(VAR IL L M N T X X1 X2 X3 X4 )
(RULES 
        a__and(tt, T) -> mark(T)
        a__isNatIList(IL) -> a__isNatList(IL)
        a__isNat(0) -> tt
        a__isNat(s(N)) -> a__isNat(N)
        a__isNat(length(L)) -> a__isNatList(L)
        a__isNatIList(zeros) -> tt
        a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL))
        a__isNatList(nil) -> tt
        a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L))
        a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL))
        a__zeros -> cons(0, zeros)
        a__take(0, IL) -> a__uTake1(a__isNatIList(IL))
        a__uTake1(tt) -> nil
        a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
        a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL))
        a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
        a__uLength(tt, L) -> s(a__length(mark(L)))
        mark(and(X1, X2)) -> a__and(mark(X1), mark(X2))
        mark(isNatIList(X)) -> a__isNatIList(X)
        mark(isNatList(X)) -> a__isNatList(X)
        mark(isNat(X)) -> a__isNat(X)
        mark(length(X)) -> a__length(mark(X))
        mark(zeros) -> a__zeros
        mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
        mark(uTake1(X)) -> a__uTake1(mark(X))
        mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4)
        mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2)
        mark(tt) -> tt
        mark(0) -> 0
        mark(s(X)) -> s(mark(X))
        mark(cons(X1, X2)) -> cons(mark(X1), X2)
        mark(nil) -> nil
        a__and(X1, X2) -> and(X1, X2)
        a__isNatIList(X) -> isNatIList(X)
        a__isNatList(X) -> isNatList(X)
        a__isNat(X) -> isNat(X)
        a__length(X) -> length(X)
        a__zeros -> zeros
        a__take(X1, X2) -> take(X1, X2)
        a__uTake1(X) -> uTake1(X)
        a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4)
        a__uLength(X1, X2) -> uLength(X1, X2)
        
)