1096.67/292.17 WORST_CASE(Omega(n^1), ?) 1096.97/292.26 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1096.97/292.26 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1096.97/292.26 1096.97/292.26 1096.97/292.26 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1096.97/292.26 1096.97/292.26 (0) CpxTRS 1096.97/292.26 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1096.97/292.26 (2) TRS for Loop Detection 1096.97/292.26 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1096.97/292.26 (4) BEST 1096.97/292.26 (5) proven lower bound 1096.97/292.26 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1096.97/292.26 (7) BOUNDS(n^1, INF) 1096.97/292.26 (8) TRS for Loop Detection 1096.97/292.26 1096.97/292.26 1096.97/292.26 ---------------------------------------- 1096.97/292.26 1096.97/292.26 (0) 1096.97/292.26 Obligation: 1096.97/292.26 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1096.97/292.26 1096.97/292.26 1096.97/292.26 The TRS R consists of the following rules: 1096.97/292.26 1096.97/292.26 a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) 1096.97/292.26 a__fst(pair(XS, YS)) -> mark(XS) 1096.97/292.26 a__snd(pair(XS, YS)) -> mark(YS) 1096.97/292.26 a__splitAt(0, XS) -> pair(nil, mark(XS)) 1096.97/292.26 a__splitAt(s(N), cons(X, XS)) -> a__u(a__splitAt(mark(N), mark(XS)), N, X, XS) 1096.97/292.26 a__u(pair(YS, ZS), N, X, XS) -> pair(cons(mark(X), YS), mark(ZS)) 1096.97/292.26 a__head(cons(N, XS)) -> mark(N) 1096.97/292.26 a__tail(cons(N, XS)) -> mark(XS) 1096.97/292.26 a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 1096.97/292.26 a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 1096.97/292.26 a__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 1096.97/292.26 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 1096.97/292.26 mark(fst(X)) -> a__fst(mark(X)) 1096.97/292.26 mark(snd(X)) -> a__snd(mark(X)) 1096.97/292.26 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 1096.97/292.26 mark(u(X1, X2, X3, X4)) -> a__u(mark(X1), X2, X3, X4) 1096.97/292.26 mark(head(X)) -> a__head(mark(X)) 1096.97/292.26 mark(tail(X)) -> a__tail(mark(X)) 1096.97/292.26 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1096.97/292.26 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 1096.97/292.26 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1096.97/292.26 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1096.97/292.26 mark(s(X)) -> s(mark(X)) 1096.97/292.26 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 1096.97/292.26 mark(0) -> 0 1096.97/292.26 mark(nil) -> nil 1096.97/292.26 a__natsFrom(X) -> natsFrom(X) 1096.97/292.26 a__fst(X) -> fst(X) 1096.97/292.26 a__snd(X) -> snd(X) 1096.97/292.26 a__splitAt(X1, X2) -> splitAt(X1, X2) 1096.97/292.26 a__u(X1, X2, X3, X4) -> u(X1, X2, X3, X4) 1096.97/292.26 a__head(X) -> head(X) 1096.97/292.26 a__tail(X) -> tail(X) 1096.97/292.26 a__sel(X1, X2) -> sel(X1, X2) 1096.97/292.26 a__afterNth(X1, X2) -> afterNth(X1, X2) 1096.97/292.26 a__take(X1, X2) -> take(X1, X2) 1096.97/292.26 1096.97/292.26 S is empty. 1096.97/292.26 Rewrite Strategy: INNERMOST 1096.97/292.26 ---------------------------------------- 1096.97/292.26 1096.97/292.26 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1096.97/292.26 Transformed a relative TRS into a decreasing-loop problem. 1096.97/292.26 ---------------------------------------- 1096.97/292.26 1096.97/292.26 (2) 1096.97/292.26 Obligation: 1096.97/292.26 Analyzing the following TRS for decreasing loops: 1096.97/292.26 1096.97/292.26 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1096.97/292.26 1096.97/292.26 1096.97/292.26 The TRS R consists of the following rules: 1096.97/292.26 1096.97/292.26 a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) 1096.97/292.26 a__fst(pair(XS, YS)) -> mark(XS) 1096.97/292.26 a__snd(pair(XS, YS)) -> mark(YS) 1096.97/292.26 a__splitAt(0, XS) -> pair(nil, mark(XS)) 1096.97/292.26 a__splitAt(s(N), cons(X, XS)) -> a__u(a__splitAt(mark(N), mark(XS)), N, X, XS) 1096.97/292.26 a__u(pair(YS, ZS), N, X, XS) -> pair(cons(mark(X), YS), mark(ZS)) 1096.97/292.26 a__head(cons(N, XS)) -> mark(N) 1096.97/292.26 a__tail(cons(N, XS)) -> mark(XS) 1096.97/292.26 a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 1096.97/292.26 a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 1096.97/292.26 a__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 1096.97/292.26 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 1096.97/292.26 mark(fst(X)) -> a__fst(mark(X)) 1096.97/292.26 mark(snd(X)) -> a__snd(mark(X)) 1096.97/292.26 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 1096.97/292.26 mark(u(X1, X2, X3, X4)) -> a__u(mark(X1), X2, X3, X4) 1096.97/292.26 mark(head(X)) -> a__head(mark(X)) 1096.97/292.26 mark(tail(X)) -> a__tail(mark(X)) 1096.97/292.26 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1096.97/292.26 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 1096.97/292.26 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1096.97/292.26 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1096.97/292.26 mark(s(X)) -> s(mark(X)) 1096.97/292.26 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 1096.97/292.26 mark(0) -> 0 1096.97/292.26 mark(nil) -> nil 1096.97/292.26 a__natsFrom(X) -> natsFrom(X) 1096.97/292.26 a__fst(X) -> fst(X) 1096.97/292.26 a__snd(X) -> snd(X) 1096.97/292.26 a__splitAt(X1, X2) -> splitAt(X1, X2) 1096.97/292.26 a__u(X1, X2, X3, X4) -> u(X1, X2, X3, X4) 1096.97/292.26 a__head(X) -> head(X) 1096.97/292.26 a__tail(X) -> tail(X) 1096.97/292.26 a__sel(X1, X2) -> sel(X1, X2) 1096.97/292.26 a__afterNth(X1, X2) -> afterNth(X1, X2) 1096.97/292.26 a__take(X1, X2) -> take(X1, X2) 1096.97/292.26 1096.97/292.26 S is empty. 1096.97/292.26 Rewrite Strategy: INNERMOST 1096.97/292.26 ---------------------------------------- 1096.97/292.26 1096.97/292.26 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1096.97/292.26 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1096.97/292.26 1096.97/292.26 The rewrite sequence 1096.97/292.26 1096.97/292.26 mark(afterNth(X1, X2)) ->^+ a__afterNth(mark(X1), mark(X2)) 1096.97/292.26 1096.97/292.26 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1096.97/292.26 1096.97/292.26 The pumping substitution is [X1 / afterNth(X1, X2)]. 1096.97/292.26 1096.97/292.26 The result substitution is [ ]. 1096.97/292.26 1096.97/292.26 1096.97/292.26 1096.97/292.26 1096.97/292.26 ---------------------------------------- 1096.97/292.26 1096.97/292.26 (4) 1096.97/292.26 Complex Obligation (BEST) 1096.97/292.26 1096.97/292.26 ---------------------------------------- 1096.97/292.26 1096.97/292.26 (5) 1096.97/292.26 Obligation: 1096.97/292.26 Proved the lower bound n^1 for the following obligation: 1096.97/292.26 1096.97/292.26 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1096.97/292.26 1096.97/292.26 1096.97/292.26 The TRS R consists of the following rules: 1096.97/292.26 1096.97/292.26 a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) 1096.97/292.26 a__fst(pair(XS, YS)) -> mark(XS) 1096.97/292.26 a__snd(pair(XS, YS)) -> mark(YS) 1096.97/292.26 a__splitAt(0, XS) -> pair(nil, mark(XS)) 1096.97/292.26 a__splitAt(s(N), cons(X, XS)) -> a__u(a__splitAt(mark(N), mark(XS)), N, X, XS) 1096.97/292.26 a__u(pair(YS, ZS), N, X, XS) -> pair(cons(mark(X), YS), mark(ZS)) 1096.97/292.26 a__head(cons(N, XS)) -> mark(N) 1096.97/292.26 a__tail(cons(N, XS)) -> mark(XS) 1096.97/292.26 a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 1096.97/292.26 a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 1096.97/292.26 a__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 1096.97/292.26 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 1096.97/292.26 mark(fst(X)) -> a__fst(mark(X)) 1096.97/292.26 mark(snd(X)) -> a__snd(mark(X)) 1096.97/292.26 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 1096.97/292.26 mark(u(X1, X2, X3, X4)) -> a__u(mark(X1), X2, X3, X4) 1096.97/292.26 mark(head(X)) -> a__head(mark(X)) 1096.97/292.26 mark(tail(X)) -> a__tail(mark(X)) 1096.97/292.26 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1096.97/292.26 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 1096.97/292.26 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1096.97/292.26 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1096.97/292.26 mark(s(X)) -> s(mark(X)) 1096.97/292.26 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 1096.97/292.26 mark(0) -> 0 1096.97/292.26 mark(nil) -> nil 1096.97/292.26 a__natsFrom(X) -> natsFrom(X) 1096.97/292.26 a__fst(X) -> fst(X) 1096.97/292.26 a__snd(X) -> snd(X) 1096.97/292.26 a__splitAt(X1, X2) -> splitAt(X1, X2) 1096.97/292.26 a__u(X1, X2, X3, X4) -> u(X1, X2, X3, X4) 1096.97/292.26 a__head(X) -> head(X) 1096.97/292.26 a__tail(X) -> tail(X) 1096.97/292.26 a__sel(X1, X2) -> sel(X1, X2) 1096.97/292.26 a__afterNth(X1, X2) -> afterNth(X1, X2) 1096.97/292.26 a__take(X1, X2) -> take(X1, X2) 1096.97/292.26 1096.97/292.26 S is empty. 1096.97/292.26 Rewrite Strategy: INNERMOST 1096.97/292.26 ---------------------------------------- 1096.97/292.26 1096.97/292.26 (6) LowerBoundPropagationProof (FINISHED) 1096.97/292.26 Propagated lower bound. 1096.97/292.26 ---------------------------------------- 1096.97/292.26 1096.97/292.26 (7) 1096.97/292.26 BOUNDS(n^1, INF) 1096.97/292.26 1096.97/292.26 ---------------------------------------- 1096.97/292.26 1096.97/292.26 (8) 1096.97/292.26 Obligation: 1096.97/292.26 Analyzing the following TRS for decreasing loops: 1096.97/292.26 1096.97/292.26 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1096.97/292.26 1096.97/292.26 1096.97/292.26 The TRS R consists of the following rules: 1096.97/292.26 1096.97/292.26 a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) 1096.97/292.26 a__fst(pair(XS, YS)) -> mark(XS) 1096.97/292.26 a__snd(pair(XS, YS)) -> mark(YS) 1096.97/292.26 a__splitAt(0, XS) -> pair(nil, mark(XS)) 1096.97/292.26 a__splitAt(s(N), cons(X, XS)) -> a__u(a__splitAt(mark(N), mark(XS)), N, X, XS) 1096.97/292.26 a__u(pair(YS, ZS), N, X, XS) -> pair(cons(mark(X), YS), mark(ZS)) 1096.97/292.26 a__head(cons(N, XS)) -> mark(N) 1096.97/292.26 a__tail(cons(N, XS)) -> mark(XS) 1096.97/292.26 a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 1096.97/292.26 a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 1096.97/292.26 a__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 1096.97/292.26 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 1096.97/292.26 mark(fst(X)) -> a__fst(mark(X)) 1096.97/292.26 mark(snd(X)) -> a__snd(mark(X)) 1096.97/292.26 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 1096.97/292.26 mark(u(X1, X2, X3, X4)) -> a__u(mark(X1), X2, X3, X4) 1096.97/292.26 mark(head(X)) -> a__head(mark(X)) 1096.97/292.26 mark(tail(X)) -> a__tail(mark(X)) 1096.97/292.26 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1096.97/292.26 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 1096.97/292.26 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1096.97/292.26 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1096.97/292.26 mark(s(X)) -> s(mark(X)) 1096.97/292.26 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 1096.97/292.26 mark(0) -> 0 1096.97/292.26 mark(nil) -> nil 1096.97/292.26 a__natsFrom(X) -> natsFrom(X) 1096.97/292.26 a__fst(X) -> fst(X) 1096.97/292.26 a__snd(X) -> snd(X) 1096.97/292.26 a__splitAt(X1, X2) -> splitAt(X1, X2) 1096.97/292.26 a__u(X1, X2, X3, X4) -> u(X1, X2, X3, X4) 1096.97/292.26 a__head(X) -> head(X) 1096.97/292.26 a__tail(X) -> tail(X) 1096.97/292.26 a__sel(X1, X2) -> sel(X1, X2) 1096.97/292.26 a__afterNth(X1, X2) -> afterNth(X1, X2) 1096.97/292.26 a__take(X1, X2) -> take(X1, X2) 1096.97/292.26 1096.97/292.26 S is empty. 1096.97/292.26 Rewrite Strategy: INNERMOST 1097.09/292.33 EOF