/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). (0) CpxTRS (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (2) TRS for Loop Detection (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (4) BEST (5) proven lower bound (6) LowerBoundPropagationProof [FINISHED, 0 ms] (7) BOUNDS(n^1, INF) (8) TRS for Loop Detection (9) DecreasingLoopProof [FINISHED, 1118 ms] (10) BOUNDS(EXP, INF) ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__U101(tt, V1, V2) -> a__U102(a__isNatural(V1), V2) a__U102(tt, V2) -> a__U103(a__isLNat(V2)) a__U103(tt) -> tt a__U11(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) a__U111(tt, V1) -> a__U112(a__isLNat(V1)) a__U112(tt) -> tt a__U121(tt, V1) -> a__U122(a__isNatural(V1)) a__U122(tt) -> tt a__U131(tt, V1, V2) -> a__U132(a__isNatural(V1), V2) a__U132(tt, V2) -> a__U133(a__isLNat(V2)) a__U133(tt) -> tt a__U141(tt, V1, V2) -> a__U142(a__isLNat(V1), V2) a__U142(tt, V2) -> a__U143(a__isLNat(V2)) a__U143(tt) -> tt a__U151(tt, V1, V2) -> a__U152(a__isNatural(V1), V2) a__U152(tt, V2) -> a__U153(a__isLNat(V2)) a__U153(tt) -> tt a__U161(tt, N) -> cons(mark(N), natsFrom(s(N))) a__U171(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) a__U181(tt, Y) -> mark(Y) a__U191(tt, XS) -> pair(nil, mark(XS)) a__U201(tt, N, X, XS) -> a__U202(a__splitAt(mark(N), mark(XS)), X) a__U202(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) a__U21(tt, X) -> mark(X) a__U211(tt, XS) -> mark(XS) a__U221(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) a__U31(tt, N) -> mark(N) a__U41(tt, V1, V2) -> a__U42(a__isNatural(V1), V2) a__U42(tt, V2) -> a__U43(a__isLNat(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNatural(V1), V2) a__U52(tt, V2) -> a__U53(a__isLNat(V2)) a__U53(tt) -> tt a__U61(tt, V1) -> a__U62(a__isPLNat(V1)) a__U62(tt) -> tt a__U71(tt, V1) -> a__U72(a__isNatural(V1)) a__U72(tt) -> tt a__U81(tt, V1) -> a__U82(a__isPLNat(V1)) a__U82(tt) -> tt a__U91(tt, V1) -> a__U92(a__isLNat(V1)) a__U92(tt) -> tt a__afterNth(N, XS) -> a__U11(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) a__and(tt, X) -> mark(X) a__fst(pair(X, Y)) -> a__U21(a__and(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X) a__head(cons(N, XS)) -> a__U31(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N) a__isLNat(nil) -> tt a__isLNat(afterNth(V1, V2)) -> a__U41(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isLNat(cons(V1, V2)) -> a__U51(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isLNat(fst(V1)) -> a__U61(a__isPLNatKind(V1), V1) a__isLNat(natsFrom(V1)) -> a__U71(a__isNaturalKind(V1), V1) a__isLNat(snd(V1)) -> a__U81(a__isPLNatKind(V1), V1) a__isLNat(tail(V1)) -> a__U91(a__isLNatKind(V1), V1) a__isLNat(take(V1, V2)) -> a__U101(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isLNatKind(nil) -> tt a__isLNatKind(afterNth(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__isLNatKind(cons(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__isLNatKind(fst(V1)) -> a__isPLNatKind(V1) a__isLNatKind(natsFrom(V1)) -> a__isNaturalKind(V1) a__isLNatKind(snd(V1)) -> a__isPLNatKind(V1) a__isLNatKind(tail(V1)) -> a__isLNatKind(V1) a__isLNatKind(take(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__isNatural(0) -> tt a__isNatural(head(V1)) -> a__U111(a__isLNatKind(V1), V1) a__isNatural(s(V1)) -> a__U121(a__isNaturalKind(V1), V1) a__isNatural(sel(V1, V2)) -> a__U131(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isNaturalKind(0) -> tt a__isNaturalKind(head(V1)) -> a__isLNatKind(V1) a__isNaturalKind(s(V1)) -> a__isNaturalKind(V1) a__isNaturalKind(sel(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__isPLNat(pair(V1, V2)) -> a__U141(a__and(a__isLNatKind(V1), isLNatKind(V2)), V1, V2) a__isPLNat(splitAt(V1, V2)) -> a__U151(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isPLNatKind(pair(V1, V2)) -> a__and(a__isLNatKind(V1), isLNatKind(V2)) a__isPLNatKind(splitAt(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__natsFrom(N) -> a__U161(a__and(a__isNatural(N), isNaturalKind(N)), N) a__sel(N, XS) -> a__U171(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) a__snd(pair(X, Y)) -> a__U181(a__and(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y) a__splitAt(0, XS) -> a__U191(a__and(a__isLNat(XS), isLNatKind(XS)), XS) a__splitAt(s(N), cons(X, XS)) -> a__U201(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS) a__tail(cons(N, XS)) -> a__U211(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS) a__take(N, XS) -> a__U221(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) mark(U102(X1, X2)) -> a__U102(mark(X1), X2) mark(isNatural(X)) -> a__isNatural(X) mark(U103(X)) -> a__U103(mark(X)) mark(isLNat(X)) -> a__isLNat(X) mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) mark(snd(X)) -> a__snd(mark(X)) mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) mark(U111(X1, X2)) -> a__U111(mark(X1), X2) mark(U112(X)) -> a__U112(mark(X)) mark(U121(X1, X2)) -> a__U121(mark(X1), X2) mark(U122(X)) -> a__U122(mark(X)) mark(U131(X1, X2, X3)) -> a__U131(mark(X1), X2, X3) mark(U132(X1, X2)) -> a__U132(mark(X1), X2) mark(U133(X)) -> a__U133(mark(X)) mark(U141(X1, X2, X3)) -> a__U141(mark(X1), X2, X3) mark(U142(X1, X2)) -> a__U142(mark(X1), X2) mark(U143(X)) -> a__U143(mark(X)) mark(U151(X1, X2, X3)) -> a__U151(mark(X1), X2, X3) mark(U152(X1, X2)) -> a__U152(mark(X1), X2) mark(U153(X)) -> a__U153(mark(X)) mark(U161(X1, X2)) -> a__U161(mark(X1), X2) mark(natsFrom(X)) -> a__natsFrom(mark(X)) mark(U171(X1, X2, X3)) -> a__U171(mark(X1), X2, X3) mark(head(X)) -> a__head(mark(X)) mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) mark(U181(X1, X2)) -> a__U181(mark(X1), X2) mark(U191(X1, X2)) -> a__U191(mark(X1), X2) mark(U201(X1, X2, X3, X4)) -> a__U201(mark(X1), X2, X3, X4) mark(U202(X1, X2)) -> a__U202(mark(X1), X2) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U211(X1, X2)) -> a__U211(mark(X1), X2) mark(U221(X1, X2, X3)) -> a__U221(mark(X1), X2, X3) mark(fst(X)) -> a__fst(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(isPLNat(X)) -> a__isPLNat(X) mark(U71(X1, X2)) -> a__U71(mark(X1), X2) mark(U72(X)) -> a__U72(mark(X)) mark(U81(X1, X2)) -> a__U81(mark(X1), X2) mark(U82(X)) -> a__U82(mark(X)) mark(U91(X1, X2)) -> a__U91(mark(X1), X2) mark(U92(X)) -> a__U92(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNaturalKind(X)) -> a__isNaturalKind(X) mark(isLNatKind(X)) -> a__isLNatKind(X) mark(isPLNatKind(X)) -> a__isPLNatKind(X) mark(tail(X)) -> a__tail(mark(X)) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(tt) -> tt mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) mark(nil) -> nil mark(0) -> 0 a__U101(X1, X2, X3) -> U101(X1, X2, X3) a__U102(X1, X2) -> U102(X1, X2) a__isNatural(X) -> isNatural(X) a__U103(X) -> U103(X) a__isLNat(X) -> isLNat(X) a__U11(X1, X2, X3) -> U11(X1, X2, X3) a__snd(X) -> snd(X) a__splitAt(X1, X2) -> splitAt(X1, X2) a__U111(X1, X2) -> U111(X1, X2) a__U112(X) -> U112(X) a__U121(X1, X2) -> U121(X1, X2) a__U122(X) -> U122(X) a__U131(X1, X2, X3) -> U131(X1, X2, X3) a__U132(X1, X2) -> U132(X1, X2) a__U133(X) -> U133(X) a__U141(X1, X2, X3) -> U141(X1, X2, X3) a__U142(X1, X2) -> U142(X1, X2) a__U143(X) -> U143(X) a__U151(X1, X2, X3) -> U151(X1, X2, X3) a__U152(X1, X2) -> U152(X1, X2) a__U153(X) -> U153(X) a__U161(X1, X2) -> U161(X1, X2) a__natsFrom(X) -> natsFrom(X) a__U171(X1, X2, X3) -> U171(X1, X2, X3) a__head(X) -> head(X) a__afterNth(X1, X2) -> afterNth(X1, X2) a__U181(X1, X2) -> U181(X1, X2) a__U191(X1, X2) -> U191(X1, X2) a__U201(X1, X2, X3, X4) -> U201(X1, X2, X3, X4) a__U202(X1, X2) -> U202(X1, X2) a__U21(X1, X2) -> U21(X1, X2) a__U211(X1, X2) -> U211(X1, X2) a__U221(X1, X2, X3) -> U221(X1, X2, X3) a__fst(X) -> fst(X) a__U31(X1, X2) -> U31(X1, X2) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__U62(X) -> U62(X) a__isPLNat(X) -> isPLNat(X) a__U71(X1, X2) -> U71(X1, X2) a__U72(X) -> U72(X) a__U81(X1, X2) -> U81(X1, X2) a__U82(X) -> U82(X) a__U91(X1, X2) -> U91(X1, X2) a__U92(X) -> U92(X) a__and(X1, X2) -> and(X1, X2) a__isNaturalKind(X) -> isNaturalKind(X) a__isLNatKind(X) -> isLNatKind(X) a__isPLNatKind(X) -> isPLNatKind(X) a__tail(X) -> tail(X) a__take(X1, X2) -> take(X1, X2) a__sel(X1, X2) -> sel(X1, X2) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (2) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__U101(tt, V1, V2) -> a__U102(a__isNatural(V1), V2) a__U102(tt, V2) -> a__U103(a__isLNat(V2)) a__U103(tt) -> tt a__U11(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) a__U111(tt, V1) -> a__U112(a__isLNat(V1)) a__U112(tt) -> tt a__U121(tt, V1) -> a__U122(a__isNatural(V1)) a__U122(tt) -> tt a__U131(tt, V1, V2) -> a__U132(a__isNatural(V1), V2) a__U132(tt, V2) -> a__U133(a__isLNat(V2)) a__U133(tt) -> tt a__U141(tt, V1, V2) -> a__U142(a__isLNat(V1), V2) a__U142(tt, V2) -> a__U143(a__isLNat(V2)) a__U143(tt) -> tt a__U151(tt, V1, V2) -> a__U152(a__isNatural(V1), V2) a__U152(tt, V2) -> a__U153(a__isLNat(V2)) a__U153(tt) -> tt a__U161(tt, N) -> cons(mark(N), natsFrom(s(N))) a__U171(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) a__U181(tt, Y) -> mark(Y) a__U191(tt, XS) -> pair(nil, mark(XS)) a__U201(tt, N, X, XS) -> a__U202(a__splitAt(mark(N), mark(XS)), X) a__U202(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) a__U21(tt, X) -> mark(X) a__U211(tt, XS) -> mark(XS) a__U221(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) a__U31(tt, N) -> mark(N) a__U41(tt, V1, V2) -> a__U42(a__isNatural(V1), V2) a__U42(tt, V2) -> a__U43(a__isLNat(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNatural(V1), V2) a__U52(tt, V2) -> a__U53(a__isLNat(V2)) a__U53(tt) -> tt a__U61(tt, V1) -> a__U62(a__isPLNat(V1)) a__U62(tt) -> tt a__U71(tt, V1) -> a__U72(a__isNatural(V1)) a__U72(tt) -> tt a__U81(tt, V1) -> a__U82(a__isPLNat(V1)) a__U82(tt) -> tt a__U91(tt, V1) -> a__U92(a__isLNat(V1)) a__U92(tt) -> tt a__afterNth(N, XS) -> a__U11(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) a__and(tt, X) -> mark(X) a__fst(pair(X, Y)) -> a__U21(a__and(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X) a__head(cons(N, XS)) -> a__U31(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N) a__isLNat(nil) -> tt a__isLNat(afterNth(V1, V2)) -> a__U41(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isLNat(cons(V1, V2)) -> a__U51(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isLNat(fst(V1)) -> a__U61(a__isPLNatKind(V1), V1) a__isLNat(natsFrom(V1)) -> a__U71(a__isNaturalKind(V1), V1) a__isLNat(snd(V1)) -> a__U81(a__isPLNatKind(V1), V1) a__isLNat(tail(V1)) -> a__U91(a__isLNatKind(V1), V1) a__isLNat(take(V1, V2)) -> a__U101(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isLNatKind(nil) -> tt a__isLNatKind(afterNth(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__isLNatKind(cons(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__isLNatKind(fst(V1)) -> a__isPLNatKind(V1) a__isLNatKind(natsFrom(V1)) -> a__isNaturalKind(V1) a__isLNatKind(snd(V1)) -> a__isPLNatKind(V1) a__isLNatKind(tail(V1)) -> a__isLNatKind(V1) a__isLNatKind(take(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__isNatural(0) -> tt a__isNatural(head(V1)) -> a__U111(a__isLNatKind(V1), V1) a__isNatural(s(V1)) -> a__U121(a__isNaturalKind(V1), V1) a__isNatural(sel(V1, V2)) -> a__U131(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isNaturalKind(0) -> tt a__isNaturalKind(head(V1)) -> a__isLNatKind(V1) a__isNaturalKind(s(V1)) -> a__isNaturalKind(V1) a__isNaturalKind(sel(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__isPLNat(pair(V1, V2)) -> a__U141(a__and(a__isLNatKind(V1), isLNatKind(V2)), V1, V2) a__isPLNat(splitAt(V1, V2)) -> a__U151(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isPLNatKind(pair(V1, V2)) -> a__and(a__isLNatKind(V1), isLNatKind(V2)) a__isPLNatKind(splitAt(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__natsFrom(N) -> a__U161(a__and(a__isNatural(N), isNaturalKind(N)), N) a__sel(N, XS) -> a__U171(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) a__snd(pair(X, Y)) -> a__U181(a__and(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y) a__splitAt(0, XS) -> a__U191(a__and(a__isLNat(XS), isLNatKind(XS)), XS) a__splitAt(s(N), cons(X, XS)) -> a__U201(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS) a__tail(cons(N, XS)) -> a__U211(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS) a__take(N, XS) -> a__U221(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) mark(U102(X1, X2)) -> a__U102(mark(X1), X2) mark(isNatural(X)) -> a__isNatural(X) mark(U103(X)) -> a__U103(mark(X)) mark(isLNat(X)) -> a__isLNat(X) mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) mark(snd(X)) -> a__snd(mark(X)) mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) mark(U111(X1, X2)) -> a__U111(mark(X1), X2) mark(U112(X)) -> a__U112(mark(X)) mark(U121(X1, X2)) -> a__U121(mark(X1), X2) mark(U122(X)) -> a__U122(mark(X)) mark(U131(X1, X2, X3)) -> a__U131(mark(X1), X2, X3) mark(U132(X1, X2)) -> a__U132(mark(X1), X2) mark(U133(X)) -> a__U133(mark(X)) mark(U141(X1, X2, X3)) -> a__U141(mark(X1), X2, X3) mark(U142(X1, X2)) -> a__U142(mark(X1), X2) mark(U143(X)) -> a__U143(mark(X)) mark(U151(X1, X2, X3)) -> a__U151(mark(X1), X2, X3) mark(U152(X1, X2)) -> a__U152(mark(X1), X2) mark(U153(X)) -> a__U153(mark(X)) mark(U161(X1, X2)) -> a__U161(mark(X1), X2) mark(natsFrom(X)) -> a__natsFrom(mark(X)) mark(U171(X1, X2, X3)) -> a__U171(mark(X1), X2, X3) mark(head(X)) -> a__head(mark(X)) mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) mark(U181(X1, X2)) -> a__U181(mark(X1), X2) mark(U191(X1, X2)) -> a__U191(mark(X1), X2) mark(U201(X1, X2, X3, X4)) -> a__U201(mark(X1), X2, X3, X4) mark(U202(X1, X2)) -> a__U202(mark(X1), X2) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U211(X1, X2)) -> a__U211(mark(X1), X2) mark(U221(X1, X2, X3)) -> a__U221(mark(X1), X2, X3) mark(fst(X)) -> a__fst(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(isPLNat(X)) -> a__isPLNat(X) mark(U71(X1, X2)) -> a__U71(mark(X1), X2) mark(U72(X)) -> a__U72(mark(X)) mark(U81(X1, X2)) -> a__U81(mark(X1), X2) mark(U82(X)) -> a__U82(mark(X)) mark(U91(X1, X2)) -> a__U91(mark(X1), X2) mark(U92(X)) -> a__U92(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNaturalKind(X)) -> a__isNaturalKind(X) mark(isLNatKind(X)) -> a__isLNatKind(X) mark(isPLNatKind(X)) -> a__isPLNatKind(X) mark(tail(X)) -> a__tail(mark(X)) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(tt) -> tt mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) mark(nil) -> nil mark(0) -> 0 a__U101(X1, X2, X3) -> U101(X1, X2, X3) a__U102(X1, X2) -> U102(X1, X2) a__isNatural(X) -> isNatural(X) a__U103(X) -> U103(X) a__isLNat(X) -> isLNat(X) a__U11(X1, X2, X3) -> U11(X1, X2, X3) a__snd(X) -> snd(X) a__splitAt(X1, X2) -> splitAt(X1, X2) a__U111(X1, X2) -> U111(X1, X2) a__U112(X) -> U112(X) a__U121(X1, X2) -> U121(X1, X2) a__U122(X) -> U122(X) a__U131(X1, X2, X3) -> U131(X1, X2, X3) a__U132(X1, X2) -> U132(X1, X2) a__U133(X) -> U133(X) a__U141(X1, X2, X3) -> U141(X1, X2, X3) a__U142(X1, X2) -> U142(X1, X2) a__U143(X) -> U143(X) a__U151(X1, X2, X3) -> U151(X1, X2, X3) a__U152(X1, X2) -> U152(X1, X2) a__U153(X) -> U153(X) a__U161(X1, X2) -> U161(X1, X2) a__natsFrom(X) -> natsFrom(X) a__U171(X1, X2, X3) -> U171(X1, X2, X3) a__head(X) -> head(X) a__afterNth(X1, X2) -> afterNth(X1, X2) a__U181(X1, X2) -> U181(X1, X2) a__U191(X1, X2) -> U191(X1, X2) a__U201(X1, X2, X3, X4) -> U201(X1, X2, X3, X4) a__U202(X1, X2) -> U202(X1, X2) a__U21(X1, X2) -> U21(X1, X2) a__U211(X1, X2) -> U211(X1, X2) a__U221(X1, X2, X3) -> U221(X1, X2, X3) a__fst(X) -> fst(X) a__U31(X1, X2) -> U31(X1, X2) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__U62(X) -> U62(X) a__isPLNat(X) -> isPLNat(X) a__U71(X1, X2) -> U71(X1, X2) a__U72(X) -> U72(X) a__U81(X1, X2) -> U81(X1, X2) a__U82(X) -> U82(X) a__U91(X1, X2) -> U91(X1, X2) a__U92(X) -> U92(X) a__and(X1, X2) -> and(X1, X2) a__isNaturalKind(X) -> isNaturalKind(X) a__isLNatKind(X) -> isLNatKind(X) a__isPLNatKind(X) -> isPLNatKind(X) a__tail(X) -> tail(X) a__take(X1, X2) -> take(X1, X2) a__sel(X1, X2) -> sel(X1, X2) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence mark(U121(X1, X2)) ->^+ a__U121(mark(X1), X2) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / U121(X1, X2)]. The result substitution is [ ]. ---------------------------------------- (4) Complex Obligation (BEST) ---------------------------------------- (5) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__U101(tt, V1, V2) -> a__U102(a__isNatural(V1), V2) a__U102(tt, V2) -> a__U103(a__isLNat(V2)) a__U103(tt) -> tt a__U11(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) a__U111(tt, V1) -> a__U112(a__isLNat(V1)) a__U112(tt) -> tt a__U121(tt, V1) -> a__U122(a__isNatural(V1)) a__U122(tt) -> tt a__U131(tt, V1, V2) -> a__U132(a__isNatural(V1), V2) a__U132(tt, V2) -> a__U133(a__isLNat(V2)) a__U133(tt) -> tt a__U141(tt, V1, V2) -> a__U142(a__isLNat(V1), V2) a__U142(tt, V2) -> a__U143(a__isLNat(V2)) a__U143(tt) -> tt a__U151(tt, V1, V2) -> a__U152(a__isNatural(V1), V2) a__U152(tt, V2) -> a__U153(a__isLNat(V2)) a__U153(tt) -> tt a__U161(tt, N) -> cons(mark(N), natsFrom(s(N))) a__U171(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) a__U181(tt, Y) -> mark(Y) a__U191(tt, XS) -> pair(nil, mark(XS)) a__U201(tt, N, X, XS) -> a__U202(a__splitAt(mark(N), mark(XS)), X) a__U202(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) a__U21(tt, X) -> mark(X) a__U211(tt, XS) -> mark(XS) a__U221(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) a__U31(tt, N) -> mark(N) a__U41(tt, V1, V2) -> a__U42(a__isNatural(V1), V2) a__U42(tt, V2) -> a__U43(a__isLNat(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNatural(V1), V2) a__U52(tt, V2) -> a__U53(a__isLNat(V2)) a__U53(tt) -> tt a__U61(tt, V1) -> a__U62(a__isPLNat(V1)) a__U62(tt) -> tt a__U71(tt, V1) -> a__U72(a__isNatural(V1)) a__U72(tt) -> tt a__U81(tt, V1) -> a__U82(a__isPLNat(V1)) a__U82(tt) -> tt a__U91(tt, V1) -> a__U92(a__isLNat(V1)) a__U92(tt) -> tt a__afterNth(N, XS) -> a__U11(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) a__and(tt, X) -> mark(X) a__fst(pair(X, Y)) -> a__U21(a__and(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X) a__head(cons(N, XS)) -> a__U31(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N) a__isLNat(nil) -> tt a__isLNat(afterNth(V1, V2)) -> a__U41(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isLNat(cons(V1, V2)) -> a__U51(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isLNat(fst(V1)) -> a__U61(a__isPLNatKind(V1), V1) a__isLNat(natsFrom(V1)) -> a__U71(a__isNaturalKind(V1), V1) a__isLNat(snd(V1)) -> a__U81(a__isPLNatKind(V1), V1) a__isLNat(tail(V1)) -> a__U91(a__isLNatKind(V1), V1) a__isLNat(take(V1, V2)) -> a__U101(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isLNatKind(nil) -> tt a__isLNatKind(afterNth(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__isLNatKind(cons(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__isLNatKind(fst(V1)) -> a__isPLNatKind(V1) a__isLNatKind(natsFrom(V1)) -> a__isNaturalKind(V1) a__isLNatKind(snd(V1)) -> a__isPLNatKind(V1) a__isLNatKind(tail(V1)) -> a__isLNatKind(V1) a__isLNatKind(take(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__isNatural(0) -> tt a__isNatural(head(V1)) -> a__U111(a__isLNatKind(V1), V1) a__isNatural(s(V1)) -> a__U121(a__isNaturalKind(V1), V1) a__isNatural(sel(V1, V2)) -> a__U131(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isNaturalKind(0) -> tt a__isNaturalKind(head(V1)) -> a__isLNatKind(V1) a__isNaturalKind(s(V1)) -> a__isNaturalKind(V1) a__isNaturalKind(sel(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__isPLNat(pair(V1, V2)) -> a__U141(a__and(a__isLNatKind(V1), isLNatKind(V2)), V1, V2) a__isPLNat(splitAt(V1, V2)) -> a__U151(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isPLNatKind(pair(V1, V2)) -> a__and(a__isLNatKind(V1), isLNatKind(V2)) a__isPLNatKind(splitAt(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__natsFrom(N) -> a__U161(a__and(a__isNatural(N), isNaturalKind(N)), N) a__sel(N, XS) -> a__U171(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) a__snd(pair(X, Y)) -> a__U181(a__and(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y) a__splitAt(0, XS) -> a__U191(a__and(a__isLNat(XS), isLNatKind(XS)), XS) a__splitAt(s(N), cons(X, XS)) -> a__U201(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS) a__tail(cons(N, XS)) -> a__U211(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS) a__take(N, XS) -> a__U221(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) mark(U102(X1, X2)) -> a__U102(mark(X1), X2) mark(isNatural(X)) -> a__isNatural(X) mark(U103(X)) -> a__U103(mark(X)) mark(isLNat(X)) -> a__isLNat(X) mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) mark(snd(X)) -> a__snd(mark(X)) mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) mark(U111(X1, X2)) -> a__U111(mark(X1), X2) mark(U112(X)) -> a__U112(mark(X)) mark(U121(X1, X2)) -> a__U121(mark(X1), X2) mark(U122(X)) -> a__U122(mark(X)) mark(U131(X1, X2, X3)) -> a__U131(mark(X1), X2, X3) mark(U132(X1, X2)) -> a__U132(mark(X1), X2) mark(U133(X)) -> a__U133(mark(X)) mark(U141(X1, X2, X3)) -> a__U141(mark(X1), X2, X3) mark(U142(X1, X2)) -> a__U142(mark(X1), X2) mark(U143(X)) -> a__U143(mark(X)) mark(U151(X1, X2, X3)) -> a__U151(mark(X1), X2, X3) mark(U152(X1, X2)) -> a__U152(mark(X1), X2) mark(U153(X)) -> a__U153(mark(X)) mark(U161(X1, X2)) -> a__U161(mark(X1), X2) mark(natsFrom(X)) -> a__natsFrom(mark(X)) mark(U171(X1, X2, X3)) -> a__U171(mark(X1), X2, X3) mark(head(X)) -> a__head(mark(X)) mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) mark(U181(X1, X2)) -> a__U181(mark(X1), X2) mark(U191(X1, X2)) -> a__U191(mark(X1), X2) mark(U201(X1, X2, X3, X4)) -> a__U201(mark(X1), X2, X3, X4) mark(U202(X1, X2)) -> a__U202(mark(X1), X2) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U211(X1, X2)) -> a__U211(mark(X1), X2) mark(U221(X1, X2, X3)) -> a__U221(mark(X1), X2, X3) mark(fst(X)) -> a__fst(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(isPLNat(X)) -> a__isPLNat(X) mark(U71(X1, X2)) -> a__U71(mark(X1), X2) mark(U72(X)) -> a__U72(mark(X)) mark(U81(X1, X2)) -> a__U81(mark(X1), X2) mark(U82(X)) -> a__U82(mark(X)) mark(U91(X1, X2)) -> a__U91(mark(X1), X2) mark(U92(X)) -> a__U92(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNaturalKind(X)) -> a__isNaturalKind(X) mark(isLNatKind(X)) -> a__isLNatKind(X) mark(isPLNatKind(X)) -> a__isPLNatKind(X) mark(tail(X)) -> a__tail(mark(X)) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(tt) -> tt mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) mark(nil) -> nil mark(0) -> 0 a__U101(X1, X2, X3) -> U101(X1, X2, X3) a__U102(X1, X2) -> U102(X1, X2) a__isNatural(X) -> isNatural(X) a__U103(X) -> U103(X) a__isLNat(X) -> isLNat(X) a__U11(X1, X2, X3) -> U11(X1, X2, X3) a__snd(X) -> snd(X) a__splitAt(X1, X2) -> splitAt(X1, X2) a__U111(X1, X2) -> U111(X1, X2) a__U112(X) -> U112(X) a__U121(X1, X2) -> U121(X1, X2) a__U122(X) -> U122(X) a__U131(X1, X2, X3) -> U131(X1, X2, X3) a__U132(X1, X2) -> U132(X1, X2) a__U133(X) -> U133(X) a__U141(X1, X2, X3) -> U141(X1, X2, X3) a__U142(X1, X2) -> U142(X1, X2) a__U143(X) -> U143(X) a__U151(X1, X2, X3) -> U151(X1, X2, X3) a__U152(X1, X2) -> U152(X1, X2) a__U153(X) -> U153(X) a__U161(X1, X2) -> U161(X1, X2) a__natsFrom(X) -> natsFrom(X) a__U171(X1, X2, X3) -> U171(X1, X2, X3) a__head(X) -> head(X) a__afterNth(X1, X2) -> afterNth(X1, X2) a__U181(X1, X2) -> U181(X1, X2) a__U191(X1, X2) -> U191(X1, X2) a__U201(X1, X2, X3, X4) -> U201(X1, X2, X3, X4) a__U202(X1, X2) -> U202(X1, X2) a__U21(X1, X2) -> U21(X1, X2) a__U211(X1, X2) -> U211(X1, X2) a__U221(X1, X2, X3) -> U221(X1, X2, X3) a__fst(X) -> fst(X) a__U31(X1, X2) -> U31(X1, X2) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__U62(X) -> U62(X) a__isPLNat(X) -> isPLNat(X) a__U71(X1, X2) -> U71(X1, X2) a__U72(X) -> U72(X) a__U81(X1, X2) -> U81(X1, X2) a__U82(X) -> U82(X) a__U91(X1, X2) -> U91(X1, X2) a__U92(X) -> U92(X) a__and(X1, X2) -> and(X1, X2) a__isNaturalKind(X) -> isNaturalKind(X) a__isLNatKind(X) -> isLNatKind(X) a__isPLNatKind(X) -> isPLNatKind(X) a__tail(X) -> tail(X) a__take(X1, X2) -> take(X1, X2) a__sel(X1, X2) -> sel(X1, X2) S is empty. Rewrite Strategy: FULL ---------------------------------------- (6) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (7) BOUNDS(n^1, INF) ---------------------------------------- (8) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__U101(tt, V1, V2) -> a__U102(a__isNatural(V1), V2) a__U102(tt, V2) -> a__U103(a__isLNat(V2)) a__U103(tt) -> tt a__U11(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) a__U111(tt, V1) -> a__U112(a__isLNat(V1)) a__U112(tt) -> tt a__U121(tt, V1) -> a__U122(a__isNatural(V1)) a__U122(tt) -> tt a__U131(tt, V1, V2) -> a__U132(a__isNatural(V1), V2) a__U132(tt, V2) -> a__U133(a__isLNat(V2)) a__U133(tt) -> tt a__U141(tt, V1, V2) -> a__U142(a__isLNat(V1), V2) a__U142(tt, V2) -> a__U143(a__isLNat(V2)) a__U143(tt) -> tt a__U151(tt, V1, V2) -> a__U152(a__isNatural(V1), V2) a__U152(tt, V2) -> a__U153(a__isLNat(V2)) a__U153(tt) -> tt a__U161(tt, N) -> cons(mark(N), natsFrom(s(N))) a__U171(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) a__U181(tt, Y) -> mark(Y) a__U191(tt, XS) -> pair(nil, mark(XS)) a__U201(tt, N, X, XS) -> a__U202(a__splitAt(mark(N), mark(XS)), X) a__U202(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) a__U21(tt, X) -> mark(X) a__U211(tt, XS) -> mark(XS) a__U221(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) a__U31(tt, N) -> mark(N) a__U41(tt, V1, V2) -> a__U42(a__isNatural(V1), V2) a__U42(tt, V2) -> a__U43(a__isLNat(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNatural(V1), V2) a__U52(tt, V2) -> a__U53(a__isLNat(V2)) a__U53(tt) -> tt a__U61(tt, V1) -> a__U62(a__isPLNat(V1)) a__U62(tt) -> tt a__U71(tt, V1) -> a__U72(a__isNatural(V1)) a__U72(tt) -> tt a__U81(tt, V1) -> a__U82(a__isPLNat(V1)) a__U82(tt) -> tt a__U91(tt, V1) -> a__U92(a__isLNat(V1)) a__U92(tt) -> tt a__afterNth(N, XS) -> a__U11(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) a__and(tt, X) -> mark(X) a__fst(pair(X, Y)) -> a__U21(a__and(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X) a__head(cons(N, XS)) -> a__U31(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N) a__isLNat(nil) -> tt a__isLNat(afterNth(V1, V2)) -> a__U41(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isLNat(cons(V1, V2)) -> a__U51(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isLNat(fst(V1)) -> a__U61(a__isPLNatKind(V1), V1) a__isLNat(natsFrom(V1)) -> a__U71(a__isNaturalKind(V1), V1) a__isLNat(snd(V1)) -> a__U81(a__isPLNatKind(V1), V1) a__isLNat(tail(V1)) -> a__U91(a__isLNatKind(V1), V1) a__isLNat(take(V1, V2)) -> a__U101(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isLNatKind(nil) -> tt a__isLNatKind(afterNth(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__isLNatKind(cons(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__isLNatKind(fst(V1)) -> a__isPLNatKind(V1) a__isLNatKind(natsFrom(V1)) -> a__isNaturalKind(V1) a__isLNatKind(snd(V1)) -> a__isPLNatKind(V1) a__isLNatKind(tail(V1)) -> a__isLNatKind(V1) a__isLNatKind(take(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__isNatural(0) -> tt a__isNatural(head(V1)) -> a__U111(a__isLNatKind(V1), V1) a__isNatural(s(V1)) -> a__U121(a__isNaturalKind(V1), V1) a__isNatural(sel(V1, V2)) -> a__U131(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isNaturalKind(0) -> tt a__isNaturalKind(head(V1)) -> a__isLNatKind(V1) a__isNaturalKind(s(V1)) -> a__isNaturalKind(V1) a__isNaturalKind(sel(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__isPLNat(pair(V1, V2)) -> a__U141(a__and(a__isLNatKind(V1), isLNatKind(V2)), V1, V2) a__isPLNat(splitAt(V1, V2)) -> a__U151(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) a__isPLNatKind(pair(V1, V2)) -> a__and(a__isLNatKind(V1), isLNatKind(V2)) a__isPLNatKind(splitAt(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) a__natsFrom(N) -> a__U161(a__and(a__isNatural(N), isNaturalKind(N)), N) a__sel(N, XS) -> a__U171(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) a__snd(pair(X, Y)) -> a__U181(a__and(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y) a__splitAt(0, XS) -> a__U191(a__and(a__isLNat(XS), isLNatKind(XS)), XS) a__splitAt(s(N), cons(X, XS)) -> a__U201(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS) a__tail(cons(N, XS)) -> a__U211(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS) a__take(N, XS) -> a__U221(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) mark(U102(X1, X2)) -> a__U102(mark(X1), X2) mark(isNatural(X)) -> a__isNatural(X) mark(U103(X)) -> a__U103(mark(X)) mark(isLNat(X)) -> a__isLNat(X) mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) mark(snd(X)) -> a__snd(mark(X)) mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) mark(U111(X1, X2)) -> a__U111(mark(X1), X2) mark(U112(X)) -> a__U112(mark(X)) mark(U121(X1, X2)) -> a__U121(mark(X1), X2) mark(U122(X)) -> a__U122(mark(X)) mark(U131(X1, X2, X3)) -> a__U131(mark(X1), X2, X3) mark(U132(X1, X2)) -> a__U132(mark(X1), X2) mark(U133(X)) -> a__U133(mark(X)) mark(U141(X1, X2, X3)) -> a__U141(mark(X1), X2, X3) mark(U142(X1, X2)) -> a__U142(mark(X1), X2) mark(U143(X)) -> a__U143(mark(X)) mark(U151(X1, X2, X3)) -> a__U151(mark(X1), X2, X3) mark(U152(X1, X2)) -> a__U152(mark(X1), X2) mark(U153(X)) -> a__U153(mark(X)) mark(U161(X1, X2)) -> a__U161(mark(X1), X2) mark(natsFrom(X)) -> a__natsFrom(mark(X)) mark(U171(X1, X2, X3)) -> a__U171(mark(X1), X2, X3) mark(head(X)) -> a__head(mark(X)) mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) mark(U181(X1, X2)) -> a__U181(mark(X1), X2) mark(U191(X1, X2)) -> a__U191(mark(X1), X2) mark(U201(X1, X2, X3, X4)) -> a__U201(mark(X1), X2, X3, X4) mark(U202(X1, X2)) -> a__U202(mark(X1), X2) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U211(X1, X2)) -> a__U211(mark(X1), X2) mark(U221(X1, X2, X3)) -> a__U221(mark(X1), X2, X3) mark(fst(X)) -> a__fst(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(isPLNat(X)) -> a__isPLNat(X) mark(U71(X1, X2)) -> a__U71(mark(X1), X2) mark(U72(X)) -> a__U72(mark(X)) mark(U81(X1, X2)) -> a__U81(mark(X1), X2) mark(U82(X)) -> a__U82(mark(X)) mark(U91(X1, X2)) -> a__U91(mark(X1), X2) mark(U92(X)) -> a__U92(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNaturalKind(X)) -> a__isNaturalKind(X) mark(isLNatKind(X)) -> a__isLNatKind(X) mark(isPLNatKind(X)) -> a__isPLNatKind(X) mark(tail(X)) -> a__tail(mark(X)) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(tt) -> tt mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) mark(nil) -> nil mark(0) -> 0 a__U101(X1, X2, X3) -> U101(X1, X2, X3) a__U102(X1, X2) -> U102(X1, X2) a__isNatural(X) -> isNatural(X) a__U103(X) -> U103(X) a__isLNat(X) -> isLNat(X) a__U11(X1, X2, X3) -> U11(X1, X2, X3) a__snd(X) -> snd(X) a__splitAt(X1, X2) -> splitAt(X1, X2) a__U111(X1, X2) -> U111(X1, X2) a__U112(X) -> U112(X) a__U121(X1, X2) -> U121(X1, X2) a__U122(X) -> U122(X) a__U131(X1, X2, X3) -> U131(X1, X2, X3) a__U132(X1, X2) -> U132(X1, X2) a__U133(X) -> U133(X) a__U141(X1, X2, X3) -> U141(X1, X2, X3) a__U142(X1, X2) -> U142(X1, X2) a__U143(X) -> U143(X) a__U151(X1, X2, X3) -> U151(X1, X2, X3) a__U152(X1, X2) -> U152(X1, X2) a__U153(X) -> U153(X) a__U161(X1, X2) -> U161(X1, X2) a__natsFrom(X) -> natsFrom(X) a__U171(X1, X2, X3) -> U171(X1, X2, X3) a__head(X) -> head(X) a__afterNth(X1, X2) -> afterNth(X1, X2) a__U181(X1, X2) -> U181(X1, X2) a__U191(X1, X2) -> U191(X1, X2) a__U201(X1, X2, X3, X4) -> U201(X1, X2, X3, X4) a__U202(X1, X2) -> U202(X1, X2) a__U21(X1, X2) -> U21(X1, X2) a__U211(X1, X2) -> U211(X1, X2) a__U221(X1, X2, X3) -> U221(X1, X2, X3) a__fst(X) -> fst(X) a__U31(X1, X2) -> U31(X1, X2) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__U62(X) -> U62(X) a__isPLNat(X) -> isPLNat(X) a__U71(X1, X2) -> U71(X1, X2) a__U72(X) -> U72(X) a__U81(X1, X2) -> U81(X1, X2) a__U82(X) -> U82(X) a__U91(X1, X2) -> U91(X1, X2) a__U92(X) -> U92(X) a__and(X1, X2) -> and(X1, X2) a__isNaturalKind(X) -> isNaturalKind(X) a__isLNatKind(X) -> isLNatKind(X) a__isPLNatKind(X) -> isPLNatKind(X) a__tail(X) -> tail(X) a__take(X1, X2) -> take(X1, X2) a__sel(X1, X2) -> sel(X1, X2) S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) DecreasingLoopProof (FINISHED) The following loop(s) give(s) rise to the lower bound EXP: The rewrite sequence mark(take(X1, X2)) ->^+ a__U221(a__and(a__and(a__isNatural(mark(X1)), isNaturalKind(mark(X1))), and(isLNat(mark(X2)), isLNatKind(mark(X2)))), mark(X1), mark(X2)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0]. The pumping substitution is [X1 / take(X1, X2)]. The result substitution is [ ]. The rewrite sequence mark(take(X1, X2)) ->^+ a__U221(a__and(a__and(a__isNatural(mark(X1)), isNaturalKind(mark(X1))), and(isLNat(mark(X2)), isLNatKind(mark(X2)))), mark(X1), mark(X2)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,1,0]. The pumping substitution is [X1 / take(X1, X2)]. The result substitution is [ ]. ---------------------------------------- (10) BOUNDS(EXP, INF)