Suppression de lemmes superflus

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@3576 85f007b7-540e-0410-9357-904b9bb8a0f7

2 files changed, 13 insertions, 22 deletions

diff --git a/theories/Reals/Ranalysis2.v b/theories/Reals/Ranalysis2.v
index 69671c551..83f9409cd 100644
--- a/theories/Reals/Ranalysis2.v
+++ b/theories/Reals/Ranalysis2.v
@@ -11,7 +11,6 @@
 Require Rbase.
 Require Rfunctions.
 Require Ranalysis1.
-Require Omega.
 
 (**********)
 Lemma formule : (x,h,l1,l2:R;f1,f2:R->R) ``h<>0`` -> ``(f2 x)<>0`` -> ``(f2 (x+h))<>0`` -> ``((f1 (x+h))/(f2 (x+h))-(f1 x)/(f2 x))/h-(l1*(f2 x)-l2*(f1 x))/(Rsqr (f2 x))`` == ``/(f2 (x+h))*(((f1 (x+h))-(f1 x))/h-l1) + l1/((f2 x)*(f2 (x+h)))*((f2 x)-(f2 (x+h))) - (f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))-(f2 x))/h-l2) + (l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))*((f2 (x+h))-(f2 x))``.
@@ -33,14 +32,6 @@ Intros; Unfold Rmin.
 Case (total_order_Rle x y); Intro; Assumption.
 Qed.
 
-Lemma Rgt_8_0 : ``0 < 8``.
-Sup0.
-Qed.
-
-Lemma Rgt_4_0 : ``0 < 4``.
-Sup0.
-Qed.
-
 Lemma maj_term1 : (x,h,eps,l1,alp_f2:R;eps_f2,alp_f1d:posreal;f1,f2:R->R) ``0 < eps`` -> ``(f2 x)<>0`` -> ``(f2 (x+h))<>0`` -> ((h:R)``h <> 0``->``(Rabsolu h) < alp_f1d``->``(Rabsolu (((f1 (x+h))-(f1 x))/h-l1)) < (Rabsolu ((eps*(f2 x))/8))``) -> ((a:R)``(Rabsolu a) < (Rmin eps_f2 alp_f2)``->``/(Rabsolu (f2 (x+a))) < 2/(Rabsolu (f2 x))``) -> ``h<>0`` -> ``(Rabsolu h)<alp_f1d`` -> ``(Rabsolu h) < (Rmin eps_f2 alp_f2)`` -> ``(Rabsolu (/(f2 (x+h))*(((f1 (x+h))-(f1 x))/h-l1))) < eps/4``.
 Intros.
 Assert H7 := (H3 h H6).
@@ -65,7 +56,7 @@ Replace (Rabsolu eps) with eps.
 Repeat Rewrite <- Rinv_r_sym; Try DiscrR Orelse (Apply Rabsolu_no_R0; Assumption).
 Ring.
 Symmetry; Apply Rabsolu_right; Left; Assumption.
-Symmetry; Apply Rabsolu_right; Left; Apply Rgt_8_0.
+Symmetry; Apply Rabsolu_right; Left; Sup.
 Qed.
 
 Lemma maj_term2 : (x,h,eps,l1,alp_f2,alp_f2t2:R;eps_f2:posreal;f2:R->R) ``0 < eps`` -> ``(f2 x)<>0`` -> ``(f2 (x+h))<>0`` -> ((a:R)``(Rabsolu a) < alp_f2t2``->``(Rabsolu ((f2 (x+a))-(f2 x))) < (Rabsolu ((eps*(Rsqr (f2 x)))/(8*l1)))``)-> ((a:R)``(Rabsolu a) < (Rmin eps_f2 alp_f2)``->``/(Rabsolu (f2 (x+a))) < 2/(Rabsolu (f2 x))``) -> ``h<>0`` -> ``(Rabsolu h)<alp_f2t2`` -> ``(Rabsolu h) < (Rmin eps_f2 alp_f2)`` -> ``l1<>0`` -> ``(Rabsolu (l1/((f2 x)*(f2 (x+h)))*((f2 x)-(f2 (x+h))))) < eps/4``.
@@ -111,7 +102,7 @@ Replace ``2*((Rabsolu l1)*(/(Rabsolu (f2 x))*/(Rabsolu (f2 x))))*(eps*((Rabsolu
 Repeat Rewrite <- Rinv_r_sym; Try (Apply Rabsolu_no_R0; Assumption) Orelse DiscrR.
 Ring.
 Symmetry; Apply Rabsolu_right; Left; Sup0.
-Symmetry; Apply Rabsolu_right; Left; Apply Rgt_8_0.
+Symmetry; Apply Rabsolu_right; Left; Sup.
 Symmetry; Apply Rabsolu_right; Left; Assumption.
 Qed.
 
@@ -158,7 +149,7 @@ Replace ``2*((Rabsolu (f1 x))*(/(Rabsolu (f2 x))*/(Rabsolu (f2 x))))*((Rabsolu (
 Repeat Rewrite <- Rinv_r_sym; Try DiscrR Orelse (Apply Rabsolu_no_R0; Assumption).
 Ring.
 Symmetry; Apply Rabsolu_right; Left; Sup0.
-Symmetry; Apply Rabsolu_right; Left; Apply Rgt_8_0.
+Symmetry; Apply Rabsolu_right; Left; Sup.
 Symmetry; Apply Rabsolu_right; Left; Assumption.
 Qed.
 
@@ -211,7 +202,7 @@ Replace ``2*(Rabsolu l2)*((Rabsolu (f1 x))*(/(Rabsolu (f2 x))*/(Rabsolu (f2 x))*
 Repeat Rewrite <- Rinv_r_sym; Try DiscrR Orelse (Apply Rabsolu_no_R0; Assumption).
 Ring.
 Symmetry; Apply Rabsolu_right; Left; Sup0.
-Symmetry; Apply Rabsolu_right; Left; Apply Rgt_8_0.
+Symmetry; Apply Rabsolu_right; Left; Sup.
 Symmetry; Apply Rabsolu_right; Left; Assumption.
 Apply prod_neq_R0; Assumption Orelse DiscrR.
 Apply prod_neq_R0; Assumption.
diff --git a/theories/Reals/Ranalysis3.v b/theories/Reals/Ranalysis3.v
index 20c02f360..9cc34059b 100644
--- a/theories/Reals/Ranalysis3.v
+++ b/theories/Reals/Ranalysis3.v
@@ -65,15 +65,15 @@ Apply Rlt_4; Assumption.
 Rewrite H8.
 Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol.
 Rewrite Rabsolu_R0; Rewrite Rmult_Ol.
-Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rgt_4_0].
+Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup].
 Rewrite H8.
 Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol.
 Rewrite Rabsolu_R0; Rewrite Rmult_Ol.
-Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rgt_4_0].
+Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup].
 Rewrite H9.
 Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol.
 Rewrite Rabsolu_R0; Rewrite Rmult_Ol.
-Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rgt_4_0].
+Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup].
 Rewrite <- Rabsolu_mult.
 Apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); Try Assumption Orelse Apply H2.
 Apply H14.
@@ -123,11 +123,11 @@ Apply Rlt_4; Assumption.
 Rewrite H8.
 Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol.
 Rewrite Rabsolu_R0; Rewrite Rmult_Ol.
-Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rgt_4_0].
+Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup].
 Rewrite H8.
 Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol.
 Rewrite Rabsolu_R0; Rewrite Rmult_Ol.
-Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rgt_4_0].
+Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup].
 Rewrite <- Rabsolu_mult.
 Apply (maj_term2 x h eps l1 alp_f2 alp_f2t2 eps_f2 f2); Try Assumption.
 Apply H2; Assumption.
@@ -200,7 +200,7 @@ Apply Rlt_4; Assumption.
 Rewrite H10.
 Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol.
 Rewrite Rabsolu_R0; Rewrite Rmult_Ol.
-Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rgt_4_0].
+Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup].
 Rewrite <- Rabsolu_mult.
 Apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); Try Assumption.
 Apply H2; Assumption.
@@ -208,7 +208,7 @@ Apply Rmin_2; Assumption.
 Rewrite H9.
 Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol.
 Rewrite Rabsolu_R0; Rewrite Rmult_Ol.
-Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rgt_4_0].
+Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup].
 Rewrite <- Rabsolu_mult.
 Apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); Assumption Orelse Idtac.
 Apply H2; Assumption.
@@ -275,7 +275,7 @@ Apply Rmin_2; Assumption.
 Rewrite H9.
 Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol.
 Rewrite Rabsolu_R0; Rewrite Rmult_Ol.
-Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rgt_4_0].
+Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup].
 Rewrite <- Rabsolu_mult.
 Apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); Try Assumption.
 Apply H2; Assumption.
@@ -379,7 +379,7 @@ Apply Rlt_4; Assumption.
 Rewrite H10.
 Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol.
 Rewrite Rabsolu_R0; Rewrite Rmult_Ol.
-Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rgt_4_0].
+Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup].
 Rewrite <- Rabsolu_mult.
 Apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); Try Assumption.
 Apply H2; Assumption.