(VAR T IL N L M X1 X2 X3 X4 X)
(RULES 
active(and(tt,T)) -> mark(T)
active(isNatIList(IL)) -> mark(isNatList(IL))
active(isNat(0)) -> mark(tt)
active(isNat(s(N))) -> mark(isNat(N))
active(isNat(length(L))) -> mark(isNatList(L))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L)))
active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL)))
active(zeros) -> mark(cons(0,zeros))
active(take(0,IL)) -> mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) -> mark(nil)
active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL))
active(uTake2(tt,M,N,IL)) -> mark(cons(N,take(M,IL)))
active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L))
active(uLength(tt,L)) -> mark(s(length(L)))
active(and(X1,X2)) -> and(active(X1),X2)
active(and(X1,X2)) -> and(X1,active(X2))
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(cons(X1,X2)) -> cons(active(X1),X2)
active(take(X1,X2)) -> take(active(X1),X2)
active(take(X1,X2)) -> take(X1,active(X2))
active(uTake1(X)) -> uTake1(active(X))
active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4)
active(uLength(X1,X2)) -> uLength(active(X1),X2)
and(mark(X1),X2) -> mark(and(X1,X2))
and(X1,mark(X2)) -> mark(and(X1,X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
cons(mark(X1),X2) -> mark(cons(X1,X2))
take(mark(X1),X2) -> mark(take(X1,X2))
take(X1,mark(X2)) -> mark(take(X1,X2))
uTake1(mark(X)) -> mark(uTake1(X))
uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4))
uLength(mark(X1),X2) -> mark(uLength(X1,X2))
proper(and(X1,X2)) -> and(proper(X1),proper(X2))
proper(tt) -> ok(tt)
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
proper(nil) -> ok(nil)
proper(take(X1,X2)) -> take(proper(X1),proper(X2))
proper(uTake1(X)) -> uTake1(proper(X))
proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4))
proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2))
and(ok(X1),ok(X2)) -> ok(and(X1,X2))
isNatIList(ok(X)) -> ok(isNatIList(X))
isNatList(ok(X)) -> ok(isNatList(X))
isNat(ok(X)) -> ok(isNat(X))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
take(ok(X1),ok(X2)) -> ok(take(X1,X2))
uTake1(ok(X)) -> ok(uTake1(X))
uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4))
uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
)