NO

Problem 1: 

(VAR v_NonEmpty:S N:S X:S XS:S Y:S YS:S ZS:S)
(RULES
U101(tt) -> fst(splitAt(N:S,XS:S))
U11(tt) -> snd(splitAt(N:S,XS:S))
U21(tt) -> X:S
U31(tt) -> N:S
U41(tt) -> cons(N:S)
U51(tt) -> head(afterNth(N:S,XS:S))
U61(tt) -> Y:S
U71(tt) -> pair(nil,XS:S)
U81(tt) -> U82(splitAt(N:S,XS:S))
U82(pair(YS:S,ZS:S)) -> pair(cons(X:S),ZS:S)
U91(tt) -> XS:S
afterNth(N:S,XS:S) -> U11(and(isNatural))
and(tt) -> X:S
fst(pair(X:S,Y:S)) -> U21(and(isLNat))
head(cons(N:S)) -> U31(and(isNatural))
isLNat -> and(isNatural)
isLNat -> isLNat
isLNat -> isNatural
isLNat -> isPLNat
isLNat -> tt
isNatural -> and(isNatural)
isNatural -> isLNat
isNatural -> isNatural
isNatural -> tt
isPLNat -> and(isLNat)
isPLNat -> and(isNatural)
natsFrom(N:S) -> U41(isNatural)
sel(N:S,XS:S) -> U51(and(isNatural))
snd(pair(X:S,Y:S)) -> U61(and(isLNat))
splitAt(0,XS:S) -> U71(isLNat)
splitAt(s(N:S),cons(X:S)) -> U81(and(isNatural))
tail(cons(N:S)) -> U91(and(isNatural))
take(N:S,XS:S) -> U101(and(isNatural))
)

Problem 1: 

Extra Variables Processor:
-> Rules:
 U101(tt) -> fst(splitAt(N:S,XS:S))
 U11(tt) -> snd(splitAt(N:S,XS:S))
 U21(tt) -> X:S
 U31(tt) -> N:S
 U41(tt) -> cons(N:S)
 U51(tt) -> head(afterNth(N:S,XS:S))
 U61(tt) -> Y:S
 U71(tt) -> pair(nil,XS:S)
 U81(tt) -> U82(splitAt(N:S,XS:S))
 U82(pair(YS:S,ZS:S)) -> pair(cons(X:S),ZS:S)
 U91(tt) -> XS:S
 afterNth(N:S,XS:S) -> U11(and(isNatural))
 and(tt) -> X:S
 fst(pair(X:S,Y:S)) -> U21(and(isLNat))
 head(cons(N:S)) -> U31(and(isNatural))
 isLNat -> and(isNatural)
 isLNat -> isLNat
 isLNat -> isNatural
 isLNat -> isPLNat
 isLNat -> tt
 isNatural -> and(isNatural)
 isNatural -> isLNat
 isNatural -> isNatural
 isNatural -> tt
 isPLNat -> and(isLNat)
 isPLNat -> and(isNatural)
 natsFrom(N:S) -> U41(isNatural)
 sel(N:S,XS:S) -> U51(and(isNatural))
 snd(pair(X:S,Y:S)) -> U61(and(isLNat))
 splitAt(0,XS:S) -> U71(isLNat)
 splitAt(s(N:S),cons(X:S)) -> U81(and(isNatural))
 tail(cons(N:S)) -> U91(and(isNatural))
 take(N:S,XS:S) -> U101(and(isNatural))
-> The system has extra variables.

The problem is infinite.