(STRATEGY
    INNERMOST)

(VAR
    x y z)
(DATATYPES
    A = µX.< 0, s(X), true, false >)
(SIGNATURES
    p :: [A] -> A
    plus :: [A x A] -> A
    times :: [A x A] -> A
    div :: [A x A] -> A
    quot :: [A x A x A] -> A
    eq :: [A x A] -> A
    divides :: [A x A] -> A
    prime :: [A] -> A
    pr :: [A x A] -> A
    if :: [A x A x A] -> A
    zero :: [A] -> A)
(RULES
    p(0()) -> 0()
    p(s(x)) -> x
    plus(x,0()) -> x
    plus(0(),y) -> y
    plus(s(x),y) -> s(plus(x,y))
    plus(s(x),y) -> s(plus(p(s(x))
                          ,y))
    plus(x,s(y)) -> s(plus(x
                          ,p(s(y))))
    times(0(),y) -> 0()
    times(s(0()),y) -> y
    times(s(x),y) -> plus(y
                         ,times(x,y))
    div(0(),y) -> 0()
    div(x,y) -> quot(x,y,y)
    quot(zero(y),s(y),z) -> 0()
    quot(s(x),s(y),z) -> quot(x,y,z)
    quot(x,0(),s(z)) -> s(div(x
                             ,s(z)))
    div(div(x,y),z) -> div(x
                          ,times(zero(y),z))
    eq(0(),0()) -> true()
    eq(s(x),0()) -> false()
    eq(0(),s(y)) -> false()
    eq(s(x),s(y)) -> eq(x,y)
    divides(y,x) -> eq(x
                      ,times(div(x,y),y))
    prime(s(s(x))) -> pr(s(s(x))
                        ,s(x))
    pr(x,s(0())) -> true()
    pr(x,s(s(y))) ->
      if(divides(s(s(y)),x),x,s(y))
    if(true(),x,y) -> false()
    if(false(),x,y) -> pr(x,y)
    zero(div(x,x)) -> x
    zero(divides(x,x)) -> x
    zero(times(x,x)) -> x
    zero(quot(x,x,x)) -> x
    zero(s(x)) -> if(eq(x,s(0()))
                    ,plus(zero(0()),0())
                    ,s(plus(0(),zero(0())))))