# Library Coq.Reals.Rbasic_fun

``` ```
 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. 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.  ```
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