path: root/theories/Reals/Rbasic_fun.v

(************************************************************************)
(*  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.