1110.20/291.86 WORST_CASE(Omega(n^1), ?) 1110.20/291.87 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1110.20/291.87 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1110.20/291.87 1110.20/291.87 1110.20/291.87 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1110.20/291.87 1110.20/291.87 (0) CpxTRS 1110.20/291.87 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1110.20/291.87 (2) TRS for Loop Detection 1110.20/291.87 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1110.20/291.87 (4) BEST 1110.20/291.87 (5) proven lower bound 1110.20/291.87 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1110.20/291.87 (7) BOUNDS(n^1, INF) 1110.20/291.87 (8) TRS for Loop Detection 1110.20/291.87 1110.20/291.87 1110.20/291.87 ---------------------------------------- 1110.20/291.87 1110.20/291.87 (0) 1110.20/291.87 Obligation: 1110.20/291.87 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1110.20/291.87 1110.20/291.87 1110.20/291.87 The TRS R consists of the following rules: 1110.20/291.87 1110.20/291.87 f(x, 0, 0) -> s(x) 1110.20/291.87 f(0, y, 0) -> s(y) 1110.20/291.87 f(0, 0, z) -> s(z) 1110.20/291.87 f(s(0), y, z) -> f(0, s(y), s(z)) 1110.20/291.87 f(s(x), s(y), 0) -> f(x, y, s(0)) 1110.20/291.87 f(s(x), 0, s(z)) -> f(x, s(0), z) 1110.20/291.87 f(0, s(0), s(0)) -> s(s(0)) 1110.20/291.87 f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) 1110.20/291.87 f(0, s(s(y)), s(0)) -> f(0, y, s(0)) 1110.20/291.87 f(0, s(0), s(s(z))) -> f(0, s(0), z) 1110.20/291.87 f(0, s(s(y)), s(s(z))) -> f(0, y, f(0, s(s(y)), s(z))) 1110.20/291.87 1110.20/291.87 S is empty. 1110.20/291.87 Rewrite Strategy: INNERMOST 1110.20/291.87 ---------------------------------------- 1110.20/291.87 1110.20/291.87 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1110.20/291.87 Transformed a relative TRS into a decreasing-loop problem. 1110.20/291.87 ---------------------------------------- 1110.20/291.87 1110.20/291.87 (2) 1110.20/291.87 Obligation: 1110.20/291.87 Analyzing the following TRS for decreasing loops: 1110.20/291.87 1110.20/291.87 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1110.20/291.87 1110.20/291.87 1110.20/291.87 The TRS R consists of the following rules: 1110.20/291.87 1110.20/291.87 f(x, 0, 0) -> s(x) 1110.20/291.87 f(0, y, 0) -> s(y) 1110.20/291.87 f(0, 0, z) -> s(z) 1110.20/291.87 f(s(0), y, z) -> f(0, s(y), s(z)) 1110.20/291.87 f(s(x), s(y), 0) -> f(x, y, s(0)) 1110.20/291.87 f(s(x), 0, s(z)) -> f(x, s(0), z) 1110.20/291.87 f(0, s(0), s(0)) -> s(s(0)) 1110.20/291.87 f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) 1110.20/291.87 f(0, s(s(y)), s(0)) -> f(0, y, s(0)) 1110.20/291.87 f(0, s(0), s(s(z))) -> f(0, s(0), z) 1110.20/291.87 f(0, s(s(y)), s(s(z))) -> f(0, y, f(0, s(s(y)), s(z))) 1110.20/291.87 1110.20/291.87 S is empty. 1110.20/291.87 Rewrite Strategy: INNERMOST 1110.20/291.87 ---------------------------------------- 1110.20/291.87 1110.20/291.87 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1110.20/291.87 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1110.20/291.87 1110.20/291.87 The rewrite sequence 1110.20/291.87 1110.20/291.87 f(s(x), s(y), s(z)) ->^+ f(x, y, f(s(x), s(y), z)) 1110.20/291.87 1110.20/291.87 gives rise to a decreasing loop by considering the right hand sides subterm at position [2]. 1110.20/291.87 1110.20/291.87 The pumping substitution is [z / s(z)]. 1110.20/291.87 1110.20/291.87 The result substitution is [ ]. 1110.20/291.87 1110.20/291.87 1110.20/291.87 1110.20/291.87 1110.20/291.87 ---------------------------------------- 1110.20/291.87 1110.20/291.87 (4) 1110.20/291.87 Complex Obligation (BEST) 1110.20/291.87 1110.20/291.87 ---------------------------------------- 1110.20/291.87 1110.20/291.87 (5) 1110.20/291.87 Obligation: 1110.20/291.87 Proved the lower bound n^1 for the following obligation: 1110.20/291.87 1110.20/291.87 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1110.20/291.87 1110.20/291.87 1110.20/291.87 The TRS R consists of the following rules: 1110.20/291.87 1110.20/291.87 f(x, 0, 0) -> s(x) 1110.20/291.87 f(0, y, 0) -> s(y) 1110.20/291.87 f(0, 0, z) -> s(z) 1110.20/291.87 f(s(0), y, z) -> f(0, s(y), s(z)) 1110.20/291.87 f(s(x), s(y), 0) -> f(x, y, s(0)) 1110.20/291.87 f(s(x), 0, s(z)) -> f(x, s(0), z) 1110.20/291.87 f(0, s(0), s(0)) -> s(s(0)) 1110.20/291.87 f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) 1110.20/291.87 f(0, s(s(y)), s(0)) -> f(0, y, s(0)) 1110.20/291.87 f(0, s(0), s(s(z))) -> f(0, s(0), z) 1110.20/291.87 f(0, s(s(y)), s(s(z))) -> f(0, y, f(0, s(s(y)), s(z))) 1110.20/291.87 1110.20/291.87 S is empty. 1110.20/291.87 Rewrite Strategy: INNERMOST 1110.20/291.87 ---------------------------------------- 1110.20/291.87 1110.20/291.87 (6) LowerBoundPropagationProof (FINISHED) 1110.20/291.87 Propagated lower bound. 1110.20/291.87 ---------------------------------------- 1110.20/291.87 1110.20/291.87 (7) 1110.20/291.87 BOUNDS(n^1, INF) 1110.20/291.87 1110.20/291.87 ---------------------------------------- 1110.20/291.87 1110.20/291.87 (8) 1110.20/291.87 Obligation: 1110.20/291.87 Analyzing the following TRS for decreasing loops: 1110.20/291.87 1110.20/291.87 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1110.20/291.87 1110.20/291.87 1110.20/291.87 The TRS R consists of the following rules: 1110.20/291.87 1110.20/291.87 f(x, 0, 0) -> s(x) 1110.20/291.87 f(0, y, 0) -> s(y) 1110.20/291.87 f(0, 0, z) -> s(z) 1110.20/291.87 f(s(0), y, z) -> f(0, s(y), s(z)) 1110.20/291.87 f(s(x), s(y), 0) -> f(x, y, s(0)) 1110.20/291.87 f(s(x), 0, s(z)) -> f(x, s(0), z) 1110.20/291.87 f(0, s(0), s(0)) -> s(s(0)) 1110.20/291.87 f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) 1110.20/291.87 f(0, s(s(y)), s(0)) -> f(0, y, s(0)) 1110.20/291.87 f(0, s(0), s(s(z))) -> f(0, s(0), z) 1110.20/291.87 f(0, s(s(y)), s(s(z))) -> f(0, y, f(0, s(s(y)), s(z))) 1110.20/291.87 1110.20/291.87 S is empty. 1110.20/291.87 Rewrite Strategy: INNERMOST 1110.39/291.94 EOF