diff options
Diffstat (limited to 'theories/Reals/Ranalysis2.v')
| -rw-r--r-- | theories/Reals/Ranalysis2.v | 17 |
1 files changed, 4 insertions, 13 deletions
diff --git a/theories/Reals/Ranalysis2.v b/theories/Reals/Ranalysis2.v index 69671c5513..83f9409cdd 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. |