348.39/291.50 WORST_CASE(Omega(n^1), ?) 348.39/291.50 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 348.39/291.50 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 348.39/291.50 348.39/291.50 348.39/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 348.39/291.50 348.39/291.50 (0) CpxTRS 348.39/291.50 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 348.39/291.50 (2) TRS for Loop Detection 348.39/291.50 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 348.39/291.50 (4) BEST 348.39/291.50 (5) proven lower bound 348.39/291.50 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 348.39/291.50 (7) BOUNDS(n^1, INF) 348.39/291.50 (8) TRS for Loop Detection 348.39/291.50 348.39/291.50 348.39/291.50 ---------------------------------------- 348.39/291.50 348.39/291.50 (0) 348.39/291.50 Obligation: 348.39/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 348.39/291.50 348.39/291.50 348.39/291.50 The TRS R consists of the following rules: 348.39/291.50 348.39/291.50 app(nil, k) -> k 348.39/291.50 app(l, nil) -> l 348.39/291.50 app(cons(x, l), k) -> cons(x, app(l, k)) 348.39/291.50 sum(cons(x, nil)) -> cons(x, nil) 348.39/291.50 sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)) 348.39/291.50 sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))) 348.39/291.50 sum(plus(cons(0, x), cons(y, l))) -> pred(sum(cons(s(x), cons(y, l)))) 348.39/291.50 pred(cons(s(x), nil)) -> cons(x, nil) 348.39/291.50 plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 348.39/291.50 plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) 348.39/291.50 plus(zero, y) -> y 348.39/291.50 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) 348.39/291.50 id(x) -> x 348.39/291.50 if(true, x, y) -> x 348.39/291.50 if(false, x, y) -> y 348.39/291.50 not(x) -> if(x, false, true) 348.39/291.50 gt(s(x), zero) -> true 348.39/291.50 gt(zero, y) -> false 348.39/291.50 gt(s(x), s(y)) -> gt(x, y) 348.39/291.50 348.39/291.50 S is empty. 348.39/291.50 Rewrite Strategy: FULL 348.39/291.50 ---------------------------------------- 348.39/291.50 348.39/291.50 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 348.39/291.50 Transformed a relative TRS into a decreasing-loop problem. 348.39/291.50 ---------------------------------------- 348.39/291.50 348.39/291.50 (2) 348.39/291.50 Obligation: 348.39/291.50 Analyzing the following TRS for decreasing loops: 348.39/291.50 348.39/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 348.39/291.50 348.39/291.50 348.39/291.50 The TRS R consists of the following rules: 348.39/291.50 348.39/291.50 app(nil, k) -> k 348.39/291.50 app(l, nil) -> l 348.39/291.50 app(cons(x, l), k) -> cons(x, app(l, k)) 348.39/291.50 sum(cons(x, nil)) -> cons(x, nil) 348.39/291.50 sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)) 348.39/291.50 sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))) 348.39/291.50 sum(plus(cons(0, x), cons(y, l))) -> pred(sum(cons(s(x), cons(y, l)))) 348.39/291.50 pred(cons(s(x), nil)) -> cons(x, nil) 348.39/291.50 plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 348.39/291.50 plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) 348.39/291.50 plus(zero, y) -> y 348.39/291.50 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) 348.39/291.50 id(x) -> x 348.39/291.50 if(true, x, y) -> x 348.39/291.50 if(false, x, y) -> y 348.39/291.50 not(x) -> if(x, false, true) 348.39/291.50 gt(s(x), zero) -> true 348.39/291.50 gt(zero, y) -> false 348.39/291.50 gt(s(x), s(y)) -> gt(x, y) 348.39/291.50 348.39/291.50 S is empty. 348.39/291.50 Rewrite Strategy: FULL 348.39/291.50 ---------------------------------------- 348.39/291.50 348.39/291.50 (3) DecreasingLoopProof (LOWER BOUND(ID)) 348.39/291.50 The following loop(s) give(s) rise to the lower bound Omega(n^1): 348.39/291.50 348.39/291.50 The rewrite sequence 348.39/291.50 348.39/291.50 app(cons(x, l), k) ->^+ cons(x, app(l, k)) 348.39/291.50 348.39/291.50 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 348.39/291.50 348.39/291.50 The pumping substitution is [l / cons(x, l)]. 348.39/291.50 348.39/291.50 The result substitution is [ ]. 348.39/291.50 348.39/291.50 348.39/291.50 348.39/291.50 348.39/291.50 ---------------------------------------- 348.39/291.50 348.39/291.50 (4) 348.39/291.50 Complex Obligation (BEST) 348.39/291.50 348.39/291.50 ---------------------------------------- 348.39/291.50 348.39/291.50 (5) 348.39/291.50 Obligation: 348.39/291.50 Proved the lower bound n^1 for the following obligation: 348.39/291.50 348.39/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 348.39/291.50 348.39/291.50 348.39/291.50 The TRS R consists of the following rules: 348.39/291.50 348.39/291.50 app(nil, k) -> k 348.39/291.50 app(l, nil) -> l 348.39/291.50 app(cons(x, l), k) -> cons(x, app(l, k)) 348.39/291.50 sum(cons(x, nil)) -> cons(x, nil) 348.39/291.50 sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)) 348.39/291.50 sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))) 348.39/291.50 sum(plus(cons(0, x), cons(y, l))) -> pred(sum(cons(s(x), cons(y, l)))) 348.39/291.50 pred(cons(s(x), nil)) -> cons(x, nil) 348.39/291.50 plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 348.39/291.50 plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) 348.39/291.50 plus(zero, y) -> y 348.39/291.50 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) 348.39/291.50 id(x) -> x 348.39/291.50 if(true, x, y) -> x 348.39/291.50 if(false, x, y) -> y 348.39/291.50 not(x) -> if(x, false, true) 348.39/291.50 gt(s(x), zero) -> true 348.39/291.50 gt(zero, y) -> false 348.39/291.50 gt(s(x), s(y)) -> gt(x, y) 348.39/291.50 348.39/291.50 S is empty. 348.39/291.50 Rewrite Strategy: FULL 348.39/291.50 ---------------------------------------- 348.39/291.50 348.39/291.50 (6) LowerBoundPropagationProof (FINISHED) 348.39/291.50 Propagated lower bound. 348.39/291.50 ---------------------------------------- 348.39/291.50 348.39/291.50 (7) 348.39/291.50 BOUNDS(n^1, INF) 348.39/291.50 348.39/291.50 ---------------------------------------- 348.39/291.50 348.39/291.50 (8) 348.39/291.50 Obligation: 348.39/291.50 Analyzing the following TRS for decreasing loops: 348.39/291.50 348.39/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 348.39/291.50 348.39/291.50 348.39/291.50 The TRS R consists of the following rules: 348.39/291.50 348.39/291.50 app(nil, k) -> k 348.39/291.50 app(l, nil) -> l 348.39/291.50 app(cons(x, l), k) -> cons(x, app(l, k)) 348.39/291.50 sum(cons(x, nil)) -> cons(x, nil) 348.39/291.50 sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)) 348.39/291.50 sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))) 348.39/291.50 sum(plus(cons(0, x), cons(y, l))) -> pred(sum(cons(s(x), cons(y, l)))) 348.39/291.50 pred(cons(s(x), nil)) -> cons(x, nil) 348.39/291.50 plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 348.39/291.50 plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) 348.39/291.50 plus(zero, y) -> y 348.39/291.50 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) 348.39/291.50 id(x) -> x 348.39/291.50 if(true, x, y) -> x 348.39/291.50 if(false, x, y) -> y 348.39/291.50 not(x) -> if(x, false, true) 348.39/291.50 gt(s(x), zero) -> true 348.39/291.50 gt(zero, y) -> false 348.39/291.50 gt(s(x), s(y)) -> gt(x, y) 348.39/291.50 348.39/291.50 S is empty. 348.39/291.50 Rewrite Strategy: FULL 348.39/291.54 EOF