] f(0) -> 0 [1] g(x) -> g(x) [1] The TRS has the following type information: f :: S:0 -> S:0 S :: S:0 -> S:0 0 :: S:0 g :: a -> g const :: g const1 :: a Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) 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(S(x)) -> x [1] f(0) -> 0 [1] g(x) -> g(x) [1] The TRS has the following type information: f :: S:0 -> S:0 S :: S:0 -> S:0 0 :: S:0 g :: a -> g const :: g const1 :: a 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: 0 => 0 const => 0 const1 => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> x :|: x >= 0, z = 1 + x f(z) -{ 1 }-> 0 :|: z = 0 g(z) -{ 1 }-> g(x) :|: x >= 0, z = x ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 g(z) -{ 1 }-> g(z) :|: z >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { g } { f } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 g(z) -{ 1 }-> g(z) :|: z >= 0 Function symbols to be analyzed: {g}, {f} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 g(z) -{ 1 }-> g(z) :|: z >= 0 Function symbols to be analyzed: {g}, {f} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 g(z) -{ 1 }-> g(z) :|: z >= 0 Function symbols to be analyzed: {g}, {f} Previous analysis results are: g: runtime: ?, size: O(1) [0] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 g(z) -{ 1 }-> g(z) :|: z >= 0 Function symbols to be analyzed: {g}, {f} Previous analysis results are: g: runtime: INF, size: O(1) [0] ---------------------------------------- (25) 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 ---------------------------------------- (26) 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(S(x)) -> x [1] f(0) -> 0 [1] g(x) -> g(x) [1] The TRS has the following type information: f :: S:0 -> S:0 S :: S:0 -> S:0 0 :: S:0 g :: a -> g const :: g const1 :: a 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: 0 => 0 const => 0 const1 => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> x :|: x >= 0, z = 1 + x f(z) -{ 1 }-> 0 :|: z = 0 g(z) -{ 1 }-> g(x) :|: x >= 0, z = x Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (29) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: f(S(z0)) -> z0 f(0) -> 0 g(z0) -> g(z0) Tuples: F(S(z0)) -> c F(0) -> c1 G(z0) -> c2(G(z0)) S tuples: F(S(z0)) -> c F(0) -> c1 G(z0) -> c2(G(z0)) K tuples:none Defined Rule Symbols: f_1, g_1 Defined Pair Symbols: F_1, G_1 Compound Symbols: c, c1, c2_1 ---------------------------------------- (31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: F(S(z0)) -> c F(0) -> c1 ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: f(S(z0)) -> z0 f(0) -> 0 g(z0) -> g(z0) Tuples: G(z0) -> c2(G(z0)) S tuples: G(z0) -> c2(G(z0)) K tuples:none Defined Rule Symbols: f_1, g_1 Defined Pair Symbols: G_1 Compound Symbols: c2_1 ---------------------------------------- (33) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: f(S(z0)) -> z0 f(0) -> 0 g(z0) -> g(z0) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: G(z0) -> c2(G(z0)) S tuples: G(z0) -> c2(G(z0)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: G_1 Compound Symbols: c2_1