İ prefer to say that show tree violates the abstraction. You can have s1=s2 but not f(s1)=f(S2). So f (show tree) is not a real function (or = is not a real equal). But showtree is a debug function, it doesn't count. İt's like saying in C that a=13 and b=13 isn't a real equality because &a and &b are not equal. There is the underlying structure and the structure after the equivalence. You have to know of what you're talking.
I'm not unsympathetic to that position, but I think it's ill advised to take it as far as "Functor means functor up to Eq". I grant that showTree in particular is a debug function that will usually be called at the end of any meaningful computation, and handwaving it in particular away is probably fine. I don't think we can say the same of something like splitRoot, which instead breaks the abstraction for the sake of efficiency.
> You have to know of what you're talking.
And the language (/ecosystem) gives you no way of specifying, so we have to be somewhat conservative. I've actually been thinking (partly driven by trains of thought such as under discussion here) that it might be very interesting to have a language that let you reason explicitly about different sorts of equality (tentatively defined, in the context of the language, as indistinguishability relative to some interface).
