/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^2)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). (0) CpxTRS (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, 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), 1 ms] (16) CpxRNTS (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) IntTrsBoundProof [UPPER BOUND(ID), 258 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 107 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 283 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 95 ms] (28) CpxRNTS (29) FinalProof [FINISHED, 0 ms] (30) BOUNDS(1, n^2) (31) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (32) TRS for Loop Detection (33) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (34) BEST (35) proven lower bound (36) LowerBoundPropagationProof [FINISHED, 0 ms] (37) BOUNDS(n^1, INF) (38) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: sum(0) -> 0 sum(s(x)) -> +(sum(x), s(x)) +(x, 0) -> x +(x, s(y)) -> s(+(x, y)) 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^2). The TRS R consists of the following rules: sum(0) -> 0 [1] sum(s(x)) -> +(sum(x), s(x)) [1] +(x, 0) -> x [1] +(x, s(y)) -> s(+(x, y)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (3) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: + => plus ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: sum(0) -> 0 [1] sum(s(x)) -> plus(sum(x), s(x)) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [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: sum(0) -> 0 [1] sum(s(x)) -> plus(sum(x), s(x)) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] The TRS has the following type information: sum :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s 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: none (c) The following functions are completely defined: sum_1 plus_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (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: sum(0) -> 0 [1] sum(s(x)) -> plus(sum(x), s(x)) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] The TRS has the following type information: sum :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s 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: sum(0) -> 0 [1] sum(s(0)) -> plus(0, s(0)) [2] sum(s(s(x'))) -> plus(plus(sum(x'), s(x')), s(s(x'))) [2] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] The TRS has the following type information: sum :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s 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: 0 => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x sum(z) -{ 2 }-> plus(plus(sum(x'), 1 + x'), 1 + (1 + x')) :|: x' >= 0, z = 1 + (1 + x') sum(z) -{ 2 }-> plus(0, 1 + 0) :|: z = 1 + 0 sum(z) -{ 1 }-> 0 :|: z = 0 ---------------------------------------- (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: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 sum(z) -{ 2 }-> plus(plus(sum(z - 2), 1 + (z - 2)), 1 + (1 + (z - 2))) :|: z - 2 >= 0 sum(z) -{ 2 }-> plus(0, 1 + 0) :|: z = 1 + 0 sum(z) -{ 1 }-> 0 :|: z = 0 ---------------------------------------- (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { plus } { sum } ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 sum(z) -{ 2 }-> plus(plus(sum(z - 2), 1 + (z - 2)), 1 + (1 + (z - 2))) :|: z - 2 >= 0 sum(z) -{ 2 }-> plus(0, 1 + 0) :|: z = 1 + 0 sum(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {plus}, {sum} ---------------------------------------- (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: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 sum(z) -{ 2 }-> plus(plus(sum(z - 2), 1 + (z - 2)), 1 + (1 + (z - 2))) :|: z - 2 >= 0 sum(z) -{ 2 }-> plus(0, 1 + 0) :|: z = 1 + 0 sum(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {plus}, {sum} ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 sum(z) -{ 2 }-> plus(plus(sum(z - 2), 1 + (z - 2)), 1 + (1 + (z - 2))) :|: z - 2 >= 0 sum(z) -{ 2 }-> plus(0, 1 + 0) :|: z = 1 + 0 sum(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {plus}, {sum} Previous analysis results are: plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (21) 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: 1 + z' ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 sum(z) -{ 2 }-> plus(plus(sum(z - 2), 1 + (z - 2)), 1 + (1 + (z - 2))) :|: z - 2 >= 0 sum(z) -{ 2 }-> plus(0, 1 + 0) :|: z = 1 + 0 sum(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {sum} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (23) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0 sum(z) -{ 4 }-> s :|: s >= 0, s <= 0 + (1 + 0), z = 1 + 0 sum(z) -{ 2 }-> plus(plus(sum(z - 2), 1 + (z - 2)), 1 + (1 + (z - 2))) :|: z - 2 >= 0 sum(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {sum} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: sum after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1 + 2*z^2 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0 sum(z) -{ 4 }-> s :|: s >= 0, s <= 0 + (1 + 0), z = 1 + 0 sum(z) -{ 2 }-> plus(plus(sum(z - 2), 1 + (z - 2)), 1 + (1 + (z - 2))) :|: z - 2 >= 0 sum(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {sum} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] sum: runtime: ?, size: O(n^2) [1 + 2*z^2] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: sum after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 5 + 3*z + 2*z^2 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0 sum(z) -{ 4 }-> s :|: s >= 0, s <= 0 + (1 + 0), z = 1 + 0 sum(z) -{ 2 }-> plus(plus(sum(z - 2), 1 + (z - 2)), 1 + (1 + (z - 2))) :|: z - 2 >= 0 sum(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] sum: runtime: O(n^2) [5 + 3*z + 2*z^2], size: O(n^2) [1 + 2*z^2] ---------------------------------------- (29) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (30) BOUNDS(1, n^2) ---------------------------------------- (31) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (32) 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^2). The TRS R consists of the following rules: sum(0) -> 0 sum(s(x)) -> +(sum(x), s(x)) +(x, 0) -> x +(x, s(y)) -> s(+(x, y)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (33) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence +(x, s(y)) ->^+ s(+(x, y)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [y / s(y)]. The result substitution is [ ]. ---------------------------------------- (34) Complex Obligation (BEST) ---------------------------------------- (35) 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^2). The TRS R consists of the following rules: sum(0) -> 0 sum(s(x)) -> +(sum(x), s(x)) +(x, 0) -> x +(x, s(y)) -> s(+(x, y)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (36) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (37) BOUNDS(n^1, INF) ---------------------------------------- (38) 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^2). The TRS R consists of the following rules: sum(0) -> 0 sum(s(x)) -> +(sum(x), s(x)) +(x, 0) -> x +(x, s(y)) -> s(+(x, y)) S is empty. Rewrite Strategy: INNERMOST