341.60/291.54 WORST_CASE(Omega(n^1), ?) 341.60/291.55 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 341.60/291.55 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 341.60/291.55 341.60/291.55 341.60/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 341.60/291.55 341.60/291.55 (0) CpxTRS 341.60/291.55 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 341.60/291.55 (2) TRS for Loop Detection 341.60/291.55 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 341.60/291.55 (4) BEST 341.60/291.55 (5) proven lower bound 341.60/291.55 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 341.60/291.55 (7) BOUNDS(n^1, INF) 341.60/291.55 (8) TRS for Loop Detection 341.60/291.55 341.60/291.55 341.60/291.55 ---------------------------------------- 341.60/291.55 341.60/291.55 (0) 341.60/291.55 Obligation: 341.60/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 341.60/291.55 341.60/291.55 341.60/291.55 The TRS R consists of the following rules: 341.60/291.55 341.60/291.55 minus(x, 0) -> x 341.60/291.55 minus(s(x), s(y)) -> minus(x, y) 341.60/291.55 quot(0, s(y)) -> 0 341.60/291.55 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 341.60/291.55 plus(0, y) -> y 341.60/291.55 plus(s(x), y) -> s(plus(x, y)) 341.60/291.55 minus(minus(x, y), z) -> minus(x, plus(y, z)) 341.60/291.55 app(nil, k) -> k 341.60/291.55 app(l, nil) -> l 341.60/291.55 app(cons(x, l), k) -> cons(x, app(l, k)) 341.60/291.55 sum(cons(x, nil)) -> cons(x, nil) 341.60/291.55 sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)) 341.60/291.55 sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))) 341.60/291.55 plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 341.60/291.55 plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) 341.60/291.55 plus(zero, y) -> y 341.60/291.55 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) 341.60/291.55 id(x) -> x 341.60/291.55 if(true, x, y) -> x 341.60/291.55 if(false, x, y) -> y 341.60/291.55 not(x) -> if(x, false, true) 341.60/291.55 gt(s(x), zero) -> true 341.60/291.55 gt(zero, y) -> false 341.60/291.55 gt(s(x), s(y)) -> gt(x, y) 341.60/291.55 341.60/291.55 S is empty. 341.60/291.55 Rewrite Strategy: FULL 341.60/291.55 ---------------------------------------- 341.60/291.55 341.60/291.55 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 341.60/291.55 Transformed a relative TRS into a decreasing-loop problem. 341.60/291.55 ---------------------------------------- 341.60/291.55 341.60/291.55 (2) 341.60/291.55 Obligation: 341.60/291.55 Analyzing the following TRS for decreasing loops: 341.60/291.55 341.60/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 341.60/291.55 341.60/291.55 341.60/291.55 The TRS R consists of the following rules: 341.60/291.55 341.60/291.55 minus(x, 0) -> x 341.60/291.55 minus(s(x), s(y)) -> minus(x, y) 341.60/291.55 quot(0, s(y)) -> 0 341.60/291.55 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 341.60/291.55 plus(0, y) -> y 341.60/291.55 plus(s(x), y) -> s(plus(x, y)) 341.60/291.55 minus(minus(x, y), z) -> minus(x, plus(y, z)) 341.60/291.55 app(nil, k) -> k 341.60/291.55 app(l, nil) -> l 341.60/291.55 app(cons(x, l), k) -> cons(x, app(l, k)) 341.60/291.55 sum(cons(x, nil)) -> cons(x, nil) 341.60/291.55 sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)) 341.60/291.55 sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))) 341.60/291.55 plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 341.60/291.55 plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) 341.60/291.55 plus(zero, y) -> y 341.60/291.55 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) 341.60/291.55 id(x) -> x 341.60/291.55 if(true, x, y) -> x 341.60/291.55 if(false, x, y) -> y 341.60/291.55 not(x) -> if(x, false, true) 341.60/291.55 gt(s(x), zero) -> true 341.60/291.55 gt(zero, y) -> false 341.60/291.55 gt(s(x), s(y)) -> gt(x, y) 341.60/291.55 341.60/291.55 S is empty. 341.60/291.55 Rewrite Strategy: FULL 341.60/291.55 ---------------------------------------- 341.60/291.55 341.60/291.55 (3) DecreasingLoopProof (LOWER BOUND(ID)) 341.60/291.55 The following loop(s) give(s) rise to the lower bound Omega(n^1): 341.60/291.55 341.60/291.55 The rewrite sequence 341.60/291.55 341.60/291.55 app(cons(x, l), k) ->^+ cons(x, app(l, k)) 341.60/291.55 341.60/291.55 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 341.60/291.55 341.60/291.55 The pumping substitution is [l / cons(x, l)]. 341.60/291.55 341.60/291.55 The result substitution is [ ]. 341.60/291.55 341.60/291.55 341.60/291.55 341.60/291.55 341.60/291.55 ---------------------------------------- 341.60/291.55 341.60/291.55 (4) 341.60/291.55 Complex Obligation (BEST) 341.60/291.55 341.60/291.55 ---------------------------------------- 341.60/291.55 341.60/291.55 (5) 341.60/291.55 Obligation: 341.60/291.55 Proved the lower bound n^1 for the following obligation: 341.60/291.55 341.60/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 341.60/291.55 341.60/291.55 341.60/291.55 The TRS R consists of the following rules: 341.60/291.55 341.60/291.55 minus(x, 0) -> x 341.60/291.55 minus(s(x), s(y)) -> minus(x, y) 341.60/291.55 quot(0, s(y)) -> 0 341.60/291.55 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 341.60/291.55 plus(0, y) -> y 341.60/291.55 plus(s(x), y) -> s(plus(x, y)) 341.60/291.55 minus(minus(x, y), z) -> minus(x, plus(y, z)) 341.60/291.55 app(nil, k) -> k 341.60/291.55 app(l, nil) -> l 341.60/291.55 app(cons(x, l), k) -> cons(x, app(l, k)) 341.60/291.55 sum(cons(x, nil)) -> cons(x, nil) 341.60/291.55 sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)) 341.60/291.55 sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))) 341.60/291.55 plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 341.60/291.55 plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) 341.60/291.55 plus(zero, y) -> y 341.60/291.55 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) 341.60/291.55 id(x) -> x 341.60/291.55 if(true, x, y) -> x 341.60/291.55 if(false, x, y) -> y 341.60/291.55 not(x) -> if(x, false, true) 341.60/291.55 gt(s(x), zero) -> true 341.60/291.55 gt(zero, y) -> false 341.60/291.55 gt(s(x), s(y)) -> gt(x, y) 341.60/291.55 341.60/291.55 S is empty. 341.60/291.55 Rewrite Strategy: FULL 341.60/291.55 ---------------------------------------- 341.60/291.55 341.60/291.55 (6) LowerBoundPropagationProof (FINISHED) 341.60/291.55 Propagated lower bound. 341.60/291.55 ---------------------------------------- 341.60/291.55 341.60/291.55 (7) 341.60/291.55 BOUNDS(n^1, INF) 341.60/291.55 341.60/291.55 ---------------------------------------- 341.60/291.55 341.60/291.55 (8) 341.60/291.55 Obligation: 341.60/291.55 Analyzing the following TRS for decreasing loops: 341.60/291.55 341.60/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 341.60/291.55 341.60/291.55 341.60/291.55 The TRS R consists of the following rules: 341.60/291.55 341.60/291.55 minus(x, 0) -> x 341.60/291.55 minus(s(x), s(y)) -> minus(x, y) 341.60/291.55 quot(0, s(y)) -> 0 341.60/291.55 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 341.60/291.55 plus(0, y) -> y 341.60/291.55 plus(s(x), y) -> s(plus(x, y)) 341.60/291.55 minus(minus(x, y), z) -> minus(x, plus(y, z)) 341.60/291.55 app(nil, k) -> k 341.60/291.55 app(l, nil) -> l 341.60/291.55 app(cons(x, l), k) -> cons(x, app(l, k)) 341.60/291.55 sum(cons(x, nil)) -> cons(x, nil) 341.60/291.55 sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)) 341.60/291.55 sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))) 341.60/291.55 plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 341.60/291.55 plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) 341.60/291.55 plus(zero, y) -> y 341.60/291.55 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) 341.60/291.55 id(x) -> x 341.60/291.55 if(true, x, y) -> x 341.60/291.55 if(false, x, y) -> y 341.60/291.55 not(x) -> if(x, false, true) 341.60/291.55 gt(s(x), zero) -> true 341.60/291.55 gt(zero, y) -> false 341.60/291.55 gt(s(x), s(y)) -> gt(x, y) 341.60/291.55 341.60/291.55 S is empty. 341.60/291.55 Rewrite Strategy: FULL 341.60/291.59 EOF