Writing an interpreter with OCaml GADTs

Question

I am writing a small interpreter in OCaml and am using GADTs to type my expressions:

type _ value =
    | Bool : bool -> bool value
    | Int : int -> int value
    | Symbol : string -> string value
    | Nil : unit value
    | Pair : 'a value * 'b value -> ('a * 'b) value
and _ exp =
    | Literal : 'a value -> 'a exp
    | Var : name -> 'a exp
    | If : bool exp * 'a exp * 'a exp -> 'a exp
and name = string

exception NotFound of string

type 'a env = (name * 'a) list
let bind (n, v, e) = (n, v)::e
let rec lookup = function
    | (n, []) -> raise (NotFound n)
    | (n, (n', v)::e') -> if n=n' then v else lookup (n, e')

let rec eval : type a. a exp -> a value env -> a value = fun e rho ->
    match e with
    | Literal v -> v
    | Var n -> lookup (n, rho)
    | If (b, l, r) ->
            let Bool b' = eval b rho in
            if b' then eval l rho else eval r rho

But I cannot get my code to compile. I get the following error:

File "gadt2.ml", line 33, characters 33-36:
Error: This expression has type a value env = (name * a value) list
       but an expression was expected of type
         bool value env = (name * bool value) list
       Type a is not compatible with type bool

My understanding is that for some reason rho is being coerced into a bool value env, but I don't know why. I also tried the following:

let rec eval : 'a. 'a exp -> 'a value env -> 'a value = fun e rho ->
    match e with
    | Literal v -> v
    | Var n -> lookup (n, rho)
    | If (b, l, r) ->
            let Bool b = eval b rho in
            if b then eval l rho else eval r rho

But I am not sure how exactly that is different, and it also gives me an error -- albeit a different one:

File "gadt2.ml", line 38, characters 56-247:
Error: This definition has type bool exp -> bool value env -> bool value
       which is less general than 'a. 'a exp -> 'a value env -> 'a value

Guidance on GADTs, differences between the two evals, and this particular problem are all appreciated. Cheers.

Answer 1

The type 'a env is intended to represent a list of name/value bindings, but the values in a list must all be the same type. Two different value types (such as bool value and int value) are not the same type. If eval b rho returns Bool b, rho must be a list of string * bool value. So eval l rho and eval r rho will return bool value. But your annotation says the function returns a value.