(VAR b x xs xs2 y ys ys2 zs zs2 )
(STRATEGY INNERMOST)
(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)
        
)