From 7d555bede1664a62ed65798b00953f50a6d453fe Mon Sep 17 00:00:00 2001 From: desmettr Date: Wed, 22 Jan 2003 13:28:03 +0000 Subject: Suppression de lemmes superflus git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@3576 85f007b7-540e-0410-9357-904b9bb8a0f7 --- theories/Reals/Ranalysis2.v | 17 ++++------------- theories/Reals/Ranalysis3.v | 18 +++++++++--------- 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) ``(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) ``(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. -- cgit v1.2.3