(STRATEGY
    INNERMOST)

(VAR
    x y z)
(DATATYPES
    A = µX.< 0, s(X), false, true >)
(SIGNATURES
    minus :: [A x A] -> A
    plus :: [A x A] -> A
    zero :: [A] -> A
    p :: [A] -> A
    div :: [A x A] -> A
    quot :: [A x A x A] -> A
    ack :: [A x A] -> A)
(RULES
    minus(minus(x,y),z) -> minus(x
                                ,plus(y,z))
    minus(0(),y) -> 0()
    minus(x,0()) -> x
    minus(s(x),s(y)) -> minus(x,y)
    plus(0(),y) -> y
    plus(s(x),y) -> plus(x,s(y))
    plus(s(x),y) -> s(plus(y,x))
    zero(s(x)) -> false()
    zero(0()) -> true()
    p(s(x)) -> x
    p(0()) -> 0()
    div(x,y) -> quot(x,y,0())
    quot(s(x),s(y),z) ->
      quot(minus(p(ack(0(),x)),y)
          ,s(y)
          ,s(z))
    quot(0(),s(y),z) -> z
    ack(0(),x) -> s(x)
    ack(0(),x) -> plus(x,s(0()))
    ack(s(x),0()) -> ack(x,s(0()))
    ack(s(x),s(y)) -> ack(x
                         ,ack(s(x),y)))