1119.16/292.13 WORST_CASE(Omega(n^1), ?) 1123.19/293.12 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1123.19/293.12 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1123.19/293.12 1123.19/293.12 1123.19/293.12 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.19/293.12 1123.19/293.12 (0) CpxTRS 1123.19/293.12 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1123.19/293.12 (2) TRS for Loop Detection 1123.19/293.12 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1123.19/293.12 (4) BEST 1123.19/293.12 (5) proven lower bound 1123.19/293.12 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1123.19/293.12 (7) BOUNDS(n^1, INF) 1123.19/293.12 (8) TRS for Loop Detection 1123.19/293.12 1123.19/293.12 1123.19/293.12 ---------------------------------------- 1123.19/293.12 1123.19/293.12 (0) 1123.19/293.12 Obligation: 1123.19/293.12 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.19/293.12 1123.19/293.12 1123.19/293.12 The TRS R consists of the following rules: 1123.19/293.12 1123.19/293.12 le(s(x), 0) -> false 1123.19/293.12 le(0, y) -> true 1123.19/293.12 le(s(x), s(y)) -> le(x, y) 1123.19/293.12 double(0) -> 0 1123.19/293.12 double(s(x)) -> s(s(double(x))) 1123.19/293.12 log(0) -> logError 1123.19/293.12 log(s(x)) -> loop(s(x), s(0), 0) 1123.19/293.12 loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) 1123.19/293.12 if(true, x, y, z) -> z 1123.19/293.12 if(false, x, y, z) -> loop(x, double(y), s(z)) 1123.19/293.12 maplog(xs) -> mapIter(xs, nil) 1123.19/293.12 mapIter(xs, ys) -> ifmap(isempty(xs), xs, ys) 1123.19/293.12 ifmap(true, xs, ys) -> ys 1123.19/293.12 ifmap(false, xs, ys) -> mapIter(droplast(xs), cons(log(last(xs)), ys)) 1123.19/293.12 isempty(nil) -> true 1123.19/293.12 isempty(cons(x, xs)) -> false 1123.19/293.12 last(nil) -> error 1123.19/293.12 last(cons(x, nil)) -> x 1123.19/293.12 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 1123.19/293.12 droplast(nil) -> nil 1123.19/293.12 droplast(cons(x, nil)) -> nil 1123.19/293.12 droplast(cons(x, cons(y, xs))) -> cons(x, droplast(cons(y, xs))) 1123.19/293.12 a -> b 1123.19/293.12 a -> c 1123.19/293.12 1123.19/293.12 S is empty. 1123.19/293.12 Rewrite Strategy: INNERMOST 1123.19/293.12 ---------------------------------------- 1123.19/293.12 1123.19/293.12 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1123.19/293.12 Transformed a relative TRS into a decreasing-loop problem. 1123.19/293.12 ---------------------------------------- 1123.19/293.12 1123.19/293.12 (2) 1123.19/293.12 Obligation: 1123.19/293.12 Analyzing the following TRS for decreasing loops: 1123.19/293.12 1123.19/293.12 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.19/293.12 1123.19/293.12 1123.19/293.12 The TRS R consists of the following rules: 1123.19/293.12 1123.19/293.12 le(s(x), 0) -> false 1123.19/293.12 le(0, y) -> true 1123.19/293.12 le(s(x), s(y)) -> le(x, y) 1123.19/293.12 double(0) -> 0 1123.19/293.12 double(s(x)) -> s(s(double(x))) 1123.19/293.12 log(0) -> logError 1123.19/293.12 log(s(x)) -> loop(s(x), s(0), 0) 1123.19/293.12 loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) 1123.19/293.12 if(true, x, y, z) -> z 1123.19/293.12 if(false, x, y, z) -> loop(x, double(y), s(z)) 1123.19/293.12 maplog(xs) -> mapIter(xs, nil) 1123.19/293.12 mapIter(xs, ys) -> ifmap(isempty(xs), xs, ys) 1123.19/293.12 ifmap(true, xs, ys) -> ys 1123.19/293.12 ifmap(false, xs, ys) -> mapIter(droplast(xs), cons(log(last(xs)), ys)) 1123.19/293.12 isempty(nil) -> true 1123.19/293.12 isempty(cons(x, xs)) -> false 1123.19/293.12 last(nil) -> error 1123.19/293.12 last(cons(x, nil)) -> x 1123.19/293.12 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 1123.19/293.12 droplast(nil) -> nil 1123.19/293.12 droplast(cons(x, nil)) -> nil 1123.19/293.12 droplast(cons(x, cons(y, xs))) -> cons(x, droplast(cons(y, xs))) 1123.19/293.12 a -> b 1123.19/293.12 a -> c 1123.19/293.12 1123.19/293.12 S is empty. 1123.19/293.12 Rewrite Strategy: INNERMOST 1123.19/293.12 ---------------------------------------- 1123.19/293.12 1123.19/293.12 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1123.19/293.12 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1123.19/293.12 1123.19/293.12 The rewrite sequence 1123.19/293.12 1123.19/293.12 droplast(cons(x, cons(y, xs))) ->^+ cons(x, droplast(cons(y, xs))) 1123.19/293.12 1123.19/293.12 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 1123.19/293.12 1123.19/293.12 The pumping substitution is [xs / cons(y, xs)]. 1123.19/293.12 1123.19/293.12 The result substitution is [x / y]. 1123.19/293.12 1123.19/293.12 1123.19/293.12 1123.19/293.12 1123.19/293.12 ---------------------------------------- 1123.19/293.12 1123.19/293.12 (4) 1123.19/293.12 Complex Obligation (BEST) 1123.19/293.12 1123.19/293.12 ---------------------------------------- 1123.19/293.12 1123.19/293.12 (5) 1123.19/293.12 Obligation: 1123.19/293.12 Proved the lower bound n^1 for the following obligation: 1123.19/293.12 1123.19/293.12 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.19/293.12 1123.19/293.12 1123.19/293.12 The TRS R consists of the following rules: 1123.19/293.12 1123.19/293.12 le(s(x), 0) -> false 1123.19/293.12 le(0, y) -> true 1123.19/293.12 le(s(x), s(y)) -> le(x, y) 1123.19/293.12 double(0) -> 0 1123.19/293.12 double(s(x)) -> s(s(double(x))) 1123.19/293.12 log(0) -> logError 1123.19/293.12 log(s(x)) -> loop(s(x), s(0), 0) 1123.19/293.12 loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) 1123.19/293.12 if(true, x, y, z) -> z 1123.19/293.12 if(false, x, y, z) -> loop(x, double(y), s(z)) 1123.19/293.12 maplog(xs) -> mapIter(xs, nil) 1123.19/293.12 mapIter(xs, ys) -> ifmap(isempty(xs), xs, ys) 1123.19/293.12 ifmap(true, xs, ys) -> ys 1123.19/293.12 ifmap(false, xs, ys) -> mapIter(droplast(xs), cons(log(last(xs)), ys)) 1123.19/293.12 isempty(nil) -> true 1123.19/293.12 isempty(cons(x, xs)) -> false 1123.19/293.12 last(nil) -> error 1123.19/293.12 last(cons(x, nil)) -> x 1123.19/293.12 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 1123.19/293.12 droplast(nil) -> nil 1123.19/293.12 droplast(cons(x, nil)) -> nil 1123.19/293.12 droplast(cons(x, cons(y, xs))) -> cons(x, droplast(cons(y, xs))) 1123.19/293.12 a -> b 1123.19/293.12 a -> c 1123.19/293.12 1123.19/293.12 S is empty. 1123.19/293.12 Rewrite Strategy: INNERMOST 1123.19/293.12 ---------------------------------------- 1123.19/293.12 1123.19/293.12 (6) LowerBoundPropagationProof (FINISHED) 1123.19/293.12 Propagated lower bound. 1123.19/293.12 ---------------------------------------- 1123.19/293.12 1123.19/293.12 (7) 1123.19/293.12 BOUNDS(n^1, INF) 1123.19/293.12 1123.19/293.12 ---------------------------------------- 1123.19/293.12 1123.19/293.12 (8) 1123.19/293.12 Obligation: 1123.19/293.12 Analyzing the following TRS for decreasing loops: 1123.19/293.12 1123.19/293.12 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.19/293.12 1123.19/293.12 1123.19/293.12 The TRS R consists of the following rules: 1123.19/293.12 1123.19/293.12 le(s(x), 0) -> false 1123.19/293.12 le(0, y) -> true 1123.19/293.12 le(s(x), s(y)) -> le(x, y) 1123.19/293.12 double(0) -> 0 1123.19/293.12 double(s(x)) -> s(s(double(x))) 1123.19/293.12 log(0) -> logError 1123.19/293.12 log(s(x)) -> loop(s(x), s(0), 0) 1123.19/293.12 loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) 1123.19/293.12 if(true, x, y, z) -> z 1123.19/293.12 if(false, x, y, z) -> loop(x, double(y), s(z)) 1123.19/293.12 maplog(xs) -> mapIter(xs, nil) 1123.19/293.12 mapIter(xs, ys) -> ifmap(isempty(xs), xs, ys) 1123.19/293.12 ifmap(true, xs, ys) -> ys 1123.19/293.12 ifmap(false, xs, ys) -> mapIter(droplast(xs), cons(log(last(xs)), ys)) 1123.19/293.12 isempty(nil) -> true 1123.19/293.12 isempty(cons(x, xs)) -> false 1123.19/293.12 last(nil) -> error 1123.19/293.12 last(cons(x, nil)) -> x 1123.19/293.12 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 1123.19/293.12 droplast(nil) -> nil 1123.19/293.12 droplast(cons(x, nil)) -> nil 1123.19/293.12 droplast(cons(x, cons(y, xs))) -> cons(x, droplast(cons(y, xs))) 1123.19/293.12 a -> b 1123.19/293.12 a -> c 1123.19/293.12 1123.19/293.12 S is empty. 1123.19/293.12 Rewrite Strategy: INNERMOST 1123.39/293.18 EOF