diff options
Diffstat (limited to 'theories/Reals/Rprod.v')
| -rw-r--r-- | theories/Reals/Rprod.v | 24 |
1 files changed, 11 insertions, 13 deletions
diff --git a/theories/Reals/Rprod.v b/theories/Reals/Rprod.v index 947dbb11..88c4de23 100644 --- a/theories/Reals/Rprod.v +++ b/theories/Reals/Rprod.v @@ -1,23 +1,21 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Rprod.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Import Compare. Require Import Rbase. Require Import Rfunctions. Require Import Rseries. Require Import PartSum. Require Import Binomial. -Open Local Scope R_scope. +Local Open Scope R_scope. (** TT Ak; 0<=k<=N *) -Boxed Fixpoint prod_f_R0 (f:nat -> R) (N:nat) : R := +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) @@ -38,7 +36,7 @@ Proof. replace (S n - k - 1)%nat with O; [rewrite H1; simpl|omega]. replace (n+1+0)%nat with (S n); ring. replace (S n - k-1)%nat with (S (n - k-1));[idtac|omega]. - simpl in |- *; replace (k + S (n - k))%nat with (S n). + simpl; replace (k + S (n - k))%nat with (S n). replace (k + 1 + S (n - k - 1))%nat with (S n). rewrite Hrecn; [ ring | assumption ]. omega. @@ -51,8 +49,8 @@ Lemma prod_SO_pos : (forall n:nat, (n <= N)%nat -> 0 <= An n) -> 0 <= prod_f_R0 An N. Proof. intros; induction N as [| N HrecN]. - simpl in |- *; apply H; trivial. - simpl in |- *; apply Rmult_le_pos. + simpl; apply H; trivial. + simpl; apply Rmult_le_pos. apply HrecN; intros; apply H; apply le_trans with N; [ assumption | apply le_n_Sn ]. apply H; apply le_n. @@ -66,7 +64,7 @@ Lemma prod_SO_Rle : Proof. intros; induction N as [| N HrecN]. elim H with O; trivial. - simpl in |- *; apply Rle_trans with (prod_f_R0 An N * Bn (S N)). + simpl; apply Rle_trans with (prod_f_R0 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. @@ -116,7 +114,7 @@ Proof. (if eq_nat_dec n 0 then 1 else INR n) = INR n). intros n; case (eq_nat_dec n 0); auto with real. intros; absurd (0 < n)%nat; omega. - intros; unfold Rsqr in |- *; repeat rewrite fact_prodSO. + intros; unfold Rsqr; repeat rewrite fact_prodSO. cut ((k=N)%nat \/ (k < N)%nat \/ (N < k)%nat). intro H2; elim H2; intro H3. rewrite H3; replace (2*N-N)%nat with N;[right; ring|omega]. @@ -166,14 +164,14 @@ Qed. (**********) Lemma INR_fact_lt_0 : forall n:nat, 0 < INR (fact n). Proof. - intro; apply lt_INR_0; apply neq_O_lt; red in |- *; intro; - elim (fact_neq_0 n); symmetry in |- *; assumption. + intro; apply lt_INR_0; apply neq_O_lt; red; intro; + elim (fact_neq_0 n); symmetry ; 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. Proof. - intros; unfold C in |- *; unfold Rdiv in |- *; apply Rmult_le_compat_l. + intros; unfold C; unfold Rdiv; 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)). |