1156.97/295.52 WORST_CASE(Omega(n^1), ?) 1157.16/295.56 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1157.16/295.56 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1157.16/295.56 1157.16/295.56 1157.16/295.56 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1157.16/295.56 1157.16/295.56 (0) CpxTRS 1157.16/295.56 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1157.16/295.56 (2) TRS for Loop Detection 1157.16/295.56 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1157.16/295.56 (4) BEST 1157.16/295.56 (5) proven lower bound 1157.16/295.56 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1157.16/295.56 (7) BOUNDS(n^1, INF) 1157.16/295.56 (8) TRS for Loop Detection 1157.16/295.56 1157.16/295.56 1157.16/295.56 ---------------------------------------- 1157.16/295.56 1157.16/295.56 (0) 1157.16/295.56 Obligation: 1157.16/295.56 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1157.16/295.56 1157.16/295.56 1157.16/295.56 The TRS R consists of the following rules: 1157.16/295.56 1157.16/295.56 a__U101(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 1157.16/295.56 a__U11(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 1157.16/295.56 a__U21(tt, X) -> mark(X) 1157.16/295.56 a__U31(tt, N) -> mark(N) 1157.16/295.56 a__U41(tt, N) -> cons(mark(N), natsFrom(s(N))) 1157.16/295.56 a__U51(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 1157.16/295.56 a__U61(tt, Y) -> mark(Y) 1157.16/295.56 a__U71(tt, XS) -> pair(nil, mark(XS)) 1157.16/295.56 a__U81(tt, N, X, XS) -> a__U82(a__splitAt(mark(N), mark(XS)), X) 1157.16/295.56 a__U82(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 1157.16/295.56 a__U91(tt, XS) -> mark(XS) 1157.16/295.56 a__afterNth(N, XS) -> a__U11(a__and(a__isNatural(N), isLNat(XS)), N, XS) 1157.16/295.56 a__and(tt, X) -> mark(X) 1157.16/295.56 a__fst(pair(X, Y)) -> a__U21(a__and(a__isLNat(X), isLNat(Y)), X) 1157.16/295.56 a__head(cons(N, XS)) -> a__U31(a__and(a__isNatural(N), isLNat(XS)), N) 1157.16/295.56 a__isLNat(nil) -> tt 1157.16/295.56 a__isLNat(afterNth(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__isLNat(cons(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__isLNat(fst(V1)) -> a__isPLNat(V1) 1157.16/295.56 a__isLNat(natsFrom(V1)) -> a__isNatural(V1) 1157.16/295.56 a__isLNat(snd(V1)) -> a__isPLNat(V1) 1157.16/295.56 a__isLNat(tail(V1)) -> a__isLNat(V1) 1157.16/295.56 a__isLNat(take(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__isNatural(0) -> tt 1157.16/295.56 a__isNatural(head(V1)) -> a__isLNat(V1) 1157.16/295.56 a__isNatural(s(V1)) -> a__isNatural(V1) 1157.16/295.56 a__isNatural(sel(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__isPLNat(pair(V1, V2)) -> a__and(a__isLNat(V1), isLNat(V2)) 1157.16/295.56 a__isPLNat(splitAt(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__natsFrom(N) -> a__U41(a__isNatural(N), N) 1157.16/295.56 a__sel(N, XS) -> a__U51(a__and(a__isNatural(N), isLNat(XS)), N, XS) 1157.16/295.56 a__snd(pair(X, Y)) -> a__U61(a__and(a__isLNat(X), isLNat(Y)), Y) 1157.16/295.56 a__splitAt(0, XS) -> a__U71(a__isLNat(XS), XS) 1157.16/295.56 a__splitAt(s(N), cons(X, XS)) -> a__U81(a__and(a__isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS) 1157.16/295.56 a__tail(cons(N, XS)) -> a__U91(a__and(a__isNatural(N), isLNat(XS)), XS) 1157.16/295.56 a__take(N, XS) -> a__U101(a__and(a__isNatural(N), isLNat(XS)), N, XS) 1157.16/295.56 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 1157.16/295.56 mark(fst(X)) -> a__fst(mark(X)) 1157.16/295.56 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 1157.16/295.56 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 1157.16/295.56 mark(snd(X)) -> a__snd(mark(X)) 1157.16/295.56 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1157.16/295.56 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1157.16/295.56 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 1157.16/295.56 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 1157.16/295.56 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 1157.16/295.56 mark(head(X)) -> a__head(mark(X)) 1157.16/295.56 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 1157.16/295.56 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 1157.16/295.56 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 1157.16/295.56 mark(U81(X1, X2, X3, X4)) -> a__U81(mark(X1), X2, X3, X4) 1157.16/295.56 mark(U82(X1, X2)) -> a__U82(mark(X1), X2) 1157.16/295.56 mark(U91(X1, X2)) -> a__U91(mark(X1), X2) 1157.16/295.56 mark(and(X1, X2)) -> a__and(mark(X1), X2) 1157.16/295.56 mark(isNatural(X)) -> a__isNatural(X) 1157.16/295.56 mark(isLNat(X)) -> a__isLNat(X) 1157.16/295.56 mark(isPLNat(X)) -> a__isPLNat(X) 1157.16/295.56 mark(tail(X)) -> a__tail(mark(X)) 1157.16/295.56 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1157.16/295.56 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1157.16/295.56 mark(tt) -> tt 1157.16/295.56 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1157.16/295.56 mark(s(X)) -> s(mark(X)) 1157.16/295.56 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 1157.16/295.56 mark(nil) -> nil 1157.16/295.56 mark(0) -> 0 1157.16/295.56 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 1157.16/295.56 a__fst(X) -> fst(X) 1157.16/295.56 a__splitAt(X1, X2) -> splitAt(X1, X2) 1157.16/295.56 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 1157.16/295.56 a__snd(X) -> snd(X) 1157.16/295.56 a__U21(X1, X2) -> U21(X1, X2) 1157.16/295.56 a__U31(X1, X2) -> U31(X1, X2) 1157.16/295.56 a__U41(X1, X2) -> U41(X1, X2) 1157.16/295.56 a__natsFrom(X) -> natsFrom(X) 1157.16/295.56 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 1157.16/295.56 a__head(X) -> head(X) 1157.16/295.56 a__afterNth(X1, X2) -> afterNth(X1, X2) 1157.16/295.56 a__U61(X1, X2) -> U61(X1, X2) 1157.16/295.56 a__U71(X1, X2) -> U71(X1, X2) 1157.16/295.56 a__U81(X1, X2, X3, X4) -> U81(X1, X2, X3, X4) 1157.16/295.56 a__U82(X1, X2) -> U82(X1, X2) 1157.16/295.56 a__U91(X1, X2) -> U91(X1, X2) 1157.16/295.56 a__and(X1, X2) -> and(X1, X2) 1157.16/295.56 a__isNatural(X) -> isNatural(X) 1157.16/295.56 a__isLNat(X) -> isLNat(X) 1157.16/295.56 a__isPLNat(X) -> isPLNat(X) 1157.16/295.56 a__tail(X) -> tail(X) 1157.16/295.56 a__take(X1, X2) -> take(X1, X2) 1157.16/295.56 a__sel(X1, X2) -> sel(X1, X2) 1157.16/295.56 1157.16/295.56 S is empty. 1157.16/295.56 Rewrite Strategy: INNERMOST 1157.16/295.56 ---------------------------------------- 1157.16/295.56 1157.16/295.56 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1157.16/295.56 Transformed a relative TRS into a decreasing-loop problem. 1157.16/295.56 ---------------------------------------- 1157.16/295.56 1157.16/295.56 (2) 1157.16/295.56 Obligation: 1157.16/295.56 Analyzing the following TRS for decreasing loops: 1157.16/295.56 1157.16/295.56 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1157.16/295.56 1157.16/295.56 1157.16/295.56 The TRS R consists of the following rules: 1157.16/295.56 1157.16/295.56 a__U101(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 1157.16/295.56 a__U11(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 1157.16/295.56 a__U21(tt, X) -> mark(X) 1157.16/295.56 a__U31(tt, N) -> mark(N) 1157.16/295.56 a__U41(tt, N) -> cons(mark(N), natsFrom(s(N))) 1157.16/295.56 a__U51(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 1157.16/295.56 a__U61(tt, Y) -> mark(Y) 1157.16/295.56 a__U71(tt, XS) -> pair(nil, mark(XS)) 1157.16/295.56 a__U81(tt, N, X, XS) -> a__U82(a__splitAt(mark(N), mark(XS)), X) 1157.16/295.56 a__U82(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 1157.16/295.56 a__U91(tt, XS) -> mark(XS) 1157.16/295.56 a__afterNth(N, XS) -> a__U11(a__and(a__isNatural(N), isLNat(XS)), N, XS) 1157.16/295.56 a__and(tt, X) -> mark(X) 1157.16/295.56 a__fst(pair(X, Y)) -> a__U21(a__and(a__isLNat(X), isLNat(Y)), X) 1157.16/295.56 a__head(cons(N, XS)) -> a__U31(a__and(a__isNatural(N), isLNat(XS)), N) 1157.16/295.56 a__isLNat(nil) -> tt 1157.16/295.56 a__isLNat(afterNth(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__isLNat(cons(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__isLNat(fst(V1)) -> a__isPLNat(V1) 1157.16/295.56 a__isLNat(natsFrom(V1)) -> a__isNatural(V1) 1157.16/295.56 a__isLNat(snd(V1)) -> a__isPLNat(V1) 1157.16/295.56 a__isLNat(tail(V1)) -> a__isLNat(V1) 1157.16/295.56 a__isLNat(take(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__isNatural(0) -> tt 1157.16/295.56 a__isNatural(head(V1)) -> a__isLNat(V1) 1157.16/295.56 a__isNatural(s(V1)) -> a__isNatural(V1) 1157.16/295.56 a__isNatural(sel(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__isPLNat(pair(V1, V2)) -> a__and(a__isLNat(V1), isLNat(V2)) 1157.16/295.56 a__isPLNat(splitAt(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__natsFrom(N) -> a__U41(a__isNatural(N), N) 1157.16/295.56 a__sel(N, XS) -> a__U51(a__and(a__isNatural(N), isLNat(XS)), N, XS) 1157.16/295.56 a__snd(pair(X, Y)) -> a__U61(a__and(a__isLNat(X), isLNat(Y)), Y) 1157.16/295.56 a__splitAt(0, XS) -> a__U71(a__isLNat(XS), XS) 1157.16/295.56 a__splitAt(s(N), cons(X, XS)) -> a__U81(a__and(a__isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS) 1157.16/295.56 a__tail(cons(N, XS)) -> a__U91(a__and(a__isNatural(N), isLNat(XS)), XS) 1157.16/295.56 a__take(N, XS) -> a__U101(a__and(a__isNatural(N), isLNat(XS)), N, XS) 1157.16/295.56 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 1157.16/295.56 mark(fst(X)) -> a__fst(mark(X)) 1157.16/295.56 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 1157.16/295.56 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 1157.16/295.56 mark(snd(X)) -> a__snd(mark(X)) 1157.16/295.56 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1157.16/295.56 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1157.16/295.56 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 1157.16/295.56 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 1157.16/295.56 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 1157.16/295.56 mark(head(X)) -> a__head(mark(X)) 1157.16/295.56 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 1157.16/295.56 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 1157.16/295.56 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 1157.16/295.56 mark(U81(X1, X2, X3, X4)) -> a__U81(mark(X1), X2, X3, X4) 1157.16/295.56 mark(U82(X1, X2)) -> a__U82(mark(X1), X2) 1157.16/295.56 mark(U91(X1, X2)) -> a__U91(mark(X1), X2) 1157.16/295.56 mark(and(X1, X2)) -> a__and(mark(X1), X2) 1157.16/295.56 mark(isNatural(X)) -> a__isNatural(X) 1157.16/295.56 mark(isLNat(X)) -> a__isLNat(X) 1157.16/295.56 mark(isPLNat(X)) -> a__isPLNat(X) 1157.16/295.56 mark(tail(X)) -> a__tail(mark(X)) 1157.16/295.56 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1157.16/295.56 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1157.16/295.56 mark(tt) -> tt 1157.16/295.56 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1157.16/295.56 mark(s(X)) -> s(mark(X)) 1157.16/295.56 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 1157.16/295.56 mark(nil) -> nil 1157.16/295.56 mark(0) -> 0 1157.16/295.56 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 1157.16/295.56 a__fst(X) -> fst(X) 1157.16/295.56 a__splitAt(X1, X2) -> splitAt(X1, X2) 1157.16/295.56 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 1157.16/295.56 a__snd(X) -> snd(X) 1157.16/295.56 a__U21(X1, X2) -> U21(X1, X2) 1157.16/295.56 a__U31(X1, X2) -> U31(X1, X2) 1157.16/295.56 a__U41(X1, X2) -> U41(X1, X2) 1157.16/295.56 a__natsFrom(X) -> natsFrom(X) 1157.16/295.56 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 1157.16/295.56 a__head(X) -> head(X) 1157.16/295.56 a__afterNth(X1, X2) -> afterNth(X1, X2) 1157.16/295.56 a__U61(X1, X2) -> U61(X1, X2) 1157.16/295.56 a__U71(X1, X2) -> U71(X1, X2) 1157.16/295.56 a__U81(X1, X2, X3, X4) -> U81(X1, X2, X3, X4) 1157.16/295.56 a__U82(X1, X2) -> U82(X1, X2) 1157.16/295.56 a__U91(X1, X2) -> U91(X1, X2) 1157.16/295.56 a__and(X1, X2) -> and(X1, X2) 1157.16/295.56 a__isNatural(X) -> isNatural(X) 1157.16/295.56 a__isLNat(X) -> isLNat(X) 1157.16/295.56 a__isPLNat(X) -> isPLNat(X) 1157.16/295.56 a__tail(X) -> tail(X) 1157.16/295.56 a__take(X1, X2) -> take(X1, X2) 1157.16/295.56 a__sel(X1, X2) -> sel(X1, X2) 1157.16/295.56 1157.16/295.56 S is empty. 1157.16/295.56 Rewrite Strategy: INNERMOST 1157.16/295.56 ---------------------------------------- 1157.16/295.56 1157.16/295.56 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1157.16/295.56 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1157.16/295.56 1157.16/295.56 The rewrite sequence 1157.16/295.56 1157.16/295.56 mark(afterNth(X1, X2)) ->^+ a__afterNth(mark(X1), mark(X2)) 1157.16/295.56 1157.16/295.56 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1157.16/295.56 1157.16/295.56 The pumping substitution is [X1 / afterNth(X1, X2)]. 1157.16/295.56 1157.16/295.56 The result substitution is [ ]. 1157.16/295.56 1157.16/295.56 1157.16/295.56 1157.16/295.56 1157.16/295.56 ---------------------------------------- 1157.16/295.56 1157.16/295.56 (4) 1157.16/295.56 Complex Obligation (BEST) 1157.16/295.56 1157.16/295.56 ---------------------------------------- 1157.16/295.56 1157.16/295.56 (5) 1157.16/295.56 Obligation: 1157.16/295.56 Proved the lower bound n^1 for the following obligation: 1157.16/295.56 1157.16/295.56 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1157.16/295.56 1157.16/295.56 1157.16/295.56 The TRS R consists of the following rules: 1157.16/295.56 1157.16/295.56 a__U101(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 1157.16/295.56 a__U11(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 1157.16/295.56 a__U21(tt, X) -> mark(X) 1157.16/295.56 a__U31(tt, N) -> mark(N) 1157.16/295.56 a__U41(tt, N) -> cons(mark(N), natsFrom(s(N))) 1157.16/295.56 a__U51(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 1157.16/295.56 a__U61(tt, Y) -> mark(Y) 1157.16/295.56 a__U71(tt, XS) -> pair(nil, mark(XS)) 1157.16/295.56 a__U81(tt, N, X, XS) -> a__U82(a__splitAt(mark(N), mark(XS)), X) 1157.16/295.56 a__U82(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 1157.16/295.56 a__U91(tt, XS) -> mark(XS) 1157.16/295.56 a__afterNth(N, XS) -> a__U11(a__and(a__isNatural(N), isLNat(XS)), N, XS) 1157.16/295.56 a__and(tt, X) -> mark(X) 1157.16/295.56 a__fst(pair(X, Y)) -> a__U21(a__and(a__isLNat(X), isLNat(Y)), X) 1157.16/295.56 a__head(cons(N, XS)) -> a__U31(a__and(a__isNatural(N), isLNat(XS)), N) 1157.16/295.56 a__isLNat(nil) -> tt 1157.16/295.56 a__isLNat(afterNth(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__isLNat(cons(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__isLNat(fst(V1)) -> a__isPLNat(V1) 1157.16/295.56 a__isLNat(natsFrom(V1)) -> a__isNatural(V1) 1157.16/295.56 a__isLNat(snd(V1)) -> a__isPLNat(V1) 1157.16/295.56 a__isLNat(tail(V1)) -> a__isLNat(V1) 1157.16/295.56 a__isLNat(take(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__isNatural(0) -> tt 1157.16/295.56 a__isNatural(head(V1)) -> a__isLNat(V1) 1157.16/295.56 a__isNatural(s(V1)) -> a__isNatural(V1) 1157.16/295.56 a__isNatural(sel(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__isPLNat(pair(V1, V2)) -> a__and(a__isLNat(V1), isLNat(V2)) 1157.16/295.56 a__isPLNat(splitAt(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__natsFrom(N) -> a__U41(a__isNatural(N), N) 1157.16/295.56 a__sel(N, XS) -> a__U51(a__and(a__isNatural(N), isLNat(XS)), N, XS) 1157.16/295.56 a__snd(pair(X, Y)) -> a__U61(a__and(a__isLNat(X), isLNat(Y)), Y) 1157.16/295.56 a__splitAt(0, XS) -> a__U71(a__isLNat(XS), XS) 1157.16/295.56 a__splitAt(s(N), cons(X, XS)) -> a__U81(a__and(a__isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS) 1157.16/295.56 a__tail(cons(N, XS)) -> a__U91(a__and(a__isNatural(N), isLNat(XS)), XS) 1157.16/295.56 a__take(N, XS) -> a__U101(a__and(a__isNatural(N), isLNat(XS)), N, XS) 1157.16/295.56 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 1157.16/295.56 mark(fst(X)) -> a__fst(mark(X)) 1157.16/295.56 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 1157.16/295.56 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 1157.16/295.56 mark(snd(X)) -> a__snd(mark(X)) 1157.16/295.56 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1157.16/295.56 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1157.16/295.56 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 1157.16/295.56 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 1157.16/295.56 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 1157.16/295.56 mark(head(X)) -> a__head(mark(X)) 1157.16/295.56 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 1157.16/295.56 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 1157.16/295.56 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 1157.16/295.56 mark(U81(X1, X2, X3, X4)) -> a__U81(mark(X1), X2, X3, X4) 1157.16/295.56 mark(U82(X1, X2)) -> a__U82(mark(X1), X2) 1157.16/295.56 mark(U91(X1, X2)) -> a__U91(mark(X1), X2) 1157.16/295.56 mark(and(X1, X2)) -> a__and(mark(X1), X2) 1157.16/295.56 mark(isNatural(X)) -> a__isNatural(X) 1157.16/295.56 mark(isLNat(X)) -> a__isLNat(X) 1157.16/295.56 mark(isPLNat(X)) -> a__isPLNat(X) 1157.16/295.56 mark(tail(X)) -> a__tail(mark(X)) 1157.16/295.56 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1157.16/295.56 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1157.16/295.56 mark(tt) -> tt 1157.16/295.56 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1157.16/295.56 mark(s(X)) -> s(mark(X)) 1157.16/295.56 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 1157.16/295.56 mark(nil) -> nil 1157.16/295.56 mark(0) -> 0 1157.16/295.56 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 1157.16/295.56 a__fst(X) -> fst(X) 1157.16/295.56 a__splitAt(X1, X2) -> splitAt(X1, X2) 1157.16/295.56 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 1157.16/295.56 a__snd(X) -> snd(X) 1157.16/295.56 a__U21(X1, X2) -> U21(X1, X2) 1157.16/295.56 a__U31(X1, X2) -> U31(X1, X2) 1157.16/295.56 a__U41(X1, X2) -> U41(X1, X2) 1157.16/295.56 a__natsFrom(X) -> natsFrom(X) 1157.16/295.56 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 1157.16/295.56 a__head(X) -> head(X) 1157.16/295.56 a__afterNth(X1, X2) -> afterNth(X1, X2) 1157.16/295.56 a__U61(X1, X2) -> U61(X1, X2) 1157.16/295.56 a__U71(X1, X2) -> U71(X1, X2) 1157.16/295.56 a__U81(X1, X2, X3, X4) -> U81(X1, X2, X3, X4) 1157.16/295.56 a__U82(X1, X2) -> U82(X1, X2) 1157.16/295.56 a__U91(X1, X2) -> U91(X1, X2) 1157.16/295.56 a__and(X1, X2) -> and(X1, X2) 1157.16/295.56 a__isNatural(X) -> isNatural(X) 1157.16/295.56 a__isLNat(X) -> isLNat(X) 1157.16/295.56 a__isPLNat(X) -> isPLNat(X) 1157.16/295.56 a__tail(X) -> tail(X) 1157.16/295.56 a__take(X1, X2) -> take(X1, X2) 1157.16/295.56 a__sel(X1, X2) -> sel(X1, X2) 1157.16/295.56 1157.16/295.56 S is empty. 1157.16/295.56 Rewrite Strategy: INNERMOST 1157.16/295.56 ---------------------------------------- 1157.16/295.56 1157.16/295.56 (6) LowerBoundPropagationProof (FINISHED) 1157.16/295.56 Propagated lower bound. 1157.16/295.56 ---------------------------------------- 1157.16/295.56 1157.16/295.56 (7) 1157.16/295.56 BOUNDS(n^1, INF) 1157.16/295.56 1157.16/295.56 ---------------------------------------- 1157.16/295.56 1157.16/295.56 (8) 1157.16/295.56 Obligation: 1157.16/295.56 Analyzing the following TRS for decreasing loops: 1157.16/295.56 1157.16/295.56 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1157.16/295.56 1157.16/295.56 1157.16/295.56 The TRS R consists of the following rules: 1157.16/295.56 1157.16/295.56 a__U101(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 1157.16/295.56 a__U11(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 1157.16/295.56 a__U21(tt, X) -> mark(X) 1157.16/295.56 a__U31(tt, N) -> mark(N) 1157.16/295.56 a__U41(tt, N) -> cons(mark(N), natsFrom(s(N))) 1157.16/295.56 a__U51(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 1157.16/295.56 a__U61(tt, Y) -> mark(Y) 1157.16/295.56 a__U71(tt, XS) -> pair(nil, mark(XS)) 1157.16/295.56 a__U81(tt, N, X, XS) -> a__U82(a__splitAt(mark(N), mark(XS)), X) 1157.16/295.56 a__U82(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 1157.16/295.56 a__U91(tt, XS) -> mark(XS) 1157.16/295.56 a__afterNth(N, XS) -> a__U11(a__and(a__isNatural(N), isLNat(XS)), N, XS) 1157.16/295.56 a__and(tt, X) -> mark(X) 1157.16/295.56 a__fst(pair(X, Y)) -> a__U21(a__and(a__isLNat(X), isLNat(Y)), X) 1157.16/295.56 a__head(cons(N, XS)) -> a__U31(a__and(a__isNatural(N), isLNat(XS)), N) 1157.16/295.56 a__isLNat(nil) -> tt 1157.16/295.56 a__isLNat(afterNth(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__isLNat(cons(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__isLNat(fst(V1)) -> a__isPLNat(V1) 1157.16/295.56 a__isLNat(natsFrom(V1)) -> a__isNatural(V1) 1157.16/295.56 a__isLNat(snd(V1)) -> a__isPLNat(V1) 1157.16/295.56 a__isLNat(tail(V1)) -> a__isLNat(V1) 1157.16/295.56 a__isLNat(take(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__isNatural(0) -> tt 1157.16/295.56 a__isNatural(head(V1)) -> a__isLNat(V1) 1157.16/295.56 a__isNatural(s(V1)) -> a__isNatural(V1) 1157.16/295.56 a__isNatural(sel(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__isPLNat(pair(V1, V2)) -> a__and(a__isLNat(V1), isLNat(V2)) 1157.16/295.56 a__isPLNat(splitAt(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2)) 1157.16/295.56 a__natsFrom(N) -> a__U41(a__isNatural(N), N) 1157.16/295.56 a__sel(N, XS) -> a__U51(a__and(a__isNatural(N), isLNat(XS)), N, XS) 1157.16/295.56 a__snd(pair(X, Y)) -> a__U61(a__and(a__isLNat(X), isLNat(Y)), Y) 1157.16/295.56 a__splitAt(0, XS) -> a__U71(a__isLNat(XS), XS) 1157.16/295.56 a__splitAt(s(N), cons(X, XS)) -> a__U81(a__and(a__isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS) 1157.16/295.56 a__tail(cons(N, XS)) -> a__U91(a__and(a__isNatural(N), isLNat(XS)), XS) 1157.16/295.56 a__take(N, XS) -> a__U101(a__and(a__isNatural(N), isLNat(XS)), N, XS) 1157.16/295.56 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 1157.16/295.56 mark(fst(X)) -> a__fst(mark(X)) 1157.16/295.56 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 1157.16/295.56 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 1157.16/295.56 mark(snd(X)) -> a__snd(mark(X)) 1157.16/295.56 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1157.16/295.56 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1157.16/295.56 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 1157.16/295.56 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 1157.16/295.56 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 1157.16/295.56 mark(head(X)) -> a__head(mark(X)) 1157.16/295.56 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 1157.16/295.56 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 1157.16/295.56 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 1157.16/295.56 mark(U81(X1, X2, X3, X4)) -> a__U81(mark(X1), X2, X3, X4) 1157.16/295.56 mark(U82(X1, X2)) -> a__U82(mark(X1), X2) 1157.16/295.56 mark(U91(X1, X2)) -> a__U91(mark(X1), X2) 1157.16/295.56 mark(and(X1, X2)) -> a__and(mark(X1), X2) 1157.16/295.56 mark(isNatural(X)) -> a__isNatural(X) 1157.16/295.56 mark(isLNat(X)) -> a__isLNat(X) 1157.16/295.56 mark(isPLNat(X)) -> a__isPLNat(X) 1157.16/295.56 mark(tail(X)) -> a__tail(mark(X)) 1157.16/295.56 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1157.16/295.56 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1157.16/295.56 mark(tt) -> tt 1157.16/295.56 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1157.16/295.56 mark(s(X)) -> s(mark(X)) 1157.16/295.56 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 1157.16/295.56 mark(nil) -> nil 1157.16/295.56 mark(0) -> 0 1157.16/295.56 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 1157.16/295.56 a__fst(X) -> fst(X) 1157.16/295.56 a__splitAt(X1, X2) -> splitAt(X1, X2) 1157.16/295.56 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 1157.16/295.56 a__snd(X) -> snd(X) 1157.16/295.56 a__U21(X1, X2) -> U21(X1, X2) 1157.16/295.56 a__U31(X1, X2) -> U31(X1, X2) 1157.16/295.56 a__U41(X1, X2) -> U41(X1, X2) 1157.16/295.56 a__natsFrom(X) -> natsFrom(X) 1157.16/295.56 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 1157.16/295.56 a__head(X) -> head(X) 1157.16/295.56 a__afterNth(X1, X2) -> afterNth(X1, X2) 1157.16/295.56 a__U61(X1, X2) -> U61(X1, X2) 1157.16/295.56 a__U71(X1, X2) -> U71(X1, X2) 1157.16/295.56 a__U81(X1, X2, X3, X4) -> U81(X1, X2, X3, X4) 1157.16/295.56 a__U82(X1, X2) -> U82(X1, X2) 1157.16/295.56 a__U91(X1, X2) -> U91(X1, X2) 1157.16/295.56 a__and(X1, X2) -> and(X1, X2) 1157.16/295.56 a__isNatural(X) -> isNatural(X) 1157.16/295.56 a__isLNat(X) -> isLNat(X) 1157.16/295.56 a__isPLNat(X) -> isPLNat(X) 1157.16/295.56 a__tail(X) -> tail(X) 1157.16/295.56 a__take(X1, X2) -> take(X1, X2) 1157.16/295.56 a__sel(X1, X2) -> sel(X1, X2) 1157.16/295.56 1157.16/295.56 S is empty. 1157.16/295.56 Rewrite Strategy: INNERMOST 1157.25/295.62 EOF