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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).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) 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), 901 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 177 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 149 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 54 ms] (28) CpxRNTS (29) FinalProof [FINISHED, 0 ms] (30) BOUNDS(1, n^1) (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^1). The TRS R consists of the following rules: -(0, y) -> 0 -(x, 0) -> x -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0) p(0) -> 0 p(s(x)) -> x 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: -(0, y) -> 0 [1] -(x, 0) -> x [1] -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0) [1] p(0) -> 0 [1] p(s(x)) -> x [1] Rewrite Strategy: INNERMOST ---------------------------------------- (3) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: - => minus ---------------------------------------- (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: minus(0, y) -> 0 [1] minus(x, 0) -> x [1] minus(x, s(y)) -> if(greater(x, s(y)), s(minus(x, p(s(y)))), 0) [1] p(0) -> 0 [1] p(s(x)) -> x [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: minus(0, y) -> 0 [1] minus(x, 0) -> x [1] minus(x, s(y)) -> if(greater(x, s(y)), s(minus(x, p(s(y)))), 0) [1] p(0) -> 0 [1] p(s(x)) -> x [1] The TRS has the following type information: minus :: 0:s:if -> 0:s:if -> 0:s:if 0 :: 0:s:if s :: 0:s:if -> 0:s:if if :: greater -> 0:s:if -> 0:s:if -> 0:s:if greater :: 0:s:if -> 0:s:if -> greater p :: 0:s:if -> 0:s:if 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: minus_2 (c) The following functions are completely defined: p_1 Due to the following rules being added: p(v0) -> 0 [0] And the following fresh constants: const ---------------------------------------- (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: minus(0, y) -> 0 [1] minus(x, 0) -> x [1] minus(x, s(y)) -> if(greater(x, s(y)), s(minus(x, p(s(y)))), 0) [1] p(0) -> 0 [1] p(s(x)) -> x [1] p(v0) -> 0 [0] The TRS has the following type information: minus :: 0:s:if -> 0:s:if -> 0:s:if 0 :: 0:s:if s :: 0:s:if -> 0:s:if if :: greater -> 0:s:if -> 0:s:if -> 0:s:if greater :: 0:s:if -> 0:s:if -> greater p :: 0:s:if -> 0:s:if const :: greater 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: minus(0, y) -> 0 [1] minus(x, 0) -> x [1] minus(x, s(y)) -> if(greater(x, s(y)), s(minus(x, y)), 0) [2] minus(x, s(y)) -> if(greater(x, s(y)), s(minus(x, 0)), 0) [1] p(0) -> 0 [1] p(s(x)) -> x [1] p(v0) -> 0 [0] The TRS has the following type information: minus :: 0:s:if -> 0:s:if -> 0:s:if 0 :: 0:s:if s :: 0:s:if -> 0:s:if if :: greater -> 0:s:if -> 0:s:if -> 0:s:if greater :: 0:s:if -> 0:s:if -> greater p :: 0:s:if -> 0:s:if const :: greater 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 const => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y minus(z, z') -{ 2 }-> 1 + (1 + x + (1 + y)) + (1 + minus(x, y)) + 0 :|: z' = 1 + y, x >= 0, y >= 0, z = x minus(z, z') -{ 1 }-> 1 + (1 + x + (1 + y)) + (1 + minus(x, 0)) + 0 :|: z' = 1 + y, x >= 0, y >= 0, z = x p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (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: minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 1 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, 0)) + 0 :|: z >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, z' - 1)) + 0 :|: z >= 0, z' - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 ---------------------------------------- (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { minus } { p } ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 1 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, 0)) + 0 :|: z >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, z' - 1)) + 0 :|: z >= 0, z' - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {minus}, {p} ---------------------------------------- (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: minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 1 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, 0)) + 0 :|: z >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, z' - 1)) + 0 :|: z >= 0, z' - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {minus}, {p} ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 3 + 2*z + z*z' + 3*z' + z'^2 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 1 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, 0)) + 0 :|: z >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, z' - 1)) + 0 :|: z >= 0, z' - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {minus}, {p} Previous analysis results are: minus: runtime: ?, size: O(n^2) [3 + 2*z + z*z' + 3*z' + z'^2] ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4 + 2*z' ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 1 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, 0)) + 0 :|: z >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, z' - 1)) + 0 :|: z >= 0, z' - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {p} Previous analysis results are: minus: runtime: O(n^1) [4 + 2*z'], size: O(n^2) [3 + 2*z + z*z' + 3*z' + z'^2] ---------------------------------------- (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: minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 4 + 2*z' }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + s) + 0 :|: s >= 0, s <= (z' - 1) * (z' - 1) + 3 * (z' - 1) + 2 * z + 3 + (z' - 1) * z, z >= 0, z' - 1 >= 0 minus(z, z') -{ 5 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + s') + 0 :|: s' >= 0, s' <= 0 * 0 + 3 * 0 + 2 * z + 3 + 0 * z, z >= 0, z' - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {p} Previous analysis results are: minus: runtime: O(n^1) [4 + 2*z'], size: O(n^2) [3 + 2*z + z*z' + 3*z' + z'^2] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: p 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: minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 4 + 2*z' }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + s) + 0 :|: s >= 0, s <= (z' - 1) * (z' - 1) + 3 * (z' - 1) + 2 * z + 3 + (z' - 1) * z, z >= 0, z' - 1 >= 0 minus(z, z') -{ 5 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + s') + 0 :|: s' >= 0, s' <= 0 * 0 + 3 * 0 + 2 * z + 3 + 0 * z, z >= 0, z' - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {p} Previous analysis results are: minus: runtime: O(n^1) [4 + 2*z'], size: O(n^2) [3 + 2*z + z*z' + 3*z' + z'^2] p: runtime: ?, size: O(n^1) [z] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: p after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 4 + 2*z' }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + s) + 0 :|: s >= 0, s <= (z' - 1) * (z' - 1) + 3 * (z' - 1) + 2 * z + 3 + (z' - 1) * z, z >= 0, z' - 1 >= 0 minus(z, z') -{ 5 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + s') + 0 :|: s' >= 0, s' <= 0 * 0 + 3 * 0 + 2 * z + 3 + 0 * z, z >= 0, z' - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: Previous analysis results are: minus: runtime: O(n^1) [4 + 2*z'], size: O(n^2) [3 + 2*z + z*z' + 3*z' + z'^2] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (29) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (30) BOUNDS(1, n^1) ---------------------------------------- (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^1). The TRS R consists of the following rules: -(0, y) -> 0 -(x, 0) -> x -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0) p(0) -> 0 p(s(x)) -> x 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)) ->^+ if(greater(x, s(y)), s(-(x, y)), 0) gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0]. 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^1). The TRS R consists of the following rules: -(0, y) -> 0 -(x, 0) -> x -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0) p(0) -> 0 p(s(x)) -> x 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^1). The TRS R consists of the following rules: -(0, y) -> 0 -(x, 0) -> x -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0) p(0) -> 0 p(s(x)) -> x S is empty. Rewrite Strategy: INNERMOST