(** Infer the type of a given term and, if exists, returns the type of the term *)
let typeof t =
  let rec infer env = function
    | Term.Var id ->
      ( (match TypeSubstitution.find id env with
         | Some ty -> Some ty
         | None -> Some (Type.Var id))
      , env )
    | Term.IntConst _ -> Some Type.Int, env
    | Term.Binop (t1, _, t2) ->
      (* Both operands must have type Int *)
      (match infer env t1, infer env t2 with
       | (Some ty1, _), (Some ty2, _) ->
         (match Unification.unify ty1 Type.Int, Unification.unify ty2 Type.Int with
          | Some env1, Some env2 -> Some Type.Int, TypeSubstitution.compose env1 env2
          | _ -> None, env)
       | _, _ -> None, env)
    | Term.Pair (t1, t2) ->
      (match infer env t1, infer env t2 with
       | (Some ty1, _), (Some ty2, _) -> Some (Type.Product (ty1, ty2)), env
       | _, _ -> None, env)
    | Term.Proj (proj, t) ->
      (* Check if the term is a pair *)
      (match infer env t with
       | Some (Type.Product (ty1, ty2)), _ ->
         (match proj with
          | Term.First -> Some ty1, env
          | Term.Second -> Some ty2, env)
       | _ -> None, env)
    | Term.Fun (id, body) ->
      (match infer env body with
       | Some ty_body, env' ->
         (match infer env' (Term.Var id) with
          | Some ty, _ -> Some (Type.Arrow (ty, ty_body)), env'
          | None, _ -> Some (Type.Arrow (Type.Var id, ty_body)), env')
       | _ -> None, env)
    | Term.App (t1, t2) ->
      (* Check if the function type matches the arguments *)
      (match infer env t1, infer env t2 with
       | (Some (Type.Arrow (ty_param, ty_fn)), _), (Some ty_args, _) ->
         (match Unification.unify ty_param ty_args with
          | Some _ -> Some ty_fn, env
          | None -> None, env)
       | _, _ -> None, env)
  in
  fst (infer TypeSubstitution.empty t)
;;