(STRATEGY
    INNERMOST)

(VAR
    b x xs xs2 y ys ys2 zs zs2)
(DATATYPES
    A = µX.< cons(X, X), false, nil, true, 0, s(X), differentLengthError >)
(SIGNATURES
    isEmpty :: [A] -> A
    isZero :: [A] -> A
    head :: [A] -> A
    tail :: [A] -> A
    append :: [A x A] -> A
    p :: [A] -> A
    inc :: [A] -> A
    addLists :: [A x A x A] -> A
    if :: [A x A x A x A x A x A x A x A x A] -> A
    addList :: [A x 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()
    append(nil(),x) -> cons(x,nil())
    append(cons(y,ys),x) -> cons(y
                                ,append(ys,x))
    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())
    addLists(xs,ys,zs) ->
      if(isEmpty(xs)
        ,isEmpty(ys)
        ,isZero(head(xs))
        ,tail(xs)
        ,tail(ys)
        ,cons(p(head(xs)),tail(xs))
        ,cons(inc(head(ys)),tail(ys))
        ,zs
        ,append(zs,head(ys)))
    if(true()
      ,true()
      ,b
      ,xs
      ,ys
      ,xs2
      ,ys2
      ,zs
      ,zs2) -> zs
    if(true()
      ,false()
      ,b
      ,xs
      ,ys
      ,xs2
      ,ys2
      ,zs
      ,zs2) -> differentLengthError()
    if(false()
      ,true()
      ,b
      ,xs
      ,ys
      ,xs2
      ,ys2
      ,zs
      ,zs2) -> differentLengthError()
    if(false()
      ,false()
      ,false()
      ,xs
      ,ys
      ,xs2
      ,ys2
      ,zs
      ,zs2) -> addLists(xs2,ys2,zs)
    if(false()
      ,false()
      ,true()
      ,xs
      ,ys
      ,xs2
      ,ys2
      ,zs
      ,zs2) -> addLists(xs,ys,zs2)
    addList(xs,ys) -> addLists(xs
                              ,ys
                              ,nil()))