1142.27/291.62 WORST_CASE(Omega(n^1), ?) 1147.66/293.03 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1147.66/293.03 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1147.66/293.03 1147.66/293.03 1147.66/293.03 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1147.66/293.03 1147.66/293.03 (0) CpxTRS 1147.66/293.03 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1147.66/293.03 (2) TRS for Loop Detection 1147.66/293.03 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1147.66/293.03 (4) BEST 1147.66/293.03 (5) proven lower bound 1147.66/293.03 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1147.66/293.03 (7) BOUNDS(n^1, INF) 1147.66/293.03 (8) TRS for Loop Detection 1147.66/293.03 1147.66/293.03 1147.66/293.03 ---------------------------------------- 1147.66/293.03 1147.66/293.03 (0) 1147.66/293.03 Obligation: 1147.66/293.03 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1147.66/293.03 1147.66/293.03 1147.66/293.03 The TRS R consists of the following rules: 1147.66/293.03 1147.66/293.03 a__U101(tt, V2) -> a__U102(a__isLNat(V2)) 1147.66/293.03 a__U102(tt) -> tt 1147.66/293.03 a__U11(tt, N, XS) -> a__U12(a__isLNat(XS), N, XS) 1147.66/293.03 a__U111(tt) -> tt 1147.66/293.03 a__U12(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 1147.66/293.03 a__U121(tt) -> tt 1147.66/293.03 a__U131(tt, V2) -> a__U132(a__isLNat(V2)) 1147.66/293.03 a__U132(tt) -> tt 1147.66/293.03 a__U141(tt, V2) -> a__U142(a__isLNat(V2)) 1147.66/293.03 a__U142(tt) -> tt 1147.66/293.03 a__U151(tt, V2) -> a__U152(a__isLNat(V2)) 1147.66/293.03 a__U152(tt) -> tt 1147.66/293.03 a__U161(tt, N) -> cons(mark(N), natsFrom(s(N))) 1147.66/293.03 a__U171(tt, N, XS) -> a__U172(a__isLNat(XS), N, XS) 1147.66/293.03 a__U172(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 1147.66/293.03 a__U181(tt, Y) -> a__U182(a__isLNat(Y), Y) 1147.66/293.03 a__U182(tt, Y) -> mark(Y) 1147.66/293.03 a__U191(tt, XS) -> pair(nil, mark(XS)) 1147.66/293.03 a__U201(tt, N, X, XS) -> a__U202(a__isNatural(X), N, X, XS) 1147.66/293.03 a__U202(tt, N, X, XS) -> a__U203(a__isLNat(XS), N, X, XS) 1147.66/293.03 a__U203(tt, N, X, XS) -> a__U204(a__splitAt(mark(N), mark(XS)), X) 1147.66/293.03 a__U204(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 1147.66/293.03 a__U21(tt, X, Y) -> a__U22(a__isLNat(Y), X) 1147.66/293.03 a__U211(tt, XS) -> a__U212(a__isLNat(XS), XS) 1147.66/293.03 a__U212(tt, XS) -> mark(XS) 1147.66/293.03 a__U22(tt, X) -> mark(X) 1147.66/293.03 a__U221(tt, N, XS) -> a__U222(a__isLNat(XS), N, XS) 1147.66/293.03 a__U222(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 1147.66/293.03 a__U31(tt, N, XS) -> a__U32(a__isLNat(XS), N) 1147.66/293.03 a__U32(tt, N) -> mark(N) 1147.66/293.03 a__U41(tt, V2) -> a__U42(a__isLNat(V2)) 1147.66/293.03 a__U42(tt) -> tt 1147.66/293.03 a__U51(tt, V2) -> a__U52(a__isLNat(V2)) 1147.66/293.03 a__U52(tt) -> tt 1147.66/293.03 a__U61(tt) -> tt 1147.66/293.03 a__U71(tt) -> tt 1147.66/293.03 a__U81(tt) -> tt 1147.66/293.03 a__U91(tt) -> tt 1147.66/293.03 a__afterNth(N, XS) -> a__U11(a__isNatural(N), N, XS) 1147.66/293.03 a__fst(pair(X, Y)) -> a__U21(a__isLNat(X), X, Y) 1147.66/293.03 a__head(cons(N, XS)) -> a__U31(a__isNatural(N), N, XS) 1147.66/293.03 a__isLNat(nil) -> tt 1147.66/293.03 a__isLNat(afterNth(V1, V2)) -> a__U41(a__isNatural(V1), V2) 1147.66/293.03 a__isLNat(cons(V1, V2)) -> a__U51(a__isNatural(V1), V2) 1147.66/293.03 a__isLNat(fst(V1)) -> a__U61(a__isPLNat(V1)) 1147.66/293.03 a__isLNat(natsFrom(V1)) -> a__U71(a__isNatural(V1)) 1147.66/293.03 a__isLNat(snd(V1)) -> a__U81(a__isPLNat(V1)) 1147.66/293.03 a__isLNat(tail(V1)) -> a__U91(a__isLNat(V1)) 1147.66/293.03 a__isLNat(take(V1, V2)) -> a__U101(a__isNatural(V1), V2) 1147.66/293.03 a__isNatural(0) -> tt 1147.66/293.03 a__isNatural(head(V1)) -> a__U111(a__isLNat(V1)) 1147.66/293.03 a__isNatural(s(V1)) -> a__U121(a__isNatural(V1)) 1147.66/293.03 a__isNatural(sel(V1, V2)) -> a__U131(a__isNatural(V1), V2) 1147.66/293.03 a__isPLNat(pair(V1, V2)) -> a__U141(a__isLNat(V1), V2) 1147.66/293.03 a__isPLNat(splitAt(V1, V2)) -> a__U151(a__isNatural(V1), V2) 1147.66/293.03 a__natsFrom(N) -> a__U161(a__isNatural(N), N) 1147.66/293.03 a__sel(N, XS) -> a__U171(a__isNatural(N), N, XS) 1147.66/293.03 a__snd(pair(X, Y)) -> a__U181(a__isLNat(X), Y) 1147.66/293.03 a__splitAt(0, XS) -> a__U191(a__isLNat(XS), XS) 1147.66/293.03 a__splitAt(s(N), cons(X, XS)) -> a__U201(a__isNatural(N), N, X, XS) 1147.66/293.03 a__tail(cons(N, XS)) -> a__U211(a__isNatural(N), XS) 1147.66/293.03 a__take(N, XS) -> a__U221(a__isNatural(N), N, XS) 1147.66/293.03 mark(U101(X1, X2)) -> a__U101(mark(X1), X2) 1147.66/293.03 mark(U102(X)) -> a__U102(mark(X)) 1147.66/293.03 mark(isLNat(X)) -> a__isLNat(X) 1147.66/293.03 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 1147.66/293.03 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 1147.66/293.03 mark(U111(X)) -> a__U111(mark(X)) 1147.66/293.03 mark(snd(X)) -> a__snd(mark(X)) 1147.66/293.03 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 1147.66/293.03 mark(U121(X)) -> a__U121(mark(X)) 1147.66/293.03 mark(U131(X1, X2)) -> a__U131(mark(X1), X2) 1147.66/293.03 mark(U132(X)) -> a__U132(mark(X)) 1147.66/293.03 mark(U141(X1, X2)) -> a__U141(mark(X1), X2) 1147.66/293.03 mark(U142(X)) -> a__U142(mark(X)) 1147.66/293.03 mark(U151(X1, X2)) -> a__U151(mark(X1), X2) 1147.66/293.03 mark(U152(X)) -> a__U152(mark(X)) 1147.66/293.03 mark(U161(X1, X2)) -> a__U161(mark(X1), X2) 1147.66/293.03 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 1147.66/293.03 mark(U171(X1, X2, X3)) -> a__U171(mark(X1), X2, X3) 1147.66/293.03 mark(U172(X1, X2, X3)) -> a__U172(mark(X1), X2, X3) 1147.66/293.03 mark(head(X)) -> a__head(mark(X)) 1147.66/293.03 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 1147.66/293.03 mark(U181(X1, X2)) -> a__U181(mark(X1), X2) 1147.66/293.03 mark(U182(X1, X2)) -> a__U182(mark(X1), X2) 1147.66/293.03 mark(U191(X1, X2)) -> a__U191(mark(X1), X2) 1147.66/293.03 mark(U201(X1, X2, X3, X4)) -> a__U201(mark(X1), X2, X3, X4) 1147.66/293.03 mark(U202(X1, X2, X3, X4)) -> a__U202(mark(X1), X2, X3, X4) 1147.66/293.03 mark(isNatural(X)) -> a__isNatural(X) 1147.66/293.03 mark(U203(X1, X2, X3, X4)) -> a__U203(mark(X1), X2, X3, X4) 1147.66/293.03 mark(U204(X1, X2)) -> a__U204(mark(X1), X2) 1147.66/293.03 mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) 1147.66/293.03 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 1147.66/293.03 mark(U211(X1, X2)) -> a__U211(mark(X1), X2) 1147.66/293.03 mark(U212(X1, X2)) -> a__U212(mark(X1), X2) 1147.66/293.03 mark(U221(X1, X2, X3)) -> a__U221(mark(X1), X2, X3) 1147.66/293.03 mark(U222(X1, X2, X3)) -> a__U222(mark(X1), X2, X3) 1147.66/293.03 mark(fst(X)) -> a__fst(mark(X)) 1147.66/293.03 mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) 1147.66/293.03 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 1147.66/293.03 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 1147.66/293.03 mark(U42(X)) -> a__U42(mark(X)) 1147.66/293.03 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 1147.66/293.03 mark(U52(X)) -> a__U52(mark(X)) 1147.66/293.03 mark(U61(X)) -> a__U61(mark(X)) 1147.66/293.03 mark(U71(X)) -> a__U71(mark(X)) 1147.66/293.03 mark(U81(X)) -> a__U81(mark(X)) 1147.66/293.03 mark(U91(X)) -> a__U91(mark(X)) 1147.66/293.03 mark(isPLNat(X)) -> a__isPLNat(X) 1147.66/293.03 mark(tail(X)) -> a__tail(mark(X)) 1147.66/293.03 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1147.66/293.03 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1147.66/293.03 mark(tt) -> tt 1147.66/293.03 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1147.66/293.03 mark(s(X)) -> s(mark(X)) 1147.66/293.03 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 1147.66/293.03 mark(nil) -> nil 1147.66/293.03 mark(0) -> 0 1147.66/293.03 a__U101(X1, X2) -> U101(X1, X2) 1147.66/293.03 a__U102(X) -> U102(X) 1147.66/293.03 a__isLNat(X) -> isLNat(X) 1147.66/293.03 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 1147.66/293.03 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 1147.66/293.03 a__U111(X) -> U111(X) 1147.66/293.03 a__snd(X) -> snd(X) 1147.66/293.03 a__splitAt(X1, X2) -> splitAt(X1, X2) 1147.66/293.03 a__U121(X) -> U121(X) 1147.66/293.03 a__U131(X1, X2) -> U131(X1, X2) 1147.66/293.03 a__U132(X) -> U132(X) 1147.66/293.03 a__U141(X1, X2) -> U141(X1, X2) 1147.66/293.03 a__U142(X) -> U142(X) 1147.66/293.03 a__U151(X1, X2) -> U151(X1, X2) 1147.66/293.03 a__U152(X) -> U152(X) 1147.66/293.03 a__U161(X1, X2) -> U161(X1, X2) 1147.66/293.03 a__natsFrom(X) -> natsFrom(X) 1147.66/293.03 a__U171(X1, X2, X3) -> U171(X1, X2, X3) 1147.66/293.03 a__U172(X1, X2, X3) -> U172(X1, X2, X3) 1147.66/293.03 a__head(X) -> head(X) 1147.66/293.03 a__afterNth(X1, X2) -> afterNth(X1, X2) 1147.66/293.03 a__U181(X1, X2) -> U181(X1, X2) 1147.66/293.03 a__U182(X1, X2) -> U182(X1, X2) 1147.66/293.03 a__U191(X1, X2) -> U191(X1, X2) 1147.66/293.03 a__U201(X1, X2, X3, X4) -> U201(X1, X2, X3, X4) 1147.66/293.03 a__U202(X1, X2, X3, X4) -> U202(X1, X2, X3, X4) 1147.66/293.03 a__isNatural(X) -> isNatural(X) 1147.66/293.03 a__U203(X1, X2, X3, X4) -> U203(X1, X2, X3, X4) 1147.66/293.03 a__U204(X1, X2) -> U204(X1, X2) 1147.66/293.03 a__U21(X1, X2, X3) -> U21(X1, X2, X3) 1147.66/293.03 a__U22(X1, X2) -> U22(X1, X2) 1147.66/293.03 a__U211(X1, X2) -> U211(X1, X2) 1147.66/293.03 a__U212(X1, X2) -> U212(X1, X2) 1147.66/293.03 a__U221(X1, X2, X3) -> U221(X1, X2, X3) 1147.66/293.03 a__U222(X1, X2, X3) -> U222(X1, X2, X3) 1147.66/293.03 a__fst(X) -> fst(X) 1147.66/293.03 a__U31(X1, X2, X3) -> U31(X1, X2, X3) 1147.66/293.03 a__U32(X1, X2) -> U32(X1, X2) 1147.66/293.03 a__U41(X1, X2) -> U41(X1, X2) 1147.66/293.03 a__U42(X) -> U42(X) 1147.66/293.03 a__U51(X1, X2) -> U51(X1, X2) 1147.66/293.03 a__U52(X) -> U52(X) 1147.66/293.03 a__U61(X) -> U61(X) 1147.66/293.03 a__U71(X) -> U71(X) 1147.66/293.03 a__U81(X) -> U81(X) 1147.66/293.03 a__U91(X) -> U91(X) 1147.66/293.03 a__isPLNat(X) -> isPLNat(X) 1147.66/293.03 a__tail(X) -> tail(X) 1147.66/293.03 a__take(X1, X2) -> take(X1, X2) 1147.66/293.03 a__sel(X1, X2) -> sel(X1, X2) 1147.66/293.03 1147.66/293.03 S is empty. 1147.66/293.03 Rewrite Strategy: INNERMOST 1147.66/293.03 ---------------------------------------- 1147.66/293.03 1147.66/293.03 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1147.66/293.03 Transformed a relative TRS into a decreasing-loop problem. 1147.66/293.03 ---------------------------------------- 1147.66/293.03 1147.66/293.03 (2) 1147.66/293.03 Obligation: 1147.66/293.03 Analyzing the following TRS for decreasing loops: 1147.66/293.03 1147.66/293.03 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1147.66/293.03 1147.66/293.03 1147.66/293.03 The TRS R consists of the following rules: 1147.66/293.03 1147.66/293.03 a__U101(tt, V2) -> a__U102(a__isLNat(V2)) 1147.66/293.03 a__U102(tt) -> tt 1147.66/293.03 a__U11(tt, N, XS) -> a__U12(a__isLNat(XS), N, XS) 1147.66/293.03 a__U111(tt) -> tt 1147.66/293.03 a__U12(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 1147.66/293.03 a__U121(tt) -> tt 1147.66/293.03 a__U131(tt, V2) -> a__U132(a__isLNat(V2)) 1147.66/293.03 a__U132(tt) -> tt 1147.66/293.03 a__U141(tt, V2) -> a__U142(a__isLNat(V2)) 1147.66/293.03 a__U142(tt) -> tt 1147.66/293.03 a__U151(tt, V2) -> a__U152(a__isLNat(V2)) 1147.66/293.03 a__U152(tt) -> tt 1147.66/293.03 a__U161(tt, N) -> cons(mark(N), natsFrom(s(N))) 1147.66/293.03 a__U171(tt, N, XS) -> a__U172(a__isLNat(XS), N, XS) 1147.66/293.03 a__U172(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 1147.66/293.03 a__U181(tt, Y) -> a__U182(a__isLNat(Y), Y) 1147.66/293.03 a__U182(tt, Y) -> mark(Y) 1147.66/293.03 a__U191(tt, XS) -> pair(nil, mark(XS)) 1147.66/293.03 a__U201(tt, N, X, XS) -> a__U202(a__isNatural(X), N, X, XS) 1147.66/293.03 a__U202(tt, N, X, XS) -> a__U203(a__isLNat(XS), N, X, XS) 1147.66/293.03 a__U203(tt, N, X, XS) -> a__U204(a__splitAt(mark(N), mark(XS)), X) 1147.66/293.03 a__U204(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 1147.66/293.03 a__U21(tt, X, Y) -> a__U22(a__isLNat(Y), X) 1147.66/293.03 a__U211(tt, XS) -> a__U212(a__isLNat(XS), XS) 1147.66/293.03 a__U212(tt, XS) -> mark(XS) 1147.66/293.03 a__U22(tt, X) -> mark(X) 1147.66/293.03 a__U221(tt, N, XS) -> a__U222(a__isLNat(XS), N, XS) 1147.66/293.03 a__U222(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 1147.66/293.03 a__U31(tt, N, XS) -> a__U32(a__isLNat(XS), N) 1147.66/293.03 a__U32(tt, N) -> mark(N) 1147.66/293.03 a__U41(tt, V2) -> a__U42(a__isLNat(V2)) 1147.66/293.03 a__U42(tt) -> tt 1147.66/293.03 a__U51(tt, V2) -> a__U52(a__isLNat(V2)) 1147.66/293.03 a__U52(tt) -> tt 1147.66/293.03 a__U61(tt) -> tt 1147.66/293.03 a__U71(tt) -> tt 1147.66/293.03 a__U81(tt) -> tt 1147.66/293.03 a__U91(tt) -> tt 1147.66/293.03 a__afterNth(N, XS) -> a__U11(a__isNatural(N), N, XS) 1147.66/293.03 a__fst(pair(X, Y)) -> a__U21(a__isLNat(X), X, Y) 1147.66/293.03 a__head(cons(N, XS)) -> a__U31(a__isNatural(N), N, XS) 1147.66/293.03 a__isLNat(nil) -> tt 1147.66/293.03 a__isLNat(afterNth(V1, V2)) -> a__U41(a__isNatural(V1), V2) 1147.66/293.03 a__isLNat(cons(V1, V2)) -> a__U51(a__isNatural(V1), V2) 1147.66/293.03 a__isLNat(fst(V1)) -> a__U61(a__isPLNat(V1)) 1147.66/293.03 a__isLNat(natsFrom(V1)) -> a__U71(a__isNatural(V1)) 1147.66/293.03 a__isLNat(snd(V1)) -> a__U81(a__isPLNat(V1)) 1147.66/293.03 a__isLNat(tail(V1)) -> a__U91(a__isLNat(V1)) 1147.66/293.03 a__isLNat(take(V1, V2)) -> a__U101(a__isNatural(V1), V2) 1147.66/293.03 a__isNatural(0) -> tt 1147.66/293.03 a__isNatural(head(V1)) -> a__U111(a__isLNat(V1)) 1147.66/293.03 a__isNatural(s(V1)) -> a__U121(a__isNatural(V1)) 1147.66/293.03 a__isNatural(sel(V1, V2)) -> a__U131(a__isNatural(V1), V2) 1147.66/293.03 a__isPLNat(pair(V1, V2)) -> a__U141(a__isLNat(V1), V2) 1147.66/293.03 a__isPLNat(splitAt(V1, V2)) -> a__U151(a__isNatural(V1), V2) 1147.66/293.03 a__natsFrom(N) -> a__U161(a__isNatural(N), N) 1147.66/293.03 a__sel(N, XS) -> a__U171(a__isNatural(N), N, XS) 1147.66/293.03 a__snd(pair(X, Y)) -> a__U181(a__isLNat(X), Y) 1147.66/293.03 a__splitAt(0, XS) -> a__U191(a__isLNat(XS), XS) 1147.66/293.03 a__splitAt(s(N), cons(X, XS)) -> a__U201(a__isNatural(N), N, X, XS) 1147.66/293.03 a__tail(cons(N, XS)) -> a__U211(a__isNatural(N), XS) 1147.66/293.03 a__take(N, XS) -> a__U221(a__isNatural(N), N, XS) 1147.66/293.03 mark(U101(X1, X2)) -> a__U101(mark(X1), X2) 1147.66/293.03 mark(U102(X)) -> a__U102(mark(X)) 1147.66/293.03 mark(isLNat(X)) -> a__isLNat(X) 1147.66/293.03 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 1147.66/293.03 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 1147.66/293.03 mark(U111(X)) -> a__U111(mark(X)) 1147.66/293.03 mark(snd(X)) -> a__snd(mark(X)) 1147.66/293.03 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 1147.66/293.03 mark(U121(X)) -> a__U121(mark(X)) 1147.66/293.03 mark(U131(X1, X2)) -> a__U131(mark(X1), X2) 1147.66/293.03 mark(U132(X)) -> a__U132(mark(X)) 1147.66/293.03 mark(U141(X1, X2)) -> a__U141(mark(X1), X2) 1147.66/293.03 mark(U142(X)) -> a__U142(mark(X)) 1147.66/293.03 mark(U151(X1, X2)) -> a__U151(mark(X1), X2) 1147.66/293.03 mark(U152(X)) -> a__U152(mark(X)) 1147.66/293.03 mark(U161(X1, X2)) -> a__U161(mark(X1), X2) 1147.66/293.03 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 1147.66/293.03 mark(U171(X1, X2, X3)) -> a__U171(mark(X1), X2, X3) 1147.66/293.03 mark(U172(X1, X2, X3)) -> a__U172(mark(X1), X2, X3) 1147.66/293.03 mark(head(X)) -> a__head(mark(X)) 1147.66/293.03 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 1147.66/293.03 mark(U181(X1, X2)) -> a__U181(mark(X1), X2) 1147.66/293.03 mark(U182(X1, X2)) -> a__U182(mark(X1), X2) 1147.66/293.03 mark(U191(X1, X2)) -> a__U191(mark(X1), X2) 1147.66/293.03 mark(U201(X1, X2, X3, X4)) -> a__U201(mark(X1), X2, X3, X4) 1147.66/293.03 mark(U202(X1, X2, X3, X4)) -> a__U202(mark(X1), X2, X3, X4) 1147.66/293.03 mark(isNatural(X)) -> a__isNatural(X) 1147.66/293.03 mark(U203(X1, X2, X3, X4)) -> a__U203(mark(X1), X2, X3, X4) 1147.66/293.03 mark(U204(X1, X2)) -> a__U204(mark(X1), X2) 1147.66/293.03 mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) 1147.66/293.03 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 1147.66/293.03 mark(U211(X1, X2)) -> a__U211(mark(X1), X2) 1147.66/293.03 mark(U212(X1, X2)) -> a__U212(mark(X1), X2) 1147.66/293.03 mark(U221(X1, X2, X3)) -> a__U221(mark(X1), X2, X3) 1147.66/293.03 mark(U222(X1, X2, X3)) -> a__U222(mark(X1), X2, X3) 1147.66/293.03 mark(fst(X)) -> a__fst(mark(X)) 1147.66/293.03 mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) 1147.66/293.03 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 1147.66/293.03 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 1147.66/293.03 mark(U42(X)) -> a__U42(mark(X)) 1147.66/293.03 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 1147.66/293.03 mark(U52(X)) -> a__U52(mark(X)) 1147.66/293.03 mark(U61(X)) -> a__U61(mark(X)) 1147.66/293.03 mark(U71(X)) -> a__U71(mark(X)) 1147.66/293.03 mark(U81(X)) -> a__U81(mark(X)) 1147.66/293.03 mark(U91(X)) -> a__U91(mark(X)) 1147.66/293.03 mark(isPLNat(X)) -> a__isPLNat(X) 1147.66/293.03 mark(tail(X)) -> a__tail(mark(X)) 1147.66/293.03 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1147.66/293.03 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1147.66/293.03 mark(tt) -> tt 1147.66/293.03 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1147.66/293.03 mark(s(X)) -> s(mark(X)) 1147.66/293.03 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 1147.66/293.03 mark(nil) -> nil 1147.66/293.03 mark(0) -> 0 1147.66/293.03 a__U101(X1, X2) -> U101(X1, X2) 1147.66/293.03 a__U102(X) -> U102(X) 1147.66/293.03 a__isLNat(X) -> isLNat(X) 1147.66/293.03 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 1147.66/293.03 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 1147.66/293.03 a__U111(X) -> U111(X) 1147.66/293.03 a__snd(X) -> snd(X) 1147.66/293.03 a__splitAt(X1, X2) -> splitAt(X1, X2) 1147.66/293.03 a__U121(X) -> U121(X) 1147.66/293.03 a__U131(X1, X2) -> U131(X1, X2) 1147.66/293.03 a__U132(X) -> U132(X) 1147.66/293.03 a__U141(X1, X2) -> U141(X1, X2) 1147.66/293.03 a__U142(X) -> U142(X) 1147.66/293.03 a__U151(X1, X2) -> U151(X1, X2) 1147.66/293.03 a__U152(X) -> U152(X) 1147.66/293.03 a__U161(X1, X2) -> U161(X1, X2) 1147.66/293.03 a__natsFrom(X) -> natsFrom(X) 1147.66/293.03 a__U171(X1, X2, X3) -> U171(X1, X2, X3) 1147.66/293.03 a__U172(X1, X2, X3) -> U172(X1, X2, X3) 1147.66/293.03 a__head(X) -> head(X) 1147.66/293.03 a__afterNth(X1, X2) -> afterNth(X1, X2) 1147.66/293.03 a__U181(X1, X2) -> U181(X1, X2) 1147.66/293.03 a__U182(X1, X2) -> U182(X1, X2) 1147.66/293.03 a__U191(X1, X2) -> U191(X1, X2) 1147.66/293.03 a__U201(X1, X2, X3, X4) -> U201(X1, X2, X3, X4) 1147.66/293.03 a__U202(X1, X2, X3, X4) -> U202(X1, X2, X3, X4) 1147.66/293.03 a__isNatural(X) -> isNatural(X) 1147.66/293.03 a__U203(X1, X2, X3, X4) -> U203(X1, X2, X3, X4) 1147.66/293.03 a__U204(X1, X2) -> U204(X1, X2) 1147.66/293.03 a__U21(X1, X2, X3) -> U21(X1, X2, X3) 1147.66/293.03 a__U22(X1, X2) -> U22(X1, X2) 1147.66/293.03 a__U211(X1, X2) -> U211(X1, X2) 1147.66/293.03 a__U212(X1, X2) -> U212(X1, X2) 1147.66/293.03 a__U221(X1, X2, X3) -> U221(X1, X2, X3) 1147.66/293.03 a__U222(X1, X2, X3) -> U222(X1, X2, X3) 1147.66/293.03 a__fst(X) -> fst(X) 1147.66/293.03 a__U31(X1, X2, X3) -> U31(X1, X2, X3) 1147.66/293.03 a__U32(X1, X2) -> U32(X1, X2) 1147.66/293.03 a__U41(X1, X2) -> U41(X1, X2) 1147.66/293.03 a__U42(X) -> U42(X) 1147.66/293.03 a__U51(X1, X2) -> U51(X1, X2) 1147.66/293.03 a__U52(X) -> U52(X) 1147.66/293.03 a__U61(X) -> U61(X) 1147.66/293.03 a__U71(X) -> U71(X) 1147.66/293.03 a__U81(X) -> U81(X) 1147.66/293.03 a__U91(X) -> U91(X) 1147.66/293.03 a__isPLNat(X) -> isPLNat(X) 1147.66/293.03 a__tail(X) -> tail(X) 1147.66/293.03 a__take(X1, X2) -> take(X1, X2) 1147.66/293.03 a__sel(X1, X2) -> sel(X1, X2) 1147.66/293.03 1147.66/293.03 S is empty. 1147.66/293.03 Rewrite Strategy: INNERMOST 1147.66/293.03 ---------------------------------------- 1147.66/293.03 1147.66/293.03 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1147.66/293.03 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1147.66/293.03 1147.66/293.03 The rewrite sequence 1147.66/293.03 1147.66/293.03 mark(afterNth(X1, X2)) ->^+ a__afterNth(mark(X1), mark(X2)) 1147.66/293.03 1147.66/293.03 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1147.66/293.03 1147.66/293.03 The pumping substitution is [X1 / afterNth(X1, X2)]. 1147.66/293.03 1147.66/293.03 The result substitution is [ ]. 1147.66/293.03 1147.66/293.03 1147.66/293.03 1147.66/293.03 1147.66/293.03 ---------------------------------------- 1147.66/293.03 1147.66/293.03 (4) 1147.66/293.03 Complex Obligation (BEST) 1147.66/293.03 1147.66/293.03 ---------------------------------------- 1147.66/293.03 1147.66/293.03 (5) 1147.66/293.03 Obligation: 1147.66/293.03 Proved the lower bound n^1 for the following obligation: 1147.66/293.03 1147.66/293.03 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1147.66/293.03 1147.66/293.03 1147.66/293.03 The TRS R consists of the following rules: 1147.66/293.03 1147.66/293.03 a__U101(tt, V2) -> a__U102(a__isLNat(V2)) 1147.66/293.03 a__U102(tt) -> tt 1147.66/293.03 a__U11(tt, N, XS) -> a__U12(a__isLNat(XS), N, XS) 1147.66/293.03 a__U111(tt) -> tt 1147.66/293.03 a__U12(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 1147.66/293.03 a__U121(tt) -> tt 1147.66/293.03 a__U131(tt, V2) -> a__U132(a__isLNat(V2)) 1147.66/293.03 a__U132(tt) -> tt 1147.66/293.03 a__U141(tt, V2) -> a__U142(a__isLNat(V2)) 1147.66/293.03 a__U142(tt) -> tt 1147.66/293.03 a__U151(tt, V2) -> a__U152(a__isLNat(V2)) 1147.66/293.03 a__U152(tt) -> tt 1147.66/293.03 a__U161(tt, N) -> cons(mark(N), natsFrom(s(N))) 1147.66/293.03 a__U171(tt, N, XS) -> a__U172(a__isLNat(XS), N, XS) 1147.66/293.03 a__U172(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 1147.66/293.03 a__U181(tt, Y) -> a__U182(a__isLNat(Y), Y) 1147.66/293.03 a__U182(tt, Y) -> mark(Y) 1147.66/293.03 a__U191(tt, XS) -> pair(nil, mark(XS)) 1147.66/293.03 a__U201(tt, N, X, XS) -> a__U202(a__isNatural(X), N, X, XS) 1147.66/293.03 a__U202(tt, N, X, XS) -> a__U203(a__isLNat(XS), N, X, XS) 1147.66/293.03 a__U203(tt, N, X, XS) -> a__U204(a__splitAt(mark(N), mark(XS)), X) 1147.66/293.03 a__U204(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 1147.66/293.03 a__U21(tt, X, Y) -> a__U22(a__isLNat(Y), X) 1147.66/293.03 a__U211(tt, XS) -> a__U212(a__isLNat(XS), XS) 1147.66/293.03 a__U212(tt, XS) -> mark(XS) 1147.66/293.03 a__U22(tt, X) -> mark(X) 1147.66/293.03 a__U221(tt, N, XS) -> a__U222(a__isLNat(XS), N, XS) 1147.66/293.03 a__U222(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 1147.66/293.03 a__U31(tt, N, XS) -> a__U32(a__isLNat(XS), N) 1147.66/293.03 a__U32(tt, N) -> mark(N) 1147.66/293.03 a__U41(tt, V2) -> a__U42(a__isLNat(V2)) 1147.66/293.03 a__U42(tt) -> tt 1147.66/293.03 a__U51(tt, V2) -> a__U52(a__isLNat(V2)) 1147.66/293.03 a__U52(tt) -> tt 1147.66/293.03 a__U61(tt) -> tt 1147.66/293.03 a__U71(tt) -> tt 1147.66/293.03 a__U81(tt) -> tt 1147.66/293.03 a__U91(tt) -> tt 1147.66/293.03 a__afterNth(N, XS) -> a__U11(a__isNatural(N), N, XS) 1147.66/293.03 a__fst(pair(X, Y)) -> a__U21(a__isLNat(X), X, Y) 1147.66/293.03 a__head(cons(N, XS)) -> a__U31(a__isNatural(N), N, XS) 1147.66/293.03 a__isLNat(nil) -> tt 1147.66/293.03 a__isLNat(afterNth(V1, V2)) -> a__U41(a__isNatural(V1), V2) 1147.66/293.03 a__isLNat(cons(V1, V2)) -> a__U51(a__isNatural(V1), V2) 1147.66/293.03 a__isLNat(fst(V1)) -> a__U61(a__isPLNat(V1)) 1147.66/293.03 a__isLNat(natsFrom(V1)) -> a__U71(a__isNatural(V1)) 1147.66/293.03 a__isLNat(snd(V1)) -> a__U81(a__isPLNat(V1)) 1147.66/293.03 a__isLNat(tail(V1)) -> a__U91(a__isLNat(V1)) 1147.66/293.03 a__isLNat(take(V1, V2)) -> a__U101(a__isNatural(V1), V2) 1147.66/293.03 a__isNatural(0) -> tt 1147.66/293.03 a__isNatural(head(V1)) -> a__U111(a__isLNat(V1)) 1147.66/293.03 a__isNatural(s(V1)) -> a__U121(a__isNatural(V1)) 1147.66/293.03 a__isNatural(sel(V1, V2)) -> a__U131(a__isNatural(V1), V2) 1147.66/293.03 a__isPLNat(pair(V1, V2)) -> a__U141(a__isLNat(V1), V2) 1147.66/293.03 a__isPLNat(splitAt(V1, V2)) -> a__U151(a__isNatural(V1), V2) 1147.66/293.03 a__natsFrom(N) -> a__U161(a__isNatural(N), N) 1147.66/293.03 a__sel(N, XS) -> a__U171(a__isNatural(N), N, XS) 1147.66/293.03 a__snd(pair(X, Y)) -> a__U181(a__isLNat(X), Y) 1147.66/293.03 a__splitAt(0, XS) -> a__U191(a__isLNat(XS), XS) 1147.66/293.03 a__splitAt(s(N), cons(X, XS)) -> a__U201(a__isNatural(N), N, X, XS) 1147.66/293.03 a__tail(cons(N, XS)) -> a__U211(a__isNatural(N), XS) 1147.66/293.03 a__take(N, XS) -> a__U221(a__isNatural(N), N, XS) 1147.66/293.03 mark(U101(X1, X2)) -> a__U101(mark(X1), X2) 1147.66/293.03 mark(U102(X)) -> a__U102(mark(X)) 1147.66/293.03 mark(isLNat(X)) -> a__isLNat(X) 1147.66/293.03 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 1147.66/293.03 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 1147.66/293.03 mark(U111(X)) -> a__U111(mark(X)) 1147.66/293.03 mark(snd(X)) -> a__snd(mark(X)) 1147.66/293.03 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 1147.66/293.03 mark(U121(X)) -> a__U121(mark(X)) 1147.66/293.03 mark(U131(X1, X2)) -> a__U131(mark(X1), X2) 1147.66/293.03 mark(U132(X)) -> a__U132(mark(X)) 1147.66/293.03 mark(U141(X1, X2)) -> a__U141(mark(X1), X2) 1147.66/293.03 mark(U142(X)) -> a__U142(mark(X)) 1147.66/293.03 mark(U151(X1, X2)) -> a__U151(mark(X1), X2) 1147.66/293.03 mark(U152(X)) -> a__U152(mark(X)) 1147.66/293.03 mark(U161(X1, X2)) -> a__U161(mark(X1), X2) 1147.66/293.03 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 1147.66/293.03 mark(U171(X1, X2, X3)) -> a__U171(mark(X1), X2, X3) 1147.66/293.03 mark(U172(X1, X2, X3)) -> a__U172(mark(X1), X2, X3) 1147.66/293.03 mark(head(X)) -> a__head(mark(X)) 1147.66/293.03 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 1147.66/293.03 mark(U181(X1, X2)) -> a__U181(mark(X1), X2) 1147.66/293.03 mark(U182(X1, X2)) -> a__U182(mark(X1), X2) 1147.66/293.03 mark(U191(X1, X2)) -> a__U191(mark(X1), X2) 1147.66/293.03 mark(U201(X1, X2, X3, X4)) -> a__U201(mark(X1), X2, X3, X4) 1147.66/293.03 mark(U202(X1, X2, X3, X4)) -> a__U202(mark(X1), X2, X3, X4) 1147.66/293.03 mark(isNatural(X)) -> a__isNatural(X) 1147.66/293.03 mark(U203(X1, X2, X3, X4)) -> a__U203(mark(X1), X2, X3, X4) 1147.66/293.03 mark(U204(X1, X2)) -> a__U204(mark(X1), X2) 1147.66/293.03 mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) 1147.66/293.03 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 1147.66/293.03 mark(U211(X1, X2)) -> a__U211(mark(X1), X2) 1147.66/293.03 mark(U212(X1, X2)) -> a__U212(mark(X1), X2) 1147.66/293.03 mark(U221(X1, X2, X3)) -> a__U221(mark(X1), X2, X3) 1147.66/293.03 mark(U222(X1, X2, X3)) -> a__U222(mark(X1), X2, X3) 1147.66/293.03 mark(fst(X)) -> a__fst(mark(X)) 1147.66/293.03 mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) 1147.66/293.03 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 1147.66/293.03 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 1147.66/293.03 mark(U42(X)) -> a__U42(mark(X)) 1147.66/293.03 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 1147.66/293.03 mark(U52(X)) -> a__U52(mark(X)) 1147.66/293.03 mark(U61(X)) -> a__U61(mark(X)) 1147.66/293.03 mark(U71(X)) -> a__U71(mark(X)) 1147.66/293.03 mark(U81(X)) -> a__U81(mark(X)) 1147.66/293.03 mark(U91(X)) -> a__U91(mark(X)) 1147.66/293.03 mark(isPLNat(X)) -> a__isPLNat(X) 1147.66/293.03 mark(tail(X)) -> a__tail(mark(X)) 1147.66/293.03 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1147.66/293.03 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1147.66/293.03 mark(tt) -> tt 1147.66/293.03 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1147.66/293.03 mark(s(X)) -> s(mark(X)) 1147.66/293.03 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 1147.66/293.03 mark(nil) -> nil 1147.66/293.03 mark(0) -> 0 1147.66/293.03 a__U101(X1, X2) -> U101(X1, X2) 1147.66/293.03 a__U102(X) -> U102(X) 1147.66/293.03 a__isLNat(X) -> isLNat(X) 1147.66/293.03 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 1147.66/293.03 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 1147.66/293.03 a__U111(X) -> U111(X) 1147.66/293.03 a__snd(X) -> snd(X) 1147.66/293.03 a__splitAt(X1, X2) -> splitAt(X1, X2) 1147.66/293.03 a__U121(X) -> U121(X) 1147.66/293.03 a__U131(X1, X2) -> U131(X1, X2) 1147.66/293.03 a__U132(X) -> U132(X) 1147.66/293.03 a__U141(X1, X2) -> U141(X1, X2) 1147.66/293.03 a__U142(X) -> U142(X) 1147.66/293.03 a__U151(X1, X2) -> U151(X1, X2) 1147.66/293.03 a__U152(X) -> U152(X) 1147.66/293.03 a__U161(X1, X2) -> U161(X1, X2) 1147.66/293.03 a__natsFrom(X) -> natsFrom(X) 1147.66/293.03 a__U171(X1, X2, X3) -> U171(X1, X2, X3) 1147.66/293.03 a__U172(X1, X2, X3) -> U172(X1, X2, X3) 1147.66/293.03 a__head(X) -> head(X) 1147.66/293.03 a__afterNth(X1, X2) -> afterNth(X1, X2) 1147.66/293.03 a__U181(X1, X2) -> U181(X1, X2) 1147.66/293.03 a__U182(X1, X2) -> U182(X1, X2) 1147.66/293.03 a__U191(X1, X2) -> U191(X1, X2) 1147.66/293.03 a__U201(X1, X2, X3, X4) -> U201(X1, X2, X3, X4) 1147.66/293.03 a__U202(X1, X2, X3, X4) -> U202(X1, X2, X3, X4) 1147.66/293.03 a__isNatural(X) -> isNatural(X) 1147.66/293.03 a__U203(X1, X2, X3, X4) -> U203(X1, X2, X3, X4) 1147.66/293.03 a__U204(X1, X2) -> U204(X1, X2) 1147.66/293.03 a__U21(X1, X2, X3) -> U21(X1, X2, X3) 1147.66/293.03 a__U22(X1, X2) -> U22(X1, X2) 1147.66/293.03 a__U211(X1, X2) -> U211(X1, X2) 1147.66/293.03 a__U212(X1, X2) -> U212(X1, X2) 1147.66/293.03 a__U221(X1, X2, X3) -> U221(X1, X2, X3) 1147.66/293.03 a__U222(X1, X2, X3) -> U222(X1, X2, X3) 1147.66/293.03 a__fst(X) -> fst(X) 1147.66/293.03 a__U31(X1, X2, X3) -> U31(X1, X2, X3) 1147.66/293.03 a__U32(X1, X2) -> U32(X1, X2) 1147.66/293.03 a__U41(X1, X2) -> U41(X1, X2) 1147.66/293.03 a__U42(X) -> U42(X) 1147.66/293.03 a__U51(X1, X2) -> U51(X1, X2) 1147.66/293.03 a__U52(X) -> U52(X) 1147.66/293.03 a__U61(X) -> U61(X) 1147.66/293.03 a__U71(X) -> U71(X) 1147.66/293.03 a__U81(X) -> U81(X) 1147.66/293.03 a__U91(X) -> U91(X) 1147.66/293.03 a__isPLNat(X) -> isPLNat(X) 1147.66/293.03 a__tail(X) -> tail(X) 1147.66/293.03 a__take(X1, X2) -> take(X1, X2) 1147.66/293.03 a__sel(X1, X2) -> sel(X1, X2) 1147.66/293.03 1147.66/293.03 S is empty. 1147.66/293.03 Rewrite Strategy: INNERMOST 1147.66/293.03 ---------------------------------------- 1147.66/293.03 1147.66/293.03 (6) LowerBoundPropagationProof (FINISHED) 1147.66/293.03 Propagated lower bound. 1147.66/293.03 ---------------------------------------- 1147.66/293.03 1147.66/293.03 (7) 1147.66/293.03 BOUNDS(n^1, INF) 1147.66/293.03 1147.66/293.03 ---------------------------------------- 1147.66/293.03 1147.66/293.03 (8) 1147.66/293.03 Obligation: 1147.66/293.03 Analyzing the following TRS for decreasing loops: 1147.66/293.03 1147.66/293.03 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1147.66/293.03 1147.66/293.03 1147.66/293.03 The TRS R consists of the following rules: 1147.66/293.03 1147.66/293.03 a__U101(tt, V2) -> a__U102(a__isLNat(V2)) 1147.66/293.03 a__U102(tt) -> tt 1147.66/293.03 a__U11(tt, N, XS) -> a__U12(a__isLNat(XS), N, XS) 1147.66/293.03 a__U111(tt) -> tt 1147.66/293.03 a__U12(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 1147.66/293.03 a__U121(tt) -> tt 1147.66/293.03 a__U131(tt, V2) -> a__U132(a__isLNat(V2)) 1147.66/293.03 a__U132(tt) -> tt 1147.66/293.03 a__U141(tt, V2) -> a__U142(a__isLNat(V2)) 1147.66/293.03 a__U142(tt) -> tt 1147.66/293.03 a__U151(tt, V2) -> a__U152(a__isLNat(V2)) 1147.66/293.04 a__U152(tt) -> tt 1147.66/293.04 a__U161(tt, N) -> cons(mark(N), natsFrom(s(N))) 1147.66/293.04 a__U171(tt, N, XS) -> a__U172(a__isLNat(XS), N, XS) 1147.66/293.04 a__U172(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 1147.66/293.04 a__U181(tt, Y) -> a__U182(a__isLNat(Y), Y) 1147.66/293.04 a__U182(tt, Y) -> mark(Y) 1147.66/293.04 a__U191(tt, XS) -> pair(nil, mark(XS)) 1147.66/293.04 a__U201(tt, N, X, XS) -> a__U202(a__isNatural(X), N, X, XS) 1147.66/293.04 a__U202(tt, N, X, XS) -> a__U203(a__isLNat(XS), N, X, XS) 1147.66/293.04 a__U203(tt, N, X, XS) -> a__U204(a__splitAt(mark(N), mark(XS)), X) 1147.66/293.04 a__U204(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 1147.66/293.04 a__U21(tt, X, Y) -> a__U22(a__isLNat(Y), X) 1147.66/293.04 a__U211(tt, XS) -> a__U212(a__isLNat(XS), XS) 1147.66/293.04 a__U212(tt, XS) -> mark(XS) 1147.66/293.04 a__U22(tt, X) -> mark(X) 1147.66/293.04 a__U221(tt, N, XS) -> a__U222(a__isLNat(XS), N, XS) 1147.66/293.04 a__U222(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 1147.66/293.04 a__U31(tt, N, XS) -> a__U32(a__isLNat(XS), N) 1147.66/293.04 a__U32(tt, N) -> mark(N) 1147.66/293.04 a__U41(tt, V2) -> a__U42(a__isLNat(V2)) 1147.66/293.04 a__U42(tt) -> tt 1147.66/293.04 a__U51(tt, V2) -> a__U52(a__isLNat(V2)) 1147.66/293.04 a__U52(tt) -> tt 1147.66/293.04 a__U61(tt) -> tt 1147.66/293.04 a__U71(tt) -> tt 1147.66/293.04 a__U81(tt) -> tt 1147.66/293.04 a__U91(tt) -> tt 1147.66/293.04 a__afterNth(N, XS) -> a__U11(a__isNatural(N), N, XS) 1147.66/293.04 a__fst(pair(X, Y)) -> a__U21(a__isLNat(X), X, Y) 1147.66/293.04 a__head(cons(N, XS)) -> a__U31(a__isNatural(N), N, XS) 1147.66/293.04 a__isLNat(nil) -> tt 1147.66/293.04 a__isLNat(afterNth(V1, V2)) -> a__U41(a__isNatural(V1), V2) 1147.66/293.04 a__isLNat(cons(V1, V2)) -> a__U51(a__isNatural(V1), V2) 1147.66/293.04 a__isLNat(fst(V1)) -> a__U61(a__isPLNat(V1)) 1147.66/293.04 a__isLNat(natsFrom(V1)) -> a__U71(a__isNatural(V1)) 1147.66/293.04 a__isLNat(snd(V1)) -> a__U81(a__isPLNat(V1)) 1147.66/293.04 a__isLNat(tail(V1)) -> a__U91(a__isLNat(V1)) 1147.66/293.04 a__isLNat(take(V1, V2)) -> a__U101(a__isNatural(V1), V2) 1147.66/293.04 a__isNatural(0) -> tt 1147.66/293.04 a__isNatural(head(V1)) -> a__U111(a__isLNat(V1)) 1147.66/293.04 a__isNatural(s(V1)) -> a__U121(a__isNatural(V1)) 1147.66/293.04 a__isNatural(sel(V1, V2)) -> a__U131(a__isNatural(V1), V2) 1147.66/293.04 a__isPLNat(pair(V1, V2)) -> a__U141(a__isLNat(V1), V2) 1147.66/293.04 a__isPLNat(splitAt(V1, V2)) -> a__U151(a__isNatural(V1), V2) 1147.66/293.04 a__natsFrom(N) -> a__U161(a__isNatural(N), N) 1147.66/293.04 a__sel(N, XS) -> a__U171(a__isNatural(N), N, XS) 1147.66/293.04 a__snd(pair(X, Y)) -> a__U181(a__isLNat(X), Y) 1147.66/293.04 a__splitAt(0, XS) -> a__U191(a__isLNat(XS), XS) 1147.66/293.04 a__splitAt(s(N), cons(X, XS)) -> a__U201(a__isNatural(N), N, X, XS) 1147.66/293.04 a__tail(cons(N, XS)) -> a__U211(a__isNatural(N), XS) 1147.66/293.04 a__take(N, XS) -> a__U221(a__isNatural(N), N, XS) 1147.66/293.04 mark(U101(X1, X2)) -> a__U101(mark(X1), X2) 1147.66/293.04 mark(U102(X)) -> a__U102(mark(X)) 1147.66/293.04 mark(isLNat(X)) -> a__isLNat(X) 1147.66/293.04 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 1147.66/293.04 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 1147.66/293.04 mark(U111(X)) -> a__U111(mark(X)) 1147.66/293.04 mark(snd(X)) -> a__snd(mark(X)) 1147.66/293.04 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 1147.66/293.04 mark(U121(X)) -> a__U121(mark(X)) 1147.66/293.04 mark(U131(X1, X2)) -> a__U131(mark(X1), X2) 1147.66/293.04 mark(U132(X)) -> a__U132(mark(X)) 1147.66/293.04 mark(U141(X1, X2)) -> a__U141(mark(X1), X2) 1147.66/293.04 mark(U142(X)) -> a__U142(mark(X)) 1147.66/293.04 mark(U151(X1, X2)) -> a__U151(mark(X1), X2) 1147.66/293.04 mark(U152(X)) -> a__U152(mark(X)) 1147.66/293.04 mark(U161(X1, X2)) -> a__U161(mark(X1), X2) 1147.66/293.04 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 1147.66/293.04 mark(U171(X1, X2, X3)) -> a__U171(mark(X1), X2, X3) 1147.66/293.04 mark(U172(X1, X2, X3)) -> a__U172(mark(X1), X2, X3) 1147.66/293.04 mark(head(X)) -> a__head(mark(X)) 1147.66/293.04 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 1147.66/293.04 mark(U181(X1, X2)) -> a__U181(mark(X1), X2) 1147.66/293.04 mark(U182(X1, X2)) -> a__U182(mark(X1), X2) 1147.66/293.04 mark(U191(X1, X2)) -> a__U191(mark(X1), X2) 1147.66/293.04 mark(U201(X1, X2, X3, X4)) -> a__U201(mark(X1), X2, X3, X4) 1147.66/293.04 mark(U202(X1, X2, X3, X4)) -> a__U202(mark(X1), X2, X3, X4) 1147.66/293.04 mark(isNatural(X)) -> a__isNatural(X) 1147.66/293.04 mark(U203(X1, X2, X3, X4)) -> a__U203(mark(X1), X2, X3, X4) 1147.66/293.04 mark(U204(X1, X2)) -> a__U204(mark(X1), X2) 1147.66/293.04 mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) 1147.66/293.04 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 1147.66/293.04 mark(U211(X1, X2)) -> a__U211(mark(X1), X2) 1147.66/293.04 mark(U212(X1, X2)) -> a__U212(mark(X1), X2) 1147.66/293.04 mark(U221(X1, X2, X3)) -> a__U221(mark(X1), X2, X3) 1147.66/293.04 mark(U222(X1, X2, X3)) -> a__U222(mark(X1), X2, X3) 1147.66/293.04 mark(fst(X)) -> a__fst(mark(X)) 1147.66/293.04 mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) 1147.66/293.04 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 1147.66/293.04 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 1147.66/293.04 mark(U42(X)) -> a__U42(mark(X)) 1147.66/293.04 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 1147.66/293.04 mark(U52(X)) -> a__U52(mark(X)) 1147.66/293.04 mark(U61(X)) -> a__U61(mark(X)) 1147.66/293.04 mark(U71(X)) -> a__U71(mark(X)) 1147.66/293.04 mark(U81(X)) -> a__U81(mark(X)) 1147.66/293.04 mark(U91(X)) -> a__U91(mark(X)) 1147.66/293.04 mark(isPLNat(X)) -> a__isPLNat(X) 1147.66/293.04 mark(tail(X)) -> a__tail(mark(X)) 1147.66/293.04 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1147.66/293.04 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1147.66/293.04 mark(tt) -> tt 1147.66/293.04 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1147.66/293.04 mark(s(X)) -> s(mark(X)) 1147.66/293.04 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 1147.66/293.04 mark(nil) -> nil 1147.66/293.04 mark(0) -> 0 1147.66/293.04 a__U101(X1, X2) -> U101(X1, X2) 1147.66/293.04 a__U102(X) -> U102(X) 1147.66/293.04 a__isLNat(X) -> isLNat(X) 1147.66/293.04 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 1147.66/293.04 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 1147.66/293.04 a__U111(X) -> U111(X) 1147.66/293.04 a__snd(X) -> snd(X) 1147.66/293.04 a__splitAt(X1, X2) -> splitAt(X1, X2) 1147.66/293.04 a__U121(X) -> U121(X) 1147.66/293.04 a__U131(X1, X2) -> U131(X1, X2) 1147.66/293.04 a__U132(X) -> U132(X) 1147.66/293.04 a__U141(X1, X2) -> U141(X1, X2) 1147.66/293.04 a__U142(X) -> U142(X) 1147.66/293.04 a__U151(X1, X2) -> U151(X1, X2) 1147.66/293.04 a__U152(X) -> U152(X) 1147.66/293.04 a__U161(X1, X2) -> U161(X1, X2) 1147.66/293.04 a__natsFrom(X) -> natsFrom(X) 1147.66/293.04 a__U171(X1, X2, X3) -> U171(X1, X2, X3) 1147.66/293.04 a__U172(X1, X2, X3) -> U172(X1, X2, X3) 1147.66/293.04 a__head(X) -> head(X) 1147.66/293.04 a__afterNth(X1, X2) -> afterNth(X1, X2) 1147.66/293.04 a__U181(X1, X2) -> U181(X1, X2) 1147.66/293.04 a__U182(X1, X2) -> U182(X1, X2) 1147.66/293.04 a__U191(X1, X2) -> U191(X1, X2) 1147.66/293.04 a__U201(X1, X2, X3, X4) -> U201(X1, X2, X3, X4) 1147.66/293.04 a__U202(X1, X2, X3, X4) -> U202(X1, X2, X3, X4) 1147.66/293.04 a__isNatural(X) -> isNatural(X) 1147.66/293.04 a__U203(X1, X2, X3, X4) -> U203(X1, X2, X3, X4) 1147.66/293.04 a__U204(X1, X2) -> U204(X1, X2) 1147.66/293.04 a__U21(X1, X2, X3) -> U21(X1, X2, X3) 1147.66/293.04 a__U22(X1, X2) -> U22(X1, X2) 1147.66/293.04 a__U211(X1, X2) -> U211(X1, X2) 1147.66/293.04 a__U212(X1, X2) -> U212(X1, X2) 1147.66/293.04 a__U221(X1, X2, X3) -> U221(X1, X2, X3) 1147.66/293.04 a__U222(X1, X2, X3) -> U222(X1, X2, X3) 1147.66/293.04 a__fst(X) -> fst(X) 1147.66/293.04 a__U31(X1, X2, X3) -> U31(X1, X2, X3) 1147.66/293.04 a__U32(X1, X2) -> U32(X1, X2) 1147.66/293.04 a__U41(X1, X2) -> U41(X1, X2) 1147.66/293.04 a__U42(X) -> U42(X) 1147.66/293.04 a__U51(X1, X2) -> U51(X1, X2) 1147.66/293.04 a__U52(X) -> U52(X) 1147.66/293.04 a__U61(X) -> U61(X) 1147.66/293.04 a__U71(X) -> U71(X) 1147.66/293.04 a__U81(X) -> U81(X) 1147.66/293.04 a__U91(X) -> U91(X) 1147.66/293.04 a__isPLNat(X) -> isPLNat(X) 1147.66/293.04 a__tail(X) -> tail(X) 1147.66/293.04 a__take(X1, X2) -> take(X1, X2) 1147.66/293.04 a__sel(X1, X2) -> sel(X1, X2) 1147.66/293.04 1147.66/293.04 S is empty. 1147.66/293.04 Rewrite Strategy: INNERMOST 1148.03/293.10 EOF