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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. *)
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.
(* Minimum *)
Definition CRmin {R : ConstructiveReals} (x y : CRcarrier R) : CRcarrier R
:= (x + y - CRabs _ (y - x)) * CR_of_Q _ (1#2).
Lemma CRmin_lt_r : forall {R : ConstructiveReals} (x y : CRcarrier R),
CRmin x y < y -> CRmin x y == x.
Proof.
intros. unfold CRmin. unfold CRmin in H.
apply (CRmult_eq_reg_r (CR_of_Q R 2)).
left; apply CR_of_Q_pos; reflexivity.
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity.
rewrite CRmult_1_r.
rewrite (CR_of_Q_plus R 1 1), CRmult_plus_distr_l, CRmult_1_r.
rewrite CRabs_right. unfold CRminus.
rewrite CRopp_plus_distr, CRplus_assoc, <- (CRplus_assoc y).
rewrite CRplus_opp_r, CRplus_0_l, CRopp_involutive. reflexivity.
apply (CRmult_lt_compat_r (CR_of_Q R 2)) in H.
2: apply CR_of_Q_pos; reflexivity.
intro abs. contradict H.
apply (CRle_trans _ (x + y - CRabs R (y - x))).
rewrite CRabs_left. 2: apply CRlt_asym, abs.
unfold CRminus. rewrite CRopp_involutive, CRplus_comm.
rewrite CRplus_assoc, <- (CRplus_assoc (-x)), CRplus_opp_l.
rewrite CRplus_0_l, (CR_of_Q_plus R 1 1), CRmult_plus_distr_l.
rewrite CRmult_1_r. apply CRle_refl.
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity.
rewrite CRmult_1_r. apply CRle_refl.
Qed.
Add Parametric Morphism {R : ConstructiveReals} : CRmin
with signature (CReq R) ==> (CReq R) ==> (CReq R)
as CRmin_morph.
Proof.
intros. unfold CRmin.
apply CRmult_morph. 2: reflexivity.
unfold CRminus.
rewrite H, H0. reflexivity.
Qed.
Instance CRmin_morphT
: forall {R : ConstructiveReals},
CMorphisms.Proper
(CMorphisms.respectful (CReq R) (CMorphisms.respectful (CReq R) (CReq R))) (@CRmin R).
Proof.
intros R x y H z t H0.
rewrite H, H0. reflexivity.
Qed.
Lemma CRmin_l : forall {R : ConstructiveReals} (x y : CRcarrier R),
CRmin x y <= x.
Proof.
intros. unfold CRmin.
apply (CRmult_le_reg_r (CR_of_Q R 2)).
apply CR_of_Q_lt; reflexivity.
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity.
rewrite CRmult_1_r.
rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r.
unfold CRminus. rewrite CRplus_assoc. apply CRplus_le_compat_l.
apply (CRplus_le_reg_r (CRabs _ (y + - x)+ -x)).
rewrite CRplus_assoc, <- (CRplus_assoc (-CRabs _ (y + - x))).
rewrite CRplus_opp_l, CRplus_0_l.
rewrite (CRplus_comm x), CRplus_assoc, CRplus_opp_l, CRplus_0_r.
apply CRle_abs.
Qed.
Lemma CRmin_r : forall {R : ConstructiveReals} (x y : CRcarrier R),
CRmin x y <= y.
Proof.
intros. unfold CRmin.
apply (CRmult_le_reg_r (CR_of_Q R 2)).
apply CR_of_Q_lt; reflexivity.
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity.
rewrite CRmult_1_r.
rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r.
rewrite (CRplus_comm x).
unfold CRminus. rewrite CRplus_assoc. apply CRplus_le_compat_l.
apply (CRplus_le_reg_l (-x)).
rewrite <- CRplus_assoc, CRplus_opp_l, CRplus_0_l.
rewrite <- (CRopp_involutive y), <- CRopp_plus_distr, <- CRopp_plus_distr.
apply CRopp_ge_le_contravar. rewrite CRabs_opp, CRplus_comm.
apply CRle_abs.
Qed.
Lemma CRnegPartAbsMin : forall {R : ConstructiveReals} (x : CRcarrier R),
CRmin 0 x == (x - CRabs _ x) * (CR_of_Q _ (1#2)).
Proof.
intros. unfold CRmin. unfold CRminus. rewrite CRplus_0_l.
apply CRmult_morph. 2: reflexivity. rewrite CRopp_0, CRplus_0_r. reflexivity.
Qed.
Lemma CRmin_sym : forall {R : ConstructiveReals} (x y : CRcarrier R),
CRmin x y == CRmin y x.
Proof.
intros. unfold CRmin. apply CRmult_morph. 2: reflexivity.
rewrite CRabs_minus_sym. unfold CRminus.
rewrite (CRplus_comm x y). reflexivity.
Qed.
Lemma CRmin_mult :
forall {R : ConstructiveReals} (p q r : CRcarrier R),
0 <= r -> CRmin (r * p) (r * q) == r * CRmin p q.
Proof.
intros R p q r H. unfold CRmin.
setoid_replace (r * q - r * p) with (r * (q - p)).
rewrite CRabs_mult.
rewrite (CRabs_right r). 2: exact H.
rewrite <- CRmult_assoc. apply CRmult_morph. 2: reflexivity.
unfold CRminus. rewrite CRopp_mult_distr_r.
do 2 rewrite <- CRmult_plus_distr_l. reflexivity.
unfold CRminus. rewrite CRopp_mult_distr_r.
rewrite <- CRmult_plus_distr_l. reflexivity.
Qed.
Lemma CRmin_plus : forall {R : ConstructiveReals} (x y z : CRcarrier R),
x + CRmin y z == CRmin (x + y) (x + z).
Proof.
intros. unfold CRmin.
unfold CRminus. setoid_replace (x + z + - (x + y)) with (z-y).
apply (CRmult_eq_reg_r (CR_of_Q _ 2)).
left. apply CR_of_Q_lt; reflexivity.
rewrite CRmult_plus_distr_r.
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity.
rewrite CRmult_1_r.
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity.
rewrite CRmult_1_r.
rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r.
do 3 rewrite (CRplus_assoc x). apply CRplus_morph. reflexivity.
do 2 rewrite <- CRplus_assoc. apply CRplus_morph. 2: reflexivity.
rewrite (CRplus_comm x). apply CRplus_assoc.
rewrite CRopp_plus_distr. rewrite <- CRplus_assoc.
apply CRplus_morph. 2: reflexivity.
rewrite CRplus_comm, <- CRplus_assoc, CRplus_opp_l.
apply CRplus_0_l.
Qed.
Lemma CRmin_left : forall {R : ConstructiveReals} (x y : CRcarrier R),
x <= y -> CRmin x y == x.
Proof.
intros. unfold CRmin.
apply (CRmult_eq_reg_r (CR_of_Q R 2)).
left. apply CR_of_Q_lt; reflexivity.
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity.
rewrite CRmult_1_r.
rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r.
rewrite CRabs_right. unfold CRminus. rewrite CRopp_plus_distr.
rewrite CRplus_assoc. apply CRplus_morph. reflexivity.
rewrite <- CRplus_assoc, CRplus_opp_r, CRplus_0_l. apply CRopp_involutive.
rewrite <- (CRplus_opp_r x). apply CRplus_le_compat.
exact H. apply CRle_refl.
Qed.
Lemma CRmin_right : forall {R : ConstructiveReals} (x y : CRcarrier R),
y <= x -> CRmin x y == y.
Proof.
intros. unfold CRmin.
apply (CRmult_eq_reg_r (CR_of_Q R 2)).
left. apply CR_of_Q_lt; reflexivity.
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity.
rewrite CRmult_1_r.
rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r.
rewrite CRabs_left. unfold CRminus. do 2 rewrite CRopp_plus_distr.
rewrite (CRplus_comm x y).
rewrite CRplus_assoc. apply CRplus_morph. reflexivity.
do 2 rewrite CRopp_involutive.
rewrite CRplus_comm, CRplus_assoc, CRplus_opp_l, CRplus_0_r. reflexivity.
rewrite <- (CRplus_opp_r x). apply CRplus_le_compat.
exact H. apply CRle_refl.
Qed.
Lemma CRmin_lt : forall {R : ConstructiveReals} (x y z : CRcarrier R),
z < x -> z < y -> z < CRmin x y.
Proof.
intros. unfold CRmin.
apply (CRmult_lt_reg_r (CR_of_Q R 2)).
apply CR_of_Q_lt; reflexivity.
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity.
rewrite CRmult_1_r.
apply (CRplus_lt_reg_l _ (CRabs _ (y - x) - (z*CR_of_Q R 2))).
unfold CRminus. rewrite CRplus_assoc. rewrite CRplus_opp_l, CRplus_0_r.
rewrite (CRplus_comm (CRabs R (y + - x))).
rewrite (CRplus_comm (x+y)), CRplus_assoc.
rewrite <- (CRplus_assoc (CRabs R (y + - x))), CRplus_opp_r, CRplus_0_l.
rewrite <- (CRplus_comm (x+y)).
apply CRabs_def1.
- unfold CRminus. rewrite <- (CRplus_comm y), CRplus_assoc.
apply CRplus_lt_compat_l.
apply (CRplus_lt_reg_l R (-x)).
rewrite CRopp_mult_distr_l.
rewrite <- CRplus_assoc, CRplus_opp_l, CRplus_0_l.
rewrite (CR_of_Q_plus R 1 1), CRmult_plus_distr_l.
rewrite CRmult_1_r. apply CRplus_le_lt_compat.
apply CRlt_asym.
apply CRopp_gt_lt_contravar, H.
apply CRopp_gt_lt_contravar, H.
- rewrite CRopp_plus_distr, CRopp_involutive.
rewrite CRplus_comm, CRplus_assoc.
apply CRplus_lt_compat_l.
apply (CRplus_lt_reg_l R (-y)).
rewrite CRopp_mult_distr_l.
rewrite <- CRplus_assoc, CRplus_opp_l, CRplus_0_l.
rewrite (CR_of_Q_plus R 1 1), CRmult_plus_distr_l.
rewrite CRmult_1_r. apply CRplus_le_lt_compat.
apply CRlt_asym.
apply CRopp_gt_lt_contravar, H0.
apply CRopp_gt_lt_contravar, H0.
Qed.
Lemma CRmin_contract : forall {R : ConstructiveReals} (x y a : CRcarrier R),
CRabs _ (CRmin x a - CRmin y a) <= CRabs _ (x - y).
Proof.
intros. unfold CRmin.
unfold CRminus. rewrite CRopp_mult_distr_l, <- CRmult_plus_distr_r.
rewrite (CRabs_morph
_ ((x - y + (CRabs _ (a - y) - CRabs _ (a - x))) * CR_of_Q R (1 # 2))).
rewrite CRabs_mult, (CRabs_right (CR_of_Q R (1 # 2))).
2: apply CR_of_Q_le; discriminate.
apply (CRle_trans _
((CRabs _ (x - y) * 1 + CRabs _ (x-y) * 1)
* CR_of_Q R (1 # 2))).
apply CRmult_le_compat_r.
apply CR_of_Q_le. discriminate.
apply (CRle_trans
_ (CRabs _ (x - y) + CRabs _ (CRabs _ (a - y) - CRabs _ (a - x)))).
apply CRabs_triang. rewrite CRmult_1_r. apply CRplus_le_compat_l.
rewrite (CRabs_morph (x-y) ((a-y)-(a-x))).
apply CRabs_triang_inv2.
unfold CRminus. rewrite (CRplus_comm (a + - y)).
rewrite <- CRplus_assoc. apply CRplus_morph. 2: reflexivity.
rewrite CRplus_comm, CRopp_plus_distr, <- CRplus_assoc.
rewrite CRplus_opp_r, CRopp_involutive, CRplus_0_l.
reflexivity.
rewrite <- CRmult_plus_distr_l.
rewrite <- (CR_of_Q_plus R 1 1).
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 + 1) * (1 # 2))%Q with 1%Q. 2: reflexivity.
rewrite CRmult_1_r. apply CRle_refl.
unfold CRminus. apply CRmult_morph. 2: reflexivity.
do 4 rewrite CRplus_assoc. apply CRplus_morph. reflexivity.
rewrite <- CRplus_assoc. rewrite CRplus_comm, CRopp_plus_distr.
rewrite CRplus_assoc. apply CRplus_morph. reflexivity.
rewrite CRopp_plus_distr. rewrite (CRplus_comm (-a)).
rewrite CRplus_assoc, <- (CRplus_assoc (-a)), CRplus_opp_l.
rewrite CRplus_0_l, CRopp_involutive. reflexivity.
Qed.
Lemma CRmin_glb : forall {R : ConstructiveReals} (x y z:CRcarrier R),
z <= x -> z <= y -> z <= CRmin x y.
Proof.
intros. unfold CRmin.
apply (CRmult_le_reg_r (CR_of_Q R 2)).
apply CR_of_Q_lt; reflexivity.
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity.
rewrite CRmult_1_r.
apply (CRplus_le_reg_l (CRabs _ (y-x) - (z*CR_of_Q R 2))).
unfold CRminus. rewrite CRplus_assoc, CRplus_opp_l, CRplus_0_r.
rewrite (CRplus_comm (CRabs R (y + - x) + - (z * CR_of_Q R 2))).
rewrite CRplus_assoc, <- (CRplus_assoc (- CRabs R (y + - x))).
rewrite CRplus_opp_l, CRplus_0_l.
apply CRabs_le. split.
- do 2 rewrite CRopp_plus_distr.
rewrite CRopp_involutive, (CRplus_comm y), CRplus_assoc.
apply CRplus_le_compat_l, (CRplus_le_reg_l y).
rewrite <- CRplus_assoc, CRplus_opp_r, CRplus_0_l.
rewrite (CR_of_Q_plus R 1 1), CRmult_plus_distr_l.
rewrite CRmult_1_r. apply CRplus_le_compat; exact H0.
- rewrite (CRplus_comm x), CRplus_assoc. apply CRplus_le_compat_l.
apply (CRplus_le_reg_l (-x)).
rewrite <- CRplus_assoc, CRplus_opp_l, CRplus_0_l.
rewrite CRopp_mult_distr_l.
rewrite (CR_of_Q_plus R 1 1), CRmult_plus_distr_l.
rewrite CRmult_1_r.
apply CRplus_le_compat; apply CRopp_ge_le_contravar; exact H.
Qed.
Lemma CRmin_assoc : forall {R : ConstructiveReals} (a b c : CRcarrier R),
CRmin a (CRmin b c) == CRmin (CRmin a b) c.
Proof.
split.
- apply CRmin_glb.
+ apply (CRle_trans _ (CRmin a b)).
apply CRmin_l. apply CRmin_l.
+ apply CRmin_glb.
apply (CRle_trans _ (CRmin a b)).
apply CRmin_l. apply CRmin_r. apply CRmin_r.
- apply CRmin_glb.
+ apply CRmin_glb. apply CRmin_l.
apply (CRle_trans _ (CRmin b c)).
apply CRmin_r. apply CRmin_l.
+ apply (CRle_trans _ (CRmin b c)).
apply CRmin_r. apply CRmin_r.
Qed.
Lemma CRlt_min : forall {R : ConstructiveReals} (x y z : CRcarrier R),
z < CRmin x y -> prod (z < x) (z < y).
Proof.
intros. destruct (CR_Q_dense R _ _ H) as [q qmaj].
destruct qmaj.
split.
- apply (CRlt_le_trans _ (CR_of_Q R q) _ c).
intro abs. apply (CRlt_asym _ _ c0).
apply (CRle_lt_trans _ x). apply CRmin_l. exact abs.
- apply (CRlt_le_trans _ (CR_of_Q R q) _ c).
intro abs. apply (CRlt_asym _ _ c0).
apply (CRle_lt_trans _ y). apply CRmin_r. exact abs.
Qed.
(* Maximum *)
Definition CRmax {R : ConstructiveReals} (x y : CRcarrier R) : CRcarrier R
:= (x + y + CRabs _ (y - x)) * CR_of_Q _ (1#2).
Add Parametric Morphism {R : ConstructiveReals} : CRmax
with signature (CReq R) ==> (CReq R) ==> (CReq R)
as CRmax_morph.
Proof.
intros. unfold CRmax.
apply CRmult_morph. 2: reflexivity. unfold CRminus.
rewrite H, H0. reflexivity.
Qed.
Instance CRmax_morphT
: forall {R : ConstructiveReals},
CMorphisms.Proper
(CMorphisms.respectful (CReq R) (CMorphisms.respectful (CReq R) (CReq R))) (@CRmax R).
Proof.
intros R x y H z t H0.
rewrite H, H0. reflexivity.
Qed.
Lemma CRmax_lub : forall {R : ConstructiveReals} (x y z:CRcarrier R),
x <= z -> y <= z -> CRmax x y <= z.
Proof.
intros. unfold CRmax.
apply (CRmult_le_reg_r (CR_of_Q _ 2)).
apply CR_of_Q_lt; reflexivity.
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity.
rewrite CRmult_1_r.
apply (CRplus_le_reg_l (-x-y)).
rewrite <- CRplus_assoc. unfold CRminus.
rewrite <- CRopp_plus_distr, CRplus_opp_l, CRplus_0_l.
apply CRabs_le. split.
- repeat rewrite CRopp_plus_distr.
do 2 rewrite CRopp_involutive.
rewrite (CRplus_comm x), CRplus_assoc. apply CRplus_le_compat_l.
apply (CRplus_le_reg_l (-x)).
rewrite <- CRplus_assoc, CRplus_opp_l, CRplus_0_l.
rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r.
rewrite CRopp_plus_distr.
apply CRplus_le_compat; apply CRopp_ge_le_contravar; assumption.
- rewrite (CRplus_comm y), CRopp_plus_distr, CRplus_assoc.
apply CRplus_le_compat_l.
apply (CRplus_le_reg_l y).
rewrite <- CRplus_assoc, CRplus_opp_r, CRplus_0_l.
rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r.
apply CRplus_le_compat; assumption.
Qed.
Lemma CRmax_l : forall {R : ConstructiveReals} (x y : CRcarrier R),
x <= CRmax x y.
Proof.
intros. unfold CRmax.
apply (CRmult_le_reg_r (CR_of_Q R 2)).
apply CR_of_Q_lt; reflexivity.
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity.
rewrite CRmult_1_r.
setoid_replace 2%Q with (1+1)%Q. rewrite CR_of_Q_plus.
rewrite CRmult_plus_distr_l, CRmult_1_r, CRplus_assoc.
apply CRplus_le_compat_l.
apply (CRplus_le_reg_l (-y)).
rewrite <- CRplus_assoc, CRplus_opp_l, CRplus_0_l.
rewrite CRabs_minus_sym, CRplus_comm.
apply CRle_abs. reflexivity.
Qed.
Lemma CRmax_r : forall {R : ConstructiveReals} (x y : CRcarrier R),
y <= CRmax x y.
Proof.
intros. unfold CRmax.
apply (CRmult_le_reg_r (CR_of_Q _ 2)).
apply CR_of_Q_lt; reflexivity.
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity.
rewrite CRmult_1_r.
rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r.
rewrite (CRplus_comm x).
rewrite CRplus_assoc. apply CRplus_le_compat_l.
apply (CRplus_le_reg_l (-x)).
rewrite <- CRplus_assoc, CRplus_opp_l, CRplus_0_l.
rewrite CRplus_comm. apply CRle_abs.
Qed.
Lemma CRposPartAbsMax : forall {R : ConstructiveReals} (x : CRcarrier R),
CRmax 0 x == (x + CRabs _ x) * (CR_of_Q R (1#2)).
Proof.
intros. unfold CRmax. unfold CRminus. rewrite CRplus_0_l.
apply CRmult_morph. 2: reflexivity. rewrite CRopp_0, CRplus_0_r. reflexivity.
Qed.
Lemma CRmax_sym : forall {R : ConstructiveReals} (x y : CRcarrier R),
CRmax x y == CRmax y x.
Proof.
intros. unfold CRmax.
rewrite CRabs_minus_sym. apply CRmult_morph.
2: reflexivity. rewrite (CRplus_comm x y). reflexivity.
Qed.
Lemma CRmax_plus : forall {R : ConstructiveReals} (x y z : CRcarrier R),
x + CRmax y z == CRmax (x + y) (x + z).
Proof.
intros. unfold CRmax.
setoid_replace (x + z - (x + y)) with (z-y).
apply (CRmult_eq_reg_r (CR_of_Q _ 2)).
left. apply CR_of_Q_lt; reflexivity.
rewrite CRmult_plus_distr_r.
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity.
rewrite CRmult_1_r.
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity.
rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r.
rewrite CRmult_1_r.
do 3 rewrite (CRplus_assoc x). apply CRplus_morph. reflexivity.
do 2 rewrite <- CRplus_assoc. apply CRplus_morph. 2: reflexivity.
rewrite (CRplus_comm x). apply CRplus_assoc.
unfold CRminus. rewrite CRopp_plus_distr. rewrite <- CRplus_assoc.
apply CRplus_morph. 2: reflexivity.
rewrite CRplus_comm, <- CRplus_assoc, CRplus_opp_l.
apply CRplus_0_l.
Qed.
Lemma CRmax_left : forall {R : ConstructiveReals} (x y : CRcarrier R),
y <= x -> CRmax x y == x.
Proof.
intros. unfold CRmax.
apply (CRmult_eq_reg_r (CR_of_Q R 2)).
left. apply CR_of_Q_lt; reflexivity.
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity.
rewrite CRmult_1_r.
rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r.
rewrite CRplus_assoc. apply CRplus_morph. reflexivity.
rewrite CRabs_left. unfold CRminus. rewrite CRopp_plus_distr, CRopp_involutive.
rewrite <- CRplus_assoc, CRplus_opp_r, CRplus_0_l. reflexivity.
rewrite <- (CRplus_opp_r x). apply CRplus_le_compat_r. exact H.
Qed.
Lemma CRmax_right : forall {R : ConstructiveReals} (x y : CRcarrier R),
x <= y -> CRmax x y == y.
Proof.
intros. unfold CRmax.
apply (CRmult_eq_reg_r (CR_of_Q R 2)).
left. apply CR_of_Q_lt; reflexivity.
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity.
rewrite CRmult_1_r.
rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r.
rewrite (CRplus_comm x y).
rewrite CRplus_assoc. apply CRplus_morph. reflexivity.
rewrite CRabs_right. unfold CRminus. rewrite CRplus_comm.
rewrite CRplus_assoc, CRplus_opp_l, CRplus_0_r. reflexivity.
rewrite <- (CRplus_opp_r x). apply CRplus_le_compat_r. exact H.
Qed.
Lemma CRmax_contract : forall {R : ConstructiveReals} (x y a : CRcarrier R),
CRabs _ (CRmax x a - CRmax y a) <= CRabs _ (x - y).
Proof.
intros. unfold CRmax.
rewrite (CRabs_morph
_ ((x - y + (CRabs _ (a - x) - CRabs _ (a - y))) * CR_of_Q R (1 # 2))).
rewrite CRabs_mult, (CRabs_right (CR_of_Q R (1 # 2))).
2: apply CR_of_Q_le; discriminate.
apply (CRle_trans
_ ((CRabs _ (x - y) * 1 + CRabs _ (x-y) * 1)
* CR_of_Q R (1 # 2))).
apply CRmult_le_compat_r.
apply CR_of_Q_le. discriminate.
apply (CRle_trans
_ (CRabs _ (x - y) + CRabs _ (CRabs _ (a - x) - CRabs _ (a - y)))).
apply CRabs_triang. rewrite CRmult_1_r. apply CRplus_le_compat_l.
rewrite (CRabs_minus_sym x y).
rewrite (CRabs_morph (y-x) ((a-x)-(a-y))).
apply CRabs_triang_inv2.
unfold CRminus. rewrite (CRplus_comm (a + - x)).
rewrite <- CRplus_assoc. apply CRplus_morph. 2: reflexivity.
rewrite CRplus_comm, CRopp_plus_distr, <- CRplus_assoc.
rewrite CRplus_opp_r, CRopp_involutive, CRplus_0_l.
reflexivity.
rewrite <- CRmult_plus_distr_l.
rewrite <- (CR_of_Q_plus R 1 1).
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 + 1) * (1 # 2))%Q with 1%Q. 2: reflexivity.
rewrite CRmult_1_r. apply CRle_refl.
unfold CRminus. rewrite CRopp_mult_distr_l.
rewrite <- CRmult_plus_distr_r. apply CRmult_morph. 2: reflexivity.
do 4 rewrite CRplus_assoc. apply CRplus_morph. reflexivity.
rewrite <- CRplus_assoc. rewrite CRplus_comm, CRopp_plus_distr.
rewrite CRplus_assoc. apply CRplus_morph. reflexivity.
rewrite CRopp_plus_distr. rewrite (CRplus_comm (-a)).
rewrite CRplus_assoc, <- (CRplus_assoc (-a)), CRplus_opp_l.
rewrite CRplus_0_l. apply CRplus_comm.
Qed.
Lemma CRmax_lub_lt : forall {R : ConstructiveReals} (x y z : CRcarrier R),
x < z -> y < z -> CRmax x y < z.
Proof.
intros. unfold CRmax.
apply (CRmult_lt_reg_r (CR_of_Q R 2)).
apply CR_of_Q_lt; reflexivity.
rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity.
rewrite CRmult_1_r.
apply (CRplus_lt_reg_l _ (-y -x)). unfold CRminus.
rewrite CRplus_assoc, <- (CRplus_assoc (-x)), <- (CRplus_assoc (-x)).
rewrite CRplus_opp_l, CRplus_0_l, <- CRplus_assoc, CRplus_opp_l, CRplus_0_l.
apply CRabs_def1.
- rewrite (CRplus_comm y), (CRplus_comm (-y)), CRplus_assoc.
apply CRplus_lt_compat_l.
apply (CRplus_lt_reg_l _ y).
rewrite <- CRplus_assoc, CRplus_opp_r, CRplus_0_l.
rewrite (CR_of_Q_plus R 1 1), CRmult_plus_distr_l.
rewrite CRmult_1_r. apply CRplus_le_lt_compat.
apply CRlt_asym, H0. exact H0.
- rewrite CRopp_plus_distr, CRopp_involutive.
rewrite CRplus_assoc. apply CRplus_lt_compat_l.
apply (CRplus_lt_reg_l _ x).
rewrite <- CRplus_assoc, CRplus_opp_r, CRplus_0_l.
rewrite (CR_of_Q_plus R 1 1), CRmult_plus_distr_l.
rewrite CRmult_1_r. apply CRplus_le_lt_compat.
apply CRlt_asym, H. exact H.
Qed.
Lemma CRmax_assoc : forall {R : ConstructiveReals} (a b c : CRcarrier R),
CRmax a (CRmax b c) == CRmax (CRmax a b) c.
Proof.
split.
- apply CRmax_lub.
+ apply CRmax_lub. apply CRmax_l.
apply (CRle_trans _ (CRmax b c)).
apply CRmax_l. apply CRmax_r.
+ apply (CRle_trans _ (CRmax b c)).
apply CRmax_r. apply CRmax_r.
- apply CRmax_lub.
+ apply (CRle_trans _ (CRmax a b)).
apply CRmax_l. apply CRmax_l.
+ apply CRmax_lub.
apply (CRle_trans _ (CRmax a b)).
apply CRmax_r. apply CRmax_l. apply CRmax_r.
Qed.
Lemma CRmax_min_mult_neg :
forall {R : ConstructiveReals} (p q r:CRcarrier R),
r <= 0 -> CRmax (r * p) (r * q) == r * CRmin p q.
Proof.
intros R p q r H. unfold CRmin, CRmax.
setoid_replace (r * q - r * p) with (r * (q - p)).
rewrite CRabs_mult.
rewrite (CRabs_left r), <- CRmult_assoc.
apply CRmult_morph. 2: reflexivity. unfold CRminus.
rewrite <- CRopp_mult_distr_l, CRopp_mult_distr_r,
CRmult_plus_distr_l, CRmult_plus_distr_l.
reflexivity. exact H.
unfold CRminus. rewrite CRmult_plus_distr_l, CRopp_mult_distr_r. reflexivity.
Qed.
Lemma CRlt_max : forall {R : ConstructiveReals} (x y z : CRcarrier R),
CRmax x y < z -> prod (x < z) (y < z).
Proof.
intros. destruct (CR_Q_dense R _ _ H) as [q qmaj].
destruct qmaj.
split.
- apply (CRlt_le_trans _ (CR_of_Q R q)).
apply (CRle_lt_trans _ (CRmax x y)). apply CRmax_l. exact c.
apply CRlt_asym, c0.
- apply (CRlt_le_trans _ (CR_of_Q R q)).
apply (CRle_lt_trans _ (CRmax x y)). apply CRmax_r. exact c.
apply CRlt_asym, c0.
Qed.
Lemma CRmax_mult :
forall {R : ConstructiveReals} (p q r:CRcarrier R),
0 <= r -> CRmax (r * p) (r * q) == r * CRmax p q.
Proof.
intros R p q r H. unfold CRmin, CRmax.
setoid_replace (r * q - r * p) with (r * (q - p)).
rewrite CRabs_mult.
rewrite (CRabs_right r), <- CRmult_assoc.
apply CRmult_morph. 2: reflexivity.
rewrite CRmult_plus_distr_l, CRmult_plus_distr_l.
reflexivity. exact H.
unfold CRminus. rewrite CRmult_plus_distr_l, CRopp_mult_distr_r. reflexivity.
Qed.
Lemma CRmin_max_mult_neg :
forall {R : ConstructiveReals} (p q r:CRcarrier R),
r <= 0 -> CRmin (r * p) (r * q) == r * CRmax p q.
Proof.
intros R p q r H. unfold CRmin, CRmax.
setoid_replace (r * q - r * p) with (r * (q - p)).
rewrite CRabs_mult.
rewrite (CRabs_left r), <- CRmult_assoc.
apply CRmult_morph. 2: reflexivity. unfold CRminus.
rewrite CRopp_mult_distr_l, CRopp_involutive,
CRmult_plus_distr_l, CRmult_plus_distr_l.
reflexivity. exact H.
unfold CRminus. rewrite CRmult_plus_distr_l, CRopp_mult_distr_r. reflexivity.
Qed.
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