Abusing the algebra of algebraic data types - why does this work?

It seems that all you're doing is expanding the recurrence relation.

L = 1 + X • L
L = 1 + X • (1 + X • (1 + X • (1 + X • ...)))
  = 1 + X + X^2 + X^3 + X^4 ...

T = 1 + X • T^2
L = 1 + X • (1 + X • (1 + X • (1 + X • ...^2)^2)^2)^2
  = 1 + X + 2 • X^2 + 5 • X^3 + 14 • X^4 + ...

And since the rules for the operations on the types work like the rules for arithmetic operations, you can use algebraic means to help you figure out how to expand the recurrence relation (since it is not obvious).

Disclaimer: A lot of this doesn't really work quite right when you account for ⊥, so I'm going to blatantly disregard that for the sake of simplicity.

A few initial points:

• Note that "union" is probably not the best term for A+B here--that's specifically a disjoint union of the two types, because the two sides are distinguished even if their types are the same. For what it's worth, the more common term is simply "sum type".

• Singleton types are, effectively, all unit types. They behave identically under algebraic manipulations and, more importantly, the amount of information present is still preserved.

• You probably want a zero type as well. Haskell provides that as Void. There are no values whose type is zero, just as there is one value whose type is one.

There's still one major operation missing here but I'll get back to that in a moment.

As you've probably noticed, Haskell tends to borrow concepts from Category Theory, and all of the above has a very straightforward interpretation as such:

• Given objects A and B in Hask, their product A×B is the unique (up to isomorphism) type that allows two projections fst : A×B → A and snd : A×B → B, where given any type C and functions f : C → A, g : C → B you can define the pairing f &&& g : C → A×B such that fst ∘ (f &&& g) = f and likewise for g. Parametricity guarantees the universal properties automatically and my less-than-subtle choice of names should give you the idea. The (&&&) operator is defined in Control.Arrow, by the way.

• The dual of the above is the coproduct A+B with injections inl : A → A+B and inr : B → A+B, where given any type C and functions f : A → C, g : B → C, you can define the copairing f ||| g : A+B → C such that the obvious equivalences hold. Again, parametricity guarantees most of the tricky parts automatically. In this case, the standard injections are simply Left and Right and the copairing is the function either.

Many of the properties of product and sum types can be derived from the above. Note that any singleton type is a terminal object of Hask and any empty type is an initial object.

Returning to the aforementioned missing operation, in a cartesian closed category you have exponential objects that correspond to arrows of the category. Our arrows are functions, our objects are types with kind *, and the type A -> B indeed behaves as B^A in the context of algebraic manipulation of types. If it's not obvious why this should hold, consider the type Bool -> A. With only two possible inputs, a function of that type is isomorphic to two values of type A, i.e. (A, A). For Maybe Bool -> A we have three possible inputs, and so on. Also, observe that if we rephrase the copairing definition above to use algebraic notation, we get the identity C^A × C^B = C^A+B.

As for why this all makes sense--and in particular why your use of the power series expansion is justified--note that much of the above refers to the "inhabitants" of a type (i.e., distinct values having that type) in order to demonstrate the algebraic behavior. To make that perspective explicit:

• The product type (A, B) represents a value each from A and B, taken independently. So for any fixed value a :: A, there is one value of type (A, B) for each inhabitant of B. This is of course the cartesian product, and the number of inhabitants of the product type is the product of the number of inhabitants of the factors.

• The sum type Either A B represents a value from either A or B, with the left and right branches distinguished. As mentioned earlier, this is a disjoint union, and the number of inhabitants of the sum type is the sum of the number of inhabitants of the summands.

• The exponential type B -> A represents a mapping from values of type B to values of type A. For any fixed argument b :: B, any value of A can be assigned to it; a value of type B -> A picks one such mapping for each input, which is equivalent to a product of as many copies of A as B has inhabitants, hence the exponentiation.

While it's tempting at first to treat types as sets, that doesn't actually work very well in this context--we have disjoint union rather than the standard union of sets, there's no obvious interpretation of intersection or many other set operations, and we don't usually care about set membership (leaving that to the type checker).

On the other hand, the constructions above spend a lot of time talking about counting inhabitants, and enumerating the possible values of a type is a useful concept here. That quickly leads us to enumerative combinatorics, and if you consult the linked Wikipedia article you'll find that one of the first things it does is define "pairs" and "unions" in exactly the same sense as product and sum types by way of generating functions, then does the same for "sequences" that are identical to Haskell's lists using exactly the same technique you did.

---

Edit: Oh, and here's a quick bonus that I think demonstrates the point strikingly. You mentioned in a comment that for a tree type T = 1 + T^2 you can derive the identity T^6 = 1, which is clearly wrong. However, T^7 = T does hold, and a bijection between trees and seven-tuples of trees can be constructed directly, cf. Andreas Blass's "Seven Trees in One".

Edit×2: On the subject of the "derivative of a type" construction mentioned in other answers, you might also enjoy this paper from the same author which builds on the idea further, including notions of division and other interesting whatnot.

Binary trees are defined by the equation T=1+XT^2 in the semiring of types. By construction, T=(1-sqrt(1-4X))/(2X) is defined by the same equation in the semiring of complex numbers. So given that we're solving the same equation in the same class of algebraic structure it actually shouldn't be surprising that we see some similarities.

The catch is that when we reason about polynomials in the semiring of complex numbers we typically use the fact that the complex numbers form a ring or even a field so we find ourselves using operations such as subtraction that don't apply to semirings. But we can often eliminate subtractions from our arguments if we have a rule that allows us to cancel from both sides of an equation. This is the kind of thing proved by Fiore and Leinster showing that many arguments about rings can be transferred to semirings.

This means that lots of your mathematical knowledge about rings can be reliably transferred to types. As a result, some arguments involving complex numbers or power series (in the ring of formal power series) can carry over to types in a completely rigorous way.

However there's more to the story than this. It's one thing to prove two types are equal (say) by showing two power series are equal. But you can also deduce information about types by inspecting the terms in the power series. I'm not sure of what the formal statement here should be. (I recommend Brent Yorgey's paper on combinatorial species for some work that's closely related but species are not the same as types.)

What I find utterly mind blowing is that what you've discovered can be extended to calculus. Theorems about calculus can be transferred over to the semiring of types. In fact, even arguments about finite differences can be transferred over and you find that classical theorems from numerical analysis have interpretations in type theory.

Have fun!