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(*         *      The Rocq Prover / The Rocq 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)         *)
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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).
Add Parametric Morphism {R : ConstructiveReals} : (CRabs R)
with signature CReq R ==> CReq R
as CRabs_morph_prop.

(* Specifications: *)
Lemma CRabs_right : forall {R : ConstructiveReals} (x : CRcarrier R),
0 <= x -> CRabs R x == x.
Lemma CRabs_opp : forall {R : ConstructiveReals} (x : CRcarrier R),
CRabs R (- x) == CRabs R x.
Lemma CRabs_minus_sym : forall {R : ConstructiveReals} (x y : CRcarrier R),
CRabs R (x - y) == CRabs R (y - x).
Lemma CRabs_left : forall {R : ConstructiveReals} (x : CRcarrier R),
x <= 0 -> CRabs R x == - x.
Lemma CRabs_triang : forall {R : ConstructiveReals} (x y : CRcarrier R),
CRabs R (x + y) <= CRabs R x + CRabs R y.
Lemma CRabs_le : forall {R : ConstructiveReals} (a b:CRcarrier R),
(-b <= a /\ a <= b) -> CRabs R a <= b.
Lemma CRabs_triang_inv : forall {R : ConstructiveReals} (x y : CRcarrier R),
CRabs R x - CRabs R y <= CRabs R (x - y).
Lemma CRabs_triang_inv2 : forall {R : ConstructiveReals} (x y : CRcarrier R),
CRabs R (CRabs R x - CRabs R y) <= CRabs R (x - y).
Lemma CR_of_Q_abs : forall {R : ConstructiveReals} (q : Q),
CRabs R (CR_of_Q R q) == CR_of_Q R (Qabs q).
Lemma CRle_abs : forall {R : ConstructiveReals} (x : CRcarrier R),
x <= CRabs R x.
Lemma CRabs_pos : forall {R : ConstructiveReals} (x : CRcarrier R),
0 <= CRabs R x.
Lemma CRabs_appart_0 : forall {R : ConstructiveReals} (x : CRcarrier R),
0 < CRabs R x -> x ≶ 0.
(* 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.
Lemma CRabs_lt : forall {R : ConstructiveReals} (x y : CRcarrier R),
CRabs _ x < y -> prod (x < y) (-x < y).
Lemma CRabs_def1 : forall {R : ConstructiveReals} (x y : CRcarrier R),
x < y -> -x < y -> CRabs _ x < y.
Lemma CRabs_def2 : forall {R : ConstructiveReals} (x a:CRcarrier R),
CRabs _ x <= a -> (x <= a) /\ (- a <= x).