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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2019 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(************************************************************************)
Require Import QArith.
Require Import Qabs.
Require Import ConstructiveReals.
Local Open Scope ConstructiveReals.
(** Properties of constructive absolute value (defined in
ConstructiveReals.CRabs).
Definition of minimum, maximum and their properties.
WARNING: this file is experimental and likely to change in future releases.
*)
#[global]
Instance CRabs_morph
: forall {R : ConstructiveReals},
CMorphisms.Proper
(CMorphisms.respectful (CReq R) (CReq R)) (CRabs R).
Proof.
intros R x y [H H0]. split.
- rewrite <- CRabs_def. split.
+ apply (CRle_trans _ x). apply H.
pose proof (CRabs_def R x (CRabs R x)) as [_ H1].
apply H1. apply CRle_refl.
+ apply (CRle_trans _ (CRopp R x)). intro abs.
apply CRopp_lt_cancel in abs. contradiction.
pose proof (CRabs_def R x (CRabs R x)) as [_ H1].
apply H1. apply CRle_refl.
- rewrite <- CRabs_def. split.
+ apply (CRle_trans _ y). apply H0.
pose proof (CRabs_def R y (CRabs R y)) as [_ H1].
apply H1. apply CRle_refl.
+ apply (CRle_trans _ (CRopp R y)). intro abs.
apply CRopp_lt_cancel in abs. contradiction.
pose proof (CRabs_def R y (CRabs R y)) as [_ H1].
apply H1. apply CRle_refl.
Qed.
Add Parametric Morphism {R : ConstructiveReals} : (CRabs R)
with signature CReq R ==> CReq R
as CRabs_morph_prop.
Proof.
intros. apply CRabs_morph, H.
Qed.
Lemma CRabs_right : forall {R : ConstructiveReals} (x : CRcarrier R),
0 <= x -> CRabs R x == x.
Proof.
intros. split.
- pose proof (CRabs_def R x (CRabs R x)) as [_ H1].
apply H1, CRle_refl.
- rewrite <- CRabs_def. split. apply CRle_refl.
apply (CRle_trans _ 0). 2: exact H.
apply (CRle_trans _ (CRopp R 0)).
intro abs. apply CRopp_lt_cancel in abs. contradiction.
apply (CRplus_le_reg_l 0).
apply (CRle_trans _ 0). apply CRplus_opp_r.
apply CRplus_0_r.
Qed.
Lemma CRabs_opp : forall {R : ConstructiveReals} (x : CRcarrier R),
CRabs R (- x) == CRabs R x.
Proof.
intros. split.
- rewrite <- CRabs_def. split.
+ pose proof (CRabs_def R (CRopp R x) (CRabs R (CRopp R x))) as [_ H1].
specialize (H1 (CRle_refl (CRabs R (CRopp R x)))) as [_ H1].
apply (CRle_trans _ (CRopp R (CRopp R x))).
2: exact H1. apply (CRopp_involutive x).
+ pose proof (CRabs_def R (CRopp R x) (CRabs R (CRopp R x))) as [_ H1].
apply H1, CRle_refl.
- rewrite <- CRabs_def. split.
+ pose proof (CRabs_def R x (CRabs R x)) as [_ H1].
apply H1, CRle_refl.
+ apply (CRle_trans _ x). apply CRopp_involutive.
pose proof (CRabs_def R x (CRabs R x)) as [_ H1].
apply H1, CRle_refl.
Qed.
Lemma CRabs_minus_sym : forall {R : ConstructiveReals} (x y : CRcarrier R),
CRabs R (x - y) == CRabs R (y - x).
Proof.
intros R x y. setoid_replace (x - y) with (-(y-x)).
rewrite CRabs_opp. reflexivity. unfold CRminus.
rewrite CRopp_plus_distr, CRplus_comm, CRopp_involutive.
reflexivity.
Qed.
Lemma CRabs_left : forall {R : ConstructiveReals} (x : CRcarrier R),
x <= 0 -> CRabs R x == - x.
Proof.
intros. rewrite <- CRabs_opp. apply CRabs_right.
rewrite <- CRopp_0. apply CRopp_ge_le_contravar, H.
Qed.
Lemma CRabs_triang : forall {R : ConstructiveReals} (x y : CRcarrier R),
CRabs R (x + y) <= CRabs R x + CRabs R y.
Proof.
intros. rewrite <- CRabs_def. split.
- apply (CRle_trans _ (CRplus R (CRabs R x) y)).
apply CRplus_le_compat_r.
pose proof (CRabs_def R x (CRabs R x)) as [_ H1].
apply H1, CRle_refl.
apply CRplus_le_compat_l.
pose proof (CRabs_def R y (CRabs R y)) as [_ H1].
apply H1, CRle_refl.
- apply (CRle_trans _ (CRplus R (CRopp R x) (CRopp R y))).
apply CRopp_plus_distr.
apply (CRle_trans _ (CRplus R (CRabs R x) (CRopp R y))).
apply CRplus_le_compat_r.
pose proof (CRabs_def R x (CRabs R x)) as [_ H1].
apply H1, CRle_refl.
apply CRplus_le_compat_l.
pose proof (CRabs_def R y (CRabs R y)) as [_ H1].
apply H1, CRle_refl.
Qed.
Lemma CRabs_le : forall {R : ConstructiveReals} (a b:CRcarrier R),
(-b <= a /\ a <= b) -> CRabs R a <= b.
Proof.
intros. pose proof (CRabs_def R a b) as [H0 _].
apply H0. split. apply H. destruct H.
rewrite <- (CRopp_involutive b).
apply CRopp_ge_le_contravar. exact H.
Qed.
Lemma CRabs_triang_inv : forall {R : ConstructiveReals} (x y : CRcarrier R),
CRabs R x - CRabs R y <= CRabs R (x - y).
Proof.
intros. apply (CRplus_le_reg_r (CRabs R y)).
unfold CRminus. rewrite CRplus_assoc, CRplus_opp_l.
rewrite CRplus_0_r.
apply (CRle_trans _ (CRabs R (x - y + y))).
setoid_replace (x - y + y) with x. apply CRle_refl.
unfold CRminus. rewrite CRplus_assoc, CRplus_opp_l.
rewrite CRplus_0_r. reflexivity.
apply CRabs_triang.
Qed.
Lemma CRabs_triang_inv2 : forall {R : ConstructiveReals} (x y : CRcarrier R),
CRabs R (CRabs R x - CRabs R y) <= CRabs R (x - y).
Proof.
intros. apply CRabs_le. split.
2: apply CRabs_triang_inv.
apply (CRplus_le_reg_r (CRabs R y)).
unfold CRminus. rewrite CRplus_assoc, CRplus_opp_l.
rewrite CRplus_0_r. fold (x - y).
rewrite CRplus_comm, CRabs_minus_sym.
apply (CRle_trans _ _ _ (CRabs_triang_inv y (y-x))).
setoid_replace (y - (y - x)) with x. apply CRle_refl.
unfold CRminus. rewrite CRopp_plus_distr, <- CRplus_assoc.
rewrite CRplus_opp_r, CRplus_0_l. apply CRopp_involutive.
Qed.
Lemma CR_of_Q_abs : forall {R : ConstructiveReals} (q : Q),
CRabs R (CR_of_Q R q) == CR_of_Q R (Qabs q).
Proof.
intros. destruct (Qlt_le_dec 0 q).
- apply (CReq_trans _ (CR_of_Q R q)).
apply CRabs_right. apply CR_of_Q_le. apply Qlt_le_weak, q0.
apply CR_of_Q_morph. symmetry. apply Qabs_pos, Qlt_le_weak, q0.
- apply (CReq_trans _ (CR_of_Q R (-q))).
apply (CReq_trans _ (CRabs R (CRopp R (CR_of_Q R q)))).
apply CReq_sym, CRabs_opp.
2: apply CR_of_Q_morph; symmetry; apply Qabs_neg, q0.
apply (CReq_trans _ (CRopp R (CR_of_Q R q))).
2: apply CReq_sym, CR_of_Q_opp.
apply CRabs_right.
apply (CRle_trans _ (CR_of_Q R (-q))). apply CR_of_Q_le.
apply (Qplus_le_l _ _ q). ring_simplify. exact q0.
apply CR_of_Q_opp.
Qed.
Lemma CRle_abs : forall {R : ConstructiveReals} (x : CRcarrier R),
x <= CRabs R x.
Proof.
intros. pose proof (CRabs_def R x (CRabs R x)) as [_ H].
apply H, CRle_refl.
Qed.
Lemma CRabs_pos : forall {R : ConstructiveReals} (x : CRcarrier R),
0 <= CRabs R x.
Proof.
intros. intro abs. destruct (CRltLinear R). clear p.
specialize (s _ x _ abs). destruct s.
exact (CRle_abs x c). rewrite CRabs_left in abs.
rewrite <- CRopp_0 in abs. apply CRopp_lt_cancel in abs.
exact (CRlt_asym _ _ abs c). apply CRlt_asym, c.
Qed.
Lemma CRabs_appart_0 : forall {R : ConstructiveReals} (x : CRcarrier R),
0 < CRabs R x -> x ≶ 0.
Proof.
intros. destruct (CRltLinear R). clear p.
pose proof (s _ x _ H) as [pos|neg].
right. exact pos. left.
destruct (CR_Q_dense R _ _ neg) as [q [H0 H1]].
destruct (Qlt_le_dec 0 q).
- destruct (s (CR_of_Q R (-q)) x 0).
apply CR_of_Q_lt.
apply (Qplus_lt_l _ _ q). ring_simplify. exact q0.
exfalso. pose proof (CRabs_def R x (CR_of_Q R q)) as [H2 _].
apply H2. clear H2. split. apply CRlt_asym, H0.
2: exact H1. rewrite <- Qopp_involutive, CR_of_Q_opp.
apply CRopp_ge_le_contravar, CRlt_asym, c. exact c.
- apply (CRlt_le_trans _ _ _ H0).
apply CR_of_Q_le. exact q0.
Qed.
(* The proof by cases on the signs of x and y applies constructively,
because of the positivity hypotheses. *)
Lemma CRabs_mult : forall {R : ConstructiveReals} (x y : CRcarrier R),
CRabs R (x * y) == CRabs R x * CRabs R y.
Proof.
intro R.
assert (forall (x y : CRcarrier R),
x ≶ 0
-> y ≶ 0
-> CRabs R (x * y) == CRabs R x * CRabs R y) as prep.
{ intros. destruct H, H0.
+ rewrite CRabs_right, CRabs_left, CRabs_left.
rewrite <- CRopp_mult_distr_l, CRopp_mult_distr_r, CRopp_involutive.
reflexivity.
apply CRlt_asym, c0. apply CRlt_asym, c.
setoid_replace (x*y) with (- x * - y).
apply CRlt_asym, CRmult_lt_0_compat.
rewrite <- CRopp_0. apply CRopp_gt_lt_contravar, c.
rewrite <- CRopp_0. apply CRopp_gt_lt_contravar, c0.
rewrite <- CRopp_mult_distr_l, CRopp_mult_distr_r, CRopp_involutive.
reflexivity.
+ rewrite CRabs_left, CRabs_left, CRabs_right.
rewrite <- CRopp_mult_distr_l. reflexivity.
apply CRlt_asym, c0. apply CRlt_asym, c.
rewrite <- (CRmult_0_l y).
apply CRmult_le_compat_r_half. exact c0.
apply CRlt_asym, c.
+ rewrite CRabs_left, CRabs_right, CRabs_left.
rewrite <- CRopp_mult_distr_r. reflexivity.
apply CRlt_asym, c0. apply CRlt_asym, c.
rewrite <- (CRmult_0_r x).
apply CRmult_le_compat_l_half.
exact c. apply CRlt_asym, c0.
+ rewrite CRabs_right, CRabs_right, CRabs_right. reflexivity.
apply CRlt_asym, c0. apply CRlt_asym, c.
apply CRlt_asym, CRmult_lt_0_compat; assumption. }
split.
- intro abs.
assert (0 < CRabs R x * CRabs R y).
{ apply (CRle_lt_trans _ (CRabs R (x*y))).
apply CRabs_pos. exact abs. }
pose proof (CRmult_pos_appart_zero _ _ H).
rewrite CRmult_comm in H.
apply CRmult_pos_appart_zero in H.
destruct H. 2: apply (CRabs_pos y c).
destruct H0. 2: apply (CRabs_pos x c0).
apply CRabs_appart_0 in c.
apply CRabs_appart_0 in c0.
rewrite (prep x y) in abs.
exact (CRlt_asym _ _ abs abs). exact c0. exact c.
- intro abs.
assert (0 < CRabs R (x * y)).
{ apply (CRle_lt_trans _ (CRabs R x * CRabs R y)).
rewrite <- (CRmult_0_l (CRabs R y)).
apply CRmult_le_compat_r.
apply CRabs_pos. apply CRabs_pos. exact abs. }
apply CRabs_appart_0 in H. destruct H.
+ apply CRopp_gt_lt_contravar in c.
rewrite CRopp_0, CRopp_mult_distr_l in c.
pose proof (CRmult_pos_appart_zero _ _ c).
rewrite CRmult_comm in c.
apply CRmult_pos_appart_zero in c.
rewrite (prep x y) in abs.
exact (CRlt_asym _ _ abs abs).
destruct H. left. apply CRopp_gt_lt_contravar in c0.
rewrite CRopp_involutive, CRopp_0 in c0. exact c0.
right. apply CRopp_gt_lt_contravar in c0.
rewrite CRopp_involutive, CRopp_0 in c0. exact c0.
destruct c. right. exact c. left. exact c.
+ pose proof (CRmult_pos_appart_zero _ _ c).
rewrite CRmult_comm in c.
apply CRmult_pos_appart_zero in c.
rewrite (prep x y) in abs.
exact (CRlt_asym _ _ abs abs).
destruct H. right. exact c0. left. exact c0.
destruct c. right. exact c. left. exact c.
Qed.
Lemma CRabs_lt : forall {R : ConstructiveReals} (x y : CRcarrier R),
CRabs _ x < y -> prod (x < y) (-x < y).
Proof.
split.
- apply (CRle_lt_trans _ _ _ (CRle_abs x)), H.
- apply (CRle_lt_trans _ _ _ (CRle_abs (-x))).
rewrite CRabs_opp. exact H.
Qed.
Lemma CRabs_def1 : forall {R : ConstructiveReals} (x y : CRcarrier R),
x < y -> -x < y -> CRabs _ x < y.
Proof.
intros. destruct (CRltLinear R), p.
destruct (s x (CRabs R x) y H). 2: exact c0.
rewrite CRabs_left. exact H0. intro abs.
rewrite CRabs_right in c0. exact (CRlt_asym x x c0 c0).
apply CRlt_asym, abs.
Qed.
Lemma CRabs_def2 : forall {R : ConstructiveReals} (x a:CRcarrier R),
CRabs _ x <= a -> (x <= a) /\ (- a <= x).
Proof.
split.
- exact (CRle_trans _ _ _ (CRle_abs _) H).
- rewrite <- (CRopp_involutive x).
apply CRopp_ge_le_contravar.
rewrite <- CRabs_opp in H.
exact (CRle_trans _ _ _ (CRle_abs _) H).
Qed.
|
