GADT's with OCaml

ChatGPT: Hello, what is a GADT with OCaml ?

A GADT (Generalized Algebraic Data Type) in OCaml is an extension of the algebraic data type concept that allows for greater expressiveness in defining types. GADTs enable the definition of data types with more precise constraints on the values they can take, allowing for data-dependent types. This means that the type of an object can depend on the value of that object, providing a more expressive way to represent complex data structures. GADTs are often used to represent data structures that are difficult to model with classic algebraic data types.

ChatGPT: Could you give me a simple example of a GADT?

Of course! Here's a simple example of a GADT in OCaml:

type _ expr =
  | Int : int -> int expr
  | Bool : bool -> bool expr
  | Add : int expr * int expr -> int expr
  | And : bool expr * bool expr -> bool expr

In this example, we define a type `expr` that can represent arithmetic and boolean expressions. The `Int` and `Bool` constructors respectively take an integer and a boolean and create an expression of type `int expr` or `bool expr`. The `Add` and `And` constructors take arithmetic or boolean expressions and create new arithmetic or boolean expressions.

Here is how you could use this GADT:

let expr1 = Add(Int 3, Int 4)  (* 3 + 4 *)
let expr2 = And(Bool true, Bool false)  (* true && false *)

This example illustrates how GADTs allow for defining data types with specific constructors for each case, providing a more precise way to represent data structures.

ChatGPT: How can I use these expressions `expr1` and `expr2`?

To use the expressions `expr1` and `expr2`, you can evaluate these expressions using a function that traverses the syntax tree structure represented by these expressions and performs the appropriate operations. Here's an example of such a function to evaluate the expressions:

let rec eval : type a. a expr -> a = function
  | Int n -> n
  | Bool b -> b
  | Add (e1, e2) -> eval e1 + eval e2
  | And (e1, e2) -> eval e1 && eval e2

Now, you can use the `eval` function to evaluate the expressions `expr1` and `expr2`:

let result1 = eval expr1  (* will evaluate to 7 *)
let result2 = eval expr2  (* will evaluate to false *)

This will evaluate the expressions `expr1` and `expr2` using the `eval` function, providing the expected results based on the operations defined in the expressions.

Copy-pasted in the top-level:

# let result1 = eval expr1 ;;
val result1 : int = 7

# let result2 = eval expr2 ;;
val result2 : bool = false