Imported Upstream version 8.0pl1upstream/8.0pl1

1 files changed, 470 insertions, 0 deletions

diff --git a/theories/Reals/Rbasic_fun.v b/theories/Reals/Rbasic_fun.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: Rbasic_fun.v,v 1.22.2.1 2004/07/16 19:31:12 herbelin Exp $ i*)
+
+(*********************************************************)
+(**          Complements for the real numbers            *)
+(*                                                       *)
+(*********************************************************)
+
+Require Import Rbase.
+Require Import R_Ifp.
+Require Import Fourier. Open Local Scope R_scope.
+
+Implicit Type r : R.
+
+(*******************************)
+(**          Rmin              *)
+(*******************************)
+
+(*********)
+Definition Rmin (x y:R) : R :=
+  match Rle_dec x y with
+  | left _ => x
+  | right _ => y
+  end.
+
+(*********)
+Lemma Rmin_Rgt_l : forall r1 r2 r, Rmin r1 r2 > r -> r1 > r /\ r2 > r.
+intros r1 r2 r; unfold Rmin in |- *; case (Rle_dec r1 r2); intros.
+split.
+assumption.
+unfold Rgt in |- *; unfold Rgt in H; exact (Rlt_le_trans r r1 r2 H r0).
+split.
+generalize (Rnot_le_lt r1 r2 n); intro; exact (Rgt_trans r1 r2 r H0 H).
+assumption.
+Qed.
+
+(*********)
+Lemma Rmin_Rgt_r : forall r1 r2 r, r1 > r /\ r2 > r -> Rmin r1 r2 > r.
+intros; unfold Rmin in |- *; case (Rle_dec r1 r2); elim H; clear H; intros;
+ assumption.
+Qed.
+
+(*********)
+Lemma Rmin_Rgt : forall r1 r2 r, Rmin r1 r2 > r <-> r1 > r /\ r2 > r.
+intros; split.
+exact (Rmin_Rgt_l r1 r2 r).
+exact (Rmin_Rgt_r r1 r2 r).
+Qed.
+
+(*********)
+Lemma Rmin_l : forall x y:R, Rmin x y <= x.
+intros; unfold Rmin in |- *; case (Rle_dec x y); intro H1;
+ [ right; reflexivity | auto with real ].
+Qed.
+ 
+(*********)
+Lemma Rmin_r : forall x y:R, Rmin x y <= y.
+intros; unfold Rmin in |- *; case (Rle_dec x y); intro H1;
+ [ assumption | auto with real ].
+Qed.
+
+(*********)
+Lemma Rmin_comm : forall a b:R, Rmin a b = Rmin b a.
+intros; unfold Rmin in |- *; case (Rle_dec a b); case (Rle_dec b a); intros;
+ try reflexivity || (apply Rle_antisym; assumption || auto with real).
+Qed.
+
+(*********)
+Lemma Rmin_stable_in_posreal : forall x y:posreal, 0 < Rmin x y.
+intros; apply Rmin_Rgt_r; split; [ apply (cond_pos x) | apply (cond_pos y) ].
+Qed.
+
+(*******************************)
+(**          Rmax              *)
+(*******************************)
+
+(*********)
+Definition Rmax (x y:R) : R :=
+  match Rle_dec x y with
+  | left _ => y
+  | right _ => x
+  end.
+
+(*********)
+Lemma Rmax_Rle : forall r1 r2 r, r <= Rmax r1 r2 <-> r <= r1 \/ r <= r2.
+intros; split.
+unfold Rmax in |- *; case (Rle_dec r1 r2); intros; auto.
+intro; unfold Rmax in |- *; case (Rle_dec r1 r2); elim H; clear H; intros;
+ auto.
+apply (Rle_trans r r1 r2); auto.
+generalize (Rnot_le_lt r1 r2 n); clear n; intro; unfold Rgt in H0;
+ apply (Rlt_le r r1 (Rle_lt_trans r r2 r1 H H0)).
+Qed.
+
+Lemma RmaxLess1 : forall r1 r2, r1 <= Rmax r1 r2.
+intros r1 r2; unfold Rmax in |- *; case (Rle_dec r1 r2); auto with real.
+Qed.
+ 
+Lemma RmaxLess2 : forall r1 r2, r2 <= Rmax r1 r2.
+intros r1 r2; unfold Rmax in |- *; case (Rle_dec r1 r2); auto with real.
+Qed.
+ 
+Lemma RmaxSym : forall p q:R, Rmax p q = Rmax q p.
+intros p q; unfold Rmax in |- *; case (Rle_dec p q); case (Rle_dec q p); auto;
+ intros H1 H2; apply Rle_antisym; auto with real.
+Qed.
+
+Lemma RmaxRmult :
+ forall (p q:R) r, 0 <= r -> Rmax (r * p) (r * q) = r * Rmax p q.
+intros p q r H; unfold Rmax in |- *.
+case (Rle_dec p q); case (Rle_dec (r * p) (r * q)); auto; intros H1 H2; auto.
+case H; intros E1.
+case H1; auto with real.
+rewrite <- E1; repeat rewrite Rmult_0_l; auto.
+case H; intros E1.
+case H2; auto with real.
+apply Rmult_le_reg_l with (r := r); auto.
+rewrite <- E1; repeat rewrite Rmult_0_l; auto.
+Qed.
+
+Lemma Rmax_stable_in_negreal : forall x y:negreal, Rmax x y < 0.
+intros; unfold Rmax in |- *; case (Rle_dec x y); intro;
+ [ apply (cond_neg y) | apply (cond_neg x) ].
+Qed.
+
+(*******************************)
+(**          Rabsolu           *)
+(*******************************)
+
+(*********)
+Lemma Rcase_abs : forall r, {r < 0} + {r >= 0}. 
+intro; generalize (Rle_dec 0 r); intro X; elim X; intro; clear X.
+right; apply (Rle_ge 0 r a).
+left; fold (0 > r) in |- *; apply (Rnot_le_lt 0 r b).
+Qed.
+
+(*********)
+Definition Rabs r : R :=
+  match Rcase_abs r with
+  | left _ => - r
+  | right _ => r
+  end.
+
+(*********)
+Lemma Rabs_R0 : Rabs 0 = 0.
+unfold Rabs in |- *; case (Rcase_abs 0); auto; intro.
+generalize (Rlt_irrefl 0); intro; elimtype False; auto.
+Qed.
+
+Lemma Rabs_R1 : Rabs 1 = 1.
+unfold Rabs in |- *; case (Rcase_abs 1); auto with real.
+intros H; absurd (1 < 0); auto with real.
+Qed.
+
+(*********)
+Lemma Rabs_no_R0 : forall r, r <> 0 -> Rabs r <> 0.
+intros; unfold Rabs in |- *; case (Rcase_abs r); intro; auto.
+apply Ropp_neq_0_compat; auto.
+Qed.
+
+(*********)
+Lemma Rabs_left : forall r, r < 0 -> Rabs r = - r.
+intros; unfold Rabs in |- *; case (Rcase_abs r); trivial; intro;
+ absurd (r >= 0).
+exact (Rlt_not_ge r 0 H).
+assumption.
+Qed.
+
+(*********)
+Lemma Rabs_right : forall r, r >= 0 -> Rabs r = r.
+intros; unfold Rabs in |- *; case (Rcase_abs r); intro.
+absurd (r >= 0).
+exact (Rlt_not_ge r 0 r0).
+assumption.
+trivial.
+Qed.
+
+Lemma Rabs_left1 : forall a:R, a <= 0 -> Rabs a = - a.
+intros a H; case H; intros H1.
+apply Rabs_left; auto.
+rewrite H1; simpl in |- *; rewrite Rabs_right; auto with real.
+Qed.
+
+(*********)
+Lemma Rabs_pos : forall x:R, 0 <= Rabs x.
+intros; unfold Rabs in |- *; case (Rcase_abs x); intro.
+generalize (Ropp_lt_gt_contravar x 0 r); intro; unfold Rgt in H;
+ rewrite Ropp_0 in H; unfold Rle in |- *; left; assumption.
+apply Rge_le; assumption.
+Qed.
+
+Lemma RRle_abs : forall x:R, x <= Rabs x.
+intro; unfold Rabs in |- *; case (Rcase_abs x); intros; fourier.
+Qed.
+
+(*********)
+Lemma Rabs_pos_eq : forall x:R, 0 <= x -> Rabs x = x.
+intros; unfold Rabs in |- *; case (Rcase_abs x); intro;
+ [ generalize (Rgt_not_le 0 x r); intro; elimtype False; auto | trivial ].
+Qed.
+
+(*********)
+Lemma Rabs_Rabsolu : forall x:R, Rabs (Rabs x) = Rabs x.
+intro; apply (Rabs_pos_eq (Rabs x) (Rabs_pos x)).
+Qed.
+
+(*********)
+Lemma Rabs_pos_lt : forall x:R, x <> 0 -> 0 < Rabs x.
+intros; generalize (Rabs_pos x); intro; unfold Rle in H0; elim H0; intro;
+ auto.
+elimtype False; clear H0; elim H; clear H; generalize H1; unfold Rabs in |- *;
+ case (Rcase_abs x); intros; auto.
+clear r H1; generalize (Rplus_eq_compat_l x 0 (- x) H0);
+ rewrite (let (H1, H2) := Rplus_ne x in H1); rewrite (Rplus_opp_r x); 
+ trivial.
+Qed.
+
+(*********)
+Lemma Rabs_minus_sym : forall x y:R, Rabs (x - y) = Rabs (y - x).
+intros; unfold Rabs in |- *; case (Rcase_abs (x - y));
+ case (Rcase_abs (y - x)); intros.
+ generalize (Rminus_lt y x r); generalize (Rminus_lt x y r0); intros;
+  generalize (Rlt_asym x y H); intro; elimtype False; 
+  auto.
+rewrite (Ropp_minus_distr x y); trivial.
+rewrite (Ropp_minus_distr y x); trivial.
+unfold Rge in r, r0; elim r; elim r0; intros; clear r r0.
+generalize (Ropp_lt_gt_0_contravar (x - y) H); rewrite (Ropp_minus_distr x y);
+ intro; unfold Rgt in H0; generalize (Rlt_asym 0 (y - x) H0); 
+ intro; elimtype False; auto.
+rewrite (Rminus_diag_uniq x y H); trivial.
+rewrite (Rminus_diag_uniq y x H0); trivial.
+rewrite (Rminus_diag_uniq y x H0); trivial.
+Qed.
+
+(*********)
+Lemma Rabs_mult : forall x y:R, Rabs (x * y) = Rabs x * Rabs y.
+intros; unfold Rabs in |- *; case (Rcase_abs (x * y)); case (Rcase_abs x);
+ case (Rcase_abs y); intros; auto.
+generalize (Rmult_lt_gt_compat_neg_l y x 0 r r0); intro;
+ rewrite (Rmult_0_r y) in H; generalize (Rlt_asym (x * y) 0 r1); 
+ intro; unfold Rgt in H; elimtype False; rewrite (Rmult_comm y x) in H; 
+ auto.
+rewrite (Ropp_mult_distr_l_reverse x y); trivial. 
+rewrite (Rmult_comm x (- y)); rewrite (Ropp_mult_distr_l_reverse y x);
+ rewrite (Rmult_comm x y); trivial.
+unfold Rge in r, r0; elim r; elim r0; clear r r0; intros; unfold Rgt in H, H0.
+generalize (Rmult_lt_compat_l x 0 y H H0); intro; rewrite (Rmult_0_r x) in H1;
+ generalize (Rlt_asym (x * y) 0 r1); intro; elimtype False; 
+ auto.
+rewrite H in r1; rewrite (Rmult_0_l y) in r1; generalize (Rlt_irrefl 0);
+ intro; elimtype False; auto.
+rewrite H0 in r1; rewrite (Rmult_0_r x) in r1; generalize (Rlt_irrefl 0);
+ intro; elimtype False; auto.
+rewrite H0 in r1; rewrite (Rmult_0_r x) in r1; generalize (Rlt_irrefl 0);
+ intro; elimtype False; auto.
+rewrite (Rmult_opp_opp x y); trivial.
+unfold Rge in r, r1; elim r; elim r1; clear r r1; intros; unfold Rgt in H0, H.
+generalize (Rmult_lt_compat_l y x 0 H0 r0); intro;
+ rewrite (Rmult_0_r y) in H1; rewrite (Rmult_comm y x) in H1;
+ generalize (Rlt_asym (x * y) 0 H1); intro; elimtype False; 
+ auto.
+generalize (Rlt_dichotomy_converse x 0 (or_introl (x > 0) r0));
+ generalize (Rlt_dichotomy_converse y 0 (or_intror (y < 0) H0)); 
+ intros; generalize (Rmult_integral x y H); intro; 
+ elim H3; intro; elimtype False; auto.  
+rewrite H0 in H; rewrite (Rmult_0_r x) in H; unfold Rgt in H;
+ generalize (Rlt_irrefl 0); intro; elimtype False; 
+ auto.
+rewrite H0; rewrite (Rmult_0_r x); rewrite (Rmult_0_r (- x)); trivial.
+unfold Rge in r0, r1; elim r0; elim r1; clear r0 r1; intros;
+ unfold Rgt in H0, H.
+generalize (Rmult_lt_compat_l x y 0 H0 r); intro; rewrite (Rmult_0_r x) in H1;
+ generalize (Rlt_asym (x * y) 0 H1); intro; elimtype False; 
+ auto.
+generalize (Rlt_dichotomy_converse y 0 (or_introl (y > 0) r));
+ generalize (Rlt_dichotomy_converse 0 x (or_introl (0 > x) H0)); 
+ intros; generalize (Rmult_integral x y H); intro; 
+ elim H3; intro; elimtype False; auto.  
+rewrite H0 in H; rewrite (Rmult_0_l y) in H; unfold Rgt in H;
+ generalize (Rlt_irrefl 0); intro; elimtype False; 
+ auto.
+rewrite H0; rewrite (Rmult_0_l y); rewrite (Rmult_0_l (- y)); trivial.
+Qed.
+
+(*********)
+Lemma Rabs_Rinv : forall r, r <> 0 -> Rabs (/ r) = / Rabs r.
+intro; unfold Rabs in |- *; case (Rcase_abs r); case (Rcase_abs (/ r)); auto;
+ intros.
+apply Ropp_inv_permute; auto.
+generalize (Rinv_lt_0_compat r r1); intro; unfold Rge in r0; elim r0; intros.
+unfold Rgt in H1; generalize (Rlt_asym 0 (/ r) H1); intro; elimtype False;
+ auto.
+generalize (Rlt_dichotomy_converse (/ r) 0 (or_introl (/ r > 0) H0)); intro;
+ elimtype False; auto.
+unfold Rge in r1; elim r1; clear r1; intro.
+unfold Rgt in H0; generalize (Rlt_asym 0 (/ r) (Rinv_0_lt_compat r H0));
+ intro; elimtype False; auto.
+elimtype False; auto.
+Qed. 
+
+Lemma Rabs_Ropp : forall x:R, Rabs (- x) = Rabs x.
+intro; cut (- x = -1 * x).
+intros; rewrite H.
+rewrite Rabs_mult.
+cut (Rabs (-1) = 1).
+intros; rewrite H0.
+ring.
+unfold Rabs in |- *; case (Rcase_abs (-1)).
+intro; ring.
+intro H0; generalize (Rge_le (-1) 0 H0); intros.
+generalize (Ropp_le_ge_contravar 0 (-1) H1).
+rewrite Ropp_involutive; rewrite Ropp_0.
+intro; generalize (Rgt_not_le 1 0 Rlt_0_1); intro; generalize (Rge_le 0 1 H2);
+ intro; elimtype False; auto.  
+ring.
+Qed.
+
+(*********)
+Lemma Rabs_triang : forall a b:R, Rabs (a + b) <= Rabs a + Rabs b.
+intros a b; unfold Rabs in |- *; case (Rcase_abs (a + b)); case (Rcase_abs a);
+ case (Rcase_abs b); intros.
+apply (Req_le (- (a + b)) (- a + - b)); rewrite (Ropp_plus_distr a b);
+ reflexivity.
+(**)
+rewrite (Ropp_plus_distr a b); apply (Rplus_le_compat_l (- a) (- b) b);
+ unfold Rle in |- *; unfold Rge in r; elim r; intro.
+left; unfold Rgt in H; generalize (Rplus_lt_compat_l (- b) 0 b H); intro;
+ elim (Rplus_ne (- b)); intros v w; rewrite v in H0; 
+ clear v w; rewrite (Rplus_opp_l b) in H0; apply (Rlt_trans (- b) 0 b H0 H).
+right; rewrite H; apply Ropp_0.
+(**)
+rewrite (Ropp_plus_distr a b); rewrite (Rplus_comm (- a) (- b));
+ rewrite (Rplus_comm a (- b)); apply (Rplus_le_compat_l (- b) (- a) a);
+ unfold Rle in |- *; unfold Rge in r0; elim r0; intro.
+left; unfold Rgt in H; generalize (Rplus_lt_compat_l (- a) 0 a H); intro;
+ elim (Rplus_ne (- a)); intros v w; rewrite v in H0; 
+ clear v w; rewrite (Rplus_opp_l a) in H0; apply (Rlt_trans (- a) 0 a H0 H).
+right; rewrite H; apply Ropp_0.
+(**)
+elimtype False; generalize (Rplus_ge_compat_l a b 0 r); intro;
+ elim (Rplus_ne a); intros v w; rewrite v in H; clear v w;
+ generalize (Rge_trans (a + b) a 0 H r0); intro; clear H; 
+ unfold Rge in H0; elim H0; intro; clear H0.
+unfold Rgt in H; generalize (Rlt_asym (a + b) 0 r1); intro; auto.
+absurd (a + b = 0); auto.
+apply (Rlt_dichotomy_converse (a + b) 0); left; assumption.
+(**)
+elimtype False; generalize (Rplus_lt_compat_l a b 0 r); intro;
+ elim (Rplus_ne a); intros v w; rewrite v in H; clear v w;
+ generalize (Rlt_trans (a + b) a 0 H r0); intro; clear H; 
+ unfold Rge in r1; elim r1; clear r1; intro.
+unfold Rgt in H; generalize (Rlt_trans (a + b) 0 (a + b) H0 H); intro;
+ apply (Rlt_irrefl (a + b)); assumption.
+rewrite H in H0; apply (Rlt_irrefl 0); assumption.
+(**)
+rewrite (Rplus_comm a b); rewrite (Rplus_comm (- a) b);
+ apply (Rplus_le_compat_l b a (- a)); apply (Rminus_le a (- a));
+ unfold Rminus in |- *; rewrite (Ropp_involutive a);
+ generalize (Rplus_lt_compat_l a a 0 r0); clear r r1; 
+ intro; elim (Rplus_ne a); intros v w; rewrite v in H; 
+ clear v w; generalize (Rlt_trans (a + a) a 0 H r0); 
+ intro; apply (Rlt_le (a + a) 0 H0).
+(**)
+apply (Rplus_le_compat_l a b (- b)); apply (Rminus_le b (- b));
+ unfold Rminus in |- *; rewrite (Ropp_involutive b);
+ generalize (Rplus_lt_compat_l b b 0 r); clear r0 r1; 
+ intro; elim (Rplus_ne b); intros v w; rewrite v in H; 
+ clear v w; generalize (Rlt_trans (b + b) b 0 H r); 
+ intro; apply (Rlt_le (b + b) 0 H0).
+(**)
+unfold Rle in |- *; right; reflexivity.
+Qed.
+
+(*********)
+Lemma Rabs_triang_inv : forall a b:R, Rabs a - Rabs b <= Rabs (a - b).
+intros; apply (Rplus_le_reg_l (Rabs b) (Rabs a - Rabs b) (Rabs (a - b)));
+ unfold Rminus in |- *; rewrite <- (Rplus_assoc (Rabs b) (Rabs a) (- Rabs b));
+ rewrite (Rplus_comm (Rabs b) (Rabs a));
+ rewrite (Rplus_assoc (Rabs a) (Rabs b) (- Rabs b));
+ rewrite (Rplus_opp_r (Rabs b)); rewrite (proj1 (Rplus_ne (Rabs a)));
+ replace (Rabs a) with (Rabs (a + 0)).
+ rewrite <- (Rplus_opp_r b); rewrite <- (Rplus_assoc a b (- b));
+  rewrite (Rplus_comm a b); rewrite (Rplus_assoc b a (- b)).
+ exact (Rabs_triang b (a + - b)).
+ rewrite (proj1 (Rplus_ne a)); trivial.
+Qed.
+
+(* ||a|-|b||<=|a-b| *)
+Lemma Rabs_triang_inv2 : forall a b:R, Rabs (Rabs a - Rabs b) <= Rabs (a - b). 
+cut
+ (forall a b:R, Rabs b <= Rabs a -> Rabs (Rabs a - Rabs b) <= Rabs (a - b)). 
+intros; destruct (Rtotal_order (Rabs a) (Rabs b)) as [Hlt| [Heq| Hgt]].
+rewrite <- (Rabs_Ropp (Rabs a - Rabs b)); rewrite <- (Rabs_Ropp (a - b));
+ do 2 rewrite Ropp_minus_distr. 
+apply H; left; assumption. 
+rewrite Heq; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
+ apply Rabs_pos. 
+apply H; left; assumption. 
+intros; replace (Rabs (Rabs a - Rabs b)) with (Rabs a - Rabs b). 
+apply Rabs_triang_inv. 
+rewrite (Rabs_right (Rabs a - Rabs b));
+ [ reflexivity
+ | apply Rle_ge; apply Rplus_le_reg_l with (Rabs b); rewrite Rplus_0_r;
+    replace (Rabs b + (Rabs a - Rabs b)) with (Rabs a); 
+    [ assumption | ring ] ]. 
+Qed. 
+
+(*********)
+Lemma Rabs_def1 : forall x a:R, x < a -> - a < x -> Rabs x < a.
+unfold Rabs in |- *; intros; case (Rcase_abs x); intro.
+generalize (Ropp_lt_gt_contravar (- a) x H0); unfold Rgt in |- *;
+ rewrite Ropp_involutive; intro; assumption.
+assumption.
+Qed.
+
+(*********)
+Lemma Rabs_def2 : forall x a:R, Rabs x < a -> x < a /\ - a < x.
+unfold Rabs in |- *; intro x; case (Rcase_abs x); intros.
+generalize (Ropp_gt_lt_0_contravar x r); unfold Rgt in |- *; intro;
+ generalize (Rlt_trans 0 (- x) a H0 H); intro; split. 
+apply (Rlt_trans x 0 a r H1).
+generalize (Ropp_lt_gt_contravar (- x) a H); rewrite (Ropp_involutive x);
+ unfold Rgt in |- *; trivial.
+fold (a > x) in H; generalize (Rgt_ge_trans a x 0 H r); intro;
+ generalize (Ropp_lt_gt_0_contravar a H0); intro; fold (0 > - a) in |- *;
+ generalize (Rge_gt_trans x 0 (- a) r H1); unfold Rgt in |- *; 
+ intro; split; assumption.
+Qed.
+
+Lemma RmaxAbs :
+ forall (p q:R) r, p <= q -> q <= r -> Rabs q <= Rmax (Rabs p) (Rabs r).
+intros p q r H' H'0; case (Rle_or_lt 0 p); intros H'1.
+repeat rewrite Rabs_right; auto with real.
+apply Rle_trans with r; auto with real.
+apply RmaxLess2; auto.
+apply Rge_trans with p; auto with real; apply Rge_trans with q;
+ auto with real.
+apply Rge_trans with p; auto with real.
+rewrite (Rabs_left p); auto.
+case (Rle_or_lt 0 q); intros H'2.
+repeat rewrite Rabs_right; auto with real.
+apply Rle_trans with r; auto.
+apply RmaxLess2; auto.
+apply Rge_trans with q; auto with real.
+rewrite (Rabs_left q); auto.
+case (Rle_or_lt 0 r); intros H'3.
+repeat rewrite Rabs_right; auto with real.
+apply Rle_trans with (- p); auto with real.
+apply RmaxLess1; auto.
+rewrite (Rabs_left r); auto.
+apply Rle_trans with (- p); auto with real.
+apply RmaxLess1; auto.
+Qed.
+
+Lemma Rabs_Zabs : forall z:Z, Rabs (IZR z) = IZR (Zabs z).
+intros z; case z; simpl in |- *; auto with real.
+apply Rabs_right; auto with real.
+intros p0; apply Rabs_right; auto with real zarith.
+intros p0; rewrite Rabs_Ropp.
+apply Rabs_right; auto with real zarith.
+Qed.
+ 
\ No newline at end of file