(STRATEGY
    INNERMOST)

(VAR
    L N V V1 V2 X X1 X2)
(DATATYPES
    A = µX.< n__zeros, tt, n__0, n__length(X), n__s(X), n__cons(X, X), n__isNatIList(X), n__nil, n__isNatList(X), n__isNat(X) >)
(SIGNATURES
    zeros :: [] -> A
    U11 :: [A x A] -> A
    and :: [A x A] -> A
    isNat :: [A] -> A
    isNatIList :: [A] -> A
    isNatList :: [A] -> A
    length :: [A] -> A
    0 :: [] -> A
    s :: [A] -> A
    cons :: [A x A] -> A
    nil :: [] -> A
    activate :: [A] -> A)
(RULES
    zeros() -> cons(0(),n__zeros())
    U11(tt(),L) ->
      s(length(activate(L)))
    and(tt(),X) -> activate(X)
    isNat(n__0()) -> tt()
    isNat(n__length(V1)) ->
      isNatList(activate(V1))
    isNat(n__s(V1)) ->
      isNat(activate(V1))
    isNatIList(V) ->
      isNatList(activate(V))
    isNatIList(n__zeros()) -> tt()
    isNatIList(n__cons(V1,V2)) ->
      and(isNat(activate(V1))
         ,n__isNatIList(activate(V2)))
    isNatList(n__nil()) -> tt()
    isNatList(n__cons(V1,V2)) ->
      and(isNat(activate(V1))
         ,n__isNatList(activate(V2)))
    length(nil()) -> 0()
    length(cons(N,L)) ->
      U11(and(isNatList(activate(L))
             ,n__isNat(N))
         ,activate(L))
    zeros() -> n__zeros()
    0() -> n__0()
    length(X) -> n__length(X)
    s(X) -> n__s(X)
    cons(X1,X2) -> n__cons(X1,X2)
    isNatIList(X) ->
      n__isNatIList(X)
    nil() -> n__nil()
    isNatList(X) -> n__isNatList(X)
    isNat(X) -> n__isNat(X)
    activate(n__zeros()) -> zeros()
    activate(n__0()) -> 0()
    activate(n__length(X)) ->
      length(activate(X))
    activate(n__s(X)) ->
      s(activate(X))
    activate(n__cons(X1,X2)) ->
      cons(activate(X1),X2)
    activate(n__isNatIList(X)) ->
      isNatIList(X)
    activate(n__nil()) -> nil()
    activate(n__isNatList(X)) ->
      isNatList(X)
    activate(n__isNat(X)) ->
      isNat(X)
    activate(X) -> X)