(STRATEGY
    INNERMOST)

(VAR
    u x xs y z)
(DATATYPES
    A = µX.< s(X), 0, true, false, nil, cons(X, X), error, b, c >)
(SIGNATURES
    prod :: [A] -> A
    prodIter :: [A x A] -> A
    ifProd :: [A x A x A] -> A
    plus :: [A x A] -> A
    times :: [A x A] -> A
    timesIter :: [A x A x A x A] -> A
    ifTimes :: [A x A x A x A x A] -> A
    isempty :: [A] -> A
    head :: [A] -> A
    tail :: [A] -> A
    ge :: [A x A] -> A
    a :: [] -> A)
(RULES
    prod(xs) -> prodIter(xs,s(0()))
    prodIter(xs,x) ->
      ifProd(isempty(xs),xs,x)
    ifProd(true(),xs,x) -> x
    ifProd(false(),xs,x) ->
      prodIter(tail(xs)
              ,times(x,head(xs)))
    plus(0(),y) -> y
    plus(s(x),y) -> s(plus(x,y))
    times(x,y) -> timesIter(x
                           ,y
                           ,0()
                           ,0())
    timesIter(x,y,z,u) ->
      ifTimes(ge(u,x),x,y,z,u)
    ifTimes(true(),x,y,z,u) -> z
    ifTimes(false(),x,y,z,u) ->
      timesIter(x,y,plus(y,z),s(u))
    isempty(nil()) -> true()
    isempty(cons(x,xs)) -> false()
    head(nil()) -> error()
    head(cons(x,xs)) -> x
    tail(nil()) -> nil()
    tail(cons(x,xs)) -> xs
    ge(x,0()) -> true()
    ge(0(),s(y)) -> false()
    ge(s(x),s(y)) -> ge(x,y)
    a() -> b()
    a() -> c())