/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).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) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxWeightedTrs (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedTrs (11) CompletionProof [UPPER BOUND(ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 4 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 199 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 107 ms] (26) CpxRNTS (27) FinalProof [FINISHED, 0 ms] (28) BOUNDS(1, n^1) (29) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (30) TRS for Loop Detection (31) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (32) BEST (33) proven lower bound (34) LowerBoundPropagationProof [FINISHED, 0 ms] (35) BOUNDS(n^1, INF) (36) 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: +(*(x, y), *(x, z)) -> *(x, +(y, z)) +(+(x, y), z) -> +(x, +(y, z)) +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) The TRS does not nest defined symbols. Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: +(+(x, y), z) -> +(x, +(y, z)) +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u) ---------------------------------------- (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: +(*(x, y), *(x, z)) -> *(x, +(y, z)) 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: +(*(x, y), *(x, z)) -> *(x, +(y, z)) 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: +(*(x, y), *(x, z)) -> *(x, +(y, z)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: + => plus ---------------------------------------- (8) 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: plus(*(x, y), *(x, z)) -> *(x, plus(y, z)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: plus(*(x, y), *(x, z)) -> *(x, plus(y, z)) [1] The TRS has the following type information: plus :: * -> * -> * * :: a -> * -> * Rewrite Strategy: INNERMOST ---------------------------------------- (11) 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: plus_2 (c) The following functions are completely defined: none Due to the following rules being added: none And the following fresh constants: const, const1 ---------------------------------------- (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: plus(*(x, y), *(x, z)) -> *(x, plus(y, z)) [1] The TRS has the following type information: plus :: * -> * -> * * :: a -> * -> * const :: * const1 :: a Rewrite Strategy: INNERMOST ---------------------------------------- (13) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (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: plus(*(x, y), *(x, z)) -> *(x, plus(y, z)) [1] The TRS has the following type information: plus :: * -> * -> * * :: a -> * -> * const :: * const1 :: a Rewrite Strategy: INNERMOST ---------------------------------------- (15) 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 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: plus(z', z'') -{ 1 }-> 1 + x + plus(y, z) :|: z >= 0, z' = 1 + x + y, z'' = 1 + x + z, x >= 0, y >= 0 ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: plus(z', z'') -{ 1 }-> 1 + x + plus(y, z) :|: z >= 0, z' = 1 + x + y, z'' = 1 + x + z, x >= 0, y >= 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { plus } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: plus(z', z'') -{ 1 }-> 1 + x + plus(y, z) :|: z >= 0, z' = 1 + x + y, z'' = 1 + x + z, x >= 0, y >= 0 Function symbols to be analyzed: {plus} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: plus(z', z'') -{ 1 }-> 1 + x + plus(y, z) :|: z >= 0, z' = 1 + x + y, z'' = 1 + x + z, x >= 0, y >= 0 Function symbols to be analyzed: {plus} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: plus(z', z'') -{ 1 }-> 1 + x + plus(y, z) :|: z >= 0, z' = 1 + x + y, z'' = 1 + x + z, x >= 0, y >= 0 Function symbols to be analyzed: {plus} Previous analysis results are: plus: runtime: ?, size: O(1) [0] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z'' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: plus(z', z'') -{ 1 }-> 1 + x + plus(y, z) :|: z >= 0, z' = 1 + x + y, z'' = 1 + x + z, x >= 0, y >= 0 Function symbols to be analyzed: Previous analysis results are: plus: runtime: O(n^1) [z''], size: O(1) [0] ---------------------------------------- (27) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (28) BOUNDS(1, n^1) ---------------------------------------- (29) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (30) 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: +(*(x, y), *(x, z)) -> *(x, +(y, z)) +(+(x, y), z) -> +(x, +(y, z)) +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u) S is empty. Rewrite Strategy: FULL ---------------------------------------- (31) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence +(*(x1_0, y), *(x1_0, z)) ->^+ *(x1_0, +(y, z)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [y / *(x1_0, y), z / *(x1_0, z)]. The result substitution is [ ]. ---------------------------------------- (32) Complex Obligation (BEST) ---------------------------------------- (33) 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: +(*(x, y), *(x, z)) -> *(x, +(y, z)) +(+(x, y), z) -> +(x, +(y, z)) +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u) S is empty. Rewrite Strategy: FULL ---------------------------------------- (34) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (35) BOUNDS(n^1, INF) ---------------------------------------- (36) 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: +(*(x, y), *(x, z)) -> *(x, +(y, z)) +(+(x, y), z) -> +(x, +(y, z)) +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u) S is empty. Rewrite Strategy: FULL