1123.12/291.53 WORST_CASE(Omega(n^1), ?) 1123.12/291.57 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1123.12/291.57 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1123.12/291.57 1123.12/291.57 1123.12/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.12/291.57 1123.12/291.57 (0) CpxTRS 1123.12/291.57 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1123.12/291.57 (2) TRS for Loop Detection 1123.12/291.57 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1123.12/291.57 (4) BEST 1123.12/291.57 (5) proven lower bound 1123.12/291.57 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1123.12/291.57 (7) BOUNDS(n^1, INF) 1123.12/291.57 (8) TRS for Loop Detection 1123.12/291.57 1123.12/291.57 1123.12/291.57 ---------------------------------------- 1123.12/291.57 1123.12/291.57 (0) 1123.12/291.57 Obligation: 1123.12/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.12/291.57 1123.12/291.57 1123.12/291.57 The TRS R consists of the following rules: 1123.12/291.57 1123.12/291.57 g(A) -> A 1123.12/291.57 g(B) -> A 1123.12/291.57 g(B) -> B 1123.12/291.57 g(C) -> A 1123.12/291.57 g(C) -> B 1123.12/291.57 g(C) -> C 1123.12/291.57 foldB(t, 0) -> t 1123.12/291.57 foldB(t, s(n)) -> f(foldB(t, n), B) 1123.12/291.57 foldC(t, 0) -> t 1123.12/291.57 foldC(t, s(n)) -> f(foldC(t, n), C) 1123.12/291.57 f(t, x) -> f'(t, g(x)) 1123.12/291.57 f'(triple(a, b, c), C) -> triple(a, b, s(c)) 1123.12/291.57 f'(triple(a, b, c), B) -> f(triple(a, b, c), A) 1123.12/291.57 f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b)) 1123.12/291.57 f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c) 1123.12/291.57 1123.12/291.57 S is empty. 1123.12/291.57 Rewrite Strategy: INNERMOST 1123.12/291.57 ---------------------------------------- 1123.12/291.57 1123.12/291.57 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1123.12/291.57 Transformed a relative TRS into a decreasing-loop problem. 1123.12/291.57 ---------------------------------------- 1123.12/291.57 1123.12/291.57 (2) 1123.12/291.57 Obligation: 1123.12/291.57 Analyzing the following TRS for decreasing loops: 1123.12/291.57 1123.12/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.12/291.57 1123.12/291.57 1123.12/291.57 The TRS R consists of the following rules: 1123.12/291.57 1123.12/291.57 g(A) -> A 1123.12/291.57 g(B) -> A 1123.12/291.57 g(B) -> B 1123.12/291.57 g(C) -> A 1123.12/291.57 g(C) -> B 1123.12/291.57 g(C) -> C 1123.12/291.57 foldB(t, 0) -> t 1123.12/291.57 foldB(t, s(n)) -> f(foldB(t, n), B) 1123.12/291.57 foldC(t, 0) -> t 1123.12/291.57 foldC(t, s(n)) -> f(foldC(t, n), C) 1123.12/291.57 f(t, x) -> f'(t, g(x)) 1123.12/291.57 f'(triple(a, b, c), C) -> triple(a, b, s(c)) 1123.12/291.57 f'(triple(a, b, c), B) -> f(triple(a, b, c), A) 1123.12/291.57 f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b)) 1123.12/291.57 f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c) 1123.12/291.57 1123.12/291.57 S is empty. 1123.12/291.57 Rewrite Strategy: INNERMOST 1123.12/291.57 ---------------------------------------- 1123.12/291.57 1123.12/291.57 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1123.12/291.57 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1123.12/291.57 1123.12/291.57 The rewrite sequence 1123.12/291.57 1123.12/291.57 foldC(t, s(n)) ->^+ f(foldC(t, n), C) 1123.12/291.57 1123.12/291.57 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1123.12/291.57 1123.12/291.57 The pumping substitution is [n / s(n)]. 1123.12/291.57 1123.12/291.57 The result substitution is [ ]. 1123.12/291.57 1123.12/291.57 1123.12/291.57 1123.12/291.57 1123.12/291.57 ---------------------------------------- 1123.12/291.57 1123.12/291.57 (4) 1123.12/291.57 Complex Obligation (BEST) 1123.12/291.57 1123.12/291.57 ---------------------------------------- 1123.12/291.57 1123.12/291.57 (5) 1123.12/291.57 Obligation: 1123.12/291.57 Proved the lower bound n^1 for the following obligation: 1123.12/291.57 1123.12/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.12/291.57 1123.12/291.57 1123.12/291.57 The TRS R consists of the following rules: 1123.12/291.57 1123.12/291.57 g(A) -> A 1123.12/291.57 g(B) -> A 1123.12/291.57 g(B) -> B 1123.12/291.57 g(C) -> A 1123.12/291.57 g(C) -> B 1123.12/291.57 g(C) -> C 1123.12/291.57 foldB(t, 0) -> t 1123.12/291.57 foldB(t, s(n)) -> f(foldB(t, n), B) 1123.12/291.57 foldC(t, 0) -> t 1123.12/291.57 foldC(t, s(n)) -> f(foldC(t, n), C) 1123.12/291.57 f(t, x) -> f'(t, g(x)) 1123.12/291.57 f'(triple(a, b, c), C) -> triple(a, b, s(c)) 1123.12/291.57 f'(triple(a, b, c), B) -> f(triple(a, b, c), A) 1123.12/291.57 f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b)) 1123.12/291.57 f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c) 1123.12/291.57 1123.12/291.57 S is empty. 1123.12/291.57 Rewrite Strategy: INNERMOST 1123.12/291.57 ---------------------------------------- 1123.12/291.57 1123.12/291.57 (6) LowerBoundPropagationProof (FINISHED) 1123.12/291.57 Propagated lower bound. 1123.12/291.57 ---------------------------------------- 1123.12/291.57 1123.12/291.57 (7) 1123.12/291.57 BOUNDS(n^1, INF) 1123.12/291.57 1123.12/291.57 ---------------------------------------- 1123.12/291.57 1123.12/291.57 (8) 1123.12/291.57 Obligation: 1123.12/291.57 Analyzing the following TRS for decreasing loops: 1123.12/291.57 1123.12/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.12/291.57 1123.12/291.57 1123.12/291.57 The TRS R consists of the following rules: 1123.12/291.57 1123.12/291.57 g(A) -> A 1123.12/291.57 g(B) -> A 1123.12/291.57 g(B) -> B 1123.12/291.57 g(C) -> A 1123.12/291.57 g(C) -> B 1123.12/291.57 g(C) -> C 1123.12/291.57 foldB(t, 0) -> t 1123.12/291.57 foldB(t, s(n)) -> f(foldB(t, n), B) 1123.12/291.57 foldC(t, 0) -> t 1123.12/291.57 foldC(t, s(n)) -> f(foldC(t, n), C) 1123.12/291.57 f(t, x) -> f'(t, g(x)) 1123.12/291.57 f'(triple(a, b, c), C) -> triple(a, b, s(c)) 1123.12/291.57 f'(triple(a, b, c), B) -> f(triple(a, b, c), A) 1123.12/291.57 f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b)) 1123.12/291.57 f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c) 1123.12/291.57 1123.12/291.57 S is empty. 1123.12/291.57 Rewrite Strategy: INNERMOST 1123.39/291.66 EOF