(** Map of an id to a type *)
module IdentifierMap = Map.Make (Identifier)

(** Map type *)
type t = Type.t IdentifierMap.t

(* Empty substitution *)
let empty = IdentifierMap.empty

(** Create a substitution with one element *)
let singleton id ty = IdentifierMap.singleton id ty

(** Apply substitution to a type *)
let rec apply subst = function
  | Type.Int -> Type.Int
  | Type.Var id as t ->
    (* Look for a substitution in the map *)
    (match IdentifierMap.find_opt id subst with
     | Some ty' -> apply subst ty'
     | None -> t)
  | Type.Product (ty1, ty2) -> Type.Product (apply subst ty1, apply subst ty2)
  | Type.Arrow (ty1, ty2) -> Type.Arrow (apply subst ty1, apply subst ty2)
;;

(** Compose two substitutions, last with priority *)
let compose s2 s1 =
  let merger =
    IdentifierMap.merge
      (* ID type_s1 type_s2 *)
        (fun _ ty1 ty2 ->
        match ty1, ty2 with
        (* Dans ce cas, on donne la priorité à s1 *)
        | Some ty1', Some _ -> Some ty1'
        (* Utilisation de la substitution que l'on a déjà *)
        | Some ty1', None -> Some ty1'
        | None, Some ty2' -> Some ty2'
        (* Variable untyped *)
        | None, None -> None)
      s1
      s2
  in
  merger
;;

let to_string map =
  let rec ty_str = function
    | Type.Var s -> "Var('" ^ s ^ "')"
    | Type.Int -> "Int"
    | Type.Product (a, b) -> "Product(" ^ ty_str a ^ ", " ^ ty_str b ^ ")"
    | Type.Arrow (a, b) -> "Arrow(" ^ ty_str a ^ ", " ^ ty_str b ^ ")"
  in
  "{"
  ^ (IdentifierMap.bindings map
     |> List.map (fun (id, ty) -> Printf.sprintf "'%s' typed as %s" id (ty_str ty))
     |> String.concat "\n")
  ^ "}"
;;

let find = IdentifierMap.find_opt