Imported Upstream version 8.0pl1upstream/8.0pl1

1 files changed, 191 insertions, 0 deletions

diff --git a/theories/Reals/Rprod.v b/theories/Reals/Rprod.v
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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+ 
+(*i $Id: Rprod.v,v 1.10.2.1 2004/07/16 19:31:13 herbelin Exp $ i*)
+
+Require Import Compare.
+Require Import Rbase.
+Require Import Rfunctions.
+Require Import Rseries.
+Require Import PartSum.
+Require Import Binomial.
+Open Local Scope R_scope.
+
+(* TT Ak; 1<=k<=N *)
+Fixpoint prod_f_SO (An:nat -> R) (N:nat) {struct N} : R :=
+  match N with
+  | O => 1
+  | S p => prod_f_SO An p * An (S p)
+  end.
+
+(**********)
+Lemma prod_SO_split :
+ forall (An:nat -> R) (n k:nat),
+   (k <= n)%nat ->
+   prod_f_SO An n =
+   prod_f_SO An k * prod_f_SO (fun l:nat => An (k + l)%nat) (n - k).
+intros; induction  n as [| n Hrecn].
+cut (k = 0%nat);
+ [ intro; rewrite H0; simpl in |- *; ring | inversion H; reflexivity ].
+cut (k = S n \/ (k <= n)%nat).
+intro; elim H0; intro.
+rewrite H1; simpl in |- *; rewrite <- minus_n_n; simpl in |- *; ring.
+replace (S n - k)%nat with (S (n - k)).
+simpl in |- *; replace (k + S (n - k))%nat with (S n).
+rewrite Hrecn; [ ring | assumption ].
+apply INR_eq; rewrite S_INR; rewrite plus_INR; rewrite S_INR;
+ rewrite minus_INR; [ ring | assumption ].
+apply INR_eq; rewrite S_INR; repeat rewrite minus_INR.
+rewrite S_INR; ring.
+apply le_trans with n; [ assumption | apply le_n_Sn ].
+assumption.
+inversion H; [ left; reflexivity | right; assumption ].
+Qed.
+
+(**********)
+Lemma prod_SO_pos :
+ forall (An:nat -> R) (N:nat),
+   (forall n:nat, (n <= N)%nat -> 0 <= An n) -> 0 <= prod_f_SO An N.
+intros; induction  N as [| N HrecN].
+simpl in |- *; left; apply Rlt_0_1.
+simpl in |- *; apply Rmult_le_pos.
+apply HrecN; intros; apply H; apply le_trans with N;
+ [ assumption | apply le_n_Sn ].
+apply H; apply le_n.
+Qed.
+
+(**********)
+Lemma prod_SO_Rle :
+ forall (An Bn:nat -> R) (N:nat),
+   (forall n:nat, (n <= N)%nat -> 0 <= An n <= Bn n) ->
+   prod_f_SO An N <= prod_f_SO Bn N.
+intros; induction  N as [| N HrecN].
+right; reflexivity.
+simpl in |- *; apply Rle_trans with (prod_f_SO An N * Bn (S N)).
+apply Rmult_le_compat_l.
+apply prod_SO_pos; intros; elim (H n (le_trans _ _ _ H0 (le_n_Sn N))); intros;
+ assumption.
+elim (H (S N) (le_n (S N))); intros; assumption.
+do 2 rewrite <- (Rmult_comm (Bn (S N))); apply Rmult_le_compat_l.
+elim (H (S N) (le_n (S N))); intros.
+apply Rle_trans with (An (S N)); assumption.
+apply HrecN; intros; elim (H n (le_trans _ _ _ H0 (le_n_Sn N))); intros;
+ split; assumption.
+Qed.
+
+(* Application to factorial *)
+Lemma fact_prodSO :
+ forall n:nat, INR (fact n) = prod_f_SO (fun k:nat => INR k) n.
+intro; induction  n as [| n Hrecn].
+reflexivity.
+change (INR (S n * fact n) = prod_f_SO (fun k:nat => INR k) (S n)) in |- *.
+rewrite mult_INR; rewrite Rmult_comm; rewrite Hrecn; reflexivity.
+Qed.
+
+Lemma le_n_2n : forall n:nat, (n <= 2 * n)%nat.
+simple induction n.
+replace (2 * 0)%nat with 0%nat; [ apply le_n | ring ].
+intros; replace (2 * S n0)%nat with (S (S (2 * n0))).
+apply le_n_S; apply le_S; assumption.
+replace (S (S (2 * n0))) with (2 * n0 + 2)%nat; [ idtac | ring ].
+replace (S n0) with (n0 + 1)%nat; [ idtac | ring ].
+ring.
+Qed. 
+
+(* We prove that (N!)²<=(2N-k)!*k! forall k in [|O;2N|] *)
+Lemma RfactN_fact2N_factk :
+ forall N k:nat,
+   (k <= 2 * N)%nat ->
+   Rsqr (INR (fact N)) <= INR (fact (2 * N - k)) * INR (fact k).
+intros; unfold Rsqr in |- *; repeat rewrite fact_prodSO.
+cut ((k <= N)%nat \/ (N <= k)%nat).
+intro; elim H0; intro.
+rewrite (prod_SO_split (fun l:nat => INR l) (2 * N - k) N).
+rewrite Rmult_assoc; apply Rmult_le_compat_l.
+apply prod_SO_pos; intros; apply pos_INR.
+replace (2 * N - k - N)%nat with (N - k)%nat.
+rewrite Rmult_comm; rewrite (prod_SO_split (fun l:nat => INR l) N k).
+apply Rmult_le_compat_l.
+apply prod_SO_pos; intros; apply pos_INR.
+apply prod_SO_Rle; intros; split.
+apply pos_INR.
+apply le_INR; apply plus_le_compat_r; assumption.
+assumption.
+apply INR_eq; repeat rewrite minus_INR.
+rewrite mult_INR; repeat rewrite S_INR; ring.
+apply le_trans with N; [ assumption | apply le_n_2n ].
+apply (fun p n m:nat => plus_le_reg_l n m p) with k; rewrite <- le_plus_minus.
+replace (2 * N)%nat with (N + N)%nat; [ idtac | ring ].
+apply plus_le_compat_r; assumption.
+assumption.
+assumption.
+apply (fun p n m:nat => plus_le_reg_l n m p) with k; rewrite <- le_plus_minus.
+replace (2 * N)%nat with (N + N)%nat; [ idtac | ring ].
+apply plus_le_compat_r; assumption.
+assumption.
+rewrite <- (Rmult_comm (prod_f_SO (fun l:nat => INR l) k));
+ rewrite (prod_SO_split (fun l:nat => INR l) k N).
+rewrite Rmult_assoc; apply Rmult_le_compat_l.
+apply prod_SO_pos; intros; apply pos_INR.
+rewrite Rmult_comm;
+ rewrite (prod_SO_split (fun l:nat => INR l) N (2 * N - k)).
+apply Rmult_le_compat_l.
+apply prod_SO_pos; intros; apply pos_INR.
+replace (N - (2 * N - k))%nat with (k - N)%nat.
+apply prod_SO_Rle; intros; split.
+apply pos_INR.
+apply le_INR; apply plus_le_compat_r.
+apply (fun p n m:nat => plus_le_reg_l n m p) with k; rewrite <- le_plus_minus.
+replace (2 * N)%nat with (N + N)%nat; [ idtac | ring ];
+ apply plus_le_compat_r; assumption.
+assumption.
+apply INR_eq; repeat rewrite minus_INR.
+rewrite mult_INR; do 2 rewrite S_INR; ring.
+assumption.
+apply (fun p n m:nat => plus_le_reg_l n m p) with k; rewrite <- le_plus_minus.
+replace (2 * N)%nat with (N + N)%nat; [ idtac | ring ];
+ apply plus_le_compat_r; assumption.
+assumption.
+assumption.
+apply (fun p n m:nat => plus_le_reg_l n m p) with k; rewrite <- le_plus_minus.
+replace (2 * N)%nat with (N + N)%nat; [ idtac | ring ];
+ apply plus_le_compat_r; assumption.
+assumption.
+assumption.
+elim (le_dec k N); intro; [ left; assumption | right; assumption ].
+Qed.
+
+(**********)
+Lemma INR_fact_lt_0 : forall n:nat, 0 < INR (fact n).
+intro; apply lt_INR_0; apply neq_O_lt; red in |- *; intro;
+ elim (fact_neq_0 n); symmetry  in |- *; assumption.
+Qed.
+
+(* We have the following inequality : (C 2N k) <= (C 2N N) forall k in [|O;2N|] *)
+Lemma C_maj : forall N k:nat, (k <= 2 * N)%nat -> C (2 * N) k <= C (2 * N) N.
+intros; unfold C in |- *; unfold Rdiv in |- *; apply Rmult_le_compat_l.
+apply pos_INR.
+replace (2 * N - N)%nat with N.
+apply Rmult_le_reg_l with (INR (fact N) * INR (fact N)).
+apply Rmult_lt_0_compat; apply INR_fact_lt_0.
+rewrite <- Rinv_r_sym.
+rewrite Rmult_comm;
+ apply Rmult_le_reg_l with (INR (fact k) * INR (fact (2 * N - k))).
+apply Rmult_lt_0_compat; apply INR_fact_lt_0.
+rewrite Rmult_1_r; rewrite <- mult_INR; rewrite <- Rmult_assoc;
+ rewrite <- Rinv_r_sym.
+rewrite Rmult_1_l; rewrite mult_INR; rewrite (Rmult_comm (INR (fact k)));
+ replace (INR (fact N) * INR (fact N)) with (Rsqr (INR (fact N))).
+apply RfactN_fact2N_factk.
+assumption.
+reflexivity.
+rewrite mult_INR; apply prod_neq_R0; apply INR_fact_neq_0.
+apply prod_neq_R0; apply INR_fact_neq_0.
+apply INR_eq; rewrite minus_INR;
+ [ rewrite mult_INR; do 2 rewrite S_INR; ring | apply le_n_2n ].
+Qed.
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