diff options
Diffstat (limited to 'theories/Reals/Ranalysis1.v')
| -rw-r--r-- | theories/Reals/Ranalysis1.v | 30 |
1 files changed, 15 insertions, 15 deletions
diff --git a/theories/Reals/Ranalysis1.v b/theories/Reals/Ranalysis1.v index 918ebfc0..6d30e291 100644 --- a/theories/Reals/Ranalysis1.v +++ b/theories/Reals/Ranalysis1.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Ranalysis1.v,v 1.21.2.1 2004/07/16 19:31:12 herbelin Exp $ i*) +(*i $Id: Ranalysis1.v 8670 2006-03-28 22:16:14Z herbelin $ i*) Require Import Rbase. Require Import Rfunctions. @@ -453,7 +453,7 @@ Qed. Theorem derivable_continuous_pt : forall f (x:R), derivable_pt f x -> continuity_pt f x. -intros. +intros f x X. generalize (derivable_derive f x X); intro. elim H; intros l H1. cut (l = fct_cte l x). @@ -468,7 +468,7 @@ unfold fct_cte in |- *; reflexivity. Qed. Theorem derivable_continuous : forall f, derivable f -> continuity f. -unfold derivable, continuity in |- *; intros. +unfold derivable, continuity in |- *; intros f X x. apply (derivable_continuous_pt f x (X x)). Qed. @@ -661,7 +661,7 @@ Qed. Lemma derivable_pt_plus : forall f1 f2 (x:R), derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 + f2) x. -unfold derivable_pt in |- *; intros. +unfold derivable_pt in |- *; intros f1 f2 x X X0. elim X; intros. elim X0; intros. apply existT with (x0 + x1). @@ -670,7 +670,7 @@ Qed. Lemma derivable_pt_opp : forall f (x:R), derivable_pt f x -> derivable_pt (- f) x. -unfold derivable_pt in |- *; intros. +unfold derivable_pt in |- *; intros f x X. elim X; intros. apply existT with (- x0). apply derivable_pt_lim_opp; assumption. @@ -679,7 +679,7 @@ Qed. Lemma derivable_pt_minus : forall f1 f2 (x:R), derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 - f2) x. -unfold derivable_pt in |- *; intros. +unfold derivable_pt in |- *; intros f1 f2 x X X0. elim X; intros. elim X0; intros. apply existT with (x0 - x1). @@ -689,7 +689,7 @@ Qed. Lemma derivable_pt_mult : forall f1 f2 (x:R), derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 * f2) x. -unfold derivable_pt in |- *; intros. +unfold derivable_pt in |- *; intros f1 f2 x X X0. elim X; intros. elim X0; intros. apply existT with (x0 * f2 x + f1 x * x1). @@ -704,7 +704,7 @@ Qed. Lemma derivable_pt_scal : forall f (a x:R), derivable_pt f x -> derivable_pt (mult_real_fct a f) x. -unfold derivable_pt in |- *; intros. +unfold derivable_pt in |- *; intros f1 a x X. elim X; intros. apply existT with (a * x0). apply derivable_pt_lim_scal; assumption. @@ -724,7 +724,7 @@ Qed. Lemma derivable_pt_comp : forall f1 f2 (x:R), derivable_pt f1 x -> derivable_pt f2 (f1 x) -> derivable_pt (f2 o f1) x. -unfold derivable_pt in |- *; intros. +unfold derivable_pt in |- *; intros f1 f2 x X X0. elim X; intros. elim X0; intros. apply existT with (x1 * x0). @@ -733,24 +733,24 @@ Qed. Lemma derivable_plus : forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 + f2). -unfold derivable in |- *; intros. +unfold derivable in |- *; intros f1 f2 X X0 x. apply (derivable_pt_plus _ _ x (X _) (X0 _)). Qed. Lemma derivable_opp : forall f, derivable f -> derivable (- f). -unfold derivable in |- *; intros. +unfold derivable in |- *; intros f X x. apply (derivable_pt_opp _ x (X _)). Qed. Lemma derivable_minus : forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 - f2). -unfold derivable in |- *; intros. +unfold derivable in |- *; intros f1 f2 X X0 x. apply (derivable_pt_minus _ _ x (X _) (X0 _)). Qed. Lemma derivable_mult : forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 * f2). -unfold derivable in |- *; intros. +unfold derivable in |- *; intros f1 f2 X X0 x. apply (derivable_pt_mult _ _ x (X _) (X0 _)). Qed. @@ -761,7 +761,7 @@ Qed. Lemma derivable_scal : forall f (a:R), derivable f -> derivable (mult_real_fct a f). -unfold derivable in |- *; intros. +unfold derivable in |- *; intros f a X x. apply (derivable_pt_scal _ a x (X _)). Qed. @@ -775,7 +775,7 @@ Qed. Lemma derivable_comp : forall f1 f2, derivable f1 -> derivable f2 -> derivable (f2 o f1). -unfold derivable in |- *; intros. +unfold derivable in |- *; intros f1 f2 X X0 x. apply (derivable_pt_comp _ _ x (X _) (X0 _)). Qed. |