OCaml isn't a language I've much studied. After installing it (v4.05) and some other prerequisites, such as jbuilder, I am able to run make, whereupon the following error presents itself:

File "stdlib/util.ml", line 684, characters 9-37: Error: The function applied to this argument has type compare:('a -> 'a -> int) -> 'a list This argument cannot be applied with label ~cmp

I'm interested in source-to-source program transformations, which I think should be reversible, implying monomorphism. To do this, one needs to be able to invert the components of these transformations, so I was happy to discover the idea of lenses, which plugged a hole in my mental compendium about a decade old.

Allow me to raise the very simple example of list decomposition, using Haskell notation. There's an obvious sense in which the inverse of the cons operation is pattern-matching on the the list head|tail, with the same notation, (:), used for both.¹ Similarly, one can say that destructuring is "the same" as the (head⨯tail) joint operation. By another abuse of notation, in the form of entropy: $H(\text{<destructuring>}) = H(\mathtt{head}) + H(\mathtt{tail})$.

Now let me "lift" head and tail into the category of very well behaved (VWB?) lenses, which I'll call lhead and ltail, as

lhead⭧ = head lhead⭨ x (_:xs) = x:xs ltail⭧ = tail ltail⭨ xs (x:_) = x:xs

where the Get operations are the original head and tail.² Intuitively, the joint information provided by lhead|ltail is "the same" as that of cons plus pattern-matching, in that we could transform programs written in one syntactic convention into the other. There's probably a symmetric argument to show that a lensified cons has equivalent expressive power as well. If I were a category theorist, I might draw a little diagram.

Unfortunately, I'm asking questions on several topics, here, about which my knowledge level is abecedarian. Are there particular subcategories of lenses which show the algebraic behavior I'm suggesting? Is there a domain-theoretic way of describing the special relationship between lhead⭨, ltail⭨ and cons? I have the impression that some kind of higher-order functor should exist to map regular functions to these lenses.³ I hope someone will know what I'm talking about, even if I don't.

Thanks,
Jonathan J-S