NO

Problem 1: 

(VAR v_NonEmpty:S IL:S L:S M:S N:S T:S)
(RULES
and(tt,T:S) -> T:S
isNat -> isNat
isNat -> isNatList
isNat -> tt
isNatIList -> and(isNat,isNatIList)
isNatIList -> isNatList
isNatIList -> tt
isNatList -> and(isNat,isNatIList)
isNatList -> and(isNat,isNatList)
isNatList -> tt
length(cons(N:S)) -> uLength(and(isNat,isNatList))
take(0,IL:S) -> uTake1(isNatIList)
take(s(M:S),cons(N:S)) -> uTake2(and(isNat,and(isNat,isNatIList)))
uLength(tt) -> s(length(L:S))
uTake1(tt) -> nil
uTake2(tt) -> cons(N:S)
zeros -> cons(0)
)

Problem 1: 

Extra Variables Processor:
-> Rules:
 and(tt,T:S) -> T:S
 isNat -> isNat
 isNat -> isNatList
 isNat -> tt
 isNatIList -> and(isNat,isNatIList)
 isNatIList -> isNatList
 isNatIList -> tt
 isNatList -> and(isNat,isNatIList)
 isNatList -> and(isNat,isNatList)
 isNatList -> tt
 length(cons(N:S)) -> uLength(and(isNat,isNatList))
 take(0,IL:S) -> uTake1(isNatIList)
 take(s(M:S),cons(N:S)) -> uTake2(and(isNat,and(isNat,isNatIList)))
 uLength(tt) -> s(length(L:S))
 uTake1(tt) -> nil
 uTake2(tt) -> cons(N:S)
 zeros -> cons(0)
-> The system has extra variables.

The problem is infinite.