NO

Problem 1: 

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

Problem 1: 

Extra Variables Processor:
-> Rules:
 U101(tt) -> U102(isNatural)
 U102(tt) -> U103(isLNat)
 U103(tt) -> tt
 U11(tt) -> snd(splitAt(N:S,XS:S))
 U111(tt) -> U112(isLNat)
 U112(tt) -> tt
 U121(tt) -> U122(isNatural)
 U122(tt) -> tt
 U131(tt) -> U132(isNatural)
 U132(tt) -> U133(isLNat)
 U133(tt) -> tt
 U141(tt) -> U142(isLNat)
 U142(tt) -> U143(isLNat)
 U143(tt) -> tt
 U151(tt) -> U152(isNatural)
 U152(tt) -> U153(isLNat)
 U153(tt) -> tt
 U161(tt) -> cons(N:S)
 U171(tt) -> head(afterNth(N:S,XS:S))
 U181(tt) -> Y:S
 U191(tt) -> pair(nil,XS:S)
 U201(tt) -> U202(splitAt(N:S,XS:S))
 U202(pair(YS:S,ZS:S)) -> pair(cons(X:S),ZS:S)
 U21(tt) -> X:S
 U211(tt) -> XS:S
 U221(tt) -> fst(splitAt(N:S,XS:S))
 U31(tt) -> N:S
 U41(tt) -> U42(isNatural)
 U42(tt) -> U43(isLNat)
 U43(tt) -> tt
 U51(tt) -> U52(isNatural)
 U52(tt) -> U53(isLNat)
 U53(tt) -> tt
 U61(tt) -> U62(isPLNat)
 U62(tt) -> tt
 U71(tt) -> U72(isNatural)
 U72(tt) -> tt
 U81(tt) -> U82(isPLNat)
 U82(tt) -> tt
 U91(tt) -> U92(isLNat)
 U92(tt) -> tt
 afterNth(N:S,XS:S) -> U11(and(and(isNatural)))
 and(tt) -> X:S
 fst(pair(X:S,Y:S)) -> U21(and(and(isLNat)))
 head(cons(N:S)) -> U31(and(and(isNatural)))
 isLNat -> U101(and(isNaturalKind))
 isLNat -> U41(and(isNaturalKind))
 isLNat -> U51(and(isNaturalKind))
 isLNat -> U61(isPLNatKind)
 isLNat -> U71(isNaturalKind)
 isLNat -> U81(isPLNatKind)
 isLNat -> U91(isLNatKind)
 isLNat -> tt
 isLNatKind -> and(isNaturalKind)
 isLNatKind -> isLNatKind
 isLNatKind -> isNaturalKind
 isLNatKind -> isPLNatKind
 isLNatKind -> tt
 isNatural -> U111(isLNatKind)
 isNatural -> U121(isNaturalKind)
 isNatural -> U131(and(isNaturalKind))
 isNatural -> tt
 isNaturalKind -> and(isNaturalKind)
 isNaturalKind -> isLNatKind
 isNaturalKind -> isNaturalKind
 isNaturalKind -> tt
 isPLNat -> U141(and(isLNatKind))
 isPLNat -> U151(and(isNaturalKind))
 isPLNatKind -> and(isLNatKind)
 isPLNatKind -> and(isNaturalKind)
 natsFrom(N:S) -> U161(and(isNatural))
 sel(N:S,XS:S) -> U171(and(and(isNatural)))
 snd(pair(X:S,Y:S)) -> U181(and(and(isLNat)))
 splitAt(0,XS:S) -> U191(and(isLNat))
 splitAt(s(N:S),cons(X:S)) -> U201(and(and(isNatural)))
 tail(cons(N:S)) -> U211(and(and(isNatural)))
 take(N:S,XS:S) -> U221(and(and(isNatural)))
-> The system has extra variables.

The problem is infinite.