309.84/291.50 WORST_CASE(Omega(n^1), ?) 309.84/291.51 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 309.84/291.51 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 309.84/291.51 309.84/291.51 309.84/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 309.84/291.51 309.84/291.51 (0) CpxTRS 309.84/291.51 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 309.84/291.51 (2) TRS for Loop Detection 309.84/291.51 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 309.84/291.51 (4) BEST 309.84/291.51 (5) proven lower bound 309.84/291.51 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 309.84/291.51 (7) BOUNDS(n^1, INF) 309.84/291.51 (8) TRS for Loop Detection 309.84/291.51 309.84/291.51 309.84/291.51 ---------------------------------------- 309.84/291.51 309.84/291.51 (0) 309.84/291.51 Obligation: 309.84/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 309.84/291.51 309.84/291.51 309.84/291.51 The TRS R consists of the following rules: 309.84/291.51 309.84/291.51 le(s(x), 0) -> false 309.84/291.51 le(0, y) -> true 309.84/291.51 le(s(x), s(y)) -> le(x, y) 309.84/291.51 double(0) -> 0 309.84/291.51 double(s(x)) -> s(s(double(x))) 309.84/291.51 log(0) -> logError 309.84/291.51 log(s(x)) -> loop(s(x), s(0), 0) 309.84/291.51 loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) 309.84/291.51 if(true, x, y, z) -> z 309.84/291.51 if(false, x, y, z) -> loop(x, double(y), s(z)) 309.84/291.51 maplog(xs) -> mapIter(xs, nil) 309.84/291.51 mapIter(xs, ys) -> ifmap(isempty(xs), xs, ys) 309.84/291.51 ifmap(true, xs, ys) -> ys 309.84/291.51 ifmap(false, xs, ys) -> mapIter(droplast(xs), cons(log(last(xs)), ys)) 309.84/291.51 isempty(nil) -> true 309.84/291.51 isempty(cons(x, xs)) -> false 309.84/291.51 last(nil) -> error 309.84/291.51 last(cons(x, nil)) -> x 309.84/291.51 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 309.84/291.51 droplast(nil) -> nil 309.84/291.51 droplast(cons(x, nil)) -> nil 309.84/291.51 droplast(cons(x, cons(y, xs))) -> cons(x, droplast(cons(y, xs))) 309.84/291.51 a -> b 309.84/291.51 a -> c 309.84/291.51 309.84/291.51 S is empty. 309.84/291.51 Rewrite Strategy: FULL 309.84/291.51 ---------------------------------------- 309.84/291.51 309.84/291.51 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 309.84/291.51 Transformed a relative TRS into a decreasing-loop problem. 309.84/291.51 ---------------------------------------- 309.84/291.51 309.84/291.51 (2) 309.84/291.51 Obligation: 309.84/291.51 Analyzing the following TRS for decreasing loops: 309.84/291.51 309.84/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 309.84/291.51 309.84/291.51 309.84/291.51 The TRS R consists of the following rules: 309.84/291.51 309.84/291.51 le(s(x), 0) -> false 309.84/291.51 le(0, y) -> true 309.84/291.51 le(s(x), s(y)) -> le(x, y) 309.84/291.51 double(0) -> 0 309.84/291.51 double(s(x)) -> s(s(double(x))) 309.84/291.51 log(0) -> logError 309.84/291.51 log(s(x)) -> loop(s(x), s(0), 0) 309.84/291.51 loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) 309.84/291.51 if(true, x, y, z) -> z 309.84/291.51 if(false, x, y, z) -> loop(x, double(y), s(z)) 309.84/291.51 maplog(xs) -> mapIter(xs, nil) 309.84/291.51 mapIter(xs, ys) -> ifmap(isempty(xs), xs, ys) 309.84/291.51 ifmap(true, xs, ys) -> ys 309.84/291.51 ifmap(false, xs, ys) -> mapIter(droplast(xs), cons(log(last(xs)), ys)) 309.84/291.51 isempty(nil) -> true 309.84/291.51 isempty(cons(x, xs)) -> false 309.84/291.51 last(nil) -> error 309.84/291.51 last(cons(x, nil)) -> x 309.84/291.51 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 309.84/291.51 droplast(nil) -> nil 309.84/291.51 droplast(cons(x, nil)) -> nil 309.84/291.51 droplast(cons(x, cons(y, xs))) -> cons(x, droplast(cons(y, xs))) 309.84/291.51 a -> b 309.84/291.51 a -> c 309.84/291.51 309.84/291.51 S is empty. 309.84/291.51 Rewrite Strategy: FULL 309.84/291.51 ---------------------------------------- 309.84/291.51 309.84/291.51 (3) DecreasingLoopProof (LOWER BOUND(ID)) 309.84/291.51 The following loop(s) give(s) rise to the lower bound Omega(n^1): 309.84/291.51 309.84/291.51 The rewrite sequence 309.84/291.51 309.84/291.51 droplast(cons(x, cons(y, xs))) ->^+ cons(x, droplast(cons(y, xs))) 309.84/291.51 309.84/291.51 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 309.84/291.51 309.84/291.51 The pumping substitution is [xs / cons(y, xs)]. 309.84/291.51 309.84/291.51 The result substitution is [x / y]. 309.84/291.51 309.84/291.51 309.84/291.51 309.84/291.51 309.84/291.51 ---------------------------------------- 309.84/291.51 309.84/291.51 (4) 309.84/291.51 Complex Obligation (BEST) 309.84/291.51 309.84/291.51 ---------------------------------------- 309.84/291.51 309.84/291.51 (5) 309.84/291.51 Obligation: 309.84/291.51 Proved the lower bound n^1 for the following obligation: 309.84/291.51 309.84/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 309.84/291.51 309.84/291.51 309.84/291.51 The TRS R consists of the following rules: 309.84/291.51 309.84/291.51 le(s(x), 0) -> false 309.84/291.51 le(0, y) -> true 309.84/291.51 le(s(x), s(y)) -> le(x, y) 309.84/291.51 double(0) -> 0 309.84/291.51 double(s(x)) -> s(s(double(x))) 309.84/291.51 log(0) -> logError 309.84/291.51 log(s(x)) -> loop(s(x), s(0), 0) 309.84/291.51 loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) 309.84/291.51 if(true, x, y, z) -> z 309.84/291.51 if(false, x, y, z) -> loop(x, double(y), s(z)) 309.84/291.51 maplog(xs) -> mapIter(xs, nil) 309.84/291.51 mapIter(xs, ys) -> ifmap(isempty(xs), xs, ys) 309.84/291.51 ifmap(true, xs, ys) -> ys 309.84/291.51 ifmap(false, xs, ys) -> mapIter(droplast(xs), cons(log(last(xs)), ys)) 309.84/291.51 isempty(nil) -> true 309.84/291.51 isempty(cons(x, xs)) -> false 309.84/291.51 last(nil) -> error 309.84/291.51 last(cons(x, nil)) -> x 309.84/291.51 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 309.84/291.51 droplast(nil) -> nil 309.84/291.51 droplast(cons(x, nil)) -> nil 309.84/291.51 droplast(cons(x, cons(y, xs))) -> cons(x, droplast(cons(y, xs))) 309.84/291.51 a -> b 309.84/291.51 a -> c 309.84/291.51 309.84/291.51 S is empty. 309.84/291.51 Rewrite Strategy: FULL 309.84/291.51 ---------------------------------------- 309.84/291.51 309.84/291.51 (6) LowerBoundPropagationProof (FINISHED) 309.84/291.51 Propagated lower bound. 309.84/291.51 ---------------------------------------- 309.84/291.51 309.84/291.51 (7) 309.84/291.51 BOUNDS(n^1, INF) 309.84/291.51 309.84/291.51 ---------------------------------------- 309.84/291.51 309.84/291.51 (8) 309.84/291.51 Obligation: 309.84/291.51 Analyzing the following TRS for decreasing loops: 309.84/291.51 309.84/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 309.84/291.51 309.84/291.51 309.84/291.51 The TRS R consists of the following rules: 309.84/291.51 309.84/291.51 le(s(x), 0) -> false 309.84/291.51 le(0, y) -> true 309.84/291.51 le(s(x), s(y)) -> le(x, y) 309.84/291.51 double(0) -> 0 309.84/291.51 double(s(x)) -> s(s(double(x))) 309.84/291.51 log(0) -> logError 309.84/291.51 log(s(x)) -> loop(s(x), s(0), 0) 309.84/291.51 loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) 309.84/291.51 if(true, x, y, z) -> z 309.84/291.51 if(false, x, y, z) -> loop(x, double(y), s(z)) 309.84/291.51 maplog(xs) -> mapIter(xs, nil) 309.84/291.51 mapIter(xs, ys) -> ifmap(isempty(xs), xs, ys) 309.84/291.51 ifmap(true, xs, ys) -> ys 309.84/291.51 ifmap(false, xs, ys) -> mapIter(droplast(xs), cons(log(last(xs)), ys)) 309.84/291.51 isempty(nil) -> true 309.84/291.51 isempty(cons(x, xs)) -> false 309.84/291.51 last(nil) -> error 309.84/291.51 last(cons(x, nil)) -> x 309.84/291.51 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 309.84/291.51 droplast(nil) -> nil 309.84/291.51 droplast(cons(x, nil)) -> nil 309.84/291.51 droplast(cons(x, cons(y, xs))) -> cons(x, droplast(cons(y, xs))) 309.84/291.51 a -> b 309.84/291.51 a -> c 309.84/291.51 309.84/291.51 S is empty. 309.84/291.51 Rewrite Strategy: FULL 309.94/291.54 EOF