Imported Upstream version 8.0pl1upstream/8.0pl1

1 files changed, 450 insertions, 0 deletions

diff --git a/theories/Reals/Ranalysis2.v b/theories/Reals/Ranalysis2.v
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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(*i $Id: Ranalysis2.v,v 1.11.2.1 2004/07/16 19:31:12 herbelin Exp $ i*)
+
+Require Import Rbase.
+Require Import Rfunctions.
+Require Import Ranalysis1. Open Local Scope R_scope.
+
+(**********)
+Lemma formule :
+ forall (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).
+intros; unfold Rdiv, Rminus, Rsqr in |- *.
+repeat rewrite Rmult_plus_distr_r; repeat rewrite Rmult_plus_distr_l;
+ repeat rewrite Rinv_mult_distr; try assumption.
+replace (l1 * f2 x * (/ f2 x * / f2 x)) with (l1 * / f2 x * (f2 x * / f2 x));
+ [ idtac | ring ].
+replace (l1 * (/ f2 x * / f2 (x + h)) * f2 x) with
+ (l1 * / f2 (x + h) * (f2 x * / f2 x)); [ idtac | ring ].
+replace (l1 * (/ f2 x * / f2 (x + h)) * - f2 (x + h)) with
+ (- (l1 * / f2 x * (f2 (x + h) * / f2 (x + h)))); [ idtac | ring ].
+replace (f1 x * (/ f2 x * / f2 (x + h)) * (f2 (x + h) * / h)) with
+ (f1 x * / f2 x * / h * (f2 (x + h) * / f2 (x + h))); 
+ [ idtac | ring ].
+replace (f1 x * (/ f2 x * / f2 (x + h)) * (- f2 x * / h)) with
+ (- (f1 x * / f2 (x + h) * / h * (f2 x * / f2 x))); 
+ [ idtac | ring ].
+replace (l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) * f2 (x + h)) with
+ (l2 * f1 x * / f2 x * / f2 x * (f2 (x + h) * / f2 (x + h)));
+ [ idtac | ring ].
+replace (l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) * - f2 x) with
+ (- (l2 * f1 x * / f2 x * / f2 (x + h) * (f2 x * / f2 x))); 
+ [ idtac | ring ].
+repeat rewrite <- Rinv_r_sym; try assumption || ring.
+apply prod_neq_R0; assumption.
+Qed.
+
+Lemma Rmin_pos : forall x y:R, 0 < x -> 0 < y -> 0 < Rmin x y.
+intros; unfold Rmin in |- *.
+case (Rle_dec x y); intro; assumption.
+Qed.
+
+Lemma maj_term1 :
+ forall (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 ->
+   (forall h:R,
+      h <> 0 ->
+      Rabs h < alp_f1d ->
+      Rabs ((f1 (x + h) - f1 x) / h - l1) < Rabs (eps * f2 x / 8)) ->
+   (forall a:R,
+      Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
+   h <> 0 ->
+   Rabs h < alp_f1d ->
+   Rabs h < Rmin eps_f2 alp_f2 ->
+   Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) < eps / 4.
+intros.
+assert (H7 := H3 h H6).
+assert (H8 := H2 h H4 H5).
+apply Rle_lt_trans with
+ (2 / Rabs (f2 x) * Rabs ((f1 (x + h) - f1 x) / h - l1)).
+rewrite Rabs_mult.
+apply Rmult_le_compat_r.
+apply Rabs_pos.
+rewrite Rabs_Rinv; [ left; exact H7 | assumption ].
+apply Rlt_le_trans with (2 / Rabs (f2 x) * Rabs (eps * f2 x / 8)).
+apply Rmult_lt_compat_l.
+unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+ [ prove_sup0 | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ].
+exact H8.
+right; unfold Rdiv in |- *.
+repeat rewrite Rabs_mult.
+rewrite Rabs_Rinv; discrR.
+replace (Rabs 8) with 8.
+replace 8 with 8; [ idtac | ring ].
+rewrite Rinv_mult_distr; [ idtac | discrR | discrR ].
+replace (2 * / Rabs (f2 x) * (Rabs eps * Rabs (f2 x) * (/ 2 * / 4))) with
+ (Rabs eps * / 4 * (2 * / 2) * (Rabs (f2 x) * / Rabs (f2 x)));
+ [ idtac | ring ].
+replace (Rabs eps) with eps.
+repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption).
+ring.
+symmetry  in |- *; apply Rabs_right; left; assumption.
+symmetry  in |- *; apply Rabs_right; left; prove_sup.
+Qed.
+
+Lemma maj_term2 :
+ forall (x h eps l1 alp_f2 alp_f2t2:R) (eps_f2:posreal) 
+   (f2:R -> R),
+   0 < eps ->
+   f2 x <> 0 ->
+   f2 (x + h) <> 0 ->
+   (forall a:R,
+      Rabs a < alp_f2t2 ->
+      Rabs (f2 (x + a) - f2 x) < Rabs (eps * Rsqr (f2 x) / (8 * l1))) ->
+   (forall a:R,
+      Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
+   h <> 0 ->
+   Rabs h < alp_f2t2 ->
+   Rabs h < Rmin eps_f2 alp_f2 ->
+   l1 <> 0 -> Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) < eps / 4.
+intros.
+assert (H8 := H3 h H6).
+assert (H9 := H2 h H5).
+apply Rle_lt_trans with
+ (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (eps * Rsqr (f2 x) / (8 * l1))).
+rewrite Rabs_mult; apply Rmult_le_compat_l.
+apply Rabs_pos.
+rewrite <- (Rabs_Ropp (f2 x - f2 (x + h))); rewrite Ropp_minus_distr.
+left; apply H9.
+apply Rlt_le_trans with
+ (Rabs (2 * (l1 / (f2 x * f2 x))) * Rabs (eps * Rsqr (f2 x) / (8 * l1))).
+apply Rmult_lt_compat_r.
+apply Rabs_pos_lt.
+unfold Rdiv in |- *; unfold Rsqr in |- *; repeat apply prod_neq_R0;
+ try assumption || discrR.
+red in |- *; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H).
+apply Rinv_neq_0_compat; apply prod_neq_R0; try assumption || discrR.
+unfold Rdiv in |- *.
+repeat rewrite Rinv_mult_distr; try assumption.
+repeat rewrite Rabs_mult.
+replace (Rabs 2) with 2.
+rewrite (Rmult_comm 2).
+replace (Rabs l1 * (Rabs (/ f2 x) * Rabs (/ f2 x)) * 2) with
+ (Rabs l1 * (Rabs (/ f2 x) * (Rabs (/ f2 x) * 2))); 
+ [ idtac | ring ].
+repeat apply Rmult_lt_compat_l.
+apply Rabs_pos_lt; assumption.
+apply Rabs_pos_lt; apply Rinv_neq_0_compat; assumption.
+repeat rewrite Rabs_Rinv; try assumption.
+rewrite <- (Rmult_comm 2).
+unfold Rdiv in H8; exact H8.
+symmetry  in |- *; apply Rabs_right; left; prove_sup0.
+right.
+unfold Rsqr, Rdiv in |- *.
+do 1 rewrite Rinv_mult_distr; try assumption || discrR.
+do 1 rewrite Rinv_mult_distr; try assumption || discrR.
+repeat rewrite Rabs_mult.
+repeat rewrite Rabs_Rinv; try assumption || discrR.
+replace (Rabs eps) with eps.
+replace (Rabs 8) with 8.
+replace (Rabs 2) with 2.
+replace 8 with (4 * 2); [ idtac | ring ].
+rewrite Rinv_mult_distr; discrR.
+replace
+ (2 * (Rabs l1 * (/ Rabs (f2 x) * / Rabs (f2 x))) *
+  (eps * (Rabs (f2 x) * Rabs (f2 x)) * (/ 4 * / 2 * / Rabs l1))) with
+ (eps * / 4 * (Rabs l1 * / Rabs l1) * (Rabs (f2 x) * / Rabs (f2 x)) *
+  (Rabs (f2 x) * / Rabs (f2 x)) * (2 * / 2)); [ idtac | ring ].
+repeat rewrite <- Rinv_r_sym; try (apply Rabs_no_R0; assumption) || discrR.
+ring.
+symmetry  in |- *; apply Rabs_right; left; prove_sup0.
+symmetry  in |- *; apply Rabs_right; left; prove_sup.
+symmetry  in |- *; apply Rabs_right; left; assumption.
+Qed.
+
+Lemma maj_term3 :
+ forall (x h eps l2 alp_f2:R) (eps_f2 alp_f2d:posreal) 
+   (f1 f2:R -> R),
+   0 < eps ->
+   f2 x <> 0 ->
+   f2 (x + h) <> 0 ->
+   (forall h:R,
+      h <> 0 ->
+      Rabs h < alp_f2d ->
+      Rabs ((f2 (x + h) - f2 x) / h - l2) <
+      Rabs (Rsqr (f2 x) * eps / (8 * f1 x))) ->
+   (forall a:R,
+      Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
+   h <> 0 ->
+   Rabs h < alp_f2d ->
+   Rabs h < Rmin eps_f2 alp_f2 ->
+   f1 x <> 0 ->
+   Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) <
+   eps / 4.
+intros.
+assert (H8 := H2 h H4 H5).
+assert (H9 := H3 h H6).
+apply Rle_lt_trans with
+ (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs (Rsqr (f2 x) * eps / (8 * f1 x))).
+rewrite Rabs_mult.
+apply Rmult_le_compat_l.
+apply Rabs_pos.
+left; apply H8.
+apply Rlt_le_trans with
+ (Rabs (2 * (f1 x / (f2 x * f2 x))) * Rabs (Rsqr (f2 x) * eps / (8 * f1 x))).
+apply Rmult_lt_compat_r.
+apply Rabs_pos_lt.
+unfold Rdiv in |- *; unfold Rsqr in |- *; repeat apply prod_neq_R0;
+ try assumption.
+red in |- *; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H).
+apply Rinv_neq_0_compat; apply prod_neq_R0; discrR || assumption.
+unfold Rdiv in |- *.
+repeat rewrite Rinv_mult_distr; try assumption.
+repeat rewrite Rabs_mult.
+replace (Rabs 2) with 2.
+rewrite (Rmult_comm 2).
+replace (Rabs (f1 x) * (Rabs (/ f2 x) * Rabs (/ f2 x)) * 2) with
+ (Rabs (f1 x) * (Rabs (/ f2 x) * (Rabs (/ f2 x) * 2))); 
+ [ idtac | ring ].
+repeat apply Rmult_lt_compat_l.
+apply Rabs_pos_lt; assumption.
+apply Rabs_pos_lt; apply Rinv_neq_0_compat; assumption.
+repeat rewrite Rabs_Rinv; assumption || idtac.
+rewrite <- (Rmult_comm 2).
+unfold Rdiv in H9; exact H9.
+symmetry  in |- *; apply Rabs_right; left; prove_sup0.
+right.
+unfold Rsqr, Rdiv in |- *.
+rewrite Rinv_mult_distr; try assumption || discrR.
+rewrite Rinv_mult_distr; try assumption || discrR.
+repeat rewrite Rabs_mult.
+repeat rewrite Rabs_Rinv; try assumption || discrR.
+replace (Rabs eps) with eps.
+replace (Rabs 8) with 8.
+replace (Rabs 2) with 2.
+replace 8 with (4 * 2); [ idtac | ring ].
+rewrite Rinv_mult_distr; discrR.
+replace
+ (2 * (Rabs (f1 x) * (/ Rabs (f2 x) * / Rabs (f2 x))) *
+  (Rabs (f2 x) * Rabs (f2 x) * eps * (/ 4 * / 2 * / Rabs (f1 x)))) with
+ (eps * / 4 * (Rabs (f2 x) * / Rabs (f2 x)) * (Rabs (f2 x) * / Rabs (f2 x)) *
+  (Rabs (f1 x) * / Rabs (f1 x)) * (2 * / 2)); [ idtac | ring ].
+repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption).
+ring.
+symmetry  in |- *; apply Rabs_right; left; prove_sup0.
+symmetry  in |- *; apply Rabs_right; left; prove_sup.
+symmetry  in |- *; apply Rabs_right; left; assumption.
+Qed.
+
+Lemma maj_term4 :
+ forall (x h eps l2 alp_f2 alp_f2c:R) (eps_f2:posreal) 
+   (f1 f2:R -> R),
+   0 < eps ->
+   f2 x <> 0 ->
+   f2 (x + h) <> 0 ->
+   (forall a:R,
+      Rabs a < alp_f2c ->
+      Rabs (f2 (x + a) - f2 x) <
+      Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))) ->
+   (forall a:R,
+      Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
+   h <> 0 ->
+   Rabs h < alp_f2c ->
+   Rabs h < Rmin eps_f2 alp_f2 ->
+   f1 x <> 0 ->
+   l2 <> 0 ->
+   Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x)) <
+   eps / 4.
+intros.
+assert (H9 := H2 h H5).
+assert (H10 := H3 h H6).
+apply Rle_lt_trans with
+ (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) *
+  Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))).
+rewrite Rabs_mult.
+apply Rmult_le_compat_l.
+apply Rabs_pos.
+left; apply H9.
+apply Rlt_le_trans with
+ (Rabs (2 * l2 * (f1 x / (Rsqr (f2 x) * f2 x))) *
+  Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))).
+apply Rmult_lt_compat_r.
+apply Rabs_pos_lt.
+unfold Rdiv in |- *; unfold Rsqr in |- *; repeat apply prod_neq_R0;
+ assumption || idtac.
+red in |- *; intro H11; rewrite H11 in H; elim (Rlt_irrefl _ H).
+apply Rinv_neq_0_compat; apply prod_neq_R0.
+apply prod_neq_R0.
+discrR.
+assumption.
+assumption.
+unfold Rdiv in |- *.
+repeat rewrite Rinv_mult_distr;
+ try assumption || (unfold Rsqr in |- *; apply prod_neq_R0; assumption).
+repeat rewrite Rabs_mult.
+replace (Rabs 2) with 2.
+replace
+ (2 * Rabs l2 * (Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * Rabs (/ f2 x)))) with
+ (Rabs l2 * (Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * (Rabs (/ f2 x) * 2))));
+ [ idtac | ring ].
+replace
+ (Rabs l2 * Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * Rabs (/ f2 (x + h)))) with
+ (Rabs l2 * (Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * Rabs (/ f2 (x + h)))));
+ [ idtac | ring ].
+repeat apply Rmult_lt_compat_l.
+apply Rabs_pos_lt; assumption.
+apply Rabs_pos_lt; assumption.
+apply Rabs_pos_lt; apply Rinv_neq_0_compat; unfold Rsqr in |- *;
+ apply prod_neq_R0; assumption.
+repeat rewrite Rabs_Rinv; [ idtac | assumption | assumption ].
+rewrite <- (Rmult_comm 2).
+unfold Rdiv in H10; exact H10.
+symmetry  in |- *; apply Rabs_right; left; prove_sup0.
+right; unfold Rsqr, Rdiv in |- *.
+rewrite Rinv_mult_distr; try assumption || discrR.
+rewrite Rinv_mult_distr; try assumption || discrR.
+rewrite Rinv_mult_distr; try assumption || discrR.
+rewrite Rinv_mult_distr; try assumption || discrR.
+repeat rewrite Rabs_mult.
+repeat rewrite Rabs_Rinv; try assumption || discrR.
+replace (Rabs eps) with eps.
+replace (Rabs 8) with 8.
+replace (Rabs 2) with 2.
+replace 8 with (4 * 2); [ idtac | ring ].
+rewrite Rinv_mult_distr; discrR.
+replace
+ (2 * Rabs l2 *
+  (Rabs (f1 x) * (/ Rabs (f2 x) * / Rabs (f2 x) * / Rabs (f2 x))) *
+  (Rabs (f2 x) * Rabs (f2 x) * Rabs (f2 x) * eps *
+   (/ 4 * / 2 * / Rabs (f1 x) * / Rabs l2))) with
+ (eps * / 4 * (Rabs l2 * / Rabs l2) * (Rabs (f1 x) * / Rabs (f1 x)) *
+  (Rabs (f2 x) * / Rabs (f2 x)) * (Rabs (f2 x) * / Rabs (f2 x)) *
+  (Rabs (f2 x) * / Rabs (f2 x)) * (2 * / 2)); [ idtac | ring ].
+repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption).
+ring.
+symmetry  in |- *; apply Rabs_right; left; prove_sup0.
+symmetry  in |- *; apply Rabs_right; left; prove_sup.
+symmetry  in |- *; apply Rabs_right; left; assumption.
+apply prod_neq_R0; assumption || discrR.
+apply prod_neq_R0; assumption.
+Qed.
+
+Lemma D_x_no_cond : forall x a:R, a <> 0 -> D_x no_cond x (x + a).
+intros.
+unfold D_x, no_cond in |- *.
+split.
+trivial.
+apply Rminus_not_eq.
+unfold Rminus in |- *.
+rewrite Ropp_plus_distr.
+rewrite <- Rplus_assoc.
+rewrite Rplus_opp_r.
+rewrite Rplus_0_l.
+apply Ropp_neq_0_compat; assumption.
+Qed.
+
+Lemma Rabs_4 :
+ forall a b c d:R, Rabs (a + b + c + d) <= Rabs a + Rabs b + Rabs c + Rabs d.
+intros.
+apply Rle_trans with (Rabs (a + b) + Rabs (c + d)).
+replace (a + b + c + d) with (a + b + (c + d)); [ apply Rabs_triang | ring ].
+apply Rle_trans with (Rabs a + Rabs b + Rabs (c + d)).
+apply Rplus_le_compat_r.
+apply Rabs_triang.
+repeat rewrite Rplus_assoc; repeat apply Rplus_le_compat_l.
+apply Rabs_triang.
+Qed.
+
+Lemma Rlt_4 :
+ forall a b c d e f g h:R,
+   a < b -> c < d -> e < f -> g < h -> a + c + e + g < b + d + f + h.
+intros; apply Rlt_trans with (b + c + e + g).
+repeat apply Rplus_lt_compat_r; assumption.
+repeat rewrite Rplus_assoc; apply Rplus_lt_compat_l.
+apply Rlt_trans with (d + e + g).
+rewrite Rplus_assoc; apply Rplus_lt_compat_r; assumption.
+rewrite Rplus_assoc; apply Rplus_lt_compat_l; apply Rlt_trans with (f + g).
+apply Rplus_lt_compat_r; assumption.
+apply Rplus_lt_compat_l; assumption.
+Qed.
+
+Lemma Rmin_2 : forall a b c:R, a < b -> a < c -> a < Rmin b c.
+intros; unfold Rmin in |- *; case (Rle_dec b c); intro; assumption.
+Qed.
+
+Lemma quadruple : forall x:R, 4 * x = x + x + x + x.
+intro; ring.
+Qed.
+
+Lemma quadruple_var : forall x:R, x = x / 4 + x / 4 + x / 4 + x / 4.
+intro; rewrite <- quadruple.
+unfold Rdiv in |- *; rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m; discrR.
+reflexivity.
+Qed.
+
+(**********)
+Lemma continuous_neq_0 :
+ forall (f:R -> R) (x0:R),
+   continuity_pt f x0 ->
+   f x0 <> 0 ->
+    exists eps : posreal, (forall h:R, Rabs h < eps -> f (x0 + h) <> 0).
+intros; unfold continuity_pt in H; unfold continue_in in H;
+ unfold limit1_in in H; unfold limit_in in H; elim (H (Rabs (f x0 / 2))).
+intros; elim H1; intros.
+exists (mkposreal x H2).
+intros; assert (H5 := H3 (x0 + h)).
+cut
+ (dist R_met (x0 + h) x0 < x ->
+  dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)).
+unfold dist in |- *; simpl in |- *; unfold R_dist in |- *;
+ replace (x0 + h - x0) with h.
+intros; assert (H7 := H6 H4).
+red in |- *; intro.
+rewrite H8 in H7; unfold Rminus in H7; rewrite Rplus_0_l in H7;
+ rewrite Rabs_Ropp in H7; unfold Rdiv in H7; rewrite Rabs_mult in H7;
+ pattern (Rabs (f x0)) at 1 in H7; rewrite <- Rmult_1_r in H7.
+cut (0 < Rabs (f x0)).
+intro; assert (H10 := Rmult_lt_reg_l _ _ _ H9 H7).
+cut (Rabs (/ 2) = / 2).
+assert (Hyp : 0 < 2).
+prove_sup0.
+intro; rewrite H11 in H10; assert (H12 := Rmult_lt_compat_l 2 _ _ Hyp H10);
+ rewrite Rmult_1_r in H12; rewrite <- Rinv_r_sym in H12; 
+ [ idtac | discrR ].
+cut (IZR 1 < IZR 2).
+unfold IZR in |- *; unfold INR, nat_of_P in |- *; simpl in |- *; intro;
+ elim (Rlt_irrefl 1 (Rlt_trans _ _ _ H13 H12)).
+apply IZR_lt; omega.
+unfold Rabs in |- *; case (Rcase_abs (/ 2)); intro.
+assert (Hyp : 0 < 2).
+prove_sup0.
+assert (H11 := Rmult_lt_compat_l 2 _ _ Hyp r); rewrite Rmult_0_r in H11;
+ rewrite <- Rinv_r_sym in H11; [ idtac | discrR ].
+elim (Rlt_irrefl 0 (Rlt_trans _ _ _ Rlt_0_1 H11)).
+reflexivity.
+apply (Rabs_pos_lt _ H0).
+ring.
+assert (H6 := Req_dec x0 (x0 + h)); elim H6; intro.
+intro; rewrite <- H7; unfold dist, R_met in |- *; unfold R_dist in |- *;
+ unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
+ apply Rabs_pos_lt.
+unfold Rdiv in |- *; apply prod_neq_R0;
+ [ assumption | apply Rinv_neq_0_compat; discrR ].
+intro; apply H5.
+split.
+unfold D_x, no_cond in |- *.
+split; trivial || assumption.
+assumption.
+change (0 < Rabs (f x0 / 2)) in |- *.
+apply Rabs_pos_lt; unfold Rdiv in |- *; apply prod_neq_R0.
+assumption.
+apply Rinv_neq_0_compat; discrR.
+Qed.
\ No newline at end of file