MAYBE proof of input_1tdIpNIHqd.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) CompletionProof [UPPER BOUND(ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxTypedWeightedCompleteTrs (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 502 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 78 ms] (28) CpxRNTS (29) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (30) CdtProblem (31) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (40) 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(0) -> cons(0, f(s(0))) f(s(0)) -> f(p(s(0))) p(s(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: f(0') -> cons(0', f(s(0'))) f(s(0')) -> f(p(s(0'))) p(s(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: f(0) -> cons(0, f(s(0))) f(s(0)) -> f(p(s(0))) p(s(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: f(0) -> cons(0, f(s(0))) [1] f(s(0)) -> f(p(s(0))) [1] p(s(0)) -> 0 [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(0) -> cons(0, f(s(0))) [1] f(s(0)) -> f(p(s(0))) [1] p(s(0)) -> 0 [1] The TRS has the following type information: f :: 0:s -> cons 0 :: 0:s cons :: 0:s -> cons -> cons s :: 0:s -> 0:s p :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (9) 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: f(v0) -> null_f [0] p(v0) -> null_p [0] And the following fresh constants: null_f, null_p ---------------------------------------- (10) 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(0) -> cons(0, f(s(0))) [1] f(s(0)) -> f(p(s(0))) [1] p(s(0)) -> 0 [1] f(v0) -> null_f [0] p(v0) -> null_p [0] The TRS has the following type information: f :: 0:s:null_p -> cons:null_f 0 :: 0:s:null_p cons :: 0:s:null_p -> cons:null_f -> cons:null_f s :: 0:s:null_p -> 0:s:null_p p :: 0:s:null_p -> 0:s:null_p null_f :: cons:null_f null_p :: 0:s:null_p Rewrite Strategy: INNERMOST ---------------------------------------- (11) 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 null_f => 0 null_p => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> f(p(1 + 0)) :|: z = 1 + 0 f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 f(z) -{ 1 }-> 1 + 0 + f(1 + 0) :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (13) 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: p_1 Due to the following rules being added: p(v0) -> 0 [0] And the following fresh constants: const ---------------------------------------- (14) 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(0) -> cons(0, f(s(0))) [1] f(s(0)) -> f(p(s(0))) [1] p(s(0)) -> 0 [1] p(v0) -> 0 [0] The TRS has the following type information: f :: 0:s -> cons 0 :: 0:s cons :: 0:s -> cons -> cons s :: 0:s -> 0:s p :: 0:s -> 0:s const :: cons Rewrite Strategy: INNERMOST ---------------------------------------- (15) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (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: f(0) -> cons(0, f(s(0))) [1] f(s(0)) -> f(0) [2] f(s(0)) -> f(0) [1] p(s(0)) -> 0 [1] p(v0) -> 0 [0] The TRS has the following type information: f :: 0:s -> cons 0 :: 0:s cons :: 0:s -> cons -> cons s :: 0:s -> 0:s p :: 0:s -> 0:s const :: cons Rewrite Strategy: INNERMOST ---------------------------------------- (17) 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 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(0) :|: z = 1 + 0 f(z) -{ 1 }-> f(0) :|: z = 1 + 0 f(z) -{ 1 }-> 1 + 0 + f(1 + 0) :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (19) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(0) :|: z = 1 + 0 f(z) -{ 1 }-> f(0) :|: z = 1 + 0 f(z) -{ 1 }-> 1 + 0 + f(1 + 0) :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 ---------------------------------------- (21) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { f } { p } ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(0) :|: z = 1 + 0 f(z) -{ 1 }-> f(0) :|: z = 1 + 0 f(z) -{ 1 }-> 1 + 0 + f(1 + 0) :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {f}, {p} ---------------------------------------- (23) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(0) :|: z = 1 + 0 f(z) -{ 1 }-> f(0) :|: z = 1 + 0 f(z) -{ 1 }-> 1 + 0 + f(1 + 0) :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {f}, {p} ---------------------------------------- (25) 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 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(0) :|: z = 1 + 0 f(z) -{ 1 }-> f(0) :|: z = 1 + 0 f(z) -{ 1 }-> 1 + 0 + f(1 + 0) :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {f}, {p} Previous analysis results are: f: runtime: ?, size: O(1) [0] ---------------------------------------- (27) 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: ? ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(0) :|: z = 1 + 0 f(z) -{ 1 }-> f(0) :|: z = 1 + 0 f(z) -{ 1 }-> 1 + 0 + f(1 + 0) :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {f}, {p} Previous analysis results are: f: runtime: INF, size: O(1) [0] ---------------------------------------- (29) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: f(0) -> cons(0, f(s(0))) f(s(0)) -> f(p(s(0))) p(s(0)) -> 0 Tuples: F(0) -> c(F(s(0))) F(s(0)) -> c1(F(p(s(0))), P(s(0))) P(s(0)) -> c2 S tuples: F(0) -> c(F(s(0))) F(s(0)) -> c1(F(p(s(0))), P(s(0))) P(s(0)) -> c2 K tuples:none Defined Rule Symbols: f_1, p_1 Defined Pair Symbols: F_1, P_1 Compound Symbols: c_1, c1_2, c2 ---------------------------------------- (31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: P(s(0)) -> c2 ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: f(0) -> cons(0, f(s(0))) f(s(0)) -> f(p(s(0))) p(s(0)) -> 0 Tuples: F(0) -> c(F(s(0))) F(s(0)) -> c1(F(p(s(0))), P(s(0))) S tuples: F(0) -> c(F(s(0))) F(s(0)) -> c1(F(p(s(0))), P(s(0))) K tuples:none Defined Rule Symbols: f_1, p_1 Defined Pair Symbols: F_1 Compound Symbols: c_1, c1_2 ---------------------------------------- (33) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: f(0) -> cons(0, f(s(0))) f(s(0)) -> f(p(s(0))) p(s(0)) -> 0 Tuples: F(0) -> c(F(s(0))) F(s(0)) -> c1(F(p(s(0)))) S tuples: F(0) -> c(F(s(0))) F(s(0)) -> c1(F(p(s(0)))) K tuples:none Defined Rule Symbols: f_1, p_1 Defined Pair Symbols: F_1 Compound Symbols: c_1, c1_1 ---------------------------------------- (35) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: f(0) -> cons(0, f(s(0))) f(s(0)) -> f(p(s(0))) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 Tuples: F(0) -> c(F(s(0))) F(s(0)) -> c1(F(p(s(0)))) S tuples: F(0) -> c(F(s(0))) F(s(0)) -> c1(F(p(s(0)))) K tuples:none Defined Rule Symbols: p_1 Defined Pair Symbols: F_1 Compound Symbols: c_1, c1_1 ---------------------------------------- (37) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0)) -> c1(F(p(s(0)))) by F(s(0)) -> c1(F(0)) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 Tuples: F(0) -> c(F(s(0))) F(s(0)) -> c1(F(0)) S tuples: F(0) -> c(F(s(0))) F(s(0)) -> c1(F(0)) K tuples:none Defined Rule Symbols: p_1 Defined Pair Symbols: F_1 Compound Symbols: c_1, c1_1 ---------------------------------------- (39) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: p(s(0)) -> 0 ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(0) -> c(F(s(0))) F(s(0)) -> c1(F(0)) S tuples: F(0) -> c(F(s(0))) F(s(0)) -> c1(F(0)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: F_1 Compound Symbols: c_1, c1_1