TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60002 ms.

The following DP Processors were used


Problem 1 remains open; application of the following processors failed [DependencyGraph (timeout)].

The following open problems remain:



Open Dependency Pair Problem 1

Dependency Pairs

active#(isLNat(afterNth(V1, V2))) → isNaturalKind#(V1)mark#(fst(X)) → active#(fst(mark(X)))
U91#(X1, active(X2)) → U91#(X1, X2)U221#(X1, active(X2), X3) → U221#(X1, X2, X3)
mark#(take(X1, X2)) → active#(take(mark(X1), mark(X2)))active#(isLNat(afterNth(V1, V2))) → mark#(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active#(isNatural(sel(V1, V2))) → and#(isNaturalKind(V1), isLNatKind(V2))active#(U191(tt, XS)) → pair#(nil, XS)
active#(U121(tt, V1)) → U122#(isNatural(V1))mark#(U132(X1, X2)) → active#(U132(mark(X1), X2))
active#(U142(tt, V2)) → U143#(isLNat(V2))active#(isLNat(cons(V1, V2))) → isNaturalKind#(V1)
active#(U202(pair(YS, ZS), X)) → cons#(X, YS)U52#(X1, active(X2)) → U52#(X1, X2)
mark#(U103(X)) → active#(U103(mark(X)))isLNatKind#(active(X)) → isLNatKind#(X)
active#(isLNatKind(fst(V1))) → isPLNatKind#(V1)active#(U71(tt, V1)) → mark#(U72(isNatural(V1)))
U21#(X1, active(X2)) → U21#(X1, X2)mark#(s(X)) → mark#(X)
U152#(X1, mark(X2)) → U152#(X1, X2)mark#(U72(X)) → active#(U72(mark(X)))
active#(U11(tt, N, XS)) → splitAt#(N, XS)mark#(U72(X)) → mark#(X)
active#(isLNat(take(V1, V2))) → isNaturalKind#(V1)afterNth#(active(X1), X2) → afterNth#(X1, X2)
U151#(X1, active(X2), X3) → U151#(X1, X2, X3)U132#(X1, active(X2)) → U132#(X1, X2)
active#(isNaturalKind(0)) → mark#(tt)U141#(mark(X1), X2, X3) → U141#(X1, X2, X3)
mark#(U112(X)) → U112#(mark(X))splitAt#(X1, mark(X2)) → splitAt#(X1, X2)
active#(U82(tt)) → mark#(tt)active#(isLNatKind(natsFrom(V1))) → mark#(isNaturalKind(V1))
U42#(mark(X1), X2) → U42#(X1, X2)active#(head(cons(N, XS))) → isLNatKind#(XS)
active#(isLNatKind(afterNth(V1, V2))) → isLNatKind#(V2)U53#(mark(X)) → U53#(X)
mark#(U41(X1, X2, X3)) → mark#(X1)active#(fst(pair(X, Y))) → isLNatKind#(X)
active#(isLNat(cons(V1, V2))) → and#(isNaturalKind(V1), isLNatKind(V2))active#(isLNat(tail(V1))) → U91#(isLNatKind(V1), V1)
mark#(U11(X1, X2, X3)) → mark#(X1)mark#(U112(X)) → active#(U112(mark(X)))
mark#(U53(X)) → U53#(mark(X))U141#(X1, X2, active(X3)) → U141#(X1, X2, X3)
mark#(U142(X1, X2)) → active#(U142(mark(X1), X2))U151#(X1, X2, mark(X3)) → U151#(X1, X2, X3)
U221#(active(X1), X2, X3) → U221#(X1, X2, X3)mark#(U62(X)) → active#(U62(mark(X)))
natsFrom#(mark(X)) → natsFrom#(X)mark#(U141(X1, X2, X3)) → mark#(X1)
active#(splitAt(s(N), cons(X, XS))) → and#(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS))))active#(U91(tt, V1)) → isLNat#(V1)
U191#(X1, mark(X2)) → U191#(X1, X2)active#(take(N, XS)) → isNatural#(N)
U82#(active(X)) → U82#(X)active#(splitAt(0, XS)) → isLNat#(XS)
mark#(U211(X1, X2)) → active#(U211(mark(X1), X2))U91#(X1, mark(X2)) → U91#(X1, X2)
active#(isPLNatKind(splitAt(V1, V2))) → isLNatKind#(V2)mark#(splitAt(X1, X2)) → mark#(X1)
mark#(U21(X1, X2)) → active#(U21(mark(X1), X2))active#(isNatural(head(V1))) → U111#(isLNatKind(V1), V1)
active#(U81(tt, V1)) → isPLNat#(V1)active#(splitAt(s(N), cons(X, XS))) → isLNat#(XS)
isNatural#(mark(X)) → isNatural#(X)mark#(isNatural(X)) → isNatural#(X)
isPLNat#(active(X)) → isPLNat#(X)U11#(active(X1), X2, X3) → U11#(X1, X2, X3)
mark#(U42(X1, X2)) → U42#(mark(X1), X2)active#(U11(tt, N, XS)) → mark#(snd(splitAt(N, XS)))
active#(take(N, XS)) → and#(isLNat(XS), isLNatKind(XS))U41#(X1, X2, active(X3)) → U41#(X1, X2, X3)
U191#(mark(X1), X2) → U191#(X1, X2)U132#(X1, mark(X2)) → U132#(X1, X2)
mark#(U42(X1, X2)) → mark#(X1)mark#(U103(X)) → mark#(X)
mark#(U101(X1, X2, X3)) → U101#(mark(X1), X2, X3)active#(snd(pair(X, Y))) → isLNatKind#(Y)
active#(U42(tt, V2)) → isLNat#(V2)cons#(active(X1), X2) → cons#(X1, X2)
mark#(natsFrom(X)) → mark#(X)mark#(afterNth(X1, X2)) → active#(afterNth(mark(X1), mark(X2)))
mark#(U201(X1, X2, X3, X4)) → mark#(X1)active#(U91(tt, V1)) → U92#(isLNat(V1))
active#(sel(N, XS)) → and#(isLNat(XS), isLNatKind(XS))active#(afterNth(N, XS)) → U11#(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS)
mark#(U41(X1, X2, X3)) → active#(U41(mark(X1), X2, X3))active#(isLNatKind(take(V1, V2))) → isNaturalKind#(V1)
mark#(U191(X1, X2)) → mark#(X1)active#(U152(tt, V2)) → isLNat#(V2)
U52#(mark(X1), X2) → U52#(X1, X2)mark#(U121(X1, X2)) → active#(U121(mark(X1), X2))
active#(U101(tt, V1, V2)) → mark#(U102(isNatural(V1), V2))active#(sel(N, XS)) → mark#(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active#(U61(tt, V1)) → U62#(isPLNat(V1))U181#(X1, active(X2)) → U181#(X1, X2)
afterNth#(X1, mark(X2)) → afterNth#(X1, X2)isNatural#(active(X)) → isNatural#(X)
active#(sel(N, XS)) → and#(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))U61#(active(X1), X2) → U61#(X1, X2)
U131#(X1, active(X2), X3) → U131#(X1, X2, X3)mark#(U133(X)) → U133#(mark(X))
active#(head(cons(N, XS))) → isNatural#(N)active#(U131(tt, V1, V2)) → U132#(isNatural(V1), V2)
active#(afterNth(N, XS)) → and#(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))mark#(U11(X1, X2, X3)) → active#(U11(mark(X1), X2, X3))
mark#(tail(X)) → active#(tail(mark(X)))active#(U52(tt, V2)) → isLNat#(V2)
U91#(mark(X1), X2) → U91#(X1, X2)mark#(U191(X1, X2)) → U191#(mark(X1), X2)
U61#(mark(X1), X2) → U61#(X1, X2)active#(isLNat(natsFrom(V1))) → U71#(isNaturalKind(V1), V1)
active#(isNaturalKind(s(V1))) → mark#(isNaturalKind(V1))active#(afterNth(N, XS)) → and#(isNatural(N), isNaturalKind(N))
mark#(isLNat(X)) → active#(isLNat(X))mark#(natsFrom(X)) → active#(natsFrom(mark(X)))
active#(U62(tt)) → mark#(tt)active#(splitAt(s(N), cons(X, XS))) → isNatural#(N)
U51#(X1, mark(X2), X3) → U51#(X1, X2, X3)isPLNat#(mark(X)) → isPLNat#(X)
mark#(isPLNat(X)) → isPLNat#(X)active#(isNatural(s(V1))) → isNaturalKind#(V1)
active#(U42(tt, V2)) → U43#(isLNat(V2))mark#(U171(X1, X2, X3)) → active#(U171(mark(X1), X2, X3))
active#(natsFrom(N)) → isNatural#(N)take#(X1, mark(X2)) → take#(X1, X2)
mark#(splitAt(X1, X2)) → active#(splitAt(mark(X1), mark(X2)))mark#(U71(X1, X2)) → mark#(X1)
U171#(X1, X2, active(X3)) → U171#(X1, X2, X3)active#(head(cons(N, XS))) → and#(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))
active#(U191(tt, XS)) → mark#(pair(nil, XS))active#(isLNatKind(snd(V1))) → isPLNatKind#(V1)
active#(U111(tt, V1)) → U112#(isLNat(V1))mark#(U121(X1, X2)) → mark#(X1)
mark#(U133(X)) → mark#(X)active#(isNaturalKind(head(V1))) → mark#(isLNatKind(V1))
mark#(U211(X1, X2)) → mark#(X1)active#(U61(tt, V1)) → isPLNat#(V1)
mark#(U161(X1, X2)) → U161#(mark(X1), X2)mark#(U151(X1, X2, X3)) → U151#(mark(X1), X2, X3)
sel#(mark(X1), X2) → sel#(X1, X2)active#(U221(tt, N, XS)) → mark#(fst(splitAt(N, XS)))
mark#(head(X)) → active#(head(mark(X)))mark#(U41(X1, X2, X3)) → U41#(mark(X1), X2, X3)
and#(X1, active(X2)) → and#(X1, X2)active#(head(cons(N, XS))) → mark#(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active#(take(N, XS)) → isLNat#(XS)active#(isNaturalKind(sel(V1, V2))) → mark#(and(isNaturalKind(V1), isLNatKind(V2)))
mark#(U181(X1, X2)) → mark#(X1)U221#(X1, X2, active(X3)) → U221#(X1, X2, X3)
mark#(U21(X1, X2)) → mark#(X1)U61#(X1, mark(X2)) → U61#(X1, X2)
splitAt#(active(X1), X2) → splitAt#(X1, X2)active#(natsFrom(N)) → and#(isNatural(N), isNaturalKind(N))
active#(splitAt(s(N), cons(X, XS))) → isLNatKind#(XS)mark#(nil) → active#(nil)
U111#(X1, mark(X2)) → U111#(X1, X2)U103#(mark(X)) → U103#(X)
take#(X1, active(X2)) → take#(X1, X2)active#(U171(tt, N, XS)) → head#(afterNth(N, XS))
U101#(X1, X2, active(X3)) → U101#(X1, X2, X3)mark#(isNatural(X)) → active#(isNatural(X))
active#(U103(tt)) → mark#(tt)mark#(cons(X1, X2)) → cons#(mark(X1), X2)
active#(take(N, XS)) → mark#(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))natsFrom#(active(X)) → natsFrom#(X)
mark#(U81(X1, X2)) → active#(U81(mark(X1), X2))U43#(mark(X)) → U43#(X)
U152#(mark(X1), X2) → U152#(X1, X2)active#(sel(N, XS)) → isLNatKind#(XS)
mark#(U153(X)) → mark#(X)active#(isPLNat(pair(V1, V2))) → mark#(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active#(isLNat(afterNth(V1, V2))) → U41#(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)mark#(splitAt(X1, X2)) → mark#(X2)
mark#(U153(X)) → U153#(mark(X))s#(mark(X)) → s#(X)
active#(fst(pair(X, Y))) → isLNat#(Y)active#(tail(cons(N, XS))) → isLNat#(XS)
active#(U52(tt, V2)) → mark#(U53(isLNat(V2)))mark#(U31(X1, X2)) → U31#(mark(X1), X2)
active#(isLNatKind(cons(V1, V2))) → mark#(and(isNaturalKind(V1), isLNatKind(V2)))active#(isLNatKind(cons(V1, V2))) → isNaturalKind#(V1)
isLNatKind#(mark(X)) → isLNatKind#(X)U122#(mark(X)) → U122#(X)
mark#(isLNatKind(X)) → isLNatKind#(X)active#(afterNth(N, XS)) → isLNatKind#(XS)
active#(U31(tt, N)) → mark#(N)U112#(active(X)) → U112#(X)
mark#(U143(X)) → U143#(mark(X))active#(sel(N, XS)) → isNaturalKind#(N)
active#(isLNat(natsFrom(V1))) → isNaturalKind#(V1)mark#(U103(X)) → U103#(mark(X))
mark#(U72(X)) → U72#(mark(X))U153#(active(X)) → U153#(X)
U71#(X1, mark(X2)) → U71#(X1, X2)active#(U122(tt)) → mark#(tt)
splitAt#(mark(X1), X2) → splitAt#(X1, X2)active#(U161(tt, N)) → natsFrom#(s(N))
active#(take(N, XS)) → isLNatKind#(XS)mark#(U181(X1, X2)) → U181#(mark(X1), X2)
isPLNatKind#(active(X)) → isPLNatKind#(X)active#(isNatural(s(V1))) → mark#(U121(isNaturalKind(V1), V1))
active#(isLNatKind(afterNth(V1, V2))) → isNaturalKind#(V1)mark#(U131(X1, X2, X3)) → U131#(mark(X1), X2, X3)
active#(isLNat(take(V1, V2))) → isLNatKind#(V2)active#(natsFrom(N)) → mark#(U161(and(isNatural(N), isNaturalKind(N)), N))
mark#(U221(X1, X2, X3)) → active#(U221(mark(X1), X2, X3))U201#(X1, mark(X2), X3, X4) → U201#(X1, X2, X3, X4)
mark#(U53(X)) → mark#(X)active#(splitAt(0, XS)) → mark#(U191(and(isLNat(XS), isLNatKind(XS)), XS))
U51#(X1, X2, active(X3)) → U51#(X1, X2, X3)mark#(U71(X1, X2)) → U71#(mark(X1), X2)
active#(U53(tt)) → mark#(tt)active#(U41(tt, V1, V2)) → U42#(isNatural(V1), V2)
active#(isLNat(snd(V1))) → isPLNatKind#(V1)mark#(cons(X1, X2)) → mark#(X1)
pair#(X1, mark(X2)) → pair#(X1, X2)U112#(mark(X)) → U112#(X)
active#(tail(cons(N, XS))) → isNatural#(N)U111#(mark(X1), X2) → U111#(X1, X2)
active#(U132(tt, V2)) → mark#(U133(isLNat(V2)))U133#(mark(X)) → U133#(X)
active#(isLNat(snd(V1))) → mark#(U81(isPLNatKind(V1), V1))active#(head(cons(N, XS))) → and#(isLNat(XS), isLNatKind(XS))
mark#(isNaturalKind(X)) → active#(isNaturalKind(X))active#(isLNat(nil)) → mark#(tt)
mark#(U202(X1, X2)) → U202#(mark(X1), X2)U201#(X1, X2, X3, active(X4)) → U201#(X1, X2, X3, X4)
active#(isPLNatKind(pair(V1, V2))) → and#(isLNatKind(V1), isLNatKind(V2))mark#(U111(X1, X2)) → active#(U111(mark(X1), X2))
active#(isLNatKind(tail(V1))) → mark#(isLNatKind(V1))U211#(mark(X1), X2) → U211#(X1, X2)
active#(take(N, XS)) → and#(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))U181#(X1, mark(X2)) → U181#(X1, X2)
mark#(tail(X)) → tail#(mark(X))active#(isNatural(0)) → mark#(tt)
mark#(splitAt(X1, X2)) → splitAt#(mark(X1), mark(X2))U81#(X1, mark(X2)) → U81#(X1, X2)
mark#(U122(X)) → active#(U122(mark(X)))active#(isLNat(fst(V1))) → U61#(isPLNatKind(V1), V1)
active#(head(cons(N, XS))) → and#(isNatural(N), isNaturalKind(N))active#(U211(tt, XS)) → mark#(XS)
active#(splitAt(s(N), cons(X, XS))) → isNaturalKind#(X)U61#(X1, active(X2)) → U61#(X1, X2)
active#(isLNat(afterNth(V1, V2))) → isLNatKind#(V2)active#(U132(tt, V2)) → isLNat#(V2)
U101#(X1, X2, mark(X3)) → U101#(X1, X2, X3)active#(splitAt(0, XS)) → U191#(and(isLNat(XS), isLNatKind(XS)), XS)
splitAt#(X1, active(X2)) → splitAt#(X1, X2)active#(fst(pair(X, Y))) → U21#(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X)
mark#(U31(X1, X2)) → active#(U31(mark(X1), X2))active#(isLNat(take(V1, V2))) → mark#(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
U202#(X1, active(X2)) → U202#(X1, X2)mark#(U91(X1, X2)) → U91#(mark(X1), X2)
U122#(active(X)) → U122#(X)mark#(0) → active#(0)
active#(U161(tt, N)) → cons#(N, natsFrom(s(N)))U221#(X1, mark(X2), X3) → U221#(X1, X2, X3)
U41#(X1, X2, mark(X3)) → U41#(X1, X2, X3)active#(isNatural(sel(V1, V2))) → U131#(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)
mark#(U111(X1, X2)) → U111#(mark(X1), X2)active#(U143(tt)) → mark#(tt)
U201#(mark(X1), X2, X3, X4) → U201#(X1, X2, X3, X4)active#(snd(pair(X, Y))) → isLNat#(Y)
mark#(U202(X1, X2)) → active#(U202(mark(X1), X2))mark#(U141(X1, X2, X3)) → U141#(mark(X1), X2, X3)
active#(take(N, XS)) → isNaturalKind#(N)mark#(U202(X1, X2)) → mark#(X1)
U211#(X1, mark(X2)) → U211#(X1, X2)U202#(active(X1), X2) → U202#(X1, X2)
mark#(U62(X)) → U62#(mark(X))mark#(U131(X1, X2, X3)) → mark#(X1)
mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3))U141#(X1, active(X2), X3) → U141#(X1, X2, X3)
active#(splitAt(s(N), cons(X, XS))) → and#(isLNat(XS), isLNatKind(XS))U142#(X1, active(X2)) → U142#(X1, X2)
U11#(X1, X2, active(X3)) → U11#(X1, X2, X3)active#(U92(tt)) → mark#(tt)
U201#(X1, X2, active(X3), X4) → U201#(X1, X2, X3, X4)active#(afterNth(N, XS)) → mark#(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark#(fst(X)) → fst#(mark(X))active#(U151(tt, V1, V2)) → isNatural#(V1)
U131#(X1, mark(X2), X3) → U131#(X1, X2, X3)U53#(active(X)) → U53#(X)
active#(isNatural(sel(V1, V2))) → mark#(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))U51#(X1, active(X2), X3) → U51#(X1, X2, X3)
U141#(X1, mark(X2), X3) → U141#(X1, X2, X3)active#(isPLNat(pair(V1, V2))) → U141#(and(isLNatKind(V1), isLNatKind(V2)), V1, V2)
active#(isPLNatKind(pair(V1, V2))) → isLNatKind#(V1)active#(take(N, XS)) → U221#(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS)
mark#(U152(X1, X2)) → active#(U152(mark(X1), X2))active#(isPLNatKind(splitAt(V1, V2))) → and#(isNaturalKind(V1), isLNatKind(V2))
mark#(U82(X)) → active#(U82(mark(X)))U143#(active(X)) → U143#(X)
active#(U81(tt, V1)) → U82#(isPLNat(V1))mark#(U201(X1, X2, X3, X4)) → active#(U201(mark(X1), X2, X3, X4))
active#(U111(tt, V1)) → isLNat#(V1)U133#(active(X)) → U133#(X)
active#(U221(tt, N, XS)) → splitAt#(N, XS)active#(U111(tt, V1)) → mark#(U112(isLNat(V1)))
sel#(active(X1), X2) → sel#(X1, X2)U142#(active(X1), X2) → U142#(X1, X2)
mark#(U161(X1, X2)) → mark#(X1)U181#(mark(X1), X2) → U181#(X1, X2)
U41#(active(X1), X2, X3) → U41#(X1, X2, X3)mark#(isLNatKind(X)) → active#(isLNatKind(X))
mark#(afterNth(X1, X2)) → afterNth#(mark(X1), mark(X2))active#(fst(pair(X, Y))) → and#(isLNat(X), isLNatKind(X))
active#(U43(tt)) → mark#(tt)and#(active(X1), X2) → and#(X1, X2)
active#(U142(tt, V2)) → mark#(U143(isLNat(V2)))mark#(U91(X1, X2)) → mark#(X1)
mark#(U52(X1, X2)) → active#(U52(mark(X1), X2))U42#(X1, active(X2)) → U42#(X1, X2)
active#(splitAt(s(N), cons(X, XS))) → and#(isNatural(X), isNaturalKind(X))active#(isLNat(tail(V1))) → isLNatKind#(V1)
active#(tail(cons(N, XS))) → mark#(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))U201#(X1, X2, mark(X3), X4) → U201#(X1, X2, X3, X4)
active#(U91(tt, V1)) → mark#(U92(isLNat(V1)))U202#(mark(X1), X2) → U202#(X1, X2)
active#(isPLNatKind(pair(V1, V2))) → isLNatKind#(V2)U91#(active(X1), X2) → U91#(X1, X2)
mark#(U142(X1, X2)) → mark#(X1)active#(isPLNatKind(splitAt(V1, V2))) → mark#(and(isNaturalKind(V1), isLNatKind(V2)))
tail#(active(X)) → tail#(X)mark#(U71(X1, X2)) → active#(U71(mark(X1), X2))
U41#(mark(X1), X2, X3) → U41#(X1, X2, X3)U201#(X1, X2, X3, mark(X4)) → U201#(X1, X2, X3, X4)
active#(isLNat(take(V1, V2))) → and#(isNaturalKind(V1), isLNatKind(V2))active#(sel(N, XS)) → isLNat#(XS)
U152#(X1, active(X2)) → U152#(X1, X2)active#(isLNat(fst(V1))) → isPLNatKind#(V1)
U191#(active(X1), X2) → U191#(X1, X2)active#(take(N, XS)) → and#(isNatural(N), isNaturalKind(N))
U81#(mark(X1), X2) → U81#(X1, X2)active#(U51(tt, V1, V2)) → U52#(isNatural(V1), V2)
mark#(U52(X1, X2)) → mark#(X1)active#(isNatural(sel(V1, V2))) → isLNatKind#(V2)
active#(snd(pair(X, Y))) → mark#(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))U81#(active(X1), X2) → U81#(X1, X2)
U101#(X1, mark(X2), X3) → U101#(X1, X2, X3)active#(isPLNat(pair(V1, V2))) → isLNatKind#(V2)
U51#(X1, X2, mark(X3)) → U51#(X1, X2, X3)U41#(X1, mark(X2), X3) → U41#(X1, X2, X3)
active#(isPLNatKind(splitAt(V1, V2))) → isNaturalKind#(V1)active#(U141(tt, V1, V2)) → mark#(U142(isLNat(V1), V2))
U71#(active(X1), X2) → U71#(X1, X2)U21#(mark(X1), X2) → U21#(X1, X2)
active#(U21(tt, X)) → mark#(X)active#(U221(tt, N, XS)) → fst#(splitAt(N, XS))
mark#(take(X1, X2)) → mark#(X2)active#(tail(cons(N, XS))) → and#(isLNat(XS), isLNatKind(XS))
mark#(natsFrom(X)) → natsFrom#(mark(X))mark#(U122(X)) → U122#(mark(X))
mark#(pair(X1, X2)) → active#(pair(mark(X1), mark(X2)))active#(U102(tt, V2)) → mark#(U103(isLNat(V2)))
U161#(X1, active(X2)) → U161#(X1, X2)active#(isLNat(cons(V1, V2))) → mark#(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active#(isPLNat(splitAt(V1, V2))) → mark#(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))active#(sel(N, XS)) → U171#(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS)
U132#(mark(X1), X2) → U132#(X1, X2)active#(head(cons(N, XS))) → U31#(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N)
U171#(X1, X2, mark(X3)) → U171#(X1, X2, X3)mark#(U11(X1, X2, X3)) → U11#(mark(X1), X2, X3)
isNaturalKind#(active(X)) → isNaturalKind#(X)U82#(mark(X)) → U82#(X)
active#(U52(tt, V2)) → U53#(isLNat(V2))active#(U42(tt, V2)) → mark#(U43(isLNat(V2)))
mark#(snd(X)) → active#(snd(mark(X)))mark#(U92(X)) → active#(U92(mark(X)))
active#(splitAt(s(N), cons(X, XS))) → isNaturalKind#(N)mark#(isPLNatKind(X)) → active#(isPLNatKind(X))
mark#(sel(X1, X2)) → mark#(X1)active#(isLNatKind(fst(V1))) → mark#(isPLNatKind(V1))
active#(U141(tt, V1, V2)) → isLNat#(V1)active#(natsFrom(N)) → isNaturalKind#(N)
active#(isLNat(fst(V1))) → mark#(U61(isPLNatKind(V1), V1))U121#(active(X1), X2) → U121#(X1, X2)
mark#(U171(X1, X2, X3)) → mark#(X1)mark#(U53(X)) → active#(U53(mark(X)))
mark#(afterNth(X1, X2)) → mark#(X1)and#(mark(X1), X2) → and#(X1, X2)
mark#(U122(X)) → mark#(X)active#(U71(tt, V1)) → isNatural#(V1)
U152#(active(X1), X2) → U152#(X1, X2)active#(U101(tt, V1, V2)) → U102#(isNatural(V1), V2)
U211#(active(X1), X2) → U211#(X1, X2)active#(U161(tt, N)) → s#(N)
U41#(X1, active(X2), X3) → U41#(X1, X2, X3)active#(fst(pair(X, Y))) → and#(isLNat(Y), isLNatKind(Y))
mark#(U111(X1, X2)) → mark#(X1)head#(active(X)) → head#(X)
mark#(U132(X1, X2)) → U132#(mark(X1), X2)U131#(X1, X2, active(X3)) → U131#(X1, X2, X3)
active#(isLNatKind(take(V1, V2))) → mark#(and(isNaturalKind(V1), isLNatKind(V2)))isLNat#(active(X)) → isLNat#(X)
U102#(mark(X1), X2) → U102#(X1, X2)active#(splitAt(s(N), cons(X, XS))) → and#(isNatural(N), isNaturalKind(N))
U132#(active(X1), X2) → U132#(X1, X2)U201#(X1, active(X2), X3, X4) → U201#(X1, X2, X3, X4)
active#(U133(tt)) → mark#(tt)active#(U201(tt, N, X, XS)) → splitAt#(N, XS)
mark#(head(X)) → mark#(X)active#(isPLNat(splitAt(V1, V2))) → U151#(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)
active#(U171(tt, N, XS)) → mark#(head(afterNth(N, XS)))active#(splitAt(s(N), cons(X, XS))) → and#(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))
isPLNatKind#(mark(X)) → isPLNatKind#(X)U171#(X1, active(X2), X3) → U171#(X1, X2, X3)
cons#(X1, mark(X2)) → cons#(X1, X2)mark#(isPLNatKind(X)) → isPLNatKind#(X)
U143#(mark(X)) → U143#(X)mark#(U143(X)) → mark#(X)
U52#(X1, mark(X2)) → U52#(X1, X2)mark#(fst(X)) → mark#(X)
active#(isLNat(natsFrom(V1))) → mark#(U71(isNaturalKind(V1), V1))active#(sel(N, XS)) → isNatural#(N)
snd#(mark(X)) → snd#(X)active#(U121(tt, V1)) → isNatural#(V1)
active#(U72(tt)) → mark#(tt)U62#(active(X)) → U62#(X)
active#(head(cons(N, XS))) → isLNat#(XS)mark#(U221(X1, X2, X3)) → mark#(X1)
U21#(active(X1), X2) → U21#(X1, X2)pair#(X1, active(X2)) → pair#(X1, X2)
mark#(snd(X)) → mark#(X)active#(tail(cons(N, XS))) → U211#(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS)
mark#(U102(X1, X2)) → mark#(X1)mark#(U82(X)) → U82#(mark(X))
active#(afterNth(N, XS)) → isLNat#(XS)U142#(X1, mark(X2)) → U142#(X1, X2)
active#(U131(tt, V1, V2)) → mark#(U132(isNatural(V1), V2))U42#(active(X1), X2) → U42#(X1, X2)
active#(isLNatKind(take(V1, V2))) → isLNatKind#(V2)mark#(take(X1, X2)) → mark#(X1)
active#(U142(tt, V2)) → isLNat#(V2)mark#(U141(X1, X2, X3)) → active#(U141(mark(X1), X2, X3))
mark#(U52(X1, X2)) → U52#(mark(X1), X2)mark#(s(X)) → s#(mark(X))
active#(isNaturalKind(sel(V1, V2))) → isLNatKind#(V2)active#(afterNth(N, XS)) → isNatural#(N)
U171#(X1, mark(X2), X3) → U171#(X1, X2, X3)active#(isPLNat(pair(V1, V2))) → and#(isLNatKind(V1), isLNatKind(V2))
active#(U181(tt, Y)) → mark#(Y)afterNth#(mark(X1), X2) → afterNth#(X1, X2)
mark#(U151(X1, X2, X3)) → active#(U151(mark(X1), X2, X3))mark#(U152(X1, X2)) → U152#(mark(X1), X2)
active#(U152(tt, V2)) → mark#(U153(isLNat(V2)))active#(isNatural(sel(V1, V2))) → isNaturalKind#(V1)
mark#(sel(X1, X2)) → sel#(mark(X1), mark(X2))U121#(X1, active(X2)) → U121#(X1, X2)
active#(U202(pair(YS, ZS), X)) → pair#(cons(X, YS), ZS)mark#(and(X1, X2)) → and#(mark(X1), X2)
mark#(U153(X)) → active#(U153(mark(X)))U11#(mark(X1), X2, X3) → U11#(X1, X2, X3)
U101#(active(X1), X2, X3) → U101#(X1, X2, X3)tail#(mark(X)) → tail#(X)
mark#(sel(X1, X2)) → active#(sel(mark(X1), mark(X2)))U102#(X1, mark(X2)) → U102#(X1, X2)
mark#(U61(X1, X2)) → active#(U61(mark(X1), X2))cons#(X1, active(X2)) → cons#(X1, X2)
active#(snd(pair(X, Y))) → isLNatKind#(X)active#(U161(tt, N)) → mark#(cons(N, natsFrom(s(N))))
active#(U201(tt, N, X, XS)) → U202#(splitAt(N, XS), X)active#(isNatural(head(V1))) → isLNatKind#(V1)
active#(U41(tt, V1, V2)) → isNatural#(V1)active#(snd(pair(X, Y))) → isLNat#(X)
mark#(U133(X)) → active#(U133(mark(X)))active#(U81(tt, V1)) → mark#(U82(isPLNat(V1)))
active#(splitAt(0, XS)) → and#(isLNat(XS), isLNatKind(XS))U51#(active(X1), X2, X3) → U51#(X1, X2, X3)
U43#(active(X)) → U43#(X)mark#(U61(X1, X2)) → mark#(X1)
active#(isNaturalKind(sel(V1, V2))) → and#(isNaturalKind(V1), isLNatKind(V2))take#(mark(X1), X2) → take#(X1, X2)
U131#(active(X1), X2, X3) → U131#(X1, X2, X3)mark#(U101(X1, X2, X3)) → active#(U101(mark(X1), X2, X3))
U101#(X1, active(X2), X3) → U101#(X1, X2, X3)U221#(mark(X1), X2, X3) → U221#(X1, X2, X3)
mark#(pair(X1, X2)) → pair#(mark(X1), mark(X2))mark#(U211(X1, X2)) → U211#(mark(X1), X2)
mark#(U21(X1, X2)) → U21#(mark(X1), X2)U11#(X1, X2, mark(X3)) → U11#(X1, X2, X3)
U103#(active(X)) → U103#(X)U111#(active(X1), X2) → U111#(X1, X2)
active#(U132(tt, V2)) → U133#(isLNat(V2))U151#(X1, X2, active(X3)) → U151#(X1, X2, X3)
mark#(sel(X1, X2)) → mark#(X2)fst#(mark(X)) → fst#(X)
active#(U171(tt, N, XS)) → afterNth#(N, XS)afterNth#(X1, active(X2)) → afterNth#(X1, X2)
U92#(active(X)) → U92#(X)U151#(mark(X1), X2, X3) → U151#(X1, X2, X3)
active#(U112(tt)) → mark#(tt)and#(X1, mark(X2)) → and#(X1, X2)
U171#(mark(X1), X2, X3) → U171#(X1, X2, X3)U121#(mark(X1), X2) → U121#(X1, X2)
active#(splitAt(s(N), cons(X, XS))) → isNatural#(X)isLNat#(mark(X)) → isLNat#(X)
mark#(isLNat(X)) → isLNat#(X)active#(isPLNat(splitAt(V1, V2))) → and#(isNaturalKind(V1), isLNatKind(V2))
mark#(head(X)) → head#(mark(X))mark#(take(X1, X2)) → take#(mark(X1), mark(X2))
active#(U131(tt, V1, V2)) → isNatural#(V1)active#(snd(pair(X, Y))) → and#(isLNat(Y), isLNatKind(Y))
U151#(active(X1), X2, X3) → U151#(X1, X2, X3)active#(isLNatKind(tail(V1))) → isLNatKind#(V1)
mark#(U102(X1, X2)) → U102#(mark(X1), X2)U141#(active(X1), X2, X3) → U141#(X1, X2, X3)
active#(fst(pair(X, Y))) → isLNat#(X)mark#(U152(X1, X2)) → mark#(X1)
U171#(active(X1), X2, X3) → U171#(X1, X2, X3)mark#(U51(X1, X2, X3)) → U51#(mark(X1), X2, X3)
active#(U153(tt)) → mark#(tt)mark#(U42(X1, X2)) → active#(U42(mark(X1), X2))
active#(afterNth(N, XS)) → isNaturalKind#(N)mark#(U102(X1, X2)) → active#(U102(mark(X1), X2))
mark#(U43(X)) → U43#(mark(X))mark#(U142(X1, X2)) → U142#(mark(X1), X2)
U151#(X1, mark(X2), X3) → U151#(X1, X2, X3)mark#(U101(X1, X2, X3)) → mark#(X1)
mark#(U81(X1, X2)) → U81#(mark(X1), X2)active#(tail(cons(N, XS))) → isNaturalKind#(N)
U141#(X1, X2, mark(X3)) → U141#(X1, X2, X3)mark#(pair(X1, X2)) → mark#(X2)
active#(isNatural(s(V1))) → U121#(isNaturalKind(V1), V1)active#(isPLNat(splitAt(V1, V2))) → isLNatKind#(V2)
active#(isNaturalKind(head(V1))) → isLNatKind#(V1)active#(sel(N, XS)) → and#(isNatural(N), isNaturalKind(N))
active#(splitAt(s(N), cons(X, XS))) → mark#(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))U11#(X1, active(X2), X3) → U11#(X1, X2, X3)
mark#(U161(X1, X2)) → active#(U161(mark(X1), X2))active#(head(cons(N, XS))) → isNaturalKind#(N)
active#(splitAt(0, XS)) → isLNatKind#(XS)fst#(active(X)) → fst#(X)
U102#(active(X1), X2) → U102#(X1, X2)sel#(X1, mark(X2)) → sel#(X1, X2)
mark#(U181(X1, X2)) → active#(U181(mark(X1), X2))U131#(mark(X1), X2, X3) → U131#(X1, X2, X3)
mark#(and(X1, X2)) → mark#(X1)active#(isNaturalKind(s(V1))) → isNaturalKind#(V1)
mark#(pair(X1, X2)) → mark#(X1)sel#(X1, active(X2)) → sel#(X1, X2)
mark#(tail(X)) → mark#(X)mark#(U62(X)) → mark#(X)
U161#(X1, mark(X2)) → U161#(X1, X2)mark#(U131(X1, X2, X3)) → active#(U131(mark(X1), X2, X3))
active#(U201(tt, N, X, XS)) → mark#(U202(splitAt(N, XS), X))active#(U51(tt, V1, V2)) → mark#(U52(isNatural(V1), V2))
mark#(afterNth(X1, X2)) → mark#(X2)active#(isNatural(head(V1))) → mark#(U111(isLNatKind(V1), V1))
pair#(mark(X1), X2) → pair#(X1, X2)active#(U61(tt, V1)) → mark#(U62(isPLNat(V1)))
mark#(U92(X)) → U92#(mark(X))cons#(mark(X1), X2) → cons#(X1, X2)
active#(isLNat(tail(V1))) → mark#(U91(isLNatKind(V1), V1))mark#(tt) → active#(tt)
active#(isLNatKind(natsFrom(V1))) → isNaturalKind#(V1)U31#(X1, active(X2)) → U31#(X1, X2)
mark#(U61(X1, X2)) → U61#(mark(X1), X2)active#(fst(pair(X, Y))) → isLNatKind#(Y)
active#(U202(pair(YS, ZS), X)) → mark#(pair(cons(X, YS), ZS))mark#(U132(X1, X2)) → mark#(X1)
U21#(X1, mark(X2)) → U21#(X1, X2)mark#(U51(X1, X2, X3)) → mark#(X1)
U62#(mark(X)) → U62#(X)U71#(mark(X1), X2) → U71#(X1, X2)
U201#(active(X1), X2, X3, X4) → U201#(X1, X2, X3, X4)U221#(X1, X2, mark(X3)) → U221#(X1, X2, X3)
U181#(active(X1), X2) → U181#(X1, X2)U131#(X1, X2, mark(X3)) → U131#(X1, X2, X3)
active#(isLNat(afterNth(V1, V2))) → and#(isNaturalKind(V1), isLNatKind(V2))U51#(mark(X1), X2, X3) → U51#(X1, X2, X3)
mark#(s(X)) → active#(s(mark(X)))head#(mark(X)) → head#(X)
active#(U102(tt, V2)) → U103#(isLNat(V2))U52#(active(X1), X2) → U52#(X1, X2)
snd#(active(X)) → snd#(X)mark#(U43(X)) → active#(U43(mark(X)))
mark#(U43(X)) → mark#(X)mark#(U201(X1, X2, X3, X4)) → U201#(mark(X1), X2, X3, X4)
U81#(X1, active(X2)) → U81#(X1, X2)active#(isNaturalKind(sel(V1, V2))) → isNaturalKind#(V1)
mark#(cons(X1, X2)) → active#(cons(mark(X1), X2))U202#(X1, mark(X2)) → U202#(X1, X2)
active#(splitAt(s(N), cons(X, XS))) → U201#(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS)active#(isLNatKind(snd(V1))) → mark#(isPLNatKind(V1))
active#(isPLNatKind(pair(V1, V2))) → mark#(and(isLNatKind(V1), isLNatKind(V2)))U92#(mark(X)) → U92#(X)
active#(U121(tt, V1)) → mark#(U122(isNatural(V1)))active#(natsFrom(N)) → U161#(and(isNatural(N), isNaturalKind(N)), N)
active#(isLNatKind(cons(V1, V2))) → and#(isNaturalKind(V1), isLNatKind(V2))U102#(X1, active(X2)) → U102#(X1, X2)
mark#(snd(X)) → snd#(mark(X))active#(isLNatKind(take(V1, V2))) → and#(isNaturalKind(V1), isLNatKind(V2))
active#(isPLNat(pair(V1, V2))) → isLNatKind#(V1)active#(U101(tt, V1, V2)) → isNatural#(V1)
U191#(X1, active(X2)) → U191#(X1, X2)active#(fst(pair(X, Y))) → mark#(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
U42#(X1, mark(X2)) → U42#(X1, X2)U31#(active(X1), X2) → U31#(X1, X2)
mark#(U143(X)) → active#(U143(mark(X)))mark#(U31(X1, X2)) → mark#(X1)
pair#(active(X1), X2) → pair#(X1, X2)U153#(mark(X)) → U153#(X)
mark#(U91(X1, X2)) → active#(U91(mark(X1), X2))active#(isLNat(snd(V1))) → U81#(isPLNatKind(V1), V1)
mark#(U81(X1, X2)) → mark#(X1)active#(isLNatKind(afterNth(V1, V2))) → mark#(and(isNaturalKind(V1), isLNatKind(V2)))
isNaturalKind#(mark(X)) → isNaturalKind#(X)mark#(isNaturalKind(X)) → isNaturalKind#(X)
active#(isPLNat(splitAt(V1, V2))) → isNaturalKind#(V1)active#(isLNat(cons(V1, V2))) → U51#(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)
active#(U141(tt, V1, V2)) → U142#(isLNat(V1), V2)active#(U41(tt, V1, V2)) → mark#(U42(isNatural(V1), V2))
U142#(mark(X1), X2) → U142#(X1, X2)active#(fst(pair(X, Y))) → and#(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y)))
mark#(U82(X)) → mark#(X)active#(U151(tt, V1, V2)) → U152#(isNatural(V1), V2)
active#(tail(cons(N, XS))) → and#(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))U72#(mark(X)) → U72#(X)
U71#(X1, active(X2)) → U71#(X1, X2)mark#(U121(X1, X2)) → U121#(mark(X1), X2)
mark#(U112(X)) → mark#(X)active#(isLNatKind(afterNth(V1, V2))) → and#(isNaturalKind(V1), isLNatKind(V2))
active#(snd(pair(X, Y))) → U181#(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y)U111#(X1, active(X2)) → U111#(X1, X2)
U161#(mark(X1), X2) → U161#(X1, X2)mark#(U151(X1, X2, X3)) → mark#(X1)
active#(isLNat(cons(V1, V2))) → isLNatKind#(V2)active#(snd(pair(X, Y))) → and#(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y)))
mark#(U171(X1, X2, X3)) → U171#(mark(X1), X2, X3)U121#(X1, mark(X2)) → U121#(X1, X2)
U101#(mark(X1), X2, X3) → U101#(X1, X2, X3)mark#(U221(X1, X2, X3)) → U221#(mark(X1), X2, X3)
U31#(X1, mark(X2)) → U31#(X1, X2)active#(afterNth(N, XS)) → and#(isLNat(XS), isLNatKind(XS))
active#(U51(tt, V1, V2)) → isNatural#(V1)mark#(and(X1, X2)) → active#(and(mark(X1), X2))
mark#(isPLNat(X)) → active#(isPLNat(X))U211#(X1, active(X2)) → U211#(X1, X2)
active#(tail(cons(N, XS))) → isLNatKind#(XS)active#(isLNatKind(nil)) → mark#(tt)
active#(isLNatKind(cons(V1, V2))) → isLNatKind#(V2)U161#(active(X1), X2) → U161#(X1, X2)
active#(U102(tt, V2)) → isLNat#(V2)mark#(U191(X1, X2)) → active#(U191(mark(X1), X2))
active#(isLNat(take(V1, V2))) → U101#(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)active#(U152(tt, V2)) → U153#(isLNat(V2))
active#(and(tt, X)) → mark#(X)U31#(mark(X1), X2) → U31#(X1, X2)
active#(snd(pair(X, Y))) → and#(isLNat(X), isLNatKind(X))active#(U71(tt, V1)) → U72#(isNatural(V1))
active#(tail(cons(N, XS))) → and#(isNatural(N), isNaturalKind(N))mark#(U92(X)) → mark#(X)
U11#(X1, mark(X2), X3) → U11#(X1, X2, X3)U72#(active(X)) → U72#(X)
s#(active(X)) → s#(X)active#(U151(tt, V1, V2)) → mark#(U152(isNatural(V1), V2))
active#(U11(tt, N, XS)) → snd#(splitAt(N, XS))take#(active(X1), X2) → take#(X1, X2)

Rewrite Rules

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)s(mark(X)) → s(X)
s(active(X)) → s(X)U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)head(mark(X)) → head(X)
head(active(X)) → head(X)afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

Original Signature

Termination of terms over the following signature is verified: natsFrom, U161, U112, fst, U111, U62, U61, U211, head, U21, U153, U151, mark, isLNatKind, isPLNat, U152, tail, and, U71, U191, U72, 0, isPLNatKind, U122, U121, U221, isLNat, active, U31, U181, pair, isNatural, U132, U131, U43, U42, U92, U133, U41, U91, snd, cons, isNaturalKind, U171, splitAt, U143, U201, U142, U202, U141, s, U51, tt, take, U82, U53, U81, U52, U11, afterNth, U102, sel, U103, U101, nil