(STRATEGY
    INNERMOST)

(VAR
    b x xs y ys)
(DATATYPES
    A = µX.< cons(X, X), false, nil, true, 0, s(X) >)
(SIGNATURES
    isEmpty :: [A] -> A
    isZero :: [A] -> A
    head :: [A] -> A
    tail :: [A] -> A
    p :: [A] -> A
    inc :: [A] -> A
    sumList :: [A x A] -> A
    if :: [A x A x A x A x A x A] -> A
    sum :: [A] -> A)
(RULES
    isEmpty(cons(x,xs)) -> false()
    isEmpty(nil()) -> true()
    isZero(0()) -> true()
    isZero(s(x)) -> false()
    head(cons(x,xs)) -> x
    tail(cons(x,xs)) -> xs
    tail(nil()) -> nil()
    p(s(s(x))) -> s(p(s(x)))
    p(s(0())) -> 0()
    p(0()) -> 0()
    inc(s(x)) -> s(inc(x))
    inc(0()) -> s(0())
    sumList(xs,y) -> if(isEmpty(xs)
                       ,isZero(head(xs))
                       ,y
                       ,tail(xs)
                       ,cons(p(head(xs)),tail(xs))
                       ,inc(y))
    if(true(),b,y,xs,ys,x) -> y
    if(false(),true(),y,xs,ys,x) ->
      sumList(xs,y)
    if(false(),false(),y,xs,ys,x) ->
      sumList(ys,x)
    sum(xs) -> sumList(xs,0()))