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(**
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This file is part of the Flocq formalization of floating-point
arithmetic in Coq: http://flocq.gforge.inria.fr/

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Copyright (C) 2010-2011 Sylvie Boldo
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#<br />#
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Copyright (C) 2010-2011 Guillaume Melquiond
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This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 3 of the License, or (at your option) any later version.

This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
COPYING file for more details.
*)

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(** * Roundings: properties and/or functions *)
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Require Import Fcore_Raux.
Require Import Fcore_defs.
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Section RND_prop.

Open Scope R_scope.

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Theorem round_val_of_pred :
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  forall rnd : R -> R -> Prop,
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  round_pred rnd ->
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  forall x, { f : R | rnd x f }.
Proof.
intros rnd (H1,H2) x.
specialize (H1 x).
(* . *)
assert (H3 : bound (rnd x)).
destruct H1 as (f, H1).
exists f.
intros g Hg.
now apply H2 with (3 := Rle_refl x).
(* . *)
exists (projT1 (completeness _ H3 H1)).
destruct completeness as (f1, (H4, H5)).
simpl.
destruct H1 as (f2, H1).
assert (f1 = f2).
apply Rle_antisym.
apply H5.
intros f3 H.
now apply H2 with (3 := Rle_refl x).
now apply H4.
now rewrite H.
Qed.

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Theorem round_fun_of_pred :
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  forall rnd : R -> R -> Prop,
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  round_pred rnd ->
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  { f : R -> R | forall x, rnd x (f x) }.
Proof.
intros rnd H.
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exists (fun x => projT1 (round_val_of_pred rnd H x)).
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intros x.
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now destruct round_val_of_pred as (f, H1).
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Qed.

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Theorem round_unicity :
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  forall rnd : R -> R -> Prop,
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  round_pred_monotone rnd ->
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  forall x f1 f2,
  rnd x f1 ->
  rnd x f2 ->
  f1 = f2.
Proof.
intros rnd Hr x f1 f2 H1 H2.
apply Rle_antisym.
now apply Hr with (3 := Rle_refl x).
now apply Hr with (3 := Rle_refl x).
Qed.

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Theorem Rnd_DN_pt_monotone :
  forall F : R -> Prop,
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  round_pred_monotone (Rnd_DN_pt F).
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Proof.
intros F x y f g (Hx1,(Hx2,_)) (Hy1,(_,Hy2)) Hxy.
apply Hy2.
apply Hx1.
now apply Rle_trans with (2 := Hxy).
Qed.

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Theorem Rnd_DN_pt_unicity :
  forall F : R -> Prop,
  forall x f1 f2 : R,
  Rnd_DN_pt F x f1 -> Rnd_DN_pt F x f2 ->
  f1 = f2.
Proof.
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intros F.
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apply round_unicity.
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apply Rnd_DN_pt_monotone.
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Qed.

Theorem Rnd_DN_unicity :
  forall F : R -> Prop,
  forall rnd1 rnd2 : R -> R,
  Rnd_DN F rnd1 -> Rnd_DN F rnd2 ->
  forall x, rnd1 x = rnd2 x.
Proof.
intros F rnd1 rnd2 H1 H2 x.
now eapply Rnd_DN_pt_unicity.
Qed.

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Theorem Rnd_UP_pt_monotone :
  forall F : R -> Prop,
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  round_pred_monotone (Rnd_UP_pt F).
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Proof.
intros F x y f g (Hx1,(_,Hx2)) (Hy1,(Hy2,_)) Hxy.
apply Hx2.
apply Hy1.
now apply Rle_trans with (1 := Hxy).
Qed.

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Theorem Rnd_UP_pt_unicity :
  forall F : R -> Prop,
  forall x f1 f2 : R,
  Rnd_UP_pt F x f1 -> Rnd_UP_pt F x f2 ->
  f1 = f2.
Proof.
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intros F.
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apply round_unicity.
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apply Rnd_UP_pt_monotone.
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Qed.

Theorem Rnd_UP_unicity :
  forall F : R -> Prop,
  forall rnd1 rnd2 : R -> R,
  Rnd_UP F rnd1 -> Rnd_UP F rnd2 ->
  forall x, rnd1 x = rnd2 x.
Proof.
intros F rnd1 rnd2 H1 H2 x.
now eapply Rnd_UP_pt_unicity.
Qed.

Theorem Rnd_DN_UP_pt_sym :
  forall F : R -> Prop,
  ( forall x, F x -> F (- x) ) ->
  forall x f : R,
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  Rnd_DN_pt F x f -> Rnd_UP_pt F (-x) (-f).
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Proof.
intros F HF x f H.
repeat split.
apply HF.
apply H.
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apply Ropp_le_contravar.
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apply H.
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intros g Hg.
rewrite <- (Ropp_involutive g).
intros Hxg.
apply Ropp_le_contravar.
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apply H.
now apply HF.
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now apply Ropp_le_cancel.
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Qed.

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Theorem Rnd_UP_DN_pt_sym :
  forall F : R -> Prop,
  ( forall x, F x -> F (- x) ) ->
  forall x f : R,
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  Rnd_UP_pt F x f -> Rnd_DN_pt F (-x) (-f).
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Proof.
intros F HF x f H.
repeat split.
apply HF.
apply H.
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apply Ropp_le_contravar.
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apply H.
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intros g Hg.
rewrite <- (Ropp_involutive g).
intros Hxg.
apply Ropp_le_contravar.
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apply H.
now apply HF.
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now apply Ropp_le_cancel.
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Qed.

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Theorem Rnd_DN_UP_sym :
  forall F : R -> Prop,
  ( forall x, F x -> F (- x) ) ->
  forall rnd1 rnd2 : R -> R,
  Rnd_DN F rnd1 -> Rnd_UP F rnd2 ->
  forall x, rnd1 (- x) = - rnd2 x.
Proof.
intros F HF rnd1 rnd2 H1 H2 x.
rewrite <- (Ropp_involutive (rnd1 (-x))).
apply f_equal.
apply (Rnd_UP_unicity F (fun x => - rnd1 (-x))) ; trivial.
intros y.
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pattern y at 1 ; rewrite <- Ropp_involutive.
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apply Rnd_DN_UP_pt_sym.
apply HF.
apply H1.
Qed.

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Theorem Rnd_DN_UP_pt_split :
  forall F : R -> Prop,
  forall x d u,
  Rnd_DN_pt F x d ->
  Rnd_UP_pt F x u ->
  forall f, F f ->
  (f <= d) \/ (u <= f).
Proof.
intros F x d u Hd Hu f Hf.
destruct (Rle_or_lt f x).
left.
now apply Hd.
right.
assert (H' := Rlt_le _ _ H).
now apply Hu.
Qed.

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Theorem Rnd_DN_pt_refl :
  forall F : R -> Prop,
  forall x : R, F x ->
  Rnd_DN_pt F x x.
Proof.
intros F x Hx.
repeat split.
exact Hx.
apply Rle_refl.
now intros.
Qed.

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Theorem Rnd_DN_pt_idempotent :
  forall F : R -> Prop,
  forall x f : R,
  Rnd_DN_pt F x f -> F x ->
  f = x.
Proof.
intros F x f (_,(Hx1,Hx2)) Hx.
apply Rle_antisym.
exact Hx1.
apply Hx2.
exact Hx.
apply Rle_refl.
Qed.

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Theorem Rnd_UP_pt_refl :
  forall F : R -> Prop,
  forall x : R, F x ->
  Rnd_UP_pt F x x.
Proof.
intros F x Hx.
repeat split.
exact Hx.
apply Rle_refl.
now intros.
Qed.

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Theorem Rnd_UP_pt_idempotent :
  forall F : R -> Prop,
  forall x f : R,
  Rnd_UP_pt F x f -> F x ->
  f = x.
Proof.
intros F x f (_,(Hx1,Hx2)) Hx.
apply Rle_antisym.
apply Hx2.
exact Hx.
apply Rle_refl.
exact Hx1.
Qed.

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Theorem Only_DN_or_UP :
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  forall F : R -> Prop,
  forall x fd fu f : R,
  Rnd_DN_pt F x fd -> Rnd_UP_pt F x fu ->
  F f -> (fd <= f <= fu)%R ->
  f = fd \/ f = fu.
Proof.
intros F x fd fu f Hd Hu Hf ([Hdf|Hdf], Hfu).
2 : now left.
destruct Hfu.
2 : now right.
destruct (Rle_or_lt x f).
elim Rlt_not_le with (1 := H).
now apply Hu.
elim Rlt_not_le with (1 := Hdf).
apply Hd ; auto with real.
Qed.

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Theorem Rnd_ZR_abs :
  forall (F : R -> Prop) (rnd: R-> R),
  Rnd_ZR F rnd ->
  forall x : R,  (Rabs (rnd x) <= Rabs x)%R.
Proof.
intros F rnd H x.
assert (F 0%R).
replace 0%R with (rnd 0%R).
eapply H.
apply Rle_refl.
destruct (H 0%R) as (H1, H2).
apply Rle_antisym.
apply H1.
apply Rle_refl.
apply H2.
apply Rle_refl.
(* . *)
destruct (Rle_or_lt 0 x).
(* positive *)
rewrite Rabs_right.
rewrite Rabs_right; auto with real.
now apply (proj1 (H x)).
apply Rle_ge.
now apply (proj1 (H x)).
(* negative *)
rewrite Rabs_left1.
rewrite Rabs_left1 ; auto with real.
apply Ropp_le_contravar.
apply (proj2 (H x)).
auto with real.
apply (proj2 (H x)) ; auto with real.
Qed.

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Theorem Rnd_ZR_pt_monotone :
  forall F : R -> Prop, F 0 ->
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  round_pred_monotone (Rnd_ZR_pt F).
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Proof.
intros F F0 x y f g (Hx1, Hx2) (Hy1, Hy2) Hxy.
destruct (Rle_or_lt 0 x) as [Hx|Hx].
(* . *)
apply Hy1.
now apply Rle_trans with x.
now apply Hx1.
apply Rle_trans with (2 := Hxy).
now apply Hx1.
(* . *)
apply Rlt_le in Hx.
destruct (Rle_or_lt 0 y) as [Hy|Hy].
apply Rle_trans with 0.
now apply Hx2.
now apply Hy1.
apply Rlt_le in Hy.
apply Hx2.
exact Hx.
now apply Hy2.
apply Rle_trans with (1 := Hxy).
now apply Hy2.
Qed.

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Theorem Rnd_N_pt_DN_or_UP :
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  forall F : R -> Prop,
  forall x f : R,
  Rnd_N_pt F x f ->
  Rnd_DN_pt F x f \/ Rnd_UP_pt F x f.
Proof.
intros F x f (Hf1,Hf2).
destruct (Rle_or_lt x f) as [Hxf|Hxf].
(* . *)
right.
repeat split ; try assumption.
intros g Hg Hxg.
specialize (Hf2 g Hg).
rewrite 2!Rabs_pos_eq in Hf2.
now apply Rplus_le_reg_r with (-x)%R.
now apply Rle_0_minus.
now apply Rle_0_minus.
(* . *)
left.
repeat split ; try assumption.
now apply Rlt_le.
intros g Hg Hxg.
specialize (Hf2 g Hg).
rewrite 2!Rabs_left1 in Hf2.
generalize (Ropp_le_cancel _ _ Hf2).
intros H.
now apply Rplus_le_reg_r with (-x)%R.
now apply Rle_minus.
apply Rlt_le.
now apply Rlt_minus.
Qed.

Theorem Rnd_N_pt_DN_or_UP_eq :
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  forall F : R -> Prop,
  forall x fd fu f : R,
  Rnd_DN_pt F x fd -> Rnd_UP_pt F x fu ->
  Rnd_N_pt F x f ->
  f = fd \/ f = fu.
Proof.
intros F x fd fu f Hd Hu Hf.
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destruct (Rnd_N_pt_DN_or_UP F x f Hf) as [H|H].
left.
apply Rnd_DN_pt_unicity with (1 := H) (2 := Hd).
right.
apply Rnd_UP_pt_unicity with (1 := H) (2 := Hu).
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Qed.

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Theorem Rnd_N_pt_sym :
  forall F : R -> Prop,
  ( forall x, F x -> F (- x) ) ->
  forall x f : R,
  Rnd_N_pt F (-x) (-f) -> Rnd_N_pt F x f.
Proof.
intros F HF x f (H1,H2).
rewrite <- (Ropp_involutive f).
repeat split.
apply HF.
apply H1.
intros g H3.
rewrite Ropp_involutive.
replace (f - x)%R with (-(-f - -x))%R by ring.
replace (g - x)%R with (-(-g - -x))%R by ring.
rewrite 2!Rabs_Ropp.
apply H2.
now apply HF.
Qed.

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Theorem Rnd_N_pt_monotone :
  forall F : R -> Prop,
  forall x y f g : R,
  Rnd_N_pt F x f -> Rnd_N_pt F y g ->
  x < y -> f <= g.
Proof.
intros F x y f g (Hf,Hx) (Hg,Hy) Hxy.
apply Rnot_lt_le.
intros Hgf.
assert (Hfgx := Hx g Hg).
assert (Hgfy := Hy f Hf).
clear F Hf Hg Hx Hy.
destruct (Rle_or_lt x g) as [Hxg|Hgx].
(* x <= g < f *)
apply Rle_not_lt with (1 := Hfgx).
rewrite 2!Rabs_pos_eq.
now apply Rplus_lt_compat_r.
apply Rle_0_minus.
apply Rlt_le.
now apply Rle_lt_trans with (1 := Hxg).
now apply Rle_0_minus.
assert (Hgy := Rlt_trans _ _ _ Hgx Hxy).
destruct (Rle_or_lt f y) as [Hfy|Hyf].
(* g < f <= y *)
apply Rle_not_lt with (1 := Hgfy).
rewrite (Rabs_left (g - y)).
2: now apply Rlt_minus.
rewrite Rabs_left1.
apply Ropp_lt_contravar.
now apply Rplus_lt_compat_r.
now apply Rle_minus.
(* g < x < y < f *)
rewrite Rabs_left, Rabs_pos_eq, Ropp_minus_distr in Hgfy.
rewrite Rabs_pos_eq, Rabs_left, Ropp_minus_distr in Hfgx.
apply Rle_not_lt with (1 := Rplus_le_compat _ _ _ _ Hfgx Hgfy).
apply Rminus_lt.
ring_simplify.
apply Rlt_minus.
apply Rmult_lt_compat_l.
now apply (Z2R_lt 0 2).
exact Hxy.
now apply Rlt_minus.
apply Rle_0_minus.
apply Rlt_le.
now apply Rlt_trans with (1 := Hxy).
apply Rle_0_minus.
now apply Rlt_le.
now apply Rlt_minus.
Qed.

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Theorem Rnd_N_pt_unicity :
  forall F : R -> Prop,
  forall x d u f1 f2 : R,
  Rnd_DN_pt F x d ->
  Rnd_UP_pt F x u ->
  x - d <> u - x ->
  Rnd_N_pt F x f1 ->
  Rnd_N_pt F x f2 ->
  f1 = f2.
Proof.
intros F x d u f1 f2 Hd Hu Hdu.
assert (forall f1 f2, Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 < f2 -> False).
clear f1 f2. intros f1 f2 Hf1 Hf2 H12.
destruct (Rnd_N_pt_DN_or_UP F x f1 Hf1) as [Hd1|Hu1] ;
  destruct (Rnd_N_pt_DN_or_UP F x f2 Hf2) as [Hd2|Hu2].
apply Rlt_not_eq with (1 := H12).
now apply Rnd_DN_pt_unicity with (1 := Hd1).
apply Hdu.
rewrite Rnd_DN_pt_unicity with (1 := Hd) (2 := Hd1).
rewrite Rnd_UP_pt_unicity with (1 := Hu) (2 := Hu2).
rewrite <- (Rabs_pos_eq (x - f1)).
rewrite <- (Rabs_pos_eq (f2 - x)).
rewrite Rabs_minus_sym.
apply Rle_antisym.
apply Hf1. apply Hf2.
apply Hf2. apply Hf1.
apply Rle_0_minus.
apply Hu2.
apply Rle_0_minus.
apply Hd1.
apply Rlt_not_le with (1 := H12).
apply Rle_trans with x.
apply Hd2.
apply Hu1.
apply Rgt_not_eq with (1 := H12).
now apply Rnd_UP_pt_unicity with (1 := Hu2).
intros Hf1 Hf2.
now apply Rle_antisym ; apply Rnot_lt_le ; refine (H _ _ _ _).
Qed.

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Theorem Rnd_N_pt_refl :
  forall F : R -> Prop,
  forall x : R, F x ->
  Rnd_N_pt F x x.
Proof.
intros F x Hx.
repeat split.
exact Hx.
intros g _.
unfold Rminus at 1.
rewrite Rplus_opp_r, Rabs_R0.
apply Rabs_pos.
Qed.

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Theorem Rnd_N_pt_idempotent :
  forall F : R -> Prop,
  forall x f : R,
  Rnd_N_pt F x f -> F x ->
  f = x.
Proof.
intros F x f (_,Hf) Hx.
apply Rminus_diag_uniq.
destruct (Req_dec (f - x) 0) as [H|H].
exact H.
elim Rabs_no_R0 with (1 := H).
apply Rle_antisym.
replace 0 with (Rabs (x - x)).
now apply Hf.
unfold Rminus.
rewrite Rplus_opp_r.
apply Rabs_R0.
apply Rabs_pos.
Qed.

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Theorem Rnd_N_pt_0 :
  forall F : R -> Prop,
  F 0 ->
  Rnd_N_pt F 0 0.
Proof.
intros F HF.
split.
exact HF.
intros g _.
rewrite 2!Rminus_0_r, Rabs_R0.
apply Rabs_pos.
Qed.

Theorem Rnd_N_pt_pos :
  forall F : R -> Prop, F 0 ->
  forall x f, 0 <= x ->
  Rnd_N_pt F x f ->
  0 <= f.
Proof.
intros F HF x f [Hx|Hx] Hxf.
eapply Rnd_N_pt_monotone ; try eassumption.
now apply Rnd_N_pt_0.
right.
apply sym_eq.
apply Rnd_N_pt_idempotent with F.
now rewrite Hx.
exact HF.
Qed.

Theorem Rnd_N_pt_neg :
  forall F : R -> Prop, F 0 ->
  forall x f, x <= 0 ->
  Rnd_N_pt F x f ->
  f <= 0.
Proof.
intros F HF x f [Hx|Hx] Hxf.
eapply Rnd_N_pt_monotone ; try eassumption.
now apply Rnd_N_pt_0.
right.
apply Rnd_N_pt_idempotent with F.
now rewrite <- Hx.
exact HF.
Qed.

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Theorem Rnd_N_pt_abs :
  forall F : R -> Prop,
  F 0 ->
  ( forall x, F x -> F (- x) ) ->
  forall x f : R,
  Rnd_N_pt F x f -> Rnd_N_pt F (Rabs x) (Rabs f).
Proof.
intros F HF0 HF x f Hxf.
unfold Rabs at 1.
destruct (Rcase_abs x) as [Hx|Hx].
rewrite Rabs_left1.
apply Rnd_N_pt_sym.
exact HF.
now rewrite 2!Ropp_involutive.
apply Rnd_N_pt_neg with (3 := Hxf).
exact HF0.
now apply Rlt_le.
rewrite Rabs_pos_eq.
exact Hxf.
apply Rnd_N_pt_pos with (3 := Hxf).
exact HF0.
now apply Rge_le.
Qed.

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Theorem Rnd_DN_UP_pt_N :
  forall F : R -> Prop,
  forall x d u f : R,
  F f ->
  Rnd_DN_pt F x d ->
  Rnd_UP_pt F x u ->
  (Rabs (f - x) <= x - d)%R ->
  (Rabs (f - x) <= u - x)%R ->
  Rnd_N_pt F x f.
Proof.
intros F x d u f Hf Hxd Hxu Hd Hu.
split.
exact Hf.
intros g Hg.
destruct (Rnd_DN_UP_pt_split F x d u Hxd Hxu g Hg) as [Hgd|Hgu].
(* g <= d *)
apply Rle_trans with (1 := Hd).
rewrite Rabs_left1.
rewrite Ropp_minus_distr.
apply Rplus_le_compat_l.
now apply Ropp_le_contravar.
apply Rle_minus.
apply Rle_trans with (1 := Hgd).
apply Hxd.
(* u <= g *)
apply Rle_trans with (1 := Hu).
rewrite Rabs_pos_eq.
now apply Rplus_le_compat_r.
apply Rle_0_minus.
apply Rle_trans with (2 := Hgu).
apply Hxu.
Qed.

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Theorem Rnd_DN_pt_N :
  forall F : R -> Prop,
  forall x d u : R,
  Rnd_DN_pt F x d ->
  Rnd_UP_pt F x u ->
  (x - d <= u - x)%R ->
  Rnd_N_pt F x d.
Proof.
intros F x d u Hd Hu Hx.
assert (Hdx: (Rabs (d - x) = x - d)%R).
rewrite Rabs_minus_sym.
apply Rabs_pos_eq.
apply Rle_0_minus.
apply Hd.
apply Rnd_DN_UP_pt_N with (2 := Hd) (3 := Hu).
apply Hd.
rewrite Hdx.
apply Rle_refl.
now rewrite Hdx.
Qed.

Theorem Rnd_UP_pt_N :
  forall F : R -> Prop,
  forall x d u : R,
  Rnd_DN_pt F x d ->
  Rnd_UP_pt F x u ->
  (u - x <= x - d)%R ->
  Rnd_N_pt F x u.
Proof.
intros F x d u Hd Hu Hx.
assert (Hux: (Rabs (u - x) = u - x)%R).
apply Rabs_pos_eq.
apply Rle_0_minus.
apply Hu.
apply Rnd_DN_UP_pt_N with (2 := Hd) (3 := Hu).
apply Hu.
now rewrite Hux.
rewrite Hux.
apply Rle_refl.
Qed.

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Definition Rnd_NG_pt_unicity_prop F P :=
  forall x d u,
  Rnd_DN_pt F x d -> Rnd_N_pt F x d ->
  Rnd_UP_pt F x u -> Rnd_N_pt F x u ->
  P x d -> P x u -> d = u.
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Theorem Rnd_NG_pt_unicity :
  forall (F : R -> Prop) (P : R -> R -> Prop),
  Rnd_NG_pt_unicity_prop F P ->
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  forall x f1 f2 : R,
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  Rnd_NG_pt F P x f1 -> Rnd_NG_pt F P x f2 ->
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  f1 = f2.
Proof.
intros F P HP x f1 f2 (H1a,H1b) (H2a,H2b).
destruct H1b as [H1b|H1b].
destruct H2b as [H2b|H2b].
destruct (Rnd_N_pt_DN_or_UP _ _ _ H1a) as [H1c|H1c] ;
  destruct (Rnd_N_pt_DN_or_UP _ _ _ H2a) as [H2c|H2c].
eapply Rnd_DN_pt_unicity ; eassumption.
now apply (HP x f1 f2).
apply sym_eq.
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now apply (HP x f2 f1 H2c H2a H1c H1a).
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eapply Rnd_UP_pt_unicity ; eassumption.
now apply H2b.
apply sym_eq.
now apply H1b.
Qed.

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Theorem Rnd_NG_pt_monotone :
  forall (F : R -> Prop) (P : R -> R -> Prop),
  Rnd_NG_pt_unicity_prop F P ->
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  round_pred_monotone (Rnd_NG_pt F P).
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Proof.
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intros F P HP x y f g (Hf,Hx) (Hg,Hy) [Hxy|Hxy].
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now apply Rnd_N_pt_monotone with F x y.
apply Req_le.
rewrite <- Hxy in Hg, Hy.
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eapply Rnd_NG_pt_unicity ; try split ; eassumption.
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Qed.

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Theorem Rnd_NG_pt_refl :
  forall (F : R -> Prop) (P : R -> R -> Prop),
  forall x, F x -> Rnd_NG_pt F P x x.
Proof.
intros F P x Hx.
split.
now apply Rnd_N_pt_refl.
right.
intros f2 Hf2.
now apply Rnd_N_pt_idempotent with F.
Qed.

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Theorem Rnd_NG_pt_sym :
  forall (F : R -> Prop) (P : R -> R -> Prop),
  ( forall x, F x -> F (-x) ) ->
  ( forall x f, P x f -> P (-x) (-f) ) ->
  forall x f : R,
  Rnd_NG_pt F P (-x) (-f) -> Rnd_NG_pt F P x f.
Proof.
intros F P HF HP x f (H1,H2).
split.
now apply Rnd_N_pt_sym.
destruct H2 as [H2|H2].
left.
rewrite <- (Ropp_involutive x), <- (Ropp_involutive f).
now apply HP.
right.
intros f2 Hxf2.
rewrite <- (Ropp_involutive f).
rewrite <- H2 with (-f2).
apply sym_eq.
apply Ropp_involutive.
apply Rnd_N_pt_sym.
exact HF.
now rewrite 2!Ropp_involutive.
Qed.
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Theorem Rnd_NG_unicity :
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  forall (F : R -> Prop) (P : R -> R -> Prop),
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  Rnd_NG_pt_unicity_prop F P ->
  forall rnd1 rnd2 : R -> R,
  Rnd_NG F P rnd1 -> Rnd_NG F P rnd2 ->
  forall x, rnd1 x = rnd2 x.
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Proof.
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intros F P HP rnd1 rnd2 H1 H2 x.
now apply Rnd_NG_pt_unicity with F P x.
Qed.

Theorem Rnd_NA_NG_pt :
  forall F : R -> Prop,
  F 0 ->
  forall x f,
  Rnd_NA_pt F x f <-> Rnd_NG_pt F (fun x f => Rabs x <= Rabs f) x f.
Proof.
intros F HF x f.
destruct (Rle_or_lt 0 x) as [Hx|Hx].
(* *)
split ; intros (H1, H2).
(* . *)
assert (Hf := Rnd_N_pt_pos F HF x f Hx H1).
split.
exact H1.
destruct (Rnd_N_pt_DN_or_UP _ _ _ H1) as [H3|H3].
(* . . *)
right.
intros f2 Hxf2.
specialize (H2 _ Hxf2).
destruct (Rnd_N_pt_DN_or_UP _ _ _ Hxf2) as [H4|H4].
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eapply Rnd_DN_pt_unicity ; eassumption.
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apply Rle_antisym.
rewrite Rabs_pos_eq with (1 := Hf) in H2.
rewrite Rabs_pos_eq in H2.
exact H2.
now apply Rnd_N_pt_pos with F x.
apply Rle_trans with x.
apply H3.
apply H4.
(* . . *)
left.
rewrite Rabs_pos_eq with (1 := Hf).
rewrite Rabs_pos_eq with (1 := Hx).
apply H3.
(* . *)
split.
exact H1.
intros f2 Hxf2.
destruct H2 as [H2|H2].
assert (Hf := Rnd_N_pt_pos F HF x f Hx H1).
assert (Hf2 := Rnd_N_pt_pos F HF x f2 Hx Hxf2).
rewrite 2!Rabs_pos_eq ; trivial.
rewrite 2!Rabs_pos_eq in H2 ; trivial.
destruct (Rnd_N_pt_DN_or_UP _ _ _ Hxf2) as [H3|H3].
apply Rle_trans with (2 := H2).
apply H3.
apply H3.
apply H1.
apply H2.
rewrite (H2 _ Hxf2).
apply Rle_refl.
(* *)
assert (Hx' := Rlt_le _ _ Hx).
clear Hx. rename Hx' into Hx.
split ; intros (H1, H2).
(* . *)
assert (Hf := Rnd_N_pt_neg F HF x f Hx H1).
split.
exact H1.
destruct (Rnd_N_pt_DN_or_UP _ _ _ H1) as [H3|H3].
(* . . *)
left.
rewrite Rabs_left1 with (1 := Hf).
rewrite Rabs_left1 with (1 := Hx).
apply Ropp_le_contravar.
apply H3.
(* . . *)
right.
intros f2 Hxf2.
specialize (H2 _ Hxf2).
destruct (Rnd_N_pt_DN_or_UP _ _ _ Hxf2) as [H4|H4].
apply Rle_antisym.
apply Rle_trans with x.
apply H4.
apply H3.
rewrite Rabs_left1 with (1 := Hf) in H2.
rewrite Rabs_left1 in H2.
now apply Ropp_le_cancel.
now apply Rnd_N_pt_neg with F x.
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eapply Rnd_UP_pt_unicity ; eassumption.
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(* . *)
split.
exact H1.
intros f2 Hxf2.
destruct H2 as [H2|H2].
assert (Hf := Rnd_N_pt_neg F HF x f Hx H1).
assert (Hf2 := Rnd_N_pt_neg F HF x f2 Hx Hxf2).
rewrite 2!Rabs_left1 ; trivial.
rewrite 2!Rabs_left1 in H2 ; trivial.
apply Ropp_le_contravar.
apply Ropp_le_cancel in H2.
destruct (Rnd_N_pt_DN_or_UP _ _ _ Hxf2) as [H3|H3].
apply H3.
apply H1.
apply H2.
apply Rle_trans with (1 := H2).
apply H3.
rewrite (H2 _ Hxf2).
apply Rle_refl.
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Qed.

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Theorem Rnd_NA_pt_unicity_prop :
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  forall F : R -> Prop,
  F 0 ->
  Rnd_NG_pt_unicity_prop F (fun a b => (Rabs a <= Rabs b)%R).
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Proof.
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intros F HF x d u Hxd1 Hxd2 Hxu1 Hxu2 Hd Hu.
apply Rle_antisym.
apply Rle_trans with x.
apply Hxd1.
apply Hxu1.
destruct (Rle_or_lt 0 x) as [Hx|Hx].
apply Hxu1.
apply Hxd1.
rewrite Rabs_pos_eq with (1 := Hx) in Hd.
rewrite Rabs_pos_eq in Hd.
exact Hd.
now apply Hxd1.
apply Hxd1.
apply Hxu1.
rewrite Rabs_left with (1 := Hx) in Hu.
rewrite Rabs_left1 in Hu.
now apply Ropp_le_cancel.
apply Hxu1.
apply HF.
now apply Rlt_le.
Qed.

Theorem Rnd_NA_pt_unicity :
  forall F : R -> Prop,
  F 0 ->
  forall x f1 f2 : R,
  Rnd_NA_pt F x f1 -> Rnd_NA_pt F x f2 ->
  f1 = f2.
Proof.
intros F HF x f1 f2 H1 H2.
apply (Rnd_NG_pt_unicity F _ (Rnd_NA_pt_unicity_prop F HF) x).
now apply -> Rnd_NA_NG_pt.
now apply -> Rnd_NA_NG_pt.
Qed.

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Theorem Rnd_NA_N_pt :
  forall F : R -> Prop,
  F 0 ->
  forall x f : R,
  Rnd_N_pt F x f ->
  (Rabs x <= Rabs f)%R ->
  Rnd_NA_pt F x f.
Proof.
intros F HF x f Rxf Hxf.
split.
apply Rxf.
intros g Rxg.
destruct (Rabs_eq_Rabs (f - x) (g - x)) as [H|H].
apply Rle_antisym.
apply Rxf.
apply Rxg.
apply Rxg.
apply Rxf.
(* *)
replace g with f.
apply Rle_refl.
apply Rplus_eq_reg_r with (1 := H).
(* *)
assert (g = 2 * x - f)%R.
replace (2 * x - f)%R with (x - (f - x))%R by ring.
rewrite H.
ring.
destruct (Rle_lt_dec 0 x) as [Hx|Hx].
(* . *)
revert Hxf.
rewrite Rabs_pos_eq with (1 := Hx).
rewrite 2!Rabs_pos_eq ; try ( apply (Rnd_N_pt_pos F HF x) ; assumption ).
intros Hxf.
rewrite H0.
apply Rplus_le_reg_r with f.
ring_simplify.
apply Rmult_le_compat_l with (2 := Hxf).
now apply (Z2R_le 0 2).
(* . *)
revert Hxf.
apply Rlt_le in Hx.
rewrite Rabs_left1 with (1 := Hx).
rewrite 2!Rabs_left1 ; try ( apply (Rnd_N_pt_neg F HF x) ; assumption ).
intros Hxf.
rewrite H0.
apply Ropp_le_contravar.
apply Rplus_le_reg_r with f.
ring_simplify.
apply Rmult_le_compat_l.
now apply (Z2R_le 0 2).
now apply Ropp_le_cancel.
Qed.

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Theorem Rnd_NA_unicity :
  forall (F : R -> Prop),
  F 0 ->
  forall rnd1 rnd2 : R -> R,
  Rnd_NA F rnd1 -> Rnd_NA F rnd2 ->
  forall x, rnd1 x = rnd2 x.
Proof.
intros F HF rnd1 rnd2 H1 H2 x.
now apply Rnd_NA_pt_unicity with F x.
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Qed.

Theorem Rnd_NA_pt_monotone :
  forall F : R -> Prop,
  F 0 ->
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  round_pred_monotone (Rnd_NA_pt F).
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Proof.
intros F HF x y f g Hxf Hyg Hxy.
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apply (Rnd_NG_pt_monotone F _ (Rnd_NA_pt_unicity_prop F HF) x y).
now apply -> Rnd_NA_NG_pt.
now apply -> Rnd_NA_NG_pt.
exact Hxy.
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Qed.

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Theorem Rnd_NA_pt_refl :
  forall F : R -> Prop,
  forall x : R, F x ->
  Rnd_NA_pt F x x.
Proof.
intros F x Hx.
split.
now apply Rnd_N_pt_refl.
intros f Hxf.
apply Req_le.
apply f_equal.
now apply Rnd_N_pt_idempotent with (1 := Hxf).
Qed.

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Theorem Rnd_NA_pt_idempotent :
  forall F : R -> Prop,
  forall x f : R,
  Rnd_NA_pt F x f -> F x ->
  f = x.
Proof.
intros F x f (Hf,_) Hx.
now apply Rnd_N_pt_idempotent with F.
Qed.

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Theorem round_pred_ge_0 :
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  forall P : R -> R -> Prop,
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  round_pred_monotone P ->
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  P 0 0 ->
  forall x f, P x f -> 0 <= x -> 0 <= f.
Proof.
intros P HP HP0 x f Hxf Hx.
now apply (HP 0 x).
Qed.

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Theorem round_pred_gt_0 :
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  forall P : R -> R -> Prop,
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  round_pred_monotone P ->
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  P 0 0 ->
  forall x f, P x f -> 0 < f -> 0 < x.
Proof.
intros P HP HP0 x f Hxf Hf.
apply Rnot_le_lt.
intros Hx.
apply Rlt_not_le with (1 := Hf).
now apply (HP x 0).
Qed.

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Theorem round_pred_le_0 :
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  forall P : R -> R -> Prop,
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  round_pred_monotone P ->
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  P 0 0 ->
  forall x f, P x f -> x <= 0 -> f <= 0.
Proof.
intros P HP HP0 x f Hxf Hx.
now apply (HP x 0).
Qed.

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Theorem round_pred_lt_0 :
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  forall P : R -> R -> Prop,
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  round_pred_monotone P ->
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  P 0 0 ->
  forall x f, P x f -> f < 0 -> x < 0.
Proof.
intros P HP HP0 x f Hxf Hf.
apply Rnot_le_lt.
intros Hx.
apply Rlt_not_le with (1 := Hf).
now apply (HP 0 x).
Qed.

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Theorem Rnd_DN_pt_equiv_format :
  forall F1 F2 : R -> Prop,
  forall a b : R,
  F1 a ->
  ( forall x, a <= x <= b -> (F1 x <-> F2 x) ) ->
  forall x f, a <= x <= b -> Rnd_DN_pt F1 x f -> Rnd_DN_pt F2 x f.
Proof.
intros F1 F2 a b Ha HF x f Hx (H1, (H2, H3)).
split.
apply -> HF.
exact H1.
split.
now apply H3.
now apply Rle_trans with (1 := H2).
split.
exact H2.
intros k Hk Hl.
destruct (Rlt_or_le k a) as [H|H].
apply Rlt_le.
apply Rlt_le_trans with (1 := H).
now apply H3.
apply H3.
apply <- HF.
exact Hk.
split.
exact H.
now apply Rle_trans with (1 := Hl).
exact Hl.
Qed.

Theorem Rnd_UP_pt_equiv_format :
  forall F1 F2 : R -> Prop,
  forall a b : R,
  F1 b ->
  ( forall x, a <= x <= b -> (F1 x <-> F2 x) ) ->
  forall x f, a <= x <= b -> Rnd_UP_pt F1 x f -> Rnd_UP_pt F2 x f.
Proof.
intros F1 F2 a b Hb HF x f Hx (H1, (H2, H3)).
split.
apply -> HF.
exact H1.
split.
now apply Rle_trans with (2 := H2).
now apply H3.
split.
exact H2.
intros k Hk Hl.
destruct (Rle_or_lt k b) as [H|H].
apply H3.
apply <- HF.
exact Hk.
split.
now apply Rle_trans with (2 := Hl).
exact H.
exact Hl.
apply Rlt_le.
apply Rle_lt_trans with (2 := H).
now apply H3.
Qed.

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(** ensures a real number can always be rounded *)
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Inductive satisfies_any (F : R -> Prop) :=
  Satisfies_any :
    F 0 -> ( forall x : R, F x -> F (-x) ) ->
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    round_pred_total (Rnd_DN_pt F) -> satisfies_any F.
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Theorem satisfies_any_eq :
  forall F1 F2 : R -> Prop,
  ( forall x, F1 x <-> F2 x ) ->
  satisfies_any F1 ->
  satisfies_any F2.
Proof.
intros F1 F2 Heq (Hzero, Hsym, Hrnd).
split.
now apply -> Heq.
intros x Hx.
apply -> Heq.
apply Hsym.
now apply <- Heq.
intros x.
destruct (Hrnd x) as (f, (H1, (H2, H3))).
exists f.
split.
now apply -> Heq.
split.
exact H2.
intros g Hg Hgx.
apply H3.
now apply <- Heq.
exact Hgx.
Qed.

Theorem satisfies_any_imp_DN :
  forall F : R -> Prop,
  satisfies_any F ->
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  round_pred (Rnd_DN_pt F).
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Proof.
intros F (_,_,Hrnd).
split.
apply Hrnd.
apply Rnd_DN_pt_monotone.
Qed.

Theorem satisfies_any_imp_UP :
  forall F : R -> Prop,
  satisfies_any F ->
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  round_pred (Rnd_UP_pt F).
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Proof.
intros F Hany.
split.
intros x.
destruct (proj1 (satisfies_any_imp_DN F Hany) (-x)) as (f, Hf).
exists (-f).
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rewrite <- (Ropp_involutive x).
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apply Rnd_DN_UP_pt_sym.
apply Hany.
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exact Hf.
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apply Rnd_UP_pt_monotone.
Qed.

Theorem satisfies_any_imp_ZR :
  forall F : R -> Prop,
  satisfies_any F ->
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  round_pred (Rnd_ZR_pt F).
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Proof.
intros F Hany.
split.
intros x.
destruct (Rle_or_lt 0 x) as [Hx|Hx].
(* positive *)
destruct (proj1 (satisfies_any_imp_DN F Hany) x) as (f, Hf).
exists f.
split.
now intros _.
intros Hx'.
(* zero *)
assert (x = 0).
now apply Rle_antisym.
rewrite H in Hf |- *.
clear H Hx Hx'.
rewrite Rnd_DN_pt_idempotent with (1 := Hf).
apply Rnd_UP_pt_refl.
apply Hany.
apply Hany.
(* negative *)
destruct (proj1 (satisfies_any_imp_UP F Hany) x) as (f, Hf).
exists f.
split.
intros Hx'.
elim (Rlt_irrefl 0).
now apply Rle_lt_trans with x.
now intros _.
(* . *)
apply Rnd_ZR_pt_monotone.
apply Hany.
Qed.

Definition NG_existence_prop (F : R -> Prop) (P : R -> R -> Prop) :=
  forall x d u, ~F x -> Rnd_DN_pt F x d -> Rnd_UP_pt F x u -> P x u \/ P x d.

Theorem satisfies_any_imp_NG :
  forall (F : R -> Prop) (P : R -> R -> Prop),
  satisfies_any F ->
  NG_existence_prop F P ->
1220
  round_pred_total (Rnd_NG_pt F P).
1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360
Proof.
intros F P Hany HP x.
destruct (proj1 (satisfies_any_imp_DN F Hany) x) as (d, Hd).
destruct (proj1 (satisfies_any_imp_UP F Hany) x) as (u, Hu).
destruct (total_order_T (Rabs (u - x)) (Rabs (d - x))) as [[H|H]|H].
(* |up(x) - x| < |dn(x) - x| *)
exists u.
split.
(* - . *)
split.
apply Hu.
intros g Hg.
destruct (Rle_or_lt x g) as [Hxg|Hxg].
rewrite 2!Rabs_pos_eq.
apply Rplus_le_compat_r.
now apply Hu.
now apply Rle_0_minus.
apply Rle_0_minus.
apply Hu.
apply Rlt_le in Hxg.
apply Rlt_le.
apply Rlt_le_trans with (1 := H).
do 2 rewrite <- (Rabs_minus_sym x).
rewrite 2!Rabs_pos_eq.
apply Rplus_le_compat_l.
apply Ropp_le_contravar.
now apply Hd.
now apply Rle_0_minus.
apply Rle_0_minus.
apply Hd.
(* - . *)
right.
intros f Hf.
destruct (Rnd_N_pt_DN_or_UP_eq F x _ _ _ Hd Hu Hf) as [K|K] ; rewrite K.
elim Rlt_not_le with (1 := H).
rewrite <- K.
apply Hf.
apply Hu.
apply refl_equal.
(* |up(x) - x| = |dn(x) - x| *)
destruct (Req_dec x d) as [He|Hne].
(* - x = d = u *)
exists x.
split.
apply Rnd_N_pt_refl.
rewrite He.
apply Hd.
right.
intros.
apply Rnd_N_pt_idempotent with (1 := H0).
rewrite He.
apply Hd.
assert (Hf : ~F x).
intros Hf.
apply Hne.
apply sym_eq.
now apply Rnd_DN_pt_idempotent with (1 := Hd).
destruct (HP x _ _ Hf Hd Hu) as [H'|H'].
(* - u >> d *)
exists u.
split.
split.
apply Hu.
intros g Hg.
destruct (Rle_or_lt x g) as [Hxg|Hxg].
rewrite 2!Rabs_pos_eq.
apply Rplus_le_compat_r.
now apply Hu.
now apply Rle_0_minus.
apply Rle_0_minus.
apply Hu.
apply Rlt_le in Hxg.
rewrite H.
rewrite 2!Rabs_left1.
apply Ropp_le_contravar.
apply Rplus_le_compat_r.
now apply Hd.
now apply Rle_minus.
apply Rle_minus.
apply Hd.
now left.
(* - d >> u *)
exists d.
split.
split.
apply Hd.
intros g Hg.
destruct (Rle_or_lt x g) as [Hxg|Hxg].
rewrite <- H.
rewrite 2!Rabs_pos_eq.
apply Rplus_le_compat_r.
now apply Hu.
now apply Rle_0_minus.
apply Rle_0_minus.
apply Hu.
apply Rlt_le in Hxg.
rewrite 2!Rabs_left1.
apply Ropp_le_contravar.
apply Rplus_le_compat_r.
now apply Hd.
now apply Rle_minus.
apply Rle_minus.
apply Hd.
now left.
(* |up(x) - x| > |dn(x) - x| *)
exists d.
split.
split.
apply Hd.
intros g Hg.
destruct (Rle_or_lt x g) as [Hxg|Hxg].
apply Rlt_le.
apply Rlt_le_trans with (1 := H).
rewrite 2!Rabs_pos_eq.
apply Rplus_le_compat_r.
now apply Hu.
now apply Rle_0_minus.
apply Rle_0_minus.
apply Hu.
apply Rlt_le in Hxg.
rewrite 2!Rabs_left1.
apply Ropp_le_contravar.
apply Rplus_le_compat_r.
now apply Hd.
now apply Rle_minus.
apply Rle_minus.
apply Hd.
right.
intros f Hf.
destruct (Rnd_N_pt_DN_or_UP_eq F x _ _ _ Hd Hu Hf) as [K|K] ; rewrite K.
apply refl_equal.
elim Rlt_not_le with (1 := H).
rewrite <- K.
apply Hf.
apply Hd.
Qed.

Theorem satisfies_any_imp_NA :
  forall F : R -> Prop,
  satisfies_any F ->
1361
  round_pred (Rnd_NA_pt F).
1362 1363 1364
Proof.
intros F Hany.
split.
1365
assert (H : round_pred_total (Rnd_NG_pt F (fun a b => (Rabs a <= Rabs b)%R))).
1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393
apply satisfies_any_imp_NG.
apply Hany.
intros x d u Hf Hd Hu.
destruct (Rle_lt_dec 0 x) as [Hx|Hx].
left.
rewrite Rabs_pos_eq with (1 := Hx).
rewrite Rabs_pos_eq.
apply Hu.
apply Rle_trans with (1 := Hx).
apply Hu.
right.
rewrite Rabs_left with (1 := Hx).
rewrite Rabs_left1.
apply Ropp_le_contravar.
apply Hd.
apply Rlt_le in Hx.
apply Rle_trans with (2 := Hx).
apply Hd.
intros x.
destruct (H x) as (f, Hf).
exists f.
apply <- Rnd_NA_NG_pt.
apply Hf.
apply Hany.
apply Rnd_NA_pt_monotone.
apply Hany.
Qed.

Guillaume Melquiond's avatar
Guillaume Melquiond committed
1394
End RND_prop.