308.95/291.48 WORST_CASE(Omega(n^1), ?) 308.95/291.49 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 308.95/291.49 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 308.95/291.49 308.95/291.49 308.95/291.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 308.95/291.49 308.95/291.49 (0) CpxTRS 308.95/291.49 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 308.95/291.49 (2) TRS for Loop Detection 308.95/291.49 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 308.95/291.49 (4) BEST 308.95/291.49 (5) proven lower bound 308.95/291.49 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 308.95/291.49 (7) BOUNDS(n^1, INF) 308.95/291.49 (8) TRS for Loop Detection 308.95/291.49 308.95/291.49 308.95/291.49 ---------------------------------------- 308.95/291.49 308.95/291.49 (0) 308.95/291.49 Obligation: 308.95/291.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 308.95/291.49 308.95/291.49 308.95/291.49 The TRS R consists of the following rules: 308.95/291.49 308.95/291.49 division(x, y) -> div(x, y, 0) 308.95/291.49 div(x, y, z) -> if(lt(x, y), x, y, inc(z)) 308.95/291.49 if(true, x, y, z) -> z 308.95/291.49 if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z) 308.95/291.49 minus(x, 0) -> x 308.95/291.49 minus(s(x), s(y)) -> minus(x, y) 308.95/291.49 lt(x, 0) -> false 308.95/291.49 lt(0, s(y)) -> true 308.95/291.49 lt(s(x), s(y)) -> lt(x, y) 308.95/291.49 inc(0) -> s(0) 308.95/291.49 inc(s(x)) -> s(inc(x)) 308.95/291.49 308.95/291.49 S is empty. 308.95/291.49 Rewrite Strategy: FULL 308.95/291.49 ---------------------------------------- 308.95/291.49 308.95/291.49 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 308.95/291.49 Transformed a relative TRS into a decreasing-loop problem. 308.95/291.49 ---------------------------------------- 308.95/291.49 308.95/291.49 (2) 308.95/291.49 Obligation: 308.95/291.49 Analyzing the following TRS for decreasing loops: 308.95/291.49 308.95/291.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 308.95/291.49 308.95/291.49 308.95/291.49 The TRS R consists of the following rules: 308.95/291.49 308.95/291.49 division(x, y) -> div(x, y, 0) 308.95/291.49 div(x, y, z) -> if(lt(x, y), x, y, inc(z)) 308.95/291.49 if(true, x, y, z) -> z 308.95/291.49 if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z) 308.95/291.49 minus(x, 0) -> x 308.95/291.49 minus(s(x), s(y)) -> minus(x, y) 308.95/291.49 lt(x, 0) -> false 308.95/291.49 lt(0, s(y)) -> true 308.95/291.49 lt(s(x), s(y)) -> lt(x, y) 308.95/291.49 inc(0) -> s(0) 308.95/291.49 inc(s(x)) -> s(inc(x)) 308.95/291.49 308.95/291.49 S is empty. 308.95/291.49 Rewrite Strategy: FULL 308.95/291.49 ---------------------------------------- 308.95/291.49 308.95/291.49 (3) DecreasingLoopProof (LOWER BOUND(ID)) 308.95/291.49 The following loop(s) give(s) rise to the lower bound Omega(n^1): 308.95/291.49 308.95/291.49 The rewrite sequence 308.95/291.49 308.95/291.49 minus(s(x), s(y)) ->^+ minus(x, y) 308.95/291.49 308.95/291.49 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 308.95/291.49 308.95/291.49 The pumping substitution is [x / s(x), y / s(y)]. 308.95/291.49 308.95/291.49 The result substitution is [ ]. 308.95/291.49 308.95/291.49 308.95/291.49 308.95/291.49 308.95/291.49 ---------------------------------------- 308.95/291.49 308.95/291.49 (4) 308.95/291.49 Complex Obligation (BEST) 308.95/291.49 308.95/291.49 ---------------------------------------- 308.95/291.49 308.95/291.49 (5) 308.95/291.49 Obligation: 308.95/291.49 Proved the lower bound n^1 for the following obligation: 308.95/291.49 308.95/291.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 308.95/291.49 308.95/291.49 308.95/291.49 The TRS R consists of the following rules: 308.95/291.49 308.95/291.49 division(x, y) -> div(x, y, 0) 308.95/291.49 div(x, y, z) -> if(lt(x, y), x, y, inc(z)) 308.95/291.49 if(true, x, y, z) -> z 308.95/291.49 if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z) 308.95/291.49 minus(x, 0) -> x 308.95/291.49 minus(s(x), s(y)) -> minus(x, y) 308.95/291.49 lt(x, 0) -> false 308.95/291.49 lt(0, s(y)) -> true 308.95/291.49 lt(s(x), s(y)) -> lt(x, y) 308.95/291.49 inc(0) -> s(0) 308.95/291.49 inc(s(x)) -> s(inc(x)) 308.95/291.49 308.95/291.49 S is empty. 308.95/291.49 Rewrite Strategy: FULL 308.95/291.49 ---------------------------------------- 308.95/291.49 308.95/291.49 (6) LowerBoundPropagationProof (FINISHED) 308.95/291.49 Propagated lower bound. 308.95/291.49 ---------------------------------------- 308.95/291.49 308.95/291.49 (7) 308.95/291.49 BOUNDS(n^1, INF) 308.95/291.49 308.95/291.49 ---------------------------------------- 308.95/291.49 308.95/291.49 (8) 308.95/291.49 Obligation: 308.95/291.49 Analyzing the following TRS for decreasing loops: 308.95/291.49 308.95/291.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 308.95/291.49 308.95/291.49 308.95/291.49 The TRS R consists of the following rules: 308.95/291.49 308.95/291.49 division(x, y) -> div(x, y, 0) 308.95/291.49 div(x, y, z) -> if(lt(x, y), x, y, inc(z)) 308.95/291.49 if(true, x, y, z) -> z 308.95/291.49 if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z) 308.95/291.49 minus(x, 0) -> x 308.95/291.49 minus(s(x), s(y)) -> minus(x, y) 308.95/291.49 lt(x, 0) -> false 308.95/291.49 lt(0, s(y)) -> true 308.95/291.49 lt(s(x), s(y)) -> lt(x, y) 308.95/291.49 inc(0) -> s(0) 308.95/291.49 inc(s(x)) -> s(inc(x)) 308.95/291.49 308.95/291.49 S is empty. 308.95/291.49 Rewrite Strategy: FULL 308.95/291.51 EOF