/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxWeightedTrs (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxTypedWeightedTrs (5) CompletionProof [UPPER BOUND(ID), 0 ms] (6) CpxTypedWeightedCompleteTrs (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (10) CpxRNTS (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxRNTS (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxRNTS (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) IntTrsBoundProof [UPPER BOUND(ID), 466 ms] (18) CpxRNTS (19) IntTrsBoundProof [UPPER BOUND(ID), 157 ms] (20) CpxRNTS (21) FinalProof [FINISHED, 0 ms] (22) BOUNDS(1, n^1) (23) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (24) TRS for Loop Detection (25) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (26) BEST (27) proven lower bound (28) LowerBoundPropagationProof [FINISHED, 0 ms] (29) BOUNDS(n^1, INF) (30) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: average(s(x), y) -> average(x, s(y)) average(x, s(s(s(y)))) -> s(average(s(x), y)) average(0, 0) -> 0 average(0, s(0)) -> 0 average(0, s(s(0))) -> s(0) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, 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: average(s(x), y) -> average(x, s(y)) [1] average(x, s(s(s(y)))) -> s(average(s(x), y)) [1] average(0, 0) -> 0 [1] average(0, s(0)) -> 0 [1] average(0, s(s(0))) -> s(0) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (4) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: average(s(x), y) -> average(x, s(y)) [1] average(x, s(s(s(y)))) -> s(average(s(x), y)) [1] average(0, 0) -> 0 [1] average(0, s(0)) -> 0 [1] average(0, s(s(0))) -> s(0) [1] The TRS has the following type information: average :: s:0 -> s:0 -> s:0 s :: s:0 -> s:0 0 :: s:0 Rewrite Strategy: INNERMOST ---------------------------------------- (5) 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: average_2 (c) The following functions are completely defined: none Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (6) 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: average(s(x), y) -> average(x, s(y)) [1] average(x, s(s(s(y)))) -> s(average(s(x), y)) [1] average(0, 0) -> 0 [1] average(0, s(0)) -> 0 [1] average(0, s(s(0))) -> s(0) [1] The TRS has the following type information: average :: s:0 -> s:0 -> s:0 s :: s:0 -> s:0 0 :: s:0 Rewrite Strategy: INNERMOST ---------------------------------------- (7) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (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: average(s(x), y) -> average(x, s(y)) [1] average(x, s(s(s(y)))) -> s(average(s(x), y)) [1] average(0, 0) -> 0 [1] average(0, s(0)) -> 0 [1] average(0, s(s(0))) -> s(0) [1] The TRS has the following type information: average :: s:0 -> s:0 -> s:0 s :: s:0 -> s:0 0 :: s:0 Rewrite Strategy: INNERMOST ---------------------------------------- (9) 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 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: average(z, z') -{ 1 }-> average(x, 1 + y) :|: x >= 0, y >= 0, z = 1 + x, z' = y average(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 average(z, z') -{ 1 }-> 0 :|: z' = 1 + 0, z = 0 average(z, z') -{ 1 }-> 1 + average(1 + x, y) :|: x >= 0, y >= 0, z' = 1 + (1 + (1 + y)), z = x average(z, z') -{ 1 }-> 1 + 0 :|: z' = 1 + (1 + 0), z = 0 ---------------------------------------- (11) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: average(z, z') -{ 1 }-> average(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0 average(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 average(z, z') -{ 1 }-> 0 :|: z' = 1 + 0, z = 0 average(z, z') -{ 1 }-> 1 + average(1 + z, z' - 3) :|: z >= 0, z' - 3 >= 0 average(z, z') -{ 1 }-> 1 + 0 :|: z' = 1 + (1 + 0), z = 0 ---------------------------------------- (13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { average } ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: average(z, z') -{ 1 }-> average(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0 average(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 average(z, z') -{ 1 }-> 0 :|: z' = 1 + 0, z = 0 average(z, z') -{ 1 }-> 1 + average(1 + z, z' - 3) :|: z >= 0, z' - 3 >= 0 average(z, z') -{ 1 }-> 1 + 0 :|: z' = 1 + (1 + 0), z = 0 Function symbols to be analyzed: {average} ---------------------------------------- (15) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: average(z, z') -{ 1 }-> average(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0 average(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 average(z, z') -{ 1 }-> 0 :|: z' = 1 + 0, z = 0 average(z, z') -{ 1 }-> 1 + average(1 + z, z' - 3) :|: z >= 0, z' - 3 >= 0 average(z, z') -{ 1 }-> 1 + 0 :|: z' = 1 + (1 + 0), z = 0 Function symbols to be analyzed: {average} ---------------------------------------- (17) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: average after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: average(z, z') -{ 1 }-> average(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0 average(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 average(z, z') -{ 1 }-> 0 :|: z' = 1 + 0, z = 0 average(z, z') -{ 1 }-> 1 + average(1 + z, z' - 3) :|: z >= 0, z' - 3 >= 0 average(z, z') -{ 1 }-> 1 + 0 :|: z' = 1 + (1 + 0), z = 0 Function symbols to be analyzed: {average} Previous analysis results are: average: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: average after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + 2*z + z' ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: average(z, z') -{ 1 }-> average(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0 average(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 average(z, z') -{ 1 }-> 0 :|: z' = 1 + 0, z = 0 average(z, z') -{ 1 }-> 1 + average(1 + z, z' - 3) :|: z >= 0, z' - 3 >= 0 average(z, z') -{ 1 }-> 1 + 0 :|: z' = 1 + (1 + 0), z = 0 Function symbols to be analyzed: Previous analysis results are: average: runtime: O(n^1) [3 + 2*z + z'], size: O(n^1) [z + z'] ---------------------------------------- (21) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (22) BOUNDS(1, n^1) ---------------------------------------- (23) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (24) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: average(s(x), y) -> average(x, s(y)) average(x, s(s(s(y)))) -> s(average(s(x), y)) average(0, 0) -> 0 average(0, s(0)) -> 0 average(0, s(s(0))) -> s(0) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (25) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence average(s(x), y) ->^+ average(x, s(y)) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [x / s(x)]. The result substitution is [y / s(y)]. ---------------------------------------- (26) Complex Obligation (BEST) ---------------------------------------- (27) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: average(s(x), y) -> average(x, s(y)) average(x, s(s(s(y)))) -> s(average(s(x), y)) average(0, 0) -> 0 average(0, s(0)) -> 0 average(0, s(s(0))) -> s(0) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (28) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (29) BOUNDS(n^1, INF) ---------------------------------------- (30) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: average(s(x), y) -> average(x, s(y)) average(x, s(s(s(y)))) -> s(average(s(x), y)) average(0, 0) -> 0 average(0, s(0)) -> 0 average(0, s(s(0))) -> s(0) S is empty. Rewrite Strategy: INNERMOST