From 83a067cae072132484cfe18a5c7b4e91676deedf Mon Sep 17 00:00:00 2001
From: desmettr
Date: Fri, 30 Nov 2001 15:31:47 +0000
Subject: Integration de nouveaux lemmes

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@2253 85f007b7-540e-0410-9357-904b9bb8a0f7
---
 theories/Reals/Rbase.v | 98 ++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 98 insertions(+)

diff --git a/theories/Reals/Rbase.v b/theories/Reals/Rbase.v
index 3a1d677db3..b37b7b1a06 100644
--- a/theories/Reals/Rbase.v
+++ b/theories/Reals/Rbase.v
@@ -1501,3 +1501,101 @@ Intros r z H1 H2 (s, (H3,(H4,H5))).
 Apply H3; Apply single_z_r_R1 with r; Trivial.
 Save.
 
+(*****************************************************************)
+(* Définitions de nouveaux types                                 *)
+(*****************************************************************)
+
+Record nonnegreal : Type := mknonnegreal {
+nonneg :> R;
+cond_nonneg : ``0<=nonneg`` }.
+
+Record posreal : Type := mkposreal {
+pos :> R;
+cond_pos : ``0<pos`` }.
+
+Record nonposreal : Type := mknonposreal {
+nonpos :> R;
+cond_nonpos : ``nonpos<=0`` }.
+
+Record negreal : Type := mknegreal {
+neg :> R;
+cond_neg : ``neg<0`` }.
+
+Record nonzeroreal : Type := mknonzeroreal {
+nonzero :> R;
+cond_nonzero : ~``nonzero==0`` }.
+
+(**********)
+Lemma prod_neq_R0 : (x,y:R) ~``x==0``->~``y==0``->~``x*y==0``.
+Intros; Red; Intro; Generalize (without_div_Od x y H1); Intro; Elim H2; Intro; [Rewrite H3 in H; Elim H | Rewrite H3 in H0; Elim H0]; Reflexivity.
+Save.
+
+(**********)
+Lemma regle_signe : (x,y:R) ``0<x`` -> ``0<y`` -> ``0<x*y``.
+Intros; Rewrite <- (Rmult_Ol x); Rewrite <- (Rmult_sym x); Apply (Rlt_monotony x R0 y H H0).
+Save.
+
+(*********)
+Lemma regle_signe_le : (x,y:R) ``0<=x`` -> ``0<=y`` -> ``0<=x*y``.
+Intros; Rewrite <- (Rmult_Ol x); Rewrite <- (Rmult_sym x); Apply (Rle_monotony x R0 y H H0).
+Save.
+
+(**********************************************************)
+(* Quelques règles concernant < et <=                     *)
+(**********************************************************)
+
+Lemma gt0_plus_gt0_is_gt0 : (x,y:R) ``0<x`` -> ``0<y`` -> ``0<x+y``.
+Intros; Apply Rlt_trans with x; [Assumption | Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rlt_compatibility; Assumption].
+Save.
+
+Lemma ge0_plus_gt0_is_gt0 : (x,y:R) ``0<=x`` -> ``0<y`` -> ``0<x+y``.
+Intros; Apply Rle_lt_trans with x; [Assumption | Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rlt_compatibility; Assumption].
+Save.
+
+Lemma  gt0_plus_ge0_is_gt0 : (x,y:R) ``0<x`` -> ``0<=y`` -> ``0<x+y``.
+Intros; Rewrite <- Rplus_sym; Apply ge0_plus_gt0_is_gt0; Assumption.
+Save.
+
+Lemma ge0_plus_ge0_is_ge0 : (x,y:R) ``0<=x`` -> ``0<=y`` -> ``0<=x+y``.
+Intros; Apply Rle_trans with x; [Assumption | Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Assumption].
+Save.
+
+Lemma ge0_plus_ge0_eq_0 : (x,y:R) ``0<=x`` -> ``0<=y`` -> ``x+y==0`` -> ``x==0``/\``y==0``.
+Intros; Split; [Elim H; Intro; [Generalize (gt0_plus_ge0_is_gt0 x y H2 H0); Intro; Rewrite H1 in H3; Elim (Rlt_antirefl ``0`` H3) | Symmetry; Assumption] | Elim H0; Intro; [Generalize (ge0_plus_gt0_is_gt0 x y H H2); Intro; Rewrite H1 in H3; Elim (Rlt_antirefl R0 H3) | Symmetry; Assumption]].
+Save.
+
+Lemma Rmult_le : (r1,r2,r3,r4:R) ``0<=r1`` -> ``0<=r3`` -> ``r1<=r2`` -> ``r3<=r4`` -> ``r1*r3<=r2*r4``.
+Intros; Apply Rle_trans with ``r2*r3``; [Apply Rle_monotony_r; Assumption | Apply Rle_monotony; [ Apply Rle_trans with r1; Assumption | Assumption]].
+Save.
+
+Lemma plus_le_is_le : (x,y,z:R) ``0<=y`` -> ``x+y<=z`` -> ``x<=z``.
+Intros; Apply Rle_trans with ``x+y``; [Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Assumption | Assumption].
+Save.
+
+Lemma le_plus_lt_is_lt : (x,y,z,t:R) ``x<=y`` -> ``z<t`` -> ``x+z<y+t``. 
+Intros; Apply Rle_lt_trans with ``y+z``; [Apply Rle_compatibility_r; Assumption | Apply Rlt_compatibility; Assumption].
+Save.
+
+Lemma plus_lt_is_lt : (x,y,z:R) ``0<=y`` -> ``x+y<z`` -> ``x<z``.
+Intros; Apply Rle_lt_trans with ``x+y``; [Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Assumption | Assumption].
+Save.
+
+Lemma Rmult_lt2 : (r1,r2,r3,r4:R) ``0<=r1`` -> ``0<=r3`` -> ``r1<r2`` -> ``r3<r4`` -> ``r1*r3<r2*r4``.
+Intros; Apply Rle_lt_trans with ``r2*r3``; [Apply Rle_monotony_r; [Assumption | Left; Assumption] | Apply Rlt_monotony; [Apply Rle_lt_trans with r1; Assumption | Assumption]].
+Save.
+
+(*****************************************************)
+(* Résultat complémentaire sur la fonction INR       *)
+(*****************************************************)
+
+Fixpoint INR2 [n:nat] : R := Cases n of
+O => ``0``
+| (S n0) => Cases n0 of
+O => ``1``
+| (S _) => ``1+(INR2 n0)``
+end
+end.
+
+Theorem INR_eq_INR2 : (n:nat) (INR n)==(INR2 n).
+Induction n; [Unfold INR INR2; Reflexivity | Intros; Unfold INR INR2; Fold INR INR2; Rewrite H; Case n0; [Reflexivity | Intros; Ring]].
+Save.
\ No newline at end of file
-- 
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