304.69/291.56 WORST_CASE(Omega(n^1), ?) 304.81/291.62 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 304.81/291.62 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 304.81/291.62 304.81/291.62 304.81/291.62 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 304.81/291.62 304.81/291.62 (0) CpxTRS 304.81/291.62 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 304.81/291.62 (2) TRS for Loop Detection 304.81/291.62 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 304.81/291.62 (4) BEST 304.81/291.62 (5) proven lower bound 304.81/291.62 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 304.81/291.62 (7) BOUNDS(n^1, INF) 304.81/291.62 (8) TRS for Loop Detection 304.81/291.62 304.81/291.62 304.81/291.62 ---------------------------------------- 304.81/291.62 304.81/291.62 (0) 304.81/291.62 Obligation: 304.81/291.62 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 304.81/291.62 304.81/291.62 304.81/291.62 The TRS R consists of the following rules: 304.81/291.62 304.81/291.62 ge(x, 0) -> true 304.81/291.62 ge(0, s(y)) -> false 304.81/291.62 ge(s(x), s(y)) -> ge(x, y) 304.81/291.62 rev(x) -> if(x, eq(0, length(x)), nil, 0, length(x)) 304.81/291.62 if(x, true, z, c, l) -> z 304.81/291.62 if(x, false, z, c, l) -> help(s(c), l, x, z) 304.81/291.62 help(c, l, cons(x, y), z) -> if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l) 304.81/291.62 append(nil, y) -> y 304.81/291.62 append(cons(x, y), z) -> cons(x, append(y, z)) 304.81/291.62 length(nil) -> 0 304.81/291.62 length(cons(x, y)) -> s(length(y)) 304.81/291.62 304.81/291.62 S is empty. 304.81/291.62 Rewrite Strategy: FULL 304.81/291.62 ---------------------------------------- 304.81/291.62 304.81/291.62 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 304.81/291.62 Transformed a relative TRS into a decreasing-loop problem. 304.81/291.62 ---------------------------------------- 304.81/291.62 304.81/291.62 (2) 304.81/291.62 Obligation: 304.81/291.62 Analyzing the following TRS for decreasing loops: 304.81/291.62 304.81/291.62 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 304.81/291.62 304.81/291.62 304.81/291.62 The TRS R consists of the following rules: 304.81/291.62 304.81/291.62 ge(x, 0) -> true 304.81/291.62 ge(0, s(y)) -> false 304.81/291.62 ge(s(x), s(y)) -> ge(x, y) 304.81/291.62 rev(x) -> if(x, eq(0, length(x)), nil, 0, length(x)) 304.81/291.62 if(x, true, z, c, l) -> z 304.81/291.62 if(x, false, z, c, l) -> help(s(c), l, x, z) 304.81/291.62 help(c, l, cons(x, y), z) -> if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l) 304.81/291.62 append(nil, y) -> y 304.81/291.62 append(cons(x, y), z) -> cons(x, append(y, z)) 304.81/291.62 length(nil) -> 0 304.81/291.62 length(cons(x, y)) -> s(length(y)) 304.81/291.62 304.81/291.62 S is empty. 304.81/291.62 Rewrite Strategy: FULL 304.81/291.62 ---------------------------------------- 304.81/291.62 304.81/291.62 (3) DecreasingLoopProof (LOWER BOUND(ID)) 304.81/291.62 The following loop(s) give(s) rise to the lower bound Omega(n^1): 304.81/291.62 304.81/291.62 The rewrite sequence 304.81/291.62 304.81/291.62 length(cons(x, y)) ->^+ s(length(y)) 304.81/291.62 304.81/291.62 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 304.81/291.62 304.81/291.62 The pumping substitution is [y / cons(x, y)]. 304.81/291.62 304.81/291.62 The result substitution is [ ]. 304.81/291.62 304.81/291.62 304.81/291.62 304.81/291.62 304.81/291.62 ---------------------------------------- 304.81/291.62 304.81/291.62 (4) 304.81/291.62 Complex Obligation (BEST) 304.81/291.62 304.81/291.62 ---------------------------------------- 304.81/291.62 304.81/291.62 (5) 304.81/291.62 Obligation: 304.81/291.62 Proved the lower bound n^1 for the following obligation: 304.81/291.62 304.81/291.62 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 304.81/291.62 304.81/291.62 304.81/291.62 The TRS R consists of the following rules: 304.81/291.62 304.81/291.62 ge(x, 0) -> true 304.81/291.62 ge(0, s(y)) -> false 304.81/291.62 ge(s(x), s(y)) -> ge(x, y) 304.81/291.62 rev(x) -> if(x, eq(0, length(x)), nil, 0, length(x)) 304.81/291.62 if(x, true, z, c, l) -> z 304.81/291.62 if(x, false, z, c, l) -> help(s(c), l, x, z) 304.81/291.62 help(c, l, cons(x, y), z) -> if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l) 304.81/291.62 append(nil, y) -> y 304.81/291.62 append(cons(x, y), z) -> cons(x, append(y, z)) 304.81/291.62 length(nil) -> 0 304.81/291.62 length(cons(x, y)) -> s(length(y)) 304.81/291.62 304.81/291.62 S is empty. 304.81/291.62 Rewrite Strategy: FULL 304.81/291.62 ---------------------------------------- 304.81/291.62 304.81/291.62 (6) LowerBoundPropagationProof (FINISHED) 304.81/291.62 Propagated lower bound. 304.81/291.62 ---------------------------------------- 304.81/291.62 304.81/291.62 (7) 304.81/291.62 BOUNDS(n^1, INF) 304.81/291.62 304.81/291.62 ---------------------------------------- 304.81/291.62 304.81/291.62 (8) 304.81/291.62 Obligation: 304.81/291.62 Analyzing the following TRS for decreasing loops: 304.81/291.62 304.81/291.62 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 304.81/291.62 304.81/291.62 304.81/291.62 The TRS R consists of the following rules: 304.81/291.62 304.81/291.62 ge(x, 0) -> true 304.81/291.62 ge(0, s(y)) -> false 304.81/291.62 ge(s(x), s(y)) -> ge(x, y) 304.81/291.62 rev(x) -> if(x, eq(0, length(x)), nil, 0, length(x)) 304.81/291.62 if(x, true, z, c, l) -> z 304.81/291.62 if(x, false, z, c, l) -> help(s(c), l, x, z) 304.81/291.62 help(c, l, cons(x, y), z) -> if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l) 304.81/291.62 append(nil, y) -> y 304.81/291.62 append(cons(x, y), z) -> cons(x, append(y, z)) 304.81/291.62 length(nil) -> 0 304.81/291.62 length(cons(x, y)) -> s(length(y)) 304.81/291.62 304.81/291.62 S is empty. 304.81/291.62 Rewrite Strategy: FULL 304.81/291.64 EOF