Library Stdlib.Reals.Rprod

Require Import Compare.
Require Import Rbase.
Require Import Rfunctions.
Require Import Rseries.
Require Import PartSum.
Require Import Binomial.
Require Import Lia.
Local Open Scope R_scope.

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 prod_SO_split :
  forall (An:nat -> R) (n k:nat),
    (k < n)%nat ->
    prod_f_R0 An n =
    prod_f_R0 An k * prod_f_R0 (fun l:nat => An (k +1+l)%nat) (n - k -1).

Lemma prod_SO_pos :
  forall (An:nat -> R) (N:nat),
    (forall n:nat, (n <= N)%nat -> 0 <= An n) -> 0 <= prod_f_R0 An N.

Lemma prod_SO_Rle :
  forall (An Bn:nat -> R) (N:nat),
    (forall n:nat, (n <= N)%nat -> 0 <= An n <= Bn n) ->
    prod_f_R0 An N <= prod_f_R0 Bn 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.

Lemma le_n_2n : forall n:nat, (n <= 2 * n)%nat.

Lemma RfactN_fact2N_factk :
  forall N k:nat,
    (k <= 2 * N)%nat ->
    Rsqr (INR (fact N)) <= INR (fact (2 * N - k)) * INR (fact k).

Lemma INR_fact_lt_0 : forall n:nat, 0 < INR (fact n).

Lemma C_maj : forall N k:nat, (k <= 2 * N)%nat -> C (2 * N) k <= C (2 * N) N.