1114.11/291.55 WORST_CASE(Omega(n^1), ?) 1114.32/291.58 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1114.32/291.58 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1114.32/291.58 1114.32/291.58 1114.32/291.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1114.32/291.58 1114.32/291.58 (0) CpxTRS 1114.32/291.58 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1114.32/291.58 (2) TRS for Loop Detection 1114.32/291.58 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1114.32/291.58 (4) BEST 1114.32/291.58 (5) proven lower bound 1114.32/291.58 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1114.32/291.58 (7) BOUNDS(n^1, INF) 1114.32/291.58 (8) TRS for Loop Detection 1114.32/291.58 1114.32/291.58 1114.32/291.58 ---------------------------------------- 1114.32/291.58 1114.32/291.58 (0) 1114.32/291.58 Obligation: 1114.32/291.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1114.32/291.58 1114.32/291.58 1114.32/291.58 The TRS R consists of the following rules: 1114.32/291.58 1114.32/291.58 even(0) -> true 1114.32/291.58 even(s(0)) -> false 1114.32/291.58 even(s(s(x))) -> even(x) 1114.32/291.58 half(0) -> 0 1114.32/291.58 half(s(s(x))) -> s(half(x)) 1114.32/291.58 plus(0, y) -> y 1114.32/291.58 plus(s(x), y) -> s(plus(x, y)) 1114.32/291.58 times(0, y) -> 0 1114.32/291.58 times(s(x), y) -> if_times(even(s(x)), s(x), y) 1114.32/291.58 if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y)) 1114.32/291.58 if_times(false, s(x), y) -> plus(y, times(x, y)) 1114.32/291.58 1114.32/291.58 S is empty. 1114.32/291.58 Rewrite Strategy: INNERMOST 1114.32/291.58 ---------------------------------------- 1114.32/291.58 1114.32/291.58 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1114.32/291.58 Transformed a relative TRS into a decreasing-loop problem. 1114.32/291.58 ---------------------------------------- 1114.32/291.58 1114.32/291.58 (2) 1114.32/291.58 Obligation: 1114.32/291.58 Analyzing the following TRS for decreasing loops: 1114.32/291.58 1114.32/291.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1114.32/291.58 1114.32/291.58 1114.32/291.58 The TRS R consists of the following rules: 1114.32/291.58 1114.32/291.58 even(0) -> true 1114.32/291.58 even(s(0)) -> false 1114.32/291.58 even(s(s(x))) -> even(x) 1114.32/291.58 half(0) -> 0 1114.32/291.58 half(s(s(x))) -> s(half(x)) 1114.32/291.58 plus(0, y) -> y 1114.32/291.58 plus(s(x), y) -> s(plus(x, y)) 1114.32/291.58 times(0, y) -> 0 1114.32/291.58 times(s(x), y) -> if_times(even(s(x)), s(x), y) 1114.32/291.58 if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y)) 1114.32/291.58 if_times(false, s(x), y) -> plus(y, times(x, y)) 1114.32/291.58 1114.32/291.58 S is empty. 1114.32/291.58 Rewrite Strategy: INNERMOST 1114.32/291.58 ---------------------------------------- 1114.32/291.58 1114.32/291.58 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1114.32/291.58 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1114.32/291.58 1114.32/291.58 The rewrite sequence 1114.32/291.58 1114.32/291.58 even(s(s(x))) ->^+ even(x) 1114.32/291.58 1114.32/291.58 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 1114.32/291.58 1114.32/291.58 The pumping substitution is [x / s(s(x))]. 1114.32/291.58 1114.32/291.58 The result substitution is [ ]. 1114.32/291.58 1114.32/291.58 1114.32/291.58 1114.32/291.58 1114.32/291.58 ---------------------------------------- 1114.32/291.58 1114.32/291.58 (4) 1114.32/291.58 Complex Obligation (BEST) 1114.32/291.58 1114.32/291.58 ---------------------------------------- 1114.32/291.58 1114.32/291.58 (5) 1114.32/291.58 Obligation: 1114.32/291.58 Proved the lower bound n^1 for the following obligation: 1114.32/291.58 1114.32/291.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1114.32/291.58 1114.32/291.58 1114.32/291.58 The TRS R consists of the following rules: 1114.32/291.58 1114.32/291.58 even(0) -> true 1114.32/291.58 even(s(0)) -> false 1114.32/291.58 even(s(s(x))) -> even(x) 1114.32/291.58 half(0) -> 0 1114.32/291.58 half(s(s(x))) -> s(half(x)) 1114.32/291.58 plus(0, y) -> y 1114.32/291.58 plus(s(x), y) -> s(plus(x, y)) 1114.32/291.58 times(0, y) -> 0 1114.32/291.58 times(s(x), y) -> if_times(even(s(x)), s(x), y) 1114.32/291.58 if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y)) 1114.32/291.58 if_times(false, s(x), y) -> plus(y, times(x, y)) 1114.32/291.58 1114.32/291.58 S is empty. 1114.32/291.58 Rewrite Strategy: INNERMOST 1114.32/291.58 ---------------------------------------- 1114.32/291.58 1114.32/291.58 (6) LowerBoundPropagationProof (FINISHED) 1114.32/291.58 Propagated lower bound. 1114.32/291.58 ---------------------------------------- 1114.32/291.58 1114.32/291.58 (7) 1114.32/291.58 BOUNDS(n^1, INF) 1114.32/291.58 1114.32/291.58 ---------------------------------------- 1114.32/291.58 1114.32/291.58 (8) 1114.32/291.58 Obligation: 1114.32/291.58 Analyzing the following TRS for decreasing loops: 1114.32/291.58 1114.32/291.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1114.32/291.58 1114.32/291.58 1114.32/291.58 The TRS R consists of the following rules: 1114.32/291.58 1114.32/291.58 even(0) -> true 1114.32/291.58 even(s(0)) -> false 1114.32/291.58 even(s(s(x))) -> even(x) 1114.32/291.58 half(0) -> 0 1114.32/291.58 half(s(s(x))) -> s(half(x)) 1114.32/291.58 plus(0, y) -> y 1114.32/291.58 plus(s(x), y) -> s(plus(x, y)) 1114.32/291.58 times(0, y) -> 0 1114.32/291.58 times(s(x), y) -> if_times(even(s(x)), s(x), y) 1114.32/291.58 if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y)) 1114.32/291.58 if_times(false, s(x), y) -> plus(y, times(x, y)) 1114.32/291.58 1114.32/291.58 S is empty. 1114.32/291.58 Rewrite Strategy: INNERMOST 1114.40/291.65 EOF