(** TT Ak; 0<=k<=N *)
Fixpoint prod_f_R0 (f:nat -> R) (N:nat) : R :=
  match N with
    | O => f O
    | S p => prod_f_R0 f p * f (S p)
  end.Notation prod_f_SO := (fun An N => prod_f_R0 (fun n => An (S n)) N).

Lemma fact_prodSO : forall n:nat, INR (fact n) = prod_f_R0 (fun k:nat => (match (eq_nat_dec k 0) with | left _ => 1%R | right _ => INR k end)) n.