(STRATEGY
    INNERMOST)

(VAR
    IL L M N T X X1 X2 X3 X4)
(DATATYPES
    A = µX.< tt, 0, s(X), length(X), zeros, cons(X, X), nil, take(X, X), and(X, X), isNatIList(X), isNatList(X), isNat(X), uTake1(X), uTake2(X, X, X, X), uLength(X, X) >)
(SIGNATURES
    a__and :: [A x A] -> A
    a__isNatIList :: [A] -> A
    a__isNat :: [A] -> A
    a__isNatList :: [A] -> A
    a__zeros :: [] -> A
    a__take :: [A x A] -> A
    a__uTake1 :: [A] -> A
    a__uTake2 :: [A x A x A x A] -> A
    a__length :: [A] -> A
    a__uLength :: [A x A] -> A
    mark :: [A] -> A)
(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))