9.71/3.29 WORST_CASE(NON_POLY, ?) 9.71/3.30 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 9.71/3.30 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.71/3.30 9.71/3.30 9.71/3.30 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 9.71/3.30 9.71/3.30 (0) CpxTRS 9.71/3.30 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 9.71/3.30 (2) TRS for Loop Detection 9.71/3.30 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 9.71/3.30 (4) BEST 9.71/3.30 (5) proven lower bound 9.71/3.30 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 9.71/3.30 (7) BOUNDS(n^1, INF) 9.71/3.30 (8) TRS for Loop Detection 9.71/3.30 (9) DecreasingLoopProof [FINISHED, 1062 ms] 9.71/3.30 (10) BOUNDS(EXP, INF) 9.71/3.30 9.71/3.30 9.71/3.30 ---------------------------------------- 9.71/3.30 9.71/3.30 (0) 9.71/3.30 Obligation: 9.71/3.30 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 9.71/3.30 9.71/3.30 9.71/3.30 The TRS R consists of the following rules: 9.71/3.30 9.71/3.30 a__U101(tt, V1, V2) -> a__U102(a__isNatural(V1), V2) 9.71/3.30 a__U102(tt, V2) -> a__U103(a__isLNat(V2)) 9.71/3.30 a__U103(tt) -> tt 9.71/3.30 a__U11(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 9.71/3.30 a__U111(tt, V1) -> a__U112(a__isLNat(V1)) 9.71/3.30 a__U112(tt) -> tt 9.71/3.30 a__U121(tt, V1) -> a__U122(a__isNatural(V1)) 9.71/3.30 a__U122(tt) -> tt 9.71/3.30 a__U131(tt, V1, V2) -> a__U132(a__isNatural(V1), V2) 9.71/3.30 a__U132(tt, V2) -> a__U133(a__isLNat(V2)) 9.71/3.30 a__U133(tt) -> tt 9.71/3.30 a__U141(tt, V1, V2) -> a__U142(a__isLNat(V1), V2) 9.71/3.30 a__U142(tt, V2) -> a__U143(a__isLNat(V2)) 9.71/3.30 a__U143(tt) -> tt 9.71/3.30 a__U151(tt, V1, V2) -> a__U152(a__isNatural(V1), V2) 9.71/3.30 a__U152(tt, V2) -> a__U153(a__isLNat(V2)) 9.71/3.30 a__U153(tt) -> tt 9.71/3.30 a__U161(tt, N) -> cons(mark(N), natsFrom(s(N))) 9.71/3.30 a__U171(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 9.71/3.30 a__U181(tt, Y) -> mark(Y) 9.71/3.30 a__U191(tt, XS) -> pair(nil, mark(XS)) 9.71/3.30 a__U201(tt, N, X, XS) -> a__U202(a__splitAt(mark(N), mark(XS)), X) 9.71/3.30 a__U202(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 9.71/3.30 a__U21(tt, X) -> mark(X) 9.71/3.30 a__U211(tt, XS) -> mark(XS) 9.71/3.30 a__U221(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 9.71/3.30 a__U31(tt, N) -> mark(N) 9.71/3.30 a__U41(tt, V1, V2) -> a__U42(a__isNatural(V1), V2) 9.71/3.30 a__U42(tt, V2) -> a__U43(a__isLNat(V2)) 9.71/3.30 a__U43(tt) -> tt 9.71/3.30 a__U51(tt, V1, V2) -> a__U52(a__isNatural(V1), V2) 9.71/3.30 a__U52(tt, V2) -> a__U53(a__isLNat(V2)) 9.71/3.30 a__U53(tt) -> tt 9.71/3.30 a__U61(tt, V1) -> a__U62(a__isPLNat(V1)) 9.71/3.30 a__U62(tt) -> tt 9.71/3.30 a__U71(tt, V1) -> a__U72(a__isNatural(V1)) 9.71/3.30 a__U72(tt) -> tt 9.71/3.30 a__U81(tt, V1) -> a__U82(a__isPLNat(V1)) 9.71/3.30 a__U82(tt) -> tt 9.71/3.30 a__U91(tt, V1) -> a__U92(a__isLNat(V1)) 9.71/3.30 a__U92(tt) -> tt 9.71/3.30 a__afterNth(N, XS) -> a__U11(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) 9.71/3.30 a__and(tt, X) -> mark(X) 9.71/3.30 a__fst(pair(X, Y)) -> a__U21(a__and(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X) 9.71/3.30 a__head(cons(N, XS)) -> a__U31(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N) 9.71/3.30 a__isLNat(nil) -> tt 9.71/3.30 a__isLNat(afterNth(V1, V2)) -> a__U41(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 9.71/3.30 a__isLNat(cons(V1, V2)) -> a__U51(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 9.71/3.30 a__isLNat(fst(V1)) -> a__U61(a__isPLNatKind(V1), V1) 9.71/3.30 a__isLNat(natsFrom(V1)) -> a__U71(a__isNaturalKind(V1), V1) 9.71/3.30 a__isLNat(snd(V1)) -> a__U81(a__isPLNatKind(V1), V1) 9.71/3.30 a__isLNat(tail(V1)) -> a__U91(a__isLNatKind(V1), V1) 9.71/3.30 a__isLNat(take(V1, V2)) -> a__U101(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 9.71/3.30 a__isLNatKind(nil) -> tt 9.71/3.30 a__isLNatKind(afterNth(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 9.71/3.30 a__isLNatKind(cons(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 9.71/3.30 a__isLNatKind(fst(V1)) -> a__isPLNatKind(V1) 9.71/3.30 a__isLNatKind(natsFrom(V1)) -> a__isNaturalKind(V1) 9.71/3.30 a__isLNatKind(snd(V1)) -> a__isPLNatKind(V1) 9.71/3.30 a__isLNatKind(tail(V1)) -> a__isLNatKind(V1) 9.71/3.30 a__isLNatKind(take(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 9.71/3.30 a__isNatural(0) -> tt 9.71/3.30 a__isNatural(head(V1)) -> a__U111(a__isLNatKind(V1), V1) 9.71/3.30 a__isNatural(s(V1)) -> a__U121(a__isNaturalKind(V1), V1) 9.71/3.30 a__isNatural(sel(V1, V2)) -> a__U131(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 9.71/3.30 a__isNaturalKind(0) -> tt 9.71/3.30 a__isNaturalKind(head(V1)) -> a__isLNatKind(V1) 9.71/3.30 a__isNaturalKind(s(V1)) -> a__isNaturalKind(V1) 9.71/3.30 a__isNaturalKind(sel(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 9.71/3.30 a__isPLNat(pair(V1, V2)) -> a__U141(a__and(a__isLNatKind(V1), isLNatKind(V2)), V1, V2) 9.71/3.30 a__isPLNat(splitAt(V1, V2)) -> a__U151(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 9.71/3.30 a__isPLNatKind(pair(V1, V2)) -> a__and(a__isLNatKind(V1), isLNatKind(V2)) 9.71/3.30 a__isPLNatKind(splitAt(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 9.71/3.30 a__natsFrom(N) -> a__U161(a__and(a__isNatural(N), isNaturalKind(N)), N) 9.71/3.30 a__sel(N, XS) -> a__U171(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) 9.71/3.30 a__snd(pair(X, Y)) -> a__U181(a__and(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y) 9.71/3.30 a__splitAt(0, XS) -> a__U191(a__and(a__isLNat(XS), isLNatKind(XS)), XS) 9.71/3.30 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) 9.71/3.30 a__tail(cons(N, XS)) -> a__U211(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS) 9.71/3.30 a__take(N, XS) -> a__U221(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) 9.71/3.30 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 9.71/3.30 mark(U102(X1, X2)) -> a__U102(mark(X1), X2) 9.71/3.30 mark(isNatural(X)) -> a__isNatural(X) 9.71/3.30 mark(U103(X)) -> a__U103(mark(X)) 9.71/3.30 mark(isLNat(X)) -> a__isLNat(X) 9.71/3.30 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 9.71/3.30 mark(snd(X)) -> a__snd(mark(X)) 9.71/3.30 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 9.71/3.30 mark(U111(X1, X2)) -> a__U111(mark(X1), X2) 9.71/3.30 mark(U112(X)) -> a__U112(mark(X)) 9.71/3.30 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 9.71/3.30 mark(U122(X)) -> a__U122(mark(X)) 9.71/3.30 mark(U131(X1, X2, X3)) -> a__U131(mark(X1), X2, X3) 9.71/3.30 mark(U132(X1, X2)) -> a__U132(mark(X1), X2) 9.71/3.30 mark(U133(X)) -> a__U133(mark(X)) 9.71/3.30 mark(U141(X1, X2, X3)) -> a__U141(mark(X1), X2, X3) 9.71/3.30 mark(U142(X1, X2)) -> a__U142(mark(X1), X2) 9.71/3.30 mark(U143(X)) -> a__U143(mark(X)) 9.71/3.30 mark(U151(X1, X2, X3)) -> a__U151(mark(X1), X2, X3) 9.71/3.30 mark(U152(X1, X2)) -> a__U152(mark(X1), X2) 9.71/3.30 mark(U153(X)) -> a__U153(mark(X)) 9.71/3.30 mark(U161(X1, X2)) -> a__U161(mark(X1), X2) 9.71/3.30 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 9.71/3.30 mark(U171(X1, X2, X3)) -> a__U171(mark(X1), X2, X3) 9.71/3.30 mark(head(X)) -> a__head(mark(X)) 9.71/3.30 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 9.71/3.30 mark(U181(X1, X2)) -> a__U181(mark(X1), X2) 9.71/3.30 mark(U191(X1, X2)) -> a__U191(mark(X1), X2) 9.71/3.30 mark(U201(X1, X2, X3, X4)) -> a__U201(mark(X1), X2, X3, X4) 9.71/3.30 mark(U202(X1, X2)) -> a__U202(mark(X1), X2) 9.71/3.30 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 9.71/3.30 mark(U211(X1, X2)) -> a__U211(mark(X1), X2) 9.71/3.30 mark(U221(X1, X2, X3)) -> a__U221(mark(X1), X2, X3) 9.71/3.30 mark(fst(X)) -> a__fst(mark(X)) 9.71/3.30 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 9.71/3.30 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 9.71/3.30 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 9.71/3.30 mark(U43(X)) -> a__U43(mark(X)) 9.71/3.30 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 9.71/3.30 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 9.71/3.30 mark(U53(X)) -> a__U53(mark(X)) 9.71/3.30 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 9.71/3.30 mark(U62(X)) -> a__U62(mark(X)) 9.71/3.30 mark(isPLNat(X)) -> a__isPLNat(X) 9.71/3.30 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 9.71/3.30 mark(U72(X)) -> a__U72(mark(X)) 9.71/3.30 mark(U81(X1, X2)) -> a__U81(mark(X1), X2) 9.71/3.30 mark(U82(X)) -> a__U82(mark(X)) 9.71/3.30 mark(U91(X1, X2)) -> a__U91(mark(X1), X2) 9.71/3.30 mark(U92(X)) -> a__U92(mark(X)) 9.71/3.30 mark(and(X1, X2)) -> a__and(mark(X1), X2) 9.71/3.30 mark(isNaturalKind(X)) -> a__isNaturalKind(X) 9.71/3.30 mark(isLNatKind(X)) -> a__isLNatKind(X) 9.71/3.30 mark(isPLNatKind(X)) -> a__isPLNatKind(X) 9.71/3.30 mark(tail(X)) -> a__tail(mark(X)) 9.71/3.30 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 9.71/3.30 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 9.71/3.30 mark(tt) -> tt 9.71/3.30 mark(cons(X1, X2)) -> cons(mark(X1), X2) 9.71/3.30 mark(s(X)) -> s(mark(X)) 9.71/3.30 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 9.71/3.30 mark(nil) -> nil 9.71/3.30 mark(0) -> 0 10.09/3.32 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 10.09/3.32 a__U102(X1, X2) -> U102(X1, X2) 10.09/3.32 a__isNatural(X) -> isNatural(X) 10.09/3.32 a__U103(X) -> U103(X) 10.09/3.32 a__isLNat(X) -> isLNat(X) 10.09/3.32 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 10.09/3.32 a__snd(X) -> snd(X) 10.09/3.32 a__splitAt(X1, X2) -> splitAt(X1, X2) 10.09/3.32 a__U111(X1, X2) -> U111(X1, X2) 10.09/3.32 a__U112(X) -> U112(X) 10.09/3.32 a__U121(X1, X2) -> U121(X1, X2) 10.09/3.32 a__U122(X) -> U122(X) 10.09/3.32 a__U131(X1, X2, X3) -> U131(X1, X2, X3) 10.09/3.32 a__U132(X1, X2) -> U132(X1, X2) 10.09/3.32 a__U133(X) -> U133(X) 10.09/3.32 a__U141(X1, X2, X3) -> U141(X1, X2, X3) 10.09/3.32 a__U142(X1, X2) -> U142(X1, X2) 10.09/3.32 a__U143(X) -> U143(X) 10.09/3.32 a__U151(X1, X2, X3) -> U151(X1, X2, X3) 10.09/3.32 a__U152(X1, X2) -> U152(X1, X2) 10.09/3.32 a__U153(X) -> U153(X) 10.09/3.32 a__U161(X1, X2) -> U161(X1, X2) 10.09/3.32 a__natsFrom(X) -> natsFrom(X) 10.09/3.32 a__U171(X1, X2, X3) -> U171(X1, X2, X3) 10.09/3.32 a__head(X) -> head(X) 10.09/3.32 a__afterNth(X1, X2) -> afterNth(X1, X2) 10.09/3.32 a__U181(X1, X2) -> U181(X1, X2) 10.09/3.32 a__U191(X1, X2) -> U191(X1, X2) 10.09/3.32 a__U201(X1, X2, X3, X4) -> U201(X1, X2, X3, X4) 10.09/3.32 a__U202(X1, X2) -> U202(X1, X2) 10.09/3.32 a__U21(X1, X2) -> U21(X1, X2) 10.09/3.32 a__U211(X1, X2) -> U211(X1, X2) 10.09/3.32 a__U221(X1, X2, X3) -> U221(X1, X2, X3) 10.09/3.32 a__fst(X) -> fst(X) 10.09/3.32 a__U31(X1, X2) -> U31(X1, X2) 10.09/3.32 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 10.09/3.32 a__U42(X1, X2) -> U42(X1, X2) 10.09/3.32 a__U43(X) -> U43(X) 10.09/3.32 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 10.09/3.32 a__U52(X1, X2) -> U52(X1, X2) 10.09/3.32 a__U53(X) -> U53(X) 10.09/3.32 a__U61(X1, X2) -> U61(X1, X2) 10.09/3.32 a__U62(X) -> U62(X) 10.09/3.32 a__isPLNat(X) -> isPLNat(X) 10.09/3.32 a__U71(X1, X2) -> U71(X1, X2) 10.09/3.32 a__U72(X) -> U72(X) 10.09/3.32 a__U81(X1, X2) -> U81(X1, X2) 10.09/3.32 a__U82(X) -> U82(X) 10.09/3.32 a__U91(X1, X2) -> U91(X1, X2) 10.09/3.32 a__U92(X) -> U92(X) 10.09/3.32 a__and(X1, X2) -> and(X1, X2) 10.09/3.32 a__isNaturalKind(X) -> isNaturalKind(X) 10.09/3.32 a__isLNatKind(X) -> isLNatKind(X) 10.09/3.32 a__isPLNatKind(X) -> isPLNatKind(X) 10.09/3.32 a__tail(X) -> tail(X) 10.09/3.32 a__take(X1, X2) -> take(X1, X2) 10.09/3.32 a__sel(X1, X2) -> sel(X1, X2) 10.09/3.32 10.09/3.32 S is empty. 10.09/3.32 Rewrite Strategy: FULL 10.09/3.32 ---------------------------------------- 10.09/3.32 10.09/3.32 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 10.09/3.32 Transformed a relative TRS into a decreasing-loop problem. 10.09/3.32 ---------------------------------------- 10.09/3.32 10.09/3.32 (2) 10.09/3.32 Obligation: 10.09/3.32 Analyzing the following TRS for decreasing loops: 10.09/3.32 10.09/3.32 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 10.09/3.32 10.09/3.32 10.09/3.32 The TRS R consists of the following rules: 10.09/3.32 10.09/3.32 a__U101(tt, V1, V2) -> a__U102(a__isNatural(V1), V2) 10.09/3.32 a__U102(tt, V2) -> a__U103(a__isLNat(V2)) 10.09/3.32 a__U103(tt) -> tt 10.09/3.32 a__U11(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 10.09/3.32 a__U111(tt, V1) -> a__U112(a__isLNat(V1)) 10.09/3.32 a__U112(tt) -> tt 10.09/3.32 a__U121(tt, V1) -> a__U122(a__isNatural(V1)) 10.09/3.32 a__U122(tt) -> tt 10.09/3.32 a__U131(tt, V1, V2) -> a__U132(a__isNatural(V1), V2) 10.09/3.32 a__U132(tt, V2) -> a__U133(a__isLNat(V2)) 10.09/3.32 a__U133(tt) -> tt 10.09/3.32 a__U141(tt, V1, V2) -> a__U142(a__isLNat(V1), V2) 10.09/3.32 a__U142(tt, V2) -> a__U143(a__isLNat(V2)) 10.09/3.32 a__U143(tt) -> tt 10.09/3.32 a__U151(tt, V1, V2) -> a__U152(a__isNatural(V1), V2) 10.09/3.32 a__U152(tt, V2) -> a__U153(a__isLNat(V2)) 10.09/3.32 a__U153(tt) -> tt 10.09/3.32 a__U161(tt, N) -> cons(mark(N), natsFrom(s(N))) 10.09/3.32 a__U171(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 10.09/3.32 a__U181(tt, Y) -> mark(Y) 10.09/3.32 a__U191(tt, XS) -> pair(nil, mark(XS)) 10.09/3.32 a__U201(tt, N, X, XS) -> a__U202(a__splitAt(mark(N), mark(XS)), X) 10.09/3.32 a__U202(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 10.09/3.32 a__U21(tt, X) -> mark(X) 10.09/3.32 a__U211(tt, XS) -> mark(XS) 10.09/3.32 a__U221(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 10.09/3.32 a__U31(tt, N) -> mark(N) 10.09/3.32 a__U41(tt, V1, V2) -> a__U42(a__isNatural(V1), V2) 10.09/3.32 a__U42(tt, V2) -> a__U43(a__isLNat(V2)) 10.09/3.32 a__U43(tt) -> tt 10.09/3.32 a__U51(tt, V1, V2) -> a__U52(a__isNatural(V1), V2) 10.09/3.32 a__U52(tt, V2) -> a__U53(a__isLNat(V2)) 10.09/3.32 a__U53(tt) -> tt 10.09/3.32 a__U61(tt, V1) -> a__U62(a__isPLNat(V1)) 10.09/3.32 a__U62(tt) -> tt 10.09/3.32 a__U71(tt, V1) -> a__U72(a__isNatural(V1)) 10.09/3.32 a__U72(tt) -> tt 10.09/3.32 a__U81(tt, V1) -> a__U82(a__isPLNat(V1)) 10.09/3.32 a__U82(tt) -> tt 10.09/3.32 a__U91(tt, V1) -> a__U92(a__isLNat(V1)) 10.09/3.32 a__U92(tt) -> tt 10.09/3.32 a__afterNth(N, XS) -> a__U11(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) 10.09/3.32 a__and(tt, X) -> mark(X) 10.09/3.32 a__fst(pair(X, Y)) -> a__U21(a__and(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X) 10.09/3.32 a__head(cons(N, XS)) -> a__U31(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N) 10.09/3.32 a__isLNat(nil) -> tt 10.09/3.32 a__isLNat(afterNth(V1, V2)) -> a__U41(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 10.09/3.32 a__isLNat(cons(V1, V2)) -> a__U51(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 10.09/3.32 a__isLNat(fst(V1)) -> a__U61(a__isPLNatKind(V1), V1) 10.09/3.32 a__isLNat(natsFrom(V1)) -> a__U71(a__isNaturalKind(V1), V1) 10.09/3.32 a__isLNat(snd(V1)) -> a__U81(a__isPLNatKind(V1), V1) 10.09/3.32 a__isLNat(tail(V1)) -> a__U91(a__isLNatKind(V1), V1) 10.09/3.32 a__isLNat(take(V1, V2)) -> a__U101(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 10.09/3.32 a__isLNatKind(nil) -> tt 10.09/3.32 a__isLNatKind(afterNth(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 10.09/3.32 a__isLNatKind(cons(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 10.09/3.32 a__isLNatKind(fst(V1)) -> a__isPLNatKind(V1) 10.09/3.32 a__isLNatKind(natsFrom(V1)) -> a__isNaturalKind(V1) 10.09/3.32 a__isLNatKind(snd(V1)) -> a__isPLNatKind(V1) 10.09/3.32 a__isLNatKind(tail(V1)) -> a__isLNatKind(V1) 10.09/3.32 a__isLNatKind(take(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 10.09/3.32 a__isNatural(0) -> tt 10.09/3.32 a__isNatural(head(V1)) -> a__U111(a__isLNatKind(V1), V1) 10.09/3.32 a__isNatural(s(V1)) -> a__U121(a__isNaturalKind(V1), V1) 10.09/3.32 a__isNatural(sel(V1, V2)) -> a__U131(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 10.09/3.32 a__isNaturalKind(0) -> tt 10.09/3.32 a__isNaturalKind(head(V1)) -> a__isLNatKind(V1) 10.09/3.32 a__isNaturalKind(s(V1)) -> a__isNaturalKind(V1) 10.09/3.32 a__isNaturalKind(sel(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 10.09/3.32 a__isPLNat(pair(V1, V2)) -> a__U141(a__and(a__isLNatKind(V1), isLNatKind(V2)), V1, V2) 10.09/3.32 a__isPLNat(splitAt(V1, V2)) -> a__U151(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 10.09/3.32 a__isPLNatKind(pair(V1, V2)) -> a__and(a__isLNatKind(V1), isLNatKind(V2)) 10.09/3.32 a__isPLNatKind(splitAt(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 10.09/3.32 a__natsFrom(N) -> a__U161(a__and(a__isNatural(N), isNaturalKind(N)), N) 10.09/3.32 a__sel(N, XS) -> a__U171(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) 10.09/3.32 a__snd(pair(X, Y)) -> a__U181(a__and(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y) 10.09/3.32 a__splitAt(0, XS) -> a__U191(a__and(a__isLNat(XS), isLNatKind(XS)), XS) 10.09/3.32 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) 10.09/3.32 a__tail(cons(N, XS)) -> a__U211(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS) 10.09/3.32 a__take(N, XS) -> a__U221(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) 10.09/3.32 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 10.09/3.32 mark(U102(X1, X2)) -> a__U102(mark(X1), X2) 10.09/3.32 mark(isNatural(X)) -> a__isNatural(X) 10.09/3.32 mark(U103(X)) -> a__U103(mark(X)) 10.09/3.32 mark(isLNat(X)) -> a__isLNat(X) 10.09/3.32 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 10.09/3.32 mark(snd(X)) -> a__snd(mark(X)) 10.09/3.32 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 10.09/3.32 mark(U111(X1, X2)) -> a__U111(mark(X1), X2) 10.09/3.32 mark(U112(X)) -> a__U112(mark(X)) 10.09/3.32 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 10.09/3.32 mark(U122(X)) -> a__U122(mark(X)) 10.09/3.32 mark(U131(X1, X2, X3)) -> a__U131(mark(X1), X2, X3) 10.09/3.32 mark(U132(X1, X2)) -> a__U132(mark(X1), X2) 10.09/3.32 mark(U133(X)) -> a__U133(mark(X)) 10.09/3.32 mark(U141(X1, X2, X3)) -> a__U141(mark(X1), X2, X3) 10.09/3.32 mark(U142(X1, X2)) -> a__U142(mark(X1), X2) 10.09/3.32 mark(U143(X)) -> a__U143(mark(X)) 10.09/3.32 mark(U151(X1, X2, X3)) -> a__U151(mark(X1), X2, X3) 10.09/3.32 mark(U152(X1, X2)) -> a__U152(mark(X1), X2) 10.09/3.32 mark(U153(X)) -> a__U153(mark(X)) 10.09/3.32 mark(U161(X1, X2)) -> a__U161(mark(X1), X2) 10.09/3.32 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 10.09/3.32 mark(U171(X1, X2, X3)) -> a__U171(mark(X1), X2, X3) 10.09/3.32 mark(head(X)) -> a__head(mark(X)) 10.09/3.32 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 10.09/3.32 mark(U181(X1, X2)) -> a__U181(mark(X1), X2) 10.09/3.32 mark(U191(X1, X2)) -> a__U191(mark(X1), X2) 10.09/3.32 mark(U201(X1, X2, X3, X4)) -> a__U201(mark(X1), X2, X3, X4) 10.09/3.32 mark(U202(X1, X2)) -> a__U202(mark(X1), X2) 10.09/3.32 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 10.09/3.32 mark(U211(X1, X2)) -> a__U211(mark(X1), X2) 10.09/3.32 mark(U221(X1, X2, X3)) -> a__U221(mark(X1), X2, X3) 10.09/3.32 mark(fst(X)) -> a__fst(mark(X)) 10.09/3.32 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 10.09/3.32 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 10.09/3.32 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 10.09/3.32 mark(U43(X)) -> a__U43(mark(X)) 10.09/3.32 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 10.09/3.32 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 10.09/3.32 mark(U53(X)) -> a__U53(mark(X)) 10.09/3.32 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 10.09/3.32 mark(U62(X)) -> a__U62(mark(X)) 10.09/3.32 mark(isPLNat(X)) -> a__isPLNat(X) 10.09/3.32 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 10.09/3.32 mark(U72(X)) -> a__U72(mark(X)) 10.09/3.32 mark(U81(X1, X2)) -> a__U81(mark(X1), X2) 10.09/3.32 mark(U82(X)) -> a__U82(mark(X)) 10.09/3.32 mark(U91(X1, X2)) -> a__U91(mark(X1), X2) 10.09/3.32 mark(U92(X)) -> a__U92(mark(X)) 10.09/3.32 mark(and(X1, X2)) -> a__and(mark(X1), X2) 10.09/3.32 mark(isNaturalKind(X)) -> a__isNaturalKind(X) 10.09/3.32 mark(isLNatKind(X)) -> a__isLNatKind(X) 10.09/3.32 mark(isPLNatKind(X)) -> a__isPLNatKind(X) 10.09/3.32 mark(tail(X)) -> a__tail(mark(X)) 10.09/3.32 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 10.09/3.32 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 10.09/3.32 mark(tt) -> tt 10.09/3.32 mark(cons(X1, X2)) -> cons(mark(X1), X2) 10.09/3.32 mark(s(X)) -> s(mark(X)) 10.09/3.32 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 10.09/3.32 mark(nil) -> nil 10.09/3.32 mark(0) -> 0 10.09/3.32 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 10.09/3.32 a__U102(X1, X2) -> U102(X1, X2) 10.09/3.32 a__isNatural(X) -> isNatural(X) 10.09/3.32 a__U103(X) -> U103(X) 10.09/3.32 a__isLNat(X) -> isLNat(X) 10.09/3.32 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 10.09/3.32 a__snd(X) -> snd(X) 10.09/3.32 a__splitAt(X1, X2) -> splitAt(X1, X2) 10.09/3.32 a__U111(X1, X2) -> U111(X1, X2) 10.09/3.32 a__U112(X) -> U112(X) 10.09/3.32 a__U121(X1, X2) -> U121(X1, X2) 10.09/3.32 a__U122(X) -> U122(X) 10.09/3.32 a__U131(X1, X2, X3) -> U131(X1, X2, X3) 10.09/3.32 a__U132(X1, X2) -> U132(X1, X2) 10.09/3.32 a__U133(X) -> U133(X) 10.09/3.32 a__U141(X1, X2, X3) -> U141(X1, X2, X3) 10.09/3.32 a__U142(X1, X2) -> U142(X1, X2) 10.09/3.32 a__U143(X) -> U143(X) 10.09/3.32 a__U151(X1, X2, X3) -> U151(X1, X2, X3) 10.09/3.32 a__U152(X1, X2) -> U152(X1, X2) 10.09/3.32 a__U153(X) -> U153(X) 10.09/3.32 a__U161(X1, X2) -> U161(X1, X2) 10.09/3.32 a__natsFrom(X) -> natsFrom(X) 10.09/3.32 a__U171(X1, X2, X3) -> U171(X1, X2, X3) 10.09/3.32 a__head(X) -> head(X) 10.09/3.32 a__afterNth(X1, X2) -> afterNth(X1, X2) 10.09/3.32 a__U181(X1, X2) -> U181(X1, X2) 10.09/3.32 a__U191(X1, X2) -> U191(X1, X2) 10.09/3.32 a__U201(X1, X2, X3, X4) -> U201(X1, X2, X3, X4) 10.09/3.32 a__U202(X1, X2) -> U202(X1, X2) 10.09/3.32 a__U21(X1, X2) -> U21(X1, X2) 10.09/3.32 a__U211(X1, X2) -> U211(X1, X2) 10.09/3.32 a__U221(X1, X2, X3) -> U221(X1, X2, X3) 10.09/3.32 a__fst(X) -> fst(X) 10.09/3.32 a__U31(X1, X2) -> U31(X1, X2) 10.09/3.32 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 10.09/3.32 a__U42(X1, X2) -> U42(X1, X2) 10.09/3.32 a__U43(X) -> U43(X) 10.09/3.32 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 10.09/3.32 a__U52(X1, X2) -> U52(X1, X2) 10.09/3.32 a__U53(X) -> U53(X) 10.09/3.32 a__U61(X1, X2) -> U61(X1, X2) 10.09/3.32 a__U62(X) -> U62(X) 10.09/3.32 a__isPLNat(X) -> isPLNat(X) 10.09/3.32 a__U71(X1, X2) -> U71(X1, X2) 10.09/3.32 a__U72(X) -> U72(X) 10.09/3.32 a__U81(X1, X2) -> U81(X1, X2) 10.09/3.32 a__U82(X) -> U82(X) 10.09/3.32 a__U91(X1, X2) -> U91(X1, X2) 10.09/3.32 a__U92(X) -> U92(X) 10.09/3.32 a__and(X1, X2) -> and(X1, X2) 10.09/3.32 a__isNaturalKind(X) -> isNaturalKind(X) 10.09/3.32 a__isLNatKind(X) -> isLNatKind(X) 10.09/3.32 a__isPLNatKind(X) -> isPLNatKind(X) 10.09/3.32 a__tail(X) -> tail(X) 10.09/3.32 a__take(X1, X2) -> take(X1, X2) 10.09/3.32 a__sel(X1, X2) -> sel(X1, X2) 10.09/3.32 10.09/3.32 S is empty. 10.09/3.32 Rewrite Strategy: FULL 10.09/3.32 ---------------------------------------- 10.09/3.32 10.09/3.32 (3) DecreasingLoopProof (LOWER BOUND(ID)) 10.09/3.32 The following loop(s) give(s) rise to the lower bound Omega(n^1): 10.09/3.32 10.09/3.32 The rewrite sequence 10.09/3.32 10.09/3.32 mark(U121(X1, X2)) ->^+ a__U121(mark(X1), X2) 10.09/3.32 10.09/3.32 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 10.09/3.32 10.09/3.32 The pumping substitution is [X1 / U121(X1, X2)]. 10.09/3.32 10.09/3.32 The result substitution is [ ]. 10.09/3.32 10.09/3.32 10.09/3.32 10.09/3.32 10.09/3.32 ---------------------------------------- 10.09/3.32 10.09/3.32 (4) 10.09/3.32 Complex Obligation (BEST) 10.09/3.32 10.09/3.32 ---------------------------------------- 10.09/3.32 10.09/3.32 (5) 10.09/3.32 Obligation: 10.09/3.32 Proved the lower bound n^1 for the following obligation: 10.09/3.32 10.09/3.32 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 10.09/3.32 10.09/3.32 10.09/3.32 The TRS R consists of the following rules: 10.09/3.32 10.09/3.32 a__U101(tt, V1, V2) -> a__U102(a__isNatural(V1), V2) 10.09/3.32 a__U102(tt, V2) -> a__U103(a__isLNat(V2)) 10.09/3.32 a__U103(tt) -> tt 10.09/3.32 a__U11(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 10.09/3.32 a__U111(tt, V1) -> a__U112(a__isLNat(V1)) 10.09/3.32 a__U112(tt) -> tt 10.09/3.32 a__U121(tt, V1) -> a__U122(a__isNatural(V1)) 10.09/3.32 a__U122(tt) -> tt 10.09/3.32 a__U131(tt, V1, V2) -> a__U132(a__isNatural(V1), V2) 10.09/3.32 a__U132(tt, V2) -> a__U133(a__isLNat(V2)) 10.09/3.32 a__U133(tt) -> tt 10.09/3.32 a__U141(tt, V1, V2) -> a__U142(a__isLNat(V1), V2) 10.09/3.32 a__U142(tt, V2) -> a__U143(a__isLNat(V2)) 10.09/3.32 a__U143(tt) -> tt 10.09/3.32 a__U151(tt, V1, V2) -> a__U152(a__isNatural(V1), V2) 10.09/3.32 a__U152(tt, V2) -> a__U153(a__isLNat(V2)) 10.09/3.32 a__U153(tt) -> tt 10.09/3.32 a__U161(tt, N) -> cons(mark(N), natsFrom(s(N))) 10.09/3.32 a__U171(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 10.09/3.32 a__U181(tt, Y) -> mark(Y) 10.09/3.32 a__U191(tt, XS) -> pair(nil, mark(XS)) 10.09/3.32 a__U201(tt, N, X, XS) -> a__U202(a__splitAt(mark(N), mark(XS)), X) 10.09/3.32 a__U202(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 10.09/3.32 a__U21(tt, X) -> mark(X) 10.09/3.32 a__U211(tt, XS) -> mark(XS) 10.09/3.32 a__U221(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 10.09/3.32 a__U31(tt, N) -> mark(N) 10.09/3.32 a__U41(tt, V1, V2) -> a__U42(a__isNatural(V1), V2) 10.09/3.32 a__U42(tt, V2) -> a__U43(a__isLNat(V2)) 10.09/3.32 a__U43(tt) -> tt 10.09/3.32 a__U51(tt, V1, V2) -> a__U52(a__isNatural(V1), V2) 10.09/3.32 a__U52(tt, V2) -> a__U53(a__isLNat(V2)) 10.09/3.32 a__U53(tt) -> tt 10.09/3.32 a__U61(tt, V1) -> a__U62(a__isPLNat(V1)) 10.09/3.32 a__U62(tt) -> tt 10.09/3.32 a__U71(tt, V1) -> a__U72(a__isNatural(V1)) 10.09/3.32 a__U72(tt) -> tt 10.09/3.32 a__U81(tt, V1) -> a__U82(a__isPLNat(V1)) 10.09/3.32 a__U82(tt) -> tt 10.09/3.32 a__U91(tt, V1) -> a__U92(a__isLNat(V1)) 10.09/3.32 a__U92(tt) -> tt 10.09/3.32 a__afterNth(N, XS) -> a__U11(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) 10.09/3.32 a__and(tt, X) -> mark(X) 10.09/3.32 a__fst(pair(X, Y)) -> a__U21(a__and(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X) 10.09/3.32 a__head(cons(N, XS)) -> a__U31(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N) 10.09/3.32 a__isLNat(nil) -> tt 10.09/3.32 a__isLNat(afterNth(V1, V2)) -> a__U41(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 10.09/3.32 a__isLNat(cons(V1, V2)) -> a__U51(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 10.09/3.32 a__isLNat(fst(V1)) -> a__U61(a__isPLNatKind(V1), V1) 10.09/3.32 a__isLNat(natsFrom(V1)) -> a__U71(a__isNaturalKind(V1), V1) 10.09/3.32 a__isLNat(snd(V1)) -> a__U81(a__isPLNatKind(V1), V1) 10.09/3.32 a__isLNat(tail(V1)) -> a__U91(a__isLNatKind(V1), V1) 10.09/3.32 a__isLNat(take(V1, V2)) -> a__U101(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 10.09/3.32 a__isLNatKind(nil) -> tt 10.09/3.32 a__isLNatKind(afterNth(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 10.09/3.32 a__isLNatKind(cons(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 10.09/3.32 a__isLNatKind(fst(V1)) -> a__isPLNatKind(V1) 10.09/3.32 a__isLNatKind(natsFrom(V1)) -> a__isNaturalKind(V1) 10.09/3.32 a__isLNatKind(snd(V1)) -> a__isPLNatKind(V1) 10.09/3.32 a__isLNatKind(tail(V1)) -> a__isLNatKind(V1) 10.09/3.32 a__isLNatKind(take(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 10.09/3.32 a__isNatural(0) -> tt 10.09/3.32 a__isNatural(head(V1)) -> a__U111(a__isLNatKind(V1), V1) 10.09/3.32 a__isNatural(s(V1)) -> a__U121(a__isNaturalKind(V1), V1) 10.09/3.32 a__isNatural(sel(V1, V2)) -> a__U131(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 10.09/3.32 a__isNaturalKind(0) -> tt 10.09/3.32 a__isNaturalKind(head(V1)) -> a__isLNatKind(V1) 10.09/3.32 a__isNaturalKind(s(V1)) -> a__isNaturalKind(V1) 10.09/3.32 a__isNaturalKind(sel(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 10.09/3.32 a__isPLNat(pair(V1, V2)) -> a__U141(a__and(a__isLNatKind(V1), isLNatKind(V2)), V1, V2) 10.09/3.32 a__isPLNat(splitAt(V1, V2)) -> a__U151(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 10.09/3.32 a__isPLNatKind(pair(V1, V2)) -> a__and(a__isLNatKind(V1), isLNatKind(V2)) 10.09/3.32 a__isPLNatKind(splitAt(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 10.09/3.32 a__natsFrom(N) -> a__U161(a__and(a__isNatural(N), isNaturalKind(N)), N) 10.09/3.32 a__sel(N, XS) -> a__U171(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) 10.09/3.32 a__snd(pair(X, Y)) -> a__U181(a__and(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y) 10.09/3.32 a__splitAt(0, XS) -> a__U191(a__and(a__isLNat(XS), isLNatKind(XS)), XS) 10.09/3.32 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) 10.09/3.32 a__tail(cons(N, XS)) -> a__U211(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS) 10.09/3.32 a__take(N, XS) -> a__U221(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) 10.09/3.32 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 10.09/3.32 mark(U102(X1, X2)) -> a__U102(mark(X1), X2) 10.09/3.32 mark(isNatural(X)) -> a__isNatural(X) 10.09/3.32 mark(U103(X)) -> a__U103(mark(X)) 10.09/3.32 mark(isLNat(X)) -> a__isLNat(X) 10.09/3.32 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 10.09/3.32 mark(snd(X)) -> a__snd(mark(X)) 10.09/3.32 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 10.09/3.32 mark(U111(X1, X2)) -> a__U111(mark(X1), X2) 10.09/3.32 mark(U112(X)) -> a__U112(mark(X)) 10.09/3.32 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 10.09/3.32 mark(U122(X)) -> a__U122(mark(X)) 10.09/3.32 mark(U131(X1, X2, X3)) -> a__U131(mark(X1), X2, X3) 10.09/3.32 mark(U132(X1, X2)) -> a__U132(mark(X1), X2) 10.09/3.32 mark(U133(X)) -> a__U133(mark(X)) 10.09/3.32 mark(U141(X1, X2, X3)) -> a__U141(mark(X1), X2, X3) 10.09/3.32 mark(U142(X1, X2)) -> a__U142(mark(X1), X2) 10.09/3.32 mark(U143(X)) -> a__U143(mark(X)) 10.09/3.32 mark(U151(X1, X2, X3)) -> a__U151(mark(X1), X2, X3) 10.09/3.32 mark(U152(X1, X2)) -> a__U152(mark(X1), X2) 10.09/3.32 mark(U153(X)) -> a__U153(mark(X)) 10.09/3.32 mark(U161(X1, X2)) -> a__U161(mark(X1), X2) 10.09/3.32 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 10.09/3.32 mark(U171(X1, X2, X3)) -> a__U171(mark(X1), X2, X3) 10.09/3.32 mark(head(X)) -> a__head(mark(X)) 10.09/3.32 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 10.09/3.32 mark(U181(X1, X2)) -> a__U181(mark(X1), X2) 10.09/3.32 mark(U191(X1, X2)) -> a__U191(mark(X1), X2) 10.09/3.32 mark(U201(X1, X2, X3, X4)) -> a__U201(mark(X1), X2, X3, X4) 10.09/3.32 mark(U202(X1, X2)) -> a__U202(mark(X1), X2) 10.09/3.32 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 10.09/3.32 mark(U211(X1, X2)) -> a__U211(mark(X1), X2) 10.09/3.32 mark(U221(X1, X2, X3)) -> a__U221(mark(X1), X2, X3) 10.09/3.32 mark(fst(X)) -> a__fst(mark(X)) 10.09/3.32 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 10.09/3.32 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 10.09/3.32 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 10.09/3.32 mark(U43(X)) -> a__U43(mark(X)) 10.09/3.32 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 10.09/3.32 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 10.09/3.32 mark(U53(X)) -> a__U53(mark(X)) 10.09/3.32 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 10.09/3.32 mark(U62(X)) -> a__U62(mark(X)) 10.09/3.32 mark(isPLNat(X)) -> a__isPLNat(X) 10.09/3.32 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 10.09/3.32 mark(U72(X)) -> a__U72(mark(X)) 10.09/3.32 mark(U81(X1, X2)) -> a__U81(mark(X1), X2) 10.09/3.32 mark(U82(X)) -> a__U82(mark(X)) 10.09/3.32 mark(U91(X1, X2)) -> a__U91(mark(X1), X2) 10.09/3.32 mark(U92(X)) -> a__U92(mark(X)) 10.09/3.32 mark(and(X1, X2)) -> a__and(mark(X1), X2) 10.09/3.32 mark(isNaturalKind(X)) -> a__isNaturalKind(X) 10.09/3.32 mark(isLNatKind(X)) -> a__isLNatKind(X) 10.09/3.32 mark(isPLNatKind(X)) -> a__isPLNatKind(X) 10.09/3.32 mark(tail(X)) -> a__tail(mark(X)) 10.09/3.32 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 10.09/3.32 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 10.09/3.32 mark(tt) -> tt 10.09/3.32 mark(cons(X1, X2)) -> cons(mark(X1), X2) 10.09/3.32 mark(s(X)) -> s(mark(X)) 10.09/3.32 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 10.09/3.32 mark(nil) -> nil 10.09/3.32 mark(0) -> 0 10.09/3.32 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 10.09/3.32 a__U102(X1, X2) -> U102(X1, X2) 10.09/3.32 a__isNatural(X) -> isNatural(X) 10.09/3.32 a__U103(X) -> U103(X) 10.09/3.32 a__isLNat(X) -> isLNat(X) 10.09/3.32 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 10.09/3.32 a__snd(X) -> snd(X) 10.09/3.32 a__splitAt(X1, X2) -> splitAt(X1, X2) 10.09/3.32 a__U111(X1, X2) -> U111(X1, X2) 10.09/3.32 a__U112(X) -> U112(X) 10.09/3.32 a__U121(X1, X2) -> U121(X1, X2) 10.09/3.32 a__U122(X) -> U122(X) 10.09/3.32 a__U131(X1, X2, X3) -> U131(X1, X2, X3) 10.09/3.32 a__U132(X1, X2) -> U132(X1, X2) 10.09/3.32 a__U133(X) -> U133(X) 10.09/3.32 a__U141(X1, X2, X3) -> U141(X1, X2, X3) 10.09/3.32 a__U142(X1, X2) -> U142(X1, X2) 10.09/3.32 a__U143(X) -> U143(X) 10.09/3.32 a__U151(X1, X2, X3) -> U151(X1, X2, X3) 10.09/3.32 a__U152(X1, X2) -> U152(X1, X2) 10.09/3.32 a__U153(X) -> U153(X) 10.09/3.32 a__U161(X1, X2) -> U161(X1, X2) 10.09/3.32 a__natsFrom(X) -> natsFrom(X) 10.09/3.32 a__U171(X1, X2, X3) -> U171(X1, X2, X3) 10.09/3.32 a__head(X) -> head(X) 10.09/3.32 a__afterNth(X1, X2) -> afterNth(X1, X2) 10.09/3.32 a__U181(X1, X2) -> U181(X1, X2) 10.09/3.32 a__U191(X1, X2) -> U191(X1, X2) 10.09/3.32 a__U201(X1, X2, X3, X4) -> U201(X1, X2, X3, X4) 10.09/3.32 a__U202(X1, X2) -> U202(X1, X2) 10.09/3.32 a__U21(X1, X2) -> U21(X1, X2) 10.09/3.32 a__U211(X1, X2) -> U211(X1, X2) 10.09/3.32 a__U221(X1, X2, X3) -> U221(X1, X2, X3) 10.09/3.32 a__fst(X) -> fst(X) 10.09/3.32 a__U31(X1, X2) -> U31(X1, X2) 10.09/3.32 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 10.09/3.32 a__U42(X1, X2) -> U42(X1, X2) 10.09/3.32 a__U43(X) -> U43(X) 10.09/3.32 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 10.09/3.32 a__U52(X1, X2) -> U52(X1, X2) 10.09/3.32 a__U53(X) -> U53(X) 10.09/3.32 a__U61(X1, X2) -> U61(X1, X2) 10.09/3.32 a__U62(X) -> U62(X) 10.09/3.32 a__isPLNat(X) -> isPLNat(X) 10.09/3.32 a__U71(X1, X2) -> U71(X1, X2) 10.09/3.32 a__U72(X) -> U72(X) 10.09/3.32 a__U81(X1, X2) -> U81(X1, X2) 10.09/3.32 a__U82(X) -> U82(X) 10.09/3.32 a__U91(X1, X2) -> U91(X1, X2) 10.09/3.32 a__U92(X) -> U92(X) 10.09/3.32 a__and(X1, X2) -> and(X1, X2) 10.09/3.32 a__isNaturalKind(X) -> isNaturalKind(X) 10.09/3.32 a__isLNatKind(X) -> isLNatKind(X) 10.09/3.32 a__isPLNatKind(X) -> isPLNatKind(X) 10.09/3.32 a__tail(X) -> tail(X) 10.09/3.32 a__take(X1, X2) -> take(X1, X2) 10.09/3.32 a__sel(X1, X2) -> sel(X1, X2) 10.09/3.32 10.09/3.32 S is empty. 10.09/3.32 Rewrite Strategy: FULL 10.09/3.32 ---------------------------------------- 10.09/3.32 10.09/3.32 (6) LowerBoundPropagationProof (FINISHED) 10.09/3.32 Propagated lower bound. 10.09/3.32 ---------------------------------------- 10.09/3.32 10.09/3.32 (7) 10.09/3.32 BOUNDS(n^1, INF) 10.09/3.32 10.09/3.32 ---------------------------------------- 10.09/3.32 10.09/3.32 (8) 10.09/3.32 Obligation: 10.09/3.32 Analyzing the following TRS for decreasing loops: 10.09/3.32 10.09/3.32 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 10.09/3.32 10.09/3.32 10.09/3.32 The TRS R consists of the following rules: 10.09/3.32 10.09/3.32 a__U101(tt, V1, V2) -> a__U102(a__isNatural(V1), V2) 10.09/3.32 a__U102(tt, V2) -> a__U103(a__isLNat(V2)) 10.09/3.32 a__U103(tt) -> tt 10.09/3.32 a__U11(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 10.09/3.32 a__U111(tt, V1) -> a__U112(a__isLNat(V1)) 10.09/3.32 a__U112(tt) -> tt 10.09/3.32 a__U121(tt, V1) -> a__U122(a__isNatural(V1)) 10.09/3.32 a__U122(tt) -> tt 10.09/3.32 a__U131(tt, V1, V2) -> a__U132(a__isNatural(V1), V2) 10.09/3.32 a__U132(tt, V2) -> a__U133(a__isLNat(V2)) 10.09/3.32 a__U133(tt) -> tt 10.09/3.32 a__U141(tt, V1, V2) -> a__U142(a__isLNat(V1), V2) 10.09/3.32 a__U142(tt, V2) -> a__U143(a__isLNat(V2)) 10.09/3.32 a__U143(tt) -> tt 10.09/3.32 a__U151(tt, V1, V2) -> a__U152(a__isNatural(V1), V2) 10.09/3.32 a__U152(tt, V2) -> a__U153(a__isLNat(V2)) 10.09/3.32 a__U153(tt) -> tt 10.09/3.32 a__U161(tt, N) -> cons(mark(N), natsFrom(s(N))) 10.09/3.32 a__U171(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 10.09/3.32 a__U181(tt, Y) -> mark(Y) 10.09/3.32 a__U191(tt, XS) -> pair(nil, mark(XS)) 10.09/3.32 a__U201(tt, N, X, XS) -> a__U202(a__splitAt(mark(N), mark(XS)), X) 10.09/3.32 a__U202(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 10.09/3.32 a__U21(tt, X) -> mark(X) 10.09/3.32 a__U211(tt, XS) -> mark(XS) 10.09/3.32 a__U221(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 10.09/3.32 a__U31(tt, N) -> mark(N) 10.09/3.32 a__U41(tt, V1, V2) -> a__U42(a__isNatural(V1), V2) 10.09/3.32 a__U42(tt, V2) -> a__U43(a__isLNat(V2)) 10.09/3.32 a__U43(tt) -> tt 10.09/3.32 a__U51(tt, V1, V2) -> a__U52(a__isNatural(V1), V2) 10.09/3.32 a__U52(tt, V2) -> a__U53(a__isLNat(V2)) 10.09/3.32 a__U53(tt) -> tt 10.09/3.32 a__U61(tt, V1) -> a__U62(a__isPLNat(V1)) 10.09/3.32 a__U62(tt) -> tt 10.09/3.32 a__U71(tt, V1) -> a__U72(a__isNatural(V1)) 10.09/3.32 a__U72(tt) -> tt 10.09/3.32 a__U81(tt, V1) -> a__U82(a__isPLNat(V1)) 10.09/3.32 a__U82(tt) -> tt 10.09/3.32 a__U91(tt, V1) -> a__U92(a__isLNat(V1)) 10.09/3.32 a__U92(tt) -> tt 10.09/3.32 a__afterNth(N, XS) -> a__U11(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) 10.09/3.32 a__and(tt, X) -> mark(X) 10.09/3.32 a__fst(pair(X, Y)) -> a__U21(a__and(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X) 10.09/3.32 a__head(cons(N, XS)) -> a__U31(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N) 10.09/3.32 a__isLNat(nil) -> tt 10.09/3.32 a__isLNat(afterNth(V1, V2)) -> a__U41(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 10.09/3.32 a__isLNat(cons(V1, V2)) -> a__U51(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 10.09/3.32 a__isLNat(fst(V1)) -> a__U61(a__isPLNatKind(V1), V1) 10.09/3.32 a__isLNat(natsFrom(V1)) -> a__U71(a__isNaturalKind(V1), V1) 10.09/3.32 a__isLNat(snd(V1)) -> a__U81(a__isPLNatKind(V1), V1) 10.09/3.32 a__isLNat(tail(V1)) -> a__U91(a__isLNatKind(V1), V1) 10.09/3.32 a__isLNat(take(V1, V2)) -> a__U101(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 10.09/3.32 a__isLNatKind(nil) -> tt 10.09/3.32 a__isLNatKind(afterNth(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 10.09/3.32 a__isLNatKind(cons(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 10.09/3.32 a__isLNatKind(fst(V1)) -> a__isPLNatKind(V1) 10.09/3.32 a__isLNatKind(natsFrom(V1)) -> a__isNaturalKind(V1) 10.09/3.32 a__isLNatKind(snd(V1)) -> a__isPLNatKind(V1) 10.09/3.32 a__isLNatKind(tail(V1)) -> a__isLNatKind(V1) 10.09/3.32 a__isLNatKind(take(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 10.09/3.32 a__isNatural(0) -> tt 10.09/3.32 a__isNatural(head(V1)) -> a__U111(a__isLNatKind(V1), V1) 10.09/3.32 a__isNatural(s(V1)) -> a__U121(a__isNaturalKind(V1), V1) 10.09/3.32 a__isNatural(sel(V1, V2)) -> a__U131(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 10.09/3.32 a__isNaturalKind(0) -> tt 10.09/3.32 a__isNaturalKind(head(V1)) -> a__isLNatKind(V1) 10.09/3.32 a__isNaturalKind(s(V1)) -> a__isNaturalKind(V1) 10.09/3.32 a__isNaturalKind(sel(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 10.09/3.32 a__isPLNat(pair(V1, V2)) -> a__U141(a__and(a__isLNatKind(V1), isLNatKind(V2)), V1, V2) 10.09/3.32 a__isPLNat(splitAt(V1, V2)) -> a__U151(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2) 10.09/3.32 a__isPLNatKind(pair(V1, V2)) -> a__and(a__isLNatKind(V1), isLNatKind(V2)) 10.09/3.32 a__isPLNatKind(splitAt(V1, V2)) -> a__and(a__isNaturalKind(V1), isLNatKind(V2)) 10.09/3.32 a__natsFrom(N) -> a__U161(a__and(a__isNatural(N), isNaturalKind(N)), N) 10.09/3.32 a__sel(N, XS) -> a__U171(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) 10.09/3.32 a__snd(pair(X, Y)) -> a__U181(a__and(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y) 10.09/3.32 a__splitAt(0, XS) -> a__U191(a__and(a__isLNat(XS), isLNatKind(XS)), XS) 10.09/3.32 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) 10.09/3.32 a__tail(cons(N, XS)) -> a__U211(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS) 10.09/3.32 a__take(N, XS) -> a__U221(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) 10.09/3.32 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 10.09/3.32 mark(U102(X1, X2)) -> a__U102(mark(X1), X2) 10.09/3.32 mark(isNatural(X)) -> a__isNatural(X) 10.09/3.32 mark(U103(X)) -> a__U103(mark(X)) 10.09/3.32 mark(isLNat(X)) -> a__isLNat(X) 10.09/3.32 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 10.09/3.32 mark(snd(X)) -> a__snd(mark(X)) 10.09/3.32 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 10.09/3.32 mark(U111(X1, X2)) -> a__U111(mark(X1), X2) 10.09/3.32 mark(U112(X)) -> a__U112(mark(X)) 10.09/3.32 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 10.09/3.32 mark(U122(X)) -> a__U122(mark(X)) 10.09/3.32 mark(U131(X1, X2, X3)) -> a__U131(mark(X1), X2, X3) 10.09/3.32 mark(U132(X1, X2)) -> a__U132(mark(X1), X2) 10.09/3.32 mark(U133(X)) -> a__U133(mark(X)) 10.09/3.32 mark(U141(X1, X2, X3)) -> a__U141(mark(X1), X2, X3) 10.09/3.32 mark(U142(X1, X2)) -> a__U142(mark(X1), X2) 10.09/3.32 mark(U143(X)) -> a__U143(mark(X)) 10.09/3.32 mark(U151(X1, X2, X3)) -> a__U151(mark(X1), X2, X3) 10.09/3.32 mark(U152(X1, X2)) -> a__U152(mark(X1), X2) 10.09/3.32 mark(U153(X)) -> a__U153(mark(X)) 10.09/3.32 mark(U161(X1, X2)) -> a__U161(mark(X1), X2) 10.09/3.32 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 10.09/3.32 mark(U171(X1, X2, X3)) -> a__U171(mark(X1), X2, X3) 10.09/3.32 mark(head(X)) -> a__head(mark(X)) 10.09/3.32 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 10.09/3.32 mark(U181(X1, X2)) -> a__U181(mark(X1), X2) 10.09/3.32 mark(U191(X1, X2)) -> a__U191(mark(X1), X2) 10.09/3.32 mark(U201(X1, X2, X3, X4)) -> a__U201(mark(X1), X2, X3, X4) 10.09/3.32 mark(U202(X1, X2)) -> a__U202(mark(X1), X2) 10.09/3.32 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 10.09/3.32 mark(U211(X1, X2)) -> a__U211(mark(X1), X2) 10.09/3.32 mark(U221(X1, X2, X3)) -> a__U221(mark(X1), X2, X3) 10.09/3.32 mark(fst(X)) -> a__fst(mark(X)) 10.09/3.32 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 10.09/3.32 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 10.09/3.32 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 10.09/3.32 mark(U43(X)) -> a__U43(mark(X)) 10.09/3.32 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 10.09/3.32 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 10.09/3.32 mark(U53(X)) -> a__U53(mark(X)) 10.09/3.32 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 10.09/3.32 mark(U62(X)) -> a__U62(mark(X)) 10.09/3.32 mark(isPLNat(X)) -> a__isPLNat(X) 10.09/3.32 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 10.09/3.32 mark(U72(X)) -> a__U72(mark(X)) 10.09/3.32 mark(U81(X1, X2)) -> a__U81(mark(X1), X2) 10.09/3.32 mark(U82(X)) -> a__U82(mark(X)) 10.09/3.32 mark(U91(X1, X2)) -> a__U91(mark(X1), X2) 10.09/3.32 mark(U92(X)) -> a__U92(mark(X)) 10.09/3.32 mark(and(X1, X2)) -> a__and(mark(X1), X2) 10.09/3.32 mark(isNaturalKind(X)) -> a__isNaturalKind(X) 10.09/3.32 mark(isLNatKind(X)) -> a__isLNatKind(X) 10.09/3.32 mark(isPLNatKind(X)) -> a__isPLNatKind(X) 10.09/3.32 mark(tail(X)) -> a__tail(mark(X)) 10.09/3.32 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 10.09/3.32 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 10.09/3.32 mark(tt) -> tt 10.09/3.32 mark(cons(X1, X2)) -> cons(mark(X1), X2) 10.09/3.32 mark(s(X)) -> s(mark(X)) 10.09/3.32 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 10.09/3.32 mark(nil) -> nil 10.09/3.32 mark(0) -> 0 10.09/3.32 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 10.09/3.32 a__U102(X1, X2) -> U102(X1, X2) 10.09/3.32 a__isNatural(X) -> isNatural(X) 10.09/3.32 a__U103(X) -> U103(X) 10.09/3.32 a__isLNat(X) -> isLNat(X) 10.09/3.32 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 10.09/3.32 a__snd(X) -> snd(X) 10.09/3.32 a__splitAt(X1, X2) -> splitAt(X1, X2) 10.09/3.32 a__U111(X1, X2) -> U111(X1, X2) 10.09/3.32 a__U112(X) -> U112(X) 10.09/3.32 a__U121(X1, X2) -> U121(X1, X2) 10.09/3.32 a__U122(X) -> U122(X) 10.09/3.32 a__U131(X1, X2, X3) -> U131(X1, X2, X3) 10.09/3.32 a__U132(X1, X2) -> U132(X1, X2) 10.09/3.32 a__U133(X) -> U133(X) 10.09/3.32 a__U141(X1, X2, X3) -> U141(X1, X2, X3) 10.09/3.32 a__U142(X1, X2) -> U142(X1, X2) 10.09/3.32 a__U143(X) -> U143(X) 10.09/3.32 a__U151(X1, X2, X3) -> U151(X1, X2, X3) 10.09/3.32 a__U152(X1, X2) -> U152(X1, X2) 10.09/3.32 a__U153(X) -> U153(X) 10.09/3.32 a__U161(X1, X2) -> U161(X1, X2) 10.09/3.32 a__natsFrom(X) -> natsFrom(X) 10.09/3.32 a__U171(X1, X2, X3) -> U171(X1, X2, X3) 10.09/3.32 a__head(X) -> head(X) 10.09/3.32 a__afterNth(X1, X2) -> afterNth(X1, X2) 10.09/3.32 a__U181(X1, X2) -> U181(X1, X2) 10.09/3.32 a__U191(X1, X2) -> U191(X1, X2) 10.09/3.32 a__U201(X1, X2, X3, X4) -> U201(X1, X2, X3, X4) 10.09/3.32 a__U202(X1, X2) -> U202(X1, X2) 10.09/3.32 a__U21(X1, X2) -> U21(X1, X2) 10.09/3.32 a__U211(X1, X2) -> U211(X1, X2) 10.09/3.32 a__U221(X1, X2, X3) -> U221(X1, X2, X3) 10.09/3.32 a__fst(X) -> fst(X) 10.09/3.32 a__U31(X1, X2) -> U31(X1, X2) 10.09/3.32 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 10.09/3.32 a__U42(X1, X2) -> U42(X1, X2) 10.09/3.32 a__U43(X) -> U43(X) 10.09/3.32 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 10.09/3.32 a__U52(X1, X2) -> U52(X1, X2) 10.09/3.32 a__U53(X) -> U53(X) 10.09/3.32 a__U61(X1, X2) -> U61(X1, X2) 10.09/3.32 a__U62(X) -> U62(X) 10.09/3.32 a__isPLNat(X) -> isPLNat(X) 10.09/3.32 a__U71(X1, X2) -> U71(X1, X2) 10.09/3.32 a__U72(X) -> U72(X) 10.09/3.32 a__U81(X1, X2) -> U81(X1, X2) 10.09/3.32 a__U82(X) -> U82(X) 10.09/3.32 a__U91(X1, X2) -> U91(X1, X2) 10.09/3.32 a__U92(X) -> U92(X) 10.09/3.32 a__and(X1, X2) -> and(X1, X2) 10.09/3.32 a__isNaturalKind(X) -> isNaturalKind(X) 10.09/3.32 a__isLNatKind(X) -> isLNatKind(X) 10.09/3.32 a__isPLNatKind(X) -> isPLNatKind(X) 10.09/3.32 a__tail(X) -> tail(X) 10.09/3.32 a__take(X1, X2) -> take(X1, X2) 10.09/3.32 a__sel(X1, X2) -> sel(X1, X2) 10.09/3.32 10.09/3.32 S is empty. 10.09/3.32 Rewrite Strategy: FULL 10.09/3.32 ---------------------------------------- 10.09/3.32 10.09/3.32 (9) DecreasingLoopProof (FINISHED) 10.09/3.32 The following loop(s) give(s) rise to the lower bound EXP: 10.09/3.32 10.09/3.32 The rewrite sequence 10.09/3.32 10.09/3.32 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)) 10.09/3.32 10.09/3.32 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0]. 10.09/3.32 10.09/3.32 The pumping substitution is [X1 / take(X1, X2)]. 10.09/3.32 10.09/3.32 The result substitution is [ ]. 10.09/3.32 10.09/3.32 10.09/3.32 10.09/3.32 The rewrite sequence 10.09/3.32 10.09/3.32 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)) 10.09/3.32 10.09/3.32 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,1,0]. 10.09/3.32 10.09/3.32 The pumping substitution is [X1 / take(X1, X2)]. 10.09/3.32 10.09/3.32 The result substitution is [ ]. 10.09/3.32 10.09/3.32 10.09/3.32 10.09/3.32 10.09/3.32 ---------------------------------------- 10.09/3.32 10.09/3.32 (10) 10.09/3.32 BOUNDS(EXP, INF) 10.33/3.50 EOF