diff options
author | filliatr <filliatr@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2002-02-14 14:39:07 +0000 |
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committer | filliatr <filliatr@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2002-02-14 14:39:07 +0000 |
commit | 67f72c93f5f364591224a86c52727867e02a8f71 (patch) | |
tree | ecf630daf8346e77e6620233d8f3e6c18a0c9b3c /theories/Reals/Ranalysis.v | |
parent | b239b208eb9a66037b0c629cf7ccb6e4b110636a (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.v | 26 |
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. |