WORST_CASE(Omega(n^1),O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).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), 1 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), 588 ms] (18) CpxRNTS (19) IntTrsBoundProof [UPPER BOUND(ID), 233 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), 17 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: f(empty, l) -> l f(cons(x, k), l) -> g(k, l, cons(x, k)) g(a, b, c) -> f(a, cons(b, c)) 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: f(empty, l) -> l [1] f(cons(x, k), l) -> g(k, l, cons(x, k)) [1] g(a, b, c) -> f(a, cons(b, c)) [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: f(empty, l) -> l [1] f(cons(x, k), l) -> g(k, l, cons(x, k)) [1] g(a, b, c) -> f(a, cons(b, c)) [1] The TRS has the following type information: f :: empty:cons -> empty:cons -> empty:cons empty :: empty:cons cons :: empty:cons -> empty:cons -> empty:cons g :: empty:cons -> empty:cons -> empty:cons -> empty:cons 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: f_2 g_3 (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: f(empty, l) -> l [1] f(cons(x, k), l) -> g(k, l, cons(x, k)) [1] g(a, b, c) -> f(a, cons(b, c)) [1] The TRS has the following type information: f :: empty:cons -> empty:cons -> empty:cons empty :: empty:cons cons :: empty:cons -> empty:cons -> empty:cons g :: empty:cons -> empty:cons -> empty:cons -> empty:cons 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: f(empty, l) -> l [1] f(cons(x, k), l) -> g(k, l, cons(x, k)) [1] g(a, b, c) -> f(a, cons(b, c)) [1] The TRS has the following type information: f :: empty:cons -> empty:cons -> empty:cons empty :: empty:cons cons :: empty:cons -> empty:cons -> empty:cons g :: empty:cons -> empty:cons -> empty:cons -> empty:cons 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: empty => 0 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> l :|: z' = l, l >= 0, z = 0 f(z, z') -{ 1 }-> g(k, l, 1 + x + k) :|: z' = l, x >= 0, l >= 0, k >= 0, z = 1 + x + k g(z, z', z'') -{ 1 }-> f(a, 1 + b + c) :|: z = a, b >= 0, a >= 0, c >= 0, z' = b, z'' = c ---------------------------------------- (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: f(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 f(z, z') -{ 1 }-> g(k, z', 1 + x + k) :|: x >= 0, z' >= 0, k >= 0, z = 1 + x + k g(z, z', z'') -{ 1 }-> f(z, 1 + z' + z'') :|: z' >= 0, z >= 0, z'' >= 0 ---------------------------------------- (13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { f, g } ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 f(z, z') -{ 1 }-> g(k, z', 1 + x + k) :|: x >= 0, z' >= 0, k >= 0, z = 1 + x + k g(z, z', z'') -{ 1 }-> f(z, 1 + z' + z'') :|: z' >= 0, z >= 0, z'' >= 0 Function symbols to be analyzed: {f,g} ---------------------------------------- (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: f(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 f(z, z') -{ 1 }-> g(k, z', 1 + x + k) :|: x >= 0, z' >= 0, k >= 0, z = 1 + x + k g(z, z', z'') -{ 1 }-> f(z, 1 + z' + z'') :|: z' >= 0, z >= 0, z'' >= 0 Function symbols to be analyzed: {f,g} ---------------------------------------- (17) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: z + z^2 + z' Computed SIZE bound using KoAT for: g after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1 + z + z^2 + z' + z'' ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 f(z, z') -{ 1 }-> g(k, z', 1 + x + k) :|: x >= 0, z' >= 0, k >= 0, z = 1 + x + k g(z, z', z'') -{ 1 }-> f(z, 1 + z' + z'') :|: z' >= 0, z >= 0, z'' >= 0 Function symbols to be analyzed: {f,g} Previous analysis results are: f: runtime: ?, size: O(n^2) [z + z^2 + z'] g: runtime: ?, size: O(n^2) [1 + z + z^2 + z' + z''] ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + 2*z Computed RUNTIME bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4 + 2*z ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 f(z, z') -{ 1 }-> g(k, z', 1 + x + k) :|: x >= 0, z' >= 0, k >= 0, z = 1 + x + k g(z, z', z'') -{ 1 }-> f(z, 1 + z' + z'') :|: z' >= 0, z >= 0, z'' >= 0 Function symbols to be analyzed: Previous analysis results are: f: runtime: O(n^1) [3 + 2*z], size: O(n^2) [z + z^2 + z'] g: runtime: O(n^1) [4 + 2*z], size: O(n^2) [1 + z + z^2 + 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: f(empty, l) -> l f(cons(x, k), l) -> g(k, l, cons(x, k)) g(a, b, c) -> f(a, cons(b, c)) 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 g(cons(x1_0, k2_0), b, c) ->^+ g(k2_0, cons(b, c), cons(x1_0, k2_0)) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [k2_0 / cons(x1_0, k2_0)]. The result substitution is [b / cons(b, c), c / cons(x1_0, k2_0)]. ---------------------------------------- (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: f(empty, l) -> l f(cons(x, k), l) -> g(k, l, cons(x, k)) g(a, b, c) -> f(a, cons(b, c)) 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: f(empty, l) -> l f(cons(x, k), l) -> g(k, l, cons(x, k)) g(a, b, c) -> f(a, cons(b, c)) S is empty. Rewrite Strategy: INNERMOST