MAYBE proof of input_h6ZrPHToo2.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (6) CdtProblem (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CdtProblem (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CdtProblem (11) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (12) CpxWeightedTrs (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxTypedWeightedTrs (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), 163 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 45 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 803 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 43 ms] (36) CpxRNTS (37) CompletionProof [UPPER BOUND(ID), 0 ms] (38) CpxTypedWeightedCompleteTrs (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS ---------------------------------------- (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: from(X) -> cons(X) length -> 0 length -> s(length1) length1 -> length 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: from(X) -> cons(X) length -> 0' length -> s(length1) length1 -> length 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: from(X) -> cons(X) length -> 0 length -> s(length1) length1 -> length S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: from(z0) -> cons(z0) length -> 0 length -> s(length1) length1 -> length Tuples: FROM(z0) -> c LENGTH -> c1 LENGTH -> c2(LENGTH1) LENGTH1 -> c3(LENGTH) S tuples: FROM(z0) -> c LENGTH -> c1 LENGTH -> c2(LENGTH1) LENGTH1 -> c3(LENGTH) K tuples:none Defined Rule Symbols: from_1, length, length1 Defined Pair Symbols: FROM_1, LENGTH, LENGTH1 Compound Symbols: c, c1, c2_1, c3_1 ---------------------------------------- (7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: LENGTH -> c1 FROM(z0) -> c ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: from(z0) -> cons(z0) length -> 0 length -> s(length1) length1 -> length Tuples: LENGTH -> c2(LENGTH1) LENGTH1 -> c3(LENGTH) S tuples: LENGTH -> c2(LENGTH1) LENGTH1 -> c3(LENGTH) K tuples:none Defined Rule Symbols: from_1, length, length1 Defined Pair Symbols: LENGTH, LENGTH1 Compound Symbols: c2_1, c3_1 ---------------------------------------- (9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: from(z0) -> cons(z0) length -> 0 length -> s(length1) length1 -> length ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH -> c2(LENGTH1) LENGTH1 -> c3(LENGTH) S tuples: LENGTH -> c2(LENGTH1) LENGTH1 -> c3(LENGTH) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH, LENGTH1 Compound Symbols: c2_1, c3_1 ---------------------------------------- (11) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (12) 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: from(X) -> cons(X) [1] length -> 0 [1] length -> s(length1) [1] length1 -> length [1] Rewrite Strategy: INNERMOST ---------------------------------------- (13) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (14) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: from(X) -> cons(X) [1] length -> 0 [1] length -> s(length1) [1] length1 -> length [1] The TRS has the following type information: from :: a -> cons cons :: a -> cons length :: 0:s 0 :: 0:s s :: 0:s -> 0:s length1 :: 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (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: from_1 length length1 (c) The following functions are completely defined: none Due to the following rules being added: none And the following fresh constants: const, const1 ---------------------------------------- (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: from(X) -> cons(X) [1] length -> 0 [1] length -> s(length1) [1] length1 -> length [1] The TRS has the following type information: from :: a -> cons cons :: a -> cons length :: 0:s 0 :: 0:s s :: 0:s -> 0:s length1 :: 0:s const :: cons const1 :: a 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: from(X) -> cons(X) [1] length -> 0 [1] length -> s(length1) [1] length1 -> length [1] The TRS has the following type information: from :: a -> cons cons :: a -> cons length :: 0:s 0 :: 0:s s :: 0:s -> 0:s length1 :: 0:s const :: cons const1 :: a 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: 0 => 0 const => 0 const1 => 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X length -{ 1 }-> 0 :|: length -{ 1 }-> 1 + length1 :|: length1 -{ 1 }-> length :|: ---------------------------------------- (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: from(z) -{ 1 }-> 1 + z :|: z >= 0 length -{ 1 }-> 0 :|: length -{ 1 }-> 1 + length1 :|: length1 -{ 1 }-> length :|: ---------------------------------------- (23) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { from } { length1, length } ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: from(z) -{ 1 }-> 1 + z :|: z >= 0 length -{ 1 }-> 0 :|: length -{ 1 }-> 1 + length1 :|: length1 -{ 1 }-> length :|: Function symbols to be analyzed: {from}, {length1,length} ---------------------------------------- (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: from(z) -{ 1 }-> 1 + z :|: z >= 0 length -{ 1 }-> 0 :|: length -{ 1 }-> 1 + length1 :|: length1 -{ 1 }-> length :|: Function symbols to be analyzed: {from}, {length1,length} ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: from after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: from(z) -{ 1 }-> 1 + z :|: z >= 0 length -{ 1 }-> 0 :|: length -{ 1 }-> 1 + length1 :|: length1 -{ 1 }-> length :|: Function symbols to be analyzed: {from}, {length1,length} Previous analysis results are: from: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: from after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: from(z) -{ 1 }-> 1 + z :|: z >= 0 length -{ 1 }-> 0 :|: length -{ 1 }-> 1 + length1 :|: length1 -{ 1 }-> length :|: Function symbols to be analyzed: {length1,length} Previous analysis results are: from: runtime: O(1) [1], size: O(n^1) [1 + z] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: from(z) -{ 1 }-> 1 + z :|: z >= 0 length -{ 1 }-> 0 :|: length -{ 1 }-> 1 + length1 :|: length1 -{ 1 }-> length :|: Function symbols to be analyzed: {length1,length} Previous analysis results are: from: runtime: O(1) [1], size: O(n^1) [1 + z] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: length1 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: length after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: from(z) -{ 1 }-> 1 + z :|: z >= 0 length -{ 1 }-> 0 :|: length -{ 1 }-> 1 + length1 :|: length1 -{ 1 }-> length :|: Function symbols to be analyzed: {length1,length} Previous analysis results are: from: runtime: O(1) [1], size: O(n^1) [1 + z] length1: runtime: ?, size: INF length: runtime: ?, size: INF ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: length1 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: from(z) -{ 1 }-> 1 + z :|: z >= 0 length -{ 1 }-> 0 :|: length -{ 1 }-> 1 + length1 :|: length1 -{ 1 }-> length :|: Function symbols to be analyzed: {length1,length} Previous analysis results are: from: runtime: O(1) [1], size: O(n^1) [1 + z] length1: runtime: INF, size: INF length: runtime: ?, size: INF ---------------------------------------- (37) 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 ---------------------------------------- (38) 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: from(X) -> cons(X) [1] length -> 0 [1] length -> s(length1) [1] length1 -> length [1] The TRS has the following type information: from :: a -> cons cons :: a -> cons length :: 0:s 0 :: 0:s s :: 0:s -> 0:s length1 :: 0:s const :: cons const1 :: a Rewrite Strategy: INNERMOST ---------------------------------------- (39) 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 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X length -{ 1 }-> 0 :|: length -{ 1 }-> 1 + length1 :|: length1 -{ 1 }-> length :|: Only complete derivations are relevant for the runtime complexity.