WORST_CASE(?,O(n^1)) proof of input_WKn5lRvwua.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). (0) CpxTRS (1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (2) CpxWeightedTrs (3) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxWeightedTrs (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxTypedWeightedTrs (7) CompletionProof [UPPER BOUND(ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxRNTS (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) IntTrsBoundProof [UPPER BOUND(ID), 238 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 87 ms] (22) CpxRNTS (23) FinalProof [FINISHED, 0 ms] (24) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: *(x, +(y, z)) -> +(*(x, y), *(x, z)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: *(x, +(y, z)) -> +(*(x, y), *(x, z)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (3) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: * => times ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: times(x, +(y, z)) -> +(times(x, y), times(x, z)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (6) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: times(x, +(y, z)) -> +(times(x, y), times(x, z)) [1] The TRS has the following type information: times :: a -> + -> + + :: + -> + -> + Rewrite Strategy: INNERMOST ---------------------------------------- (7) 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: times_2 (c) The following functions are completely defined: none Due to the following rules being added: none And the following fresh constants: const, const1 ---------------------------------------- (8) 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, z)) -> +(times(x, y), times(x, z)) [1] The TRS has the following type information: times :: a -> + -> + + :: + -> + -> + const :: + const1 :: a Rewrite Strategy: INNERMOST ---------------------------------------- (9) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (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: times(x, +(y, z)) -> +(times(x, y), times(x, z)) [1] The TRS has the following type information: times :: a -> + -> + + :: + -> + -> + const :: + const1 :: a 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: const => 0 const1 => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: times(z', z'') -{ 1 }-> 1 + times(x, y) + times(x, z) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z ---------------------------------------- (13) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: times(z', z'') -{ 1 }-> 1 + times(z', y) + times(z', z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z ---------------------------------------- (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { times } ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: times(z', z'') -{ 1 }-> 1 + times(z', y) + times(z', z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z Function symbols to be analyzed: {times} ---------------------------------------- (17) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: times(z', z'') -{ 1 }-> 1 + times(z', y) + times(z', z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z Function symbols to be analyzed: {times} ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: times after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: times(z', z'') -{ 1 }-> 1 + times(z', y) + times(z', z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z Function symbols to be analyzed: {times} Previous analysis results are: times: runtime: ?, size: O(1) [0] ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z'' ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: times(z', z'') -{ 1 }-> 1 + times(z', y) + times(z', z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z Function symbols to be analyzed: Previous analysis results are: times: runtime: O(n^1) [z''], size: O(1) [0] ---------------------------------------- (23) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (24) BOUNDS(1, n^1)