MAYBE proof of input_guQST80GpN.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 185 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 34 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: zeros -> cons(0, zeros) tail(cons(X, XS)) -> XS 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: zeros -> cons(0', zeros) tail(cons(X, XS)) -> XS 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: zeros -> cons(0, zeros) tail(cons(X, XS)) -> XS 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: zeros -> cons(0, zeros) [1] tail(cons(X, XS)) -> XS [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: zeros -> cons(0, zeros) [1] tail(cons(X, XS)) -> XS [1] The TRS has the following type information: zeros :: cons cons :: 0 -> cons -> cons 0 :: 0 tail :: cons -> cons 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: zeros tail_1 (c) The following functions are completely defined: none Due to the following rules being added: none And the following fresh constants: const ---------------------------------------- (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: zeros -> cons(0, zeros) [1] tail(cons(X, XS)) -> XS [1] The TRS has the following type information: zeros :: cons cons :: 0 -> cons -> cons 0 :: 0 tail :: cons -> cons const :: cons 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: zeros -> cons(0, zeros) [1] tail(cons(X, XS)) -> XS [1] The TRS has the following type information: zeros :: cons cons :: 0 -> cons -> cons 0 :: 0 tail :: cons -> cons const :: cons 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 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: tail(z) -{ 1 }-> XS :|: z = 1 + X + XS, X >= 0, XS >= 0 zeros -{ 1 }-> 1 + 0 + zeros :|: ---------------------------------------- (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: tail(z) -{ 1 }-> XS :|: z = 1 + X + XS, X >= 0, XS >= 0 zeros -{ 1 }-> 1 + 0 + zeros :|: ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { zeros } { tail } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: tail(z) -{ 1 }-> XS :|: z = 1 + X + XS, X >= 0, XS >= 0 zeros -{ 1 }-> 1 + 0 + zeros :|: Function symbols to be analyzed: {zeros}, {tail} ---------------------------------------- (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: tail(z) -{ 1 }-> XS :|: z = 1 + X + XS, X >= 0, XS >= 0 zeros -{ 1 }-> 1 + 0 + zeros :|: Function symbols to be analyzed: {zeros}, {tail} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: zeros 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: tail(z) -{ 1 }-> XS :|: z = 1 + X + XS, X >= 0, XS >= 0 zeros -{ 1 }-> 1 + 0 + zeros :|: Function symbols to be analyzed: {zeros}, {tail} Previous analysis results are: zeros: runtime: ?, size: O(1) [0] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: zeros after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: tail(z) -{ 1 }-> XS :|: z = 1 + X + XS, X >= 0, XS >= 0 zeros -{ 1 }-> 1 + 0 + zeros :|: Function symbols to be analyzed: {zeros}, {tail} Previous analysis results are: zeros: 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: tail(v0) -> null_tail [0] And the following fresh constants: null_tail ---------------------------------------- (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: zeros -> cons(0, zeros) [1] tail(cons(X, XS)) -> XS [1] tail(v0) -> null_tail [0] The TRS has the following type information: zeros :: cons:null_tail cons :: 0 -> cons:null_tail -> cons:null_tail 0 :: 0 tail :: cons:null_tail -> cons:null_tail null_tail :: cons:null_tail 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 null_tail => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: tail(z) -{ 1 }-> XS :|: z = 1 + X + XS, X >= 0, XS >= 0 tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 zeros -{ 1 }-> 1 + 0 + zeros :|: 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: zeros -> cons(0, zeros) tail(cons(z0, z1)) -> z1 Tuples: ZEROS -> c(ZEROS) TAIL(cons(z0, z1)) -> c1 S tuples: ZEROS -> c(ZEROS) TAIL(cons(z0, z1)) -> c1 K tuples:none Defined Rule Symbols: zeros, tail_1 Defined Pair Symbols: ZEROS, TAIL_1 Compound Symbols: c_1, c1 ---------------------------------------- (31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: TAIL(cons(z0, z1)) -> c1 ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: zeros -> cons(0, zeros) tail(cons(z0, z1)) -> z1 Tuples: ZEROS -> c(ZEROS) S tuples: ZEROS -> c(ZEROS) K tuples:none Defined Rule Symbols: zeros, tail_1 Defined Pair Symbols: ZEROS Compound Symbols: c_1 ---------------------------------------- (33) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: zeros -> cons(0, zeros) tail(cons(z0, z1)) -> z1 ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: ZEROS -> c(ZEROS) S tuples: ZEROS -> c(ZEROS) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: ZEROS Compound Symbols: c_1