/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] (2) CpxTRS (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, 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), 2 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), 202 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 76 ms] (24) CpxRNTS (25) FinalProof [FINISHED, 0 ms] (26) BOUNDS(1, n^1) (27) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (28) TRS for Loop Detection (29) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (30) BEST (31) proven lower bound (32) LowerBoundPropagationProof [FINISHED, 0 ms] (33) BOUNDS(n^1, INF) (34) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: plus(plus(X, Y), Z) -> plus(X, plus(Y, Z)) times(X, s(Y)) -> plus(X, times(Y, X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) The following defined symbols can occur below the 1th argument of plus: plus, times Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: plus(plus(X, Y), Z) -> plus(X, plus(Y, Z)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: times(X, s(Y)) -> plus(X, times(Y, X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) RcToIrcProof (BOTH BOUNDS(ID, ID)) Converted rc-obligation to irc-obligation. As the TRS does not nest defined symbols, we have rc = irc. ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: times(X, s(Y)) -> plus(X, times(Y, X)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) 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, s(Y)) -> plus(X, times(Y, 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: times(X, s(Y)) -> plus(X, times(Y, X)) [1] The TRS has the following type information: times :: s -> s -> plus s :: s -> s plus :: s -> plus -> plus 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: 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 ---------------------------------------- (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, s(Y)) -> plus(X, times(Y, X)) [1] The TRS has the following type information: times :: s -> s -> plus s :: s -> s plus :: s -> plus -> plus const :: plus const1 :: s 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: times(X, s(Y)) -> plus(X, times(Y, X)) [1] The TRS has the following type information: times :: s -> s -> plus s :: s -> s plus :: s -> plus -> plus const :: plus const1 :: s 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: const => 0 const1 => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: times(z, z') -{ 1 }-> 1 + X + times(Y, X) :|: Y >= 0, z' = 1 + Y, 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: times(z, z') -{ 1 }-> 1 + z + times(z' - 1, z) :|: z' - 1 >= 0, z >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { times } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: times(z, z') -{ 1 }-> 1 + z + times(z' - 1, z) :|: z' - 1 >= 0, z >= 0 Function symbols to be analyzed: {times} ---------------------------------------- (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: times(z, z') -{ 1 }-> 1 + z + times(z' - 1, z) :|: z' - 1 >= 0, z >= 0 Function symbols to be analyzed: {times} ---------------------------------------- (21) 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 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: times(z, z') -{ 1 }-> 1 + z + times(z' - 1, z) :|: z' - 1 >= 0, z >= 0 Function symbols to be analyzed: {times} Previous analysis results are: times: runtime: ?, size: O(1) [0] ---------------------------------------- (23) 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 + z' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: times(z, z') -{ 1 }-> 1 + z + times(z' - 1, z) :|: z' - 1 >= 0, z >= 0 Function symbols to be analyzed: Previous analysis results are: times: runtime: O(n^1) [z + z'], size: O(1) [0] ---------------------------------------- (25) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (26) BOUNDS(1, n^1) ---------------------------------------- (27) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (28) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: plus(plus(X, Y), Z) -> plus(X, plus(Y, Z)) times(X, s(Y)) -> plus(X, times(Y, X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (29) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence times(s(Y2_0), s(Y)) ->^+ plus(s(Y2_0), plus(Y, times(Y2_0, Y))) gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1]. The pumping substitution is [Y2_0 / s(Y2_0), Y / s(Y)]. The result substitution is [ ]. ---------------------------------------- (30) Complex Obligation (BEST) ---------------------------------------- (31) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: plus(plus(X, Y), Z) -> plus(X, plus(Y, Z)) times(X, s(Y)) -> plus(X, times(Y, X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (32) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (33) BOUNDS(n^1, INF) ---------------------------------------- (34) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: plus(plus(X, Y), Z) -> plus(X, plus(Y, Z)) times(X, s(Y)) -> plus(X, times(Y, X)) S is empty. Rewrite Strategy: FULL