Combinators over Discrimated Unions in ML

Discriminated unions or sum types are a natural way to model logical OR. Often you have a property that distributes over OR. Say, in F# (used throughout the article, though the ideas should apply equally well to any ML), you can write a combinator of the type:

P<'T1> → P<'T2> → P<Choice<'T1,'T2>>

How to go from here to a nice set of combinators that would handle an arbitrary union? This question has been on my mind for a while, and finally I have an acceptable solution.

As a disclaimer, at the level of theory the question is completely trivial. The interest is in how to find a user-friendly ML interface. Even this, I suspect, has been solved before, and at a much more general level. I am aware for instance of Vesa Karvonen's Generics for the working ML'er, and a more recent Oleg Kiselyov's presentation of generics in OCaml. I am looking here at a much simpler setting, hopefully taking it to a more accessible, for-dummies-like-myself level.

Suppose you are designing binary format combinators to let users of your library construct values of the form:

type Format<'T> =
    {
        Read : BinaryReader → 'T
        Write : BinaryWriter → 'T → unit
    }

Given an arbitrary union type and some primitive Format values, the user should be able to compose them. This involves projecting from sub-types to the parent union type, and matching backwards. After some experimentation, the design I have looks like this:

type U =
    | A of int
    | B of float
    | C of string

let UFormat =
    Union (fun a b c x →
        match x with
        | A x → a x
        | B x → b x
        | C x → c x)
    << Case A IntFormat
    << Case B FloatFormat
    << Case C StringFormat
    <| End

That's pretty much it. The magic is in the type of Case combinator that accumulates types so that the Union combinator can present N-way pattern matching as a single reasonably convenient to write function. Full code:

	open System.IO
	type Format<'T> =
	{
	Read : BinaryReader -> 'T
	Write : BinaryWriter -> 'T -> unit
	}
	type FormatUnionPart<'X,'U> =
	{
	R : list<BinaryReader -> 'U>
	W : 'X -> 'U -> BinaryWriter -> unit
	}
	let Case (ctor: 'C -> 'U) (fmt: Format<'C>)
	(part: FormatUnionPart<'X,'U>) :
	FormatUnionPart<('C -> unit) -> 'X,'U> =
	let tag = char part.R.Length
	let write (w: BinaryWriter) x =
	w.Write(tag)
	fmt.Write w x
	{
	R = (fmt.Read >> ctor) :: part.R
	W = fun f u w -> part.W (f (write w)) u w
	}
	let End<'U> : FormatUnionPart<('U -> unit),'U> =
	{
	R = []
	W = fun f u w -> f u
	}
	let Union (proof: 'X) (part: FormatUnionPart<'X,'U>) : Format<'U> =
	let rs = Array.rev (List.toArray part.R)
	{
	Read = fun r ->
	let tag = int (r.ReadChar())
	rs.[tag] r
	Write = fun w x -> part.W proof x w
	}
	let IntFormat : Format<int> =
	{
	Read = fun r -> r.ReadInt32()
	Write = fun w x -> w.Write(x)
	}
	let FloatFormat : Format<float> =
	{
	Read = fun r -> r.ReadDouble()
	Write = fun w x -> w.Write(x)
	}
	let StringFormat : Format<string> =
	{
	Read = fun r -> r.ReadString()
	Write = fun w x -> w.Write(x)
	}
	type U =
	| A of int
	| B of float
	| C of string
	let UFormat =
	Union (fun a b c x ->
	match x with
	| A x -> a x
	| B x -> b x
	| C x -> c x)
	<< Case A IntFormat
	<< Case B FloatFormat
	<< Case C StringFormat
	<| End
	let test (fmt: Format<'T>) (value: 'T) =
	let bytes =
	use s = new MemoryStream()
	use w = new BinaryWriter(s)
	fmt.Write w value
	s.ToArray()
	let roundtrip =
	use s = new MemoryStream(bytes, false)
	use r = new BinaryReader(s)
	fmt.Read r
	roundtrip = value

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