1132.33/291.57 WORST_CASE(Omega(n^1), ?) 1133.19/291.81 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1133.19/291.81 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1133.19/291.81 1133.19/291.81 1133.19/291.81 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1133.19/291.81 1133.19/291.81 (0) CpxTRS 1133.19/291.81 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1133.19/291.81 (2) TRS for Loop Detection 1133.19/291.81 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1133.19/291.81 (4) BEST 1133.19/291.81 (5) proven lower bound 1133.19/291.81 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1133.19/291.81 (7) BOUNDS(n^1, INF) 1133.19/291.81 (8) TRS for Loop Detection 1133.19/291.81 1133.19/291.81 1133.19/291.81 ---------------------------------------- 1133.19/291.81 1133.19/291.81 (0) 1133.19/291.81 Obligation: 1133.19/291.81 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1133.19/291.81 1133.19/291.81 1133.19/291.81 The TRS R consists of the following rules: 1133.19/291.81 1133.19/291.81 a__U11(tt, N, X, XS) -> a__U12(a__splitAt(mark(N), mark(XS)), X) 1133.19/291.81 a__U12(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 1133.19/291.81 a__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 1133.19/291.81 a__and(tt, X) -> mark(X) 1133.19/291.81 a__fst(pair(X, Y)) -> mark(X) 1133.19/291.81 a__head(cons(N, XS)) -> mark(N) 1133.19/291.81 a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) 1133.19/291.81 a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 1133.19/291.81 a__snd(pair(X, Y)) -> mark(Y) 1133.19/291.81 a__splitAt(0, XS) -> pair(nil, mark(XS)) 1133.19/291.81 a__splitAt(s(N), cons(X, XS)) -> a__U11(tt, N, X, XS) 1133.19/291.81 a__tail(cons(N, XS)) -> mark(XS) 1133.19/291.81 a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 1133.19/291.81 mark(U11(X1, X2, X3, X4)) -> a__U11(mark(X1), X2, X3, X4) 1133.19/291.81 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 1133.19/291.81 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 1133.19/291.81 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 1133.19/291.81 mark(snd(X)) -> a__snd(mark(X)) 1133.19/291.81 mark(and(X1, X2)) -> a__and(mark(X1), X2) 1133.19/291.81 mark(fst(X)) -> a__fst(mark(X)) 1133.19/291.81 mark(head(X)) -> a__head(mark(X)) 1133.19/291.81 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 1133.19/291.81 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1133.19/291.81 mark(tail(X)) -> a__tail(mark(X)) 1133.19/291.81 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1133.19/291.81 mark(tt) -> tt 1133.19/291.81 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 1133.19/291.81 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1133.19/291.81 mark(s(X)) -> s(mark(X)) 1133.19/291.81 mark(0) -> 0 1133.19/291.81 mark(nil) -> nil 1133.19/291.81 a__U11(X1, X2, X3, X4) -> U11(X1, X2, X3, X4) 1133.19/291.81 a__U12(X1, X2) -> U12(X1, X2) 1133.19/291.81 a__splitAt(X1, X2) -> splitAt(X1, X2) 1133.19/291.81 a__afterNth(X1, X2) -> afterNth(X1, X2) 1133.19/291.81 a__snd(X) -> snd(X) 1133.19/291.81 a__and(X1, X2) -> and(X1, X2) 1133.19/291.81 a__fst(X) -> fst(X) 1133.19/291.81 a__head(X) -> head(X) 1133.19/291.81 a__natsFrom(X) -> natsFrom(X) 1133.19/291.81 a__sel(X1, X2) -> sel(X1, X2) 1133.19/291.81 a__tail(X) -> tail(X) 1133.19/291.81 a__take(X1, X2) -> take(X1, X2) 1133.19/291.81 1133.19/291.81 S is empty. 1133.19/291.81 Rewrite Strategy: INNERMOST 1133.19/291.81 ---------------------------------------- 1133.19/291.81 1133.19/291.81 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1133.19/291.81 Transformed a relative TRS into a decreasing-loop problem. 1133.19/291.81 ---------------------------------------- 1133.19/291.81 1133.19/291.81 (2) 1133.19/291.81 Obligation: 1133.19/291.81 Analyzing the following TRS for decreasing loops: 1133.19/291.81 1133.19/291.81 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1133.19/291.81 1133.19/291.81 1133.19/291.81 The TRS R consists of the following rules: 1133.19/291.81 1133.19/291.81 a__U11(tt, N, X, XS) -> a__U12(a__splitAt(mark(N), mark(XS)), X) 1133.19/291.81 a__U12(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 1133.19/291.81 a__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 1133.19/291.81 a__and(tt, X) -> mark(X) 1133.19/291.81 a__fst(pair(X, Y)) -> mark(X) 1133.19/291.81 a__head(cons(N, XS)) -> mark(N) 1133.19/291.81 a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) 1133.19/291.81 a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 1133.19/291.81 a__snd(pair(X, Y)) -> mark(Y) 1133.19/291.81 a__splitAt(0, XS) -> pair(nil, mark(XS)) 1133.19/291.81 a__splitAt(s(N), cons(X, XS)) -> a__U11(tt, N, X, XS) 1133.19/291.81 a__tail(cons(N, XS)) -> mark(XS) 1133.19/291.81 a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 1133.19/291.81 mark(U11(X1, X2, X3, X4)) -> a__U11(mark(X1), X2, X3, X4) 1133.19/291.81 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 1133.19/291.81 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 1133.19/291.81 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 1133.19/291.81 mark(snd(X)) -> a__snd(mark(X)) 1133.19/291.81 mark(and(X1, X2)) -> a__and(mark(X1), X2) 1133.19/291.81 mark(fst(X)) -> a__fst(mark(X)) 1133.19/291.81 mark(head(X)) -> a__head(mark(X)) 1133.19/291.81 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 1133.19/291.81 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1133.19/291.81 mark(tail(X)) -> a__tail(mark(X)) 1133.19/291.81 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1133.19/291.81 mark(tt) -> tt 1133.19/291.81 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 1133.19/291.81 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1133.19/291.81 mark(s(X)) -> s(mark(X)) 1133.19/291.81 mark(0) -> 0 1133.19/291.81 mark(nil) -> nil 1133.19/291.81 a__U11(X1, X2, X3, X4) -> U11(X1, X2, X3, X4) 1133.19/291.81 a__U12(X1, X2) -> U12(X1, X2) 1133.19/291.81 a__splitAt(X1, X2) -> splitAt(X1, X2) 1133.19/291.81 a__afterNth(X1, X2) -> afterNth(X1, X2) 1133.19/291.81 a__snd(X) -> snd(X) 1133.19/291.81 a__and(X1, X2) -> and(X1, X2) 1133.19/291.81 a__fst(X) -> fst(X) 1133.19/291.81 a__head(X) -> head(X) 1133.19/291.81 a__natsFrom(X) -> natsFrom(X) 1133.19/291.81 a__sel(X1, X2) -> sel(X1, X2) 1133.19/291.81 a__tail(X) -> tail(X) 1133.19/291.81 a__take(X1, X2) -> take(X1, X2) 1133.19/291.81 1133.19/291.81 S is empty. 1133.19/291.81 Rewrite Strategy: INNERMOST 1133.19/291.81 ---------------------------------------- 1133.19/291.81 1133.19/291.81 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1133.19/291.81 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1133.19/291.81 1133.19/291.81 The rewrite sequence 1133.19/291.81 1133.19/291.81 mark(afterNth(X1, X2)) ->^+ a__afterNth(mark(X1), mark(X2)) 1133.19/291.81 1133.19/291.81 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1133.19/291.81 1133.19/291.81 The pumping substitution is [X1 / afterNth(X1, X2)]. 1133.19/291.81 1133.19/291.81 The result substitution is [ ]. 1133.19/291.81 1133.19/291.81 1133.19/291.81 1133.19/291.81 1133.19/291.81 ---------------------------------------- 1133.19/291.81 1133.19/291.81 (4) 1133.19/291.81 Complex Obligation (BEST) 1133.19/291.81 1133.19/291.81 ---------------------------------------- 1133.19/291.81 1133.19/291.81 (5) 1133.19/291.81 Obligation: 1133.19/291.81 Proved the lower bound n^1 for the following obligation: 1133.19/291.81 1133.19/291.81 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1133.19/291.81 1133.19/291.81 1133.19/291.81 The TRS R consists of the following rules: 1133.19/291.81 1133.19/291.81 a__U11(tt, N, X, XS) -> a__U12(a__splitAt(mark(N), mark(XS)), X) 1133.19/291.81 a__U12(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 1133.19/291.81 a__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 1133.19/291.81 a__and(tt, X) -> mark(X) 1133.19/291.81 a__fst(pair(X, Y)) -> mark(X) 1133.19/291.81 a__head(cons(N, XS)) -> mark(N) 1133.19/291.81 a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) 1133.19/291.81 a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 1133.19/291.81 a__snd(pair(X, Y)) -> mark(Y) 1133.19/291.81 a__splitAt(0, XS) -> pair(nil, mark(XS)) 1133.19/291.81 a__splitAt(s(N), cons(X, XS)) -> a__U11(tt, N, X, XS) 1133.19/291.81 a__tail(cons(N, XS)) -> mark(XS) 1133.19/291.81 a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 1133.19/291.81 mark(U11(X1, X2, X3, X4)) -> a__U11(mark(X1), X2, X3, X4) 1133.19/291.81 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 1133.19/291.81 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 1133.19/291.81 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 1133.19/291.81 mark(snd(X)) -> a__snd(mark(X)) 1133.19/291.81 mark(and(X1, X2)) -> a__and(mark(X1), X2) 1133.19/291.81 mark(fst(X)) -> a__fst(mark(X)) 1133.19/291.81 mark(head(X)) -> a__head(mark(X)) 1133.19/291.81 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 1133.19/291.81 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1133.19/291.81 mark(tail(X)) -> a__tail(mark(X)) 1133.19/291.81 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1133.19/291.81 mark(tt) -> tt 1133.19/291.81 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 1133.19/291.81 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1133.19/291.81 mark(s(X)) -> s(mark(X)) 1133.19/291.81 mark(0) -> 0 1133.19/291.81 mark(nil) -> nil 1133.19/291.81 a__U11(X1, X2, X3, X4) -> U11(X1, X2, X3, X4) 1133.19/291.81 a__U12(X1, X2) -> U12(X1, X2) 1133.19/291.81 a__splitAt(X1, X2) -> splitAt(X1, X2) 1133.19/291.81 a__afterNth(X1, X2) -> afterNth(X1, X2) 1133.19/291.81 a__snd(X) -> snd(X) 1133.19/291.81 a__and(X1, X2) -> and(X1, X2) 1133.19/291.81 a__fst(X) -> fst(X) 1133.19/291.81 a__head(X) -> head(X) 1133.19/291.81 a__natsFrom(X) -> natsFrom(X) 1133.19/291.81 a__sel(X1, X2) -> sel(X1, X2) 1133.19/291.81 a__tail(X) -> tail(X) 1133.19/291.81 a__take(X1, X2) -> take(X1, X2) 1133.19/291.81 1133.19/291.81 S is empty. 1133.19/291.81 Rewrite Strategy: INNERMOST 1133.19/291.81 ---------------------------------------- 1133.19/291.81 1133.19/291.81 (6) LowerBoundPropagationProof (FINISHED) 1133.19/291.81 Propagated lower bound. 1133.19/291.81 ---------------------------------------- 1133.19/291.81 1133.19/291.81 (7) 1133.19/291.81 BOUNDS(n^1, INF) 1133.19/291.81 1133.19/291.81 ---------------------------------------- 1133.19/291.81 1133.19/291.81 (8) 1133.19/291.81 Obligation: 1133.19/291.81 Analyzing the following TRS for decreasing loops: 1133.19/291.81 1133.19/291.81 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1133.19/291.81 1133.19/291.81 1133.19/291.81 The TRS R consists of the following rules: 1133.19/291.81 1133.19/291.81 a__U11(tt, N, X, XS) -> a__U12(a__splitAt(mark(N), mark(XS)), X) 1133.19/291.81 a__U12(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 1133.19/291.81 a__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 1133.19/291.81 a__and(tt, X) -> mark(X) 1133.19/291.81 a__fst(pair(X, Y)) -> mark(X) 1133.19/291.81 a__head(cons(N, XS)) -> mark(N) 1133.19/291.81 a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) 1133.19/291.81 a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 1133.19/291.81 a__snd(pair(X, Y)) -> mark(Y) 1133.19/291.81 a__splitAt(0, XS) -> pair(nil, mark(XS)) 1133.19/291.81 a__splitAt(s(N), cons(X, XS)) -> a__U11(tt, N, X, XS) 1133.19/291.81 a__tail(cons(N, XS)) -> mark(XS) 1133.19/291.81 a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 1133.19/291.81 mark(U11(X1, X2, X3, X4)) -> a__U11(mark(X1), X2, X3, X4) 1133.19/291.81 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 1133.19/291.81 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 1133.19/291.81 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 1133.19/291.81 mark(snd(X)) -> a__snd(mark(X)) 1133.19/291.81 mark(and(X1, X2)) -> a__and(mark(X1), X2) 1133.19/291.81 mark(fst(X)) -> a__fst(mark(X)) 1133.19/291.81 mark(head(X)) -> a__head(mark(X)) 1133.19/291.81 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 1133.19/291.81 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1133.19/291.81 mark(tail(X)) -> a__tail(mark(X)) 1133.19/291.81 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1133.19/291.81 mark(tt) -> tt 1133.19/291.81 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 1133.19/291.81 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1133.19/291.81 mark(s(X)) -> s(mark(X)) 1133.19/291.81 mark(0) -> 0 1133.19/291.81 mark(nil) -> nil 1133.19/291.81 a__U11(X1, X2, X3, X4) -> U11(X1, X2, X3, X4) 1133.19/291.81 a__U12(X1, X2) -> U12(X1, X2) 1133.19/291.81 a__splitAt(X1, X2) -> splitAt(X1, X2) 1133.19/291.81 a__afterNth(X1, X2) -> afterNth(X1, X2) 1133.19/291.81 a__snd(X) -> snd(X) 1133.19/291.81 a__and(X1, X2) -> and(X1, X2) 1133.19/291.81 a__fst(X) -> fst(X) 1133.19/291.81 a__head(X) -> head(X) 1133.19/291.81 a__natsFrom(X) -> natsFrom(X) 1133.19/291.81 a__sel(X1, X2) -> sel(X1, X2) 1133.19/291.81 a__tail(X) -> tail(X) 1133.19/291.81 a__take(X1, X2) -> take(X1, X2) 1133.19/291.81 1133.19/291.81 S is empty. 1133.19/291.81 Rewrite Strategy: INNERMOST 1133.38/291.88 EOF