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Diffstat (limited to 'theories/Reals/Rprod.v')
| -rw-r--r-- | theories/Reals/Rprod.v | 191 |
1 files changed, 191 insertions, 0 deletions
diff --git a/theories/Reals/Rprod.v b/theories/Reals/Rprod.v new file mode 100644 index 00000000..6577146f --- /dev/null +++ b/theories/Reals/Rprod.v @@ -0,0 +1,191 @@ +(************************************************************************) +(* 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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