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authorGravatar filliatr <filliatr@85f007b7-540e-0410-9357-904b9bb8a0f7>2002-02-14 14:39:07 +0000
committerGravatar filliatr <filliatr@85f007b7-540e-0410-9357-904b9bb8a0f7>2002-02-14 14:39:07 +0000
commit67f72c93f5f364591224a86c52727867e02a8f71 (patch)
treeecf630daf8346e77e6620233d8f3e6c18a0c9b3c /theories/Reals/Ranalysis.v
parentb239b208eb9a66037b0c629cf7ccb6e4b110636a (diff)
option -dump-glob pour coqdoc
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@2474 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/Reals/Ranalysis.v')
-rw-r--r--theories/Reals/Ranalysis.v26
1 files changed, 13 insertions, 13 deletions
diff --git a/theories/Reals/Ranalysis.v b/theories/Reals/Ranalysis.v
index 38b31d7b6..11a9fe9b8 100644
--- a/theories/Reals/Ranalysis.v
+++ b/theories/Reals/Ranalysis.v
@@ -17,7 +17,7 @@ Require DiscrR.
Require Rtrigo.
(****************************************************)
-(* Basic operations on functions *)
+(** Basic operations on functions *)
(****************************************************)
Definition plus_fct [f1,f2:R->R] : R->R := [x:R] ``(f1 x)+(f2 x)``.
Definition opp_fct [f:R->R] : R->R := [x:R] ``-(f x)``.
@@ -29,7 +29,7 @@ Definition div_real_fct [a:R;f:R->R] : R->R := [x:R] ``a/(f x)``.
Definition comp [f1,f2:R->R] : R->R := [x:R] ``(f1 (f2 x))``.
(****************************************************)
-(* Variations of functions *)
+(** Variations of functions *)
(****************************************************)
Definition increasing [f:R->R] : Prop := (x,y:R) ``x<=y``->``(f x)<=(f y)``.
Definition decreasing [f:R->R] : Prop := (x,y:R) ``x<=y``->``(f y)<=(f x)``.
@@ -44,7 +44,7 @@ Axiom fct_eq : (A,B:Type) (f1,f2:A->B) ((x:A)(f1 x)==(f2 x))->f1==f2.
Definition no_cond : R->Prop := [x:R] True.
(***************************************************)
-(* Definition of continuity as a limit *)
+(** Definition of continuity as a limit *)
(***************************************************)
(**********)
@@ -135,7 +135,7 @@ Unfold continuity; Intros; Apply (inv_continuous f x (H x) (H0 x)).
Save.
(*****************************************************)
-(* Derivative's definition using Landau's kernel *)
+(** Derivative's definition using Landau's kernel *)
(*****************************************************)
Definition derivable_pt [f:R->R; x:R] : Prop := (EXT l : R | ((eps:R) ``0<eps``->(EXT delta : posreal | ((h:R) ~``h==0``->``(Rabsolu h)<delta`` -> ``(Rabsolu ((((f (x+h))-(f x))/h)-l))<eps``)))).
@@ -158,9 +158,9 @@ Save.
(**********)
Definition derive [f:R->R] := [x:R] (derive_pt f x).
-(***********************************)
-(* Class of differential functions *)
-(***********************************)
+(************************************)
+(** Class of differential functions *)
+(************************************)
Record Differential : Type := mkDifferential {
d1 :> R->R;
cond_diff : (derivable d1) }.
@@ -180,9 +180,9 @@ Lemma derive_derivable : (f:R->R;x,l:R) (derive_pt f x)==l -> (derivable_pt f x)
Intros; Unfold derivable_pt; Generalize (derive_pt_def_1 f x l H); Intro H0; Exists l; Assumption.
Save.
-(*******************************************************************)
-(* Equivalence of this definition with the one using limit concept *)
-(*******************************************************************)
+(********************************************************************)
+(** Equivalence of this definition with the one using limit concept *)
+(********************************************************************)
Lemma derive_pt_D_in : (f,df:R->R;x:R) (D_in f df no_cond x) <-> (derive_pt
f x)==(df x).
Intros; Split.
@@ -201,7 +201,7 @@ Save.
Definition fct_cte [a:R] : R->R := [x:R]a.
(***********************************)
-(* derivability -> continuity *)
+(** derivability -> continuity *)
(***********************************)
Theorem derivable_continuous_pt : (f:R->R;x:R) (derivable_pt f x) -> (continuity_pt f x).
Intros.
@@ -223,7 +223,7 @@ Unfold derivable continuity; Intros; Apply (derivable_continuous_pt f x (H x)).
Save.
(****************************************************************)
-(* Main rules *)
+(** Main rules *)
(****************************************************************)
(* Addition *)
@@ -492,7 +492,7 @@ Intro; Generalize deriv_cos; Unfold derive; Intro; Unfold opp_fct in H; Apply (e
Save.
(************************************************************)
-(* Local extremum's condition *)
+(** Local extremum's condition *)
(************************************************************)
Theorem deriv_maximum : (f:R->R;a,b,c:R) ``a<c``->``c<b``->(derivable_pt f c)->((x:R) ``a<x``->``x<b``->``(f x)<=(f c)``)->``(derive_pt f c)==0``.
Intros; Case (total_order R0 (derive_pt f c)); Intro.