Equivalent definitions of applicative functor

applicative.ml

module type PA = sig
  type 'a t
  val pure : 'a -> 'a t
  val apply  : ('a -> 'b) t -> 'a t -> 'b t (* <*> *)
end

module type NMP = sig
  type 'a t
  val nil : unit t
  val map : ('a -> 'b) -> 'a t -> 'b t (* <$> *)
  val product : 'a t -> 'b t -> ('a * 'b) t
end

module PA_of_NMP (S : NMP) : PA = struct
  type 'a t = 'a S.t
  open S
  let pure x = map (fun () -> x) nil
  let apply f x = map (fun (f, x) -> f x) (product f x)
end

module NMP_of_PA (S : PA) : NMP = struct
  type 'a t = 'a S.t
  open S
  let nil = pure ()
  let map f = apply (pure f)
  let product a b = apply (map (fun x y -> x, y) a) b
end