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Roundings: properties and/or functions

Require Import Raux Defs.

Section RND_prop.

Open Scope R_scope.

Definition Rnd_DN (F : R -> Prop) (rnd : R -> R) :=
  forall x : R, Rnd_DN_pt F x (rnd x).

Definition Rnd_UP (F : R -> Prop) (rnd : R -> R) :=
  forall x : R, Rnd_UP_pt F x (rnd x).

Definition Rnd_ZR (F : R -> Prop) (rnd : R -> R) :=
  forall x : R, Rnd_ZR_pt F x (rnd x).

Definition Rnd_N (F : R -> Prop) (rnd : R -> R) :=
  forall x : R, Rnd_N_pt F x (rnd x).

Definition Rnd_NG (F : R -> Prop) (P : R -> R -> Prop) (rnd : R -> R) :=
  forall x : R, Rnd_NG_pt F P x (rnd x).

Definition Rnd_NA (F : R -> Prop) (rnd : R -> R) :=
  forall x : R, Rnd_NA_pt F x (rnd x).

Definition Rnd_N0 (F : R -> Prop) (rnd : R -> R) :=
  forall x : R, Rnd_N0_pt F x (rnd x).

Theorem round_val_of_pred :
  forall rnd : R -> R -> Prop,
  round_pred rnd ->
  forall x, { f : R | rnd x f }.forall rnd : R -> R -> Prop,
round_pred rnd -> forall x : R, {f : R | rnd x f}
Proof.forall rnd : R -> R -> Prop,
round_pred rnd -> forall x : R, {f : R | rnd x f}
intros rnd (H1,H2) x.rnd:R -> R -> Prop
H1:round_pred_total rnd
H2:round_pred_monotone rnd
x:R
{f : R | rnd x f}
specialize (H1 x).rnd:R -> R -> Prop
x:R
H1:exists f : R, rnd x f
H2:round_pred_monotone rnd
{f : R | rnd x f}
(* . *)
assert (H3 : bound (rnd x)).rnd:R -> R -> Prop
x:R
H1:exists f : R, rnd x f
H2:round_pred_monotone rnd
bound (rnd x)
rnd:R -> R -> Prop
x:R
H1:exists f : R, rnd x f
H2:round_pred_monotone rnd
H3:bound (rnd x)
{f : R | rnd x f}
destruct H1 as (f, H1).rnd:R -> R -> Prop
x, f:R
H1:rnd x f
H2:round_pred_monotone rnd
bound (rnd x)
rnd:R -> R -> Prop
x:R
H1:exists f : R, rnd x f
H2:round_pred_monotone rnd
H3:bound (rnd x)
{f : R | rnd x f}
exists f.rnd:R -> R -> Prop
x, f:R
H1:rnd x f
H2:round_pred_monotone rnd
is_upper_bound (rnd x) f
rnd:R -> R -> Prop
x:R
H1:exists f : R, rnd x f
H2:round_pred_monotone rnd
H3:bound (rnd x)
{f : R | rnd x f}
intros g Hg.rnd:R -> R -> Prop
x, f:R
H1:rnd x f
H2:round_pred_monotone rnd
g:R
Hg:rnd x g
g <= f
rnd:R -> R -> Prop
x:R
H1:exists f : R, rnd x f
H2:round_pred_monotone rnd
H3:bound (rnd x)
{f : R | rnd x f}
now apply H2 with (3 := Rle_refl x).rnd:R -> R -> Prop
x:R
H1:exists f : R, rnd x f
H2:round_pred_monotone rnd
H3:bound (rnd x)
{f : R | rnd x f}
(* . *)
exists (proj1_sig (completeness _ H3 H1)).rnd:R -> R -> Prop
x:R
H1:exists f : R, rnd x f
H2:round_pred_monotone rnd
H3:bound (rnd x)
rnd x (proj1_sig (completeness (rnd x) H3 H1))
destruct completeness as (f1, (H4, H5)).rnd:R -> R -> Prop
x:R
H1:exists f : R, rnd x f
H2:round_pred_monotone rnd
H3:bound (rnd x)
f1:R
H4:is_upper_bound (rnd x) f1
H5:forall b : R, is_upper_bound (rnd x) b -> f1 <= b
rnd x
  (proj1_sig
     (exist (fun m : R => is_lub (rnd x) m) f1
        (conj H4 H5)))
simpl.rnd:R -> R -> Prop
x:R
H1:exists f : R, rnd x f
H2:round_pred_monotone rnd
H3:bound (rnd x)
f1:R
H4:is_upper_bound (rnd x) f1
H5:forall b : R, is_upper_bound (rnd x) b -> f1 <= b
rnd x f1
destruct H1 as (f2, H1).rnd:R -> R -> Prop
x, f2:R
H1:rnd x f2
H2:round_pred_monotone rnd
H3:bound (rnd x)
f1:R
H4:is_upper_bound (rnd x) f1
H5:forall b : R, is_upper_bound (rnd x) b -> f1 <= b
rnd x f1
assert (f1 = f2).rnd:R -> R -> Prop
x, f2:R
H1:rnd x f2
H2:round_pred_monotone rnd
H3:bound (rnd x)
f1:R
H4:is_upper_bound (rnd x) f1
H5:forall b : R, is_upper_bound (rnd x) b -> f1 <= b
f1 = f2
rnd:R -> R -> Prop
x, f2:R
H1:rnd x f2
H2:round_pred_monotone rnd
H3:bound (rnd x)
f1:R
H4:is_upper_bound (rnd x) f1
H5:forall b : R, is_upper_bound (rnd x) b -> f1 <= b
H:f1 = f2
rnd x f1
apply Rle_antisym.rnd:R -> R -> Prop
x, f2:R
H1:rnd x f2
H2:round_pred_monotone rnd
H3:bound (rnd x)
f1:R
H4:is_upper_bound (rnd x) f1
H5:forall b : R, is_upper_bound (rnd x) b -> f1 <= b
f1 <= f2
rnd:R -> R -> Prop
x, f2:R
H1:rnd x f2
H2:round_pred_monotone rnd
H3:bound (rnd x)
f1:R
H4:is_upper_bound (rnd x) f1
H5:forall b : R, is_upper_bound (rnd x) b -> f1 <= b
f2 <= f1
rnd:R -> R -> Prop
x, f2:R
H1:rnd x f2
H2:round_pred_monotone rnd
H3:bound (rnd x)
f1:R
H4:is_upper_bound (rnd x) f1
H5:forall b : R, is_upper_bound (rnd x) b -> f1 <= b
H:f1 = f2
rnd x f1
apply H5.rnd:R -> R -> Prop
x, f2:R
H1:rnd x f2
H2:round_pred_monotone rnd
H3:bound (rnd x)
f1:R
H4:is_upper_bound (rnd x) f1
H5:forall b : R, is_upper_bound (rnd x) b -> f1 <= b
is_upper_bound (rnd x) f2
rnd:R -> R -> Prop
x, f2:R
H1:rnd x f2
H2:round_pred_monotone rnd
H3:bound (rnd x)
f1:R
H4:is_upper_bound (rnd x) f1
H5:forall b : R, is_upper_bound (rnd x) b -> f1 <= b
f2 <= f1
rnd:R -> R -> Prop
x, f2:R
H1:rnd x f2
H2:round_pred_monotone rnd
H3:bound (rnd x)
f1:R
H4:is_upper_bound (rnd x) f1
H5:forall b : R, is_upper_bound (rnd x) b -> f1 <= b
H:f1 = f2
rnd x f1
intros f3 H.rnd:R -> R -> Prop
x, f2:R
H1:rnd x f2
H2:round_pred_monotone rnd
H3:bound (rnd x)
f1:R
H4:is_upper_bound (rnd x) f1
H5:forall b : R, is_upper_bound (rnd x) b -> f1 <= b
f3:R
H:rnd x f3
f3 <= f2
rnd:R -> R -> Prop
x, f2:R
H1:rnd x f2
H2:round_pred_monotone rnd
H3:bound (rnd x)
f1:R
H4:is_upper_bound (rnd x) f1
H5:forall b : R, is_upper_bound (rnd x) b -> f1 <= b
f2 <= f1
rnd:R -> R -> Prop
x, f2:R
H1:rnd x f2
H2:round_pred_monotone rnd
H3:bound (rnd x)
f1:R
H4:is_upper_bound (rnd x) f1
H5:forall b : R, is_upper_bound (rnd x) b -> f1 <= b
H:f1 = f2
rnd x f1
now apply H2 with (3 := Rle_refl x).rnd:R -> R -> Prop
x, f2:R
H1:rnd x f2
H2:round_pred_monotone rnd
H3:bound (rnd x)
f1:R
H4:is_upper_bound (rnd x) f1
H5:forall b : R, is_upper_bound (rnd x) b -> f1 <= b
f2 <= f1
rnd:R -> R -> Prop
x, f2:R
H1:rnd x f2
H2:round_pred_monotone rnd
H3:bound (rnd x)
f1:R
H4:is_upper_bound (rnd x) f1
H5:forall b : R, is_upper_bound (rnd x) b -> f1 <= b
H:f1 = f2
rnd x f1
now apply H4.rnd:R -> R -> Prop
x, f2:R
H1:rnd x f2
H2:round_pred_monotone rnd
H3:bound (rnd x)
f1:R
H4:is_upper_bound (rnd x) f1
H5:forall b : R, is_upper_bound (rnd x) b -> f1 <= b
H:f1 = f2
rnd x f1
now rewrite H.
Qed.

Theorem round_fun_of_pred :
  forall rnd : R -> R -> Prop,
  round_pred rnd ->
  { f : R -> R | forall x, rnd x (f x) }.forall rnd : R -> R -> Prop,
round_pred rnd ->
{f : R -> R | forall x : R, rnd x (f x)}
Proof.forall rnd : R -> R -> Prop,
round_pred rnd ->
{f : R -> R | forall x : R, rnd x (f x)}
intros rnd H.rnd:R -> R -> Prop
H:round_pred rnd
{f : R -> R | forall x : R, rnd x (f x)}
exists (fun x => proj1_sig (round_val_of_pred rnd H x)).rnd:R -> R -> Prop
H:round_pred rnd
forall x : R,
rnd x (proj1_sig (round_val_of_pred rnd H x))
intros x.rnd:R -> R -> Prop
H:round_pred rnd
x:R
rnd x (proj1_sig (round_val_of_pred rnd H x))
now destruct round_val_of_pred as (f, H1).
Qed.

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

Theorem Rnd_DN_pt_monotone :
  forall F : R -> Prop,
  round_pred_monotone (Rnd_DN_pt F).forall F : R -> Prop,
round_pred_monotone (Rnd_DN_pt F)
Proof.forall F : R -> Prop,
round_pred_monotone (Rnd_DN_pt F)
intros F x y f g (Hx1,(Hx2,_)) (Hy1,(_,Hy2)) Hxy.F:R -> Prop
x, y, f, g:R
Hx1:F f
Hx2:f <= x
Hy1:F g
Hy2:forall g0 : R, F g0 -> g0 <= y -> g0 <= g
Hxy:x <= y
f <= g
apply Hy2.F:R -> Prop
x, y, f, g:R
Hx1:F f
Hx2:f <= x
Hy1:F g
Hy2:forall g0 : R, F g0 -> g0 <= y -> g0 <= g
Hxy:x <= y
F f
F:R -> Prop
x, y, f, g:R
Hx1:F f
Hx2:f <= x
Hy1:F g
Hy2:forall g0 : R, F g0 -> g0 <= y -> g0 <= g
Hxy:x <= y
f <= y
apply Hx1.F:R -> Prop
x, y, f, g:R
Hx1:F f
Hx2:f <= x
Hy1:F g
Hy2:forall g0 : R, F g0 -> g0 <= y -> g0 <= g
Hxy:x <= y
f <= y
now apply Rle_trans with (2 := Hxy).
Qed.

Theorem Rnd_DN_pt_unique :
  forall F : R -> Prop,
  forall x f1 f2 : R,
  Rnd_DN_pt F x f1 -> Rnd_DN_pt F x f2 ->
  f1 = f2.forall (F : R -> Prop) (x f1 f2 : R),
Rnd_DN_pt F x f1 -> Rnd_DN_pt F x f2 -> f1 = f2
Proof.forall (F : R -> Prop) (x f1 f2 : R),
Rnd_DN_pt F x f1 -> Rnd_DN_pt F x f2 -> f1 = f2
intros F.F:R -> Prop
forall x f1 f2 : R,
Rnd_DN_pt F x f1 -> Rnd_DN_pt F x f2 -> f1 = f2
apply round_unique.F:R -> Prop
round_pred_monotone (Rnd_DN_pt F)
apply Rnd_DN_pt_monotone.
Qed.

Theorem Rnd_DN_unique :
  forall F : R -> Prop,
  forall rnd1 rnd2 : R -> R,
  Rnd_DN F rnd1 -> Rnd_DN F rnd2 ->
  forall x, rnd1 x = rnd2 x.forall (F : R -> Prop) (rnd1 rnd2 : R -> R),
Rnd_DN F rnd1 ->
Rnd_DN F rnd2 -> forall x : R, rnd1 x = rnd2 x
Proof.forall (F : R -> Prop) (rnd1 rnd2 : R -> R),
Rnd_DN F rnd1 ->
Rnd_DN F rnd2 -> forall x : R, rnd1 x = rnd2 x
intros F rnd1 rnd2 H1 H2 x.F:R -> Prop
rnd1, rnd2:R -> R
H1:Rnd_DN F rnd1
H2:Rnd_DN F rnd2
x:R
rnd1 x = rnd2 x
now eapply Rnd_DN_pt_unique.
Qed.

Theorem Rnd_UP_pt_monotone :
  forall F : R -> Prop,
  round_pred_monotone (Rnd_UP_pt F).forall F : R -> Prop,
round_pred_monotone (Rnd_UP_pt F)
Proof.forall F : R -> Prop,
round_pred_monotone (Rnd_UP_pt F)
intros F x y f g (Hx1,(_,Hx2)) (Hy1,(Hy2,_)) Hxy.F:R -> Prop
x, y, f, g:R
Hx1:F f
Hx2:forall g0 : R, F g0 -> x <= g0 -> f <= g0
Hy1:F g
Hy2:y <= g
Hxy:x <= y
f <= g
apply Hx2.F:R -> Prop
x, y, f, g:R
Hx1:F f
Hx2:forall g0 : R, F g0 -> x <= g0 -> f <= g0
Hy1:F g
Hy2:y <= g
Hxy:x <= y
F g
F:R -> Prop
x, y, f, g:R
Hx1:F f
Hx2:forall g0 : R, F g0 -> x <= g0 -> f <= g0
Hy1:F g
Hy2:y <= g
Hxy:x <= y
x <= g
apply Hy1.F:R -> Prop
x, y, f, g:R
Hx1:F f
Hx2:forall g0 : R, F g0 -> x <= g0 -> f <= g0
Hy1:F g
Hy2:y <= g
Hxy:x <= y
x <= g
now apply Rle_trans with (1 := Hxy).
Qed.

Theorem Rnd_UP_pt_unique :
  forall F : R -> Prop,
  forall x f1 f2 : R,
  Rnd_UP_pt F x f1 -> Rnd_UP_pt F x f2 ->
  f1 = f2.forall (F : R -> Prop) (x f1 f2 : R),
Rnd_UP_pt F x f1 -> Rnd_UP_pt F x f2 -> f1 = f2
Proof.forall (F : R -> Prop) (x f1 f2 : R),
Rnd_UP_pt F x f1 -> Rnd_UP_pt F x f2 -> f1 = f2
intros F.F:R -> Prop
forall x f1 f2 : R,
Rnd_UP_pt F x f1 -> Rnd_UP_pt F x f2 -> f1 = f2
apply round_unique.F:R -> Prop
round_pred_monotone (Rnd_UP_pt F)
apply Rnd_UP_pt_monotone.
Qed.

Theorem Rnd_UP_unique :
  forall F : R -> Prop,
  forall rnd1 rnd2 : R -> R,
  Rnd_UP F rnd1 -> Rnd_UP F rnd2 ->
  forall x, rnd1 x = rnd2 x.forall (F : R -> Prop) (rnd1 rnd2 : R -> R),
Rnd_UP F rnd1 ->
Rnd_UP F rnd2 -> forall x : R, rnd1 x = rnd2 x
Proof.forall (F : R -> Prop) (rnd1 rnd2 : R -> R),
Rnd_UP F rnd1 ->
Rnd_UP F rnd2 -> forall x : R, rnd1 x = rnd2 x
intros F rnd1 rnd2 H1 H2 x.F:R -> Prop
rnd1, rnd2:R -> R
H1:Rnd_UP F rnd1
H2:Rnd_UP F rnd2
x:R
rnd1 x = rnd2 x
now eapply Rnd_UP_pt_unique.
Qed.

Theorem Rnd_UP_pt_opp :
  forall F : R -> Prop,
  ( forall x, F x -> F (- x) ) ->
  forall x f : R,
  Rnd_DN_pt F x f -> Rnd_UP_pt F (-x) (-f).forall F : R -> Prop,
(forall x : R, F x -> F (- x)) ->
forall x f : R,
Rnd_DN_pt F x f -> Rnd_UP_pt F (- x) (- f)
Proof.forall F : R -> Prop,
(forall x : R, F x -> F (- x)) ->
forall x f : R,
Rnd_DN_pt F x f -> Rnd_UP_pt F (- x) (- f)
intros F HF x f H.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_DN_pt F x f
Rnd_UP_pt F (- x) (- f)
repeat split.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_DN_pt F x f
F (- f)
F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_DN_pt F x f
- x <= - f
F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_DN_pt F x f
forall g : R, F g -> - x <= g -> - f <= g
apply HF.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_DN_pt F x f
F f
F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_DN_pt F x f
- x <= - f
F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_DN_pt F x f
forall g : R, F g -> - x <= g -> - f <= g
apply H.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_DN_pt F x f
- x <= - f
F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_DN_pt F x f
forall g : R, F g -> - x <= g -> - f <= g
apply Ropp_le_contravar.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_DN_pt F x f
f <= x
F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_DN_pt F x f
forall g : R, F g -> - x <= g -> - f <= g
apply H.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_DN_pt F x f
forall g : R, F g -> - x <= g -> - f <= g
intros g Hg.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_DN_pt F x f
g:R
Hg:F g
- x <= g -> - f <= g
rewrite <- (Ropp_involutive g).F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_DN_pt F x f
g:R
Hg:F g
- x <= - - g -> - f <= - - g
intros Hxg.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_DN_pt F x f
g:R
Hg:F g
Hxg:- x <= - - g
- f <= - - g
apply Ropp_le_contravar.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_DN_pt F x f
g:R
Hg:F g
Hxg:- x <= - - g
- g <= f
apply H.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_DN_pt F x f
g:R
Hg:F g
Hxg:- x <= - - g
F (- g)
F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_DN_pt F x f
g:R
Hg:F g
Hxg:- x <= - - g
- g <= x
now apply HF.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_DN_pt F x f
g:R
Hg:F g
Hxg:- x <= - - g
- g <= x
now apply Ropp_le_cancel.
Qed.

Theorem Rnd_DN_pt_opp :
  forall F : R -> Prop,
  ( forall x, F x -> F (- x) ) ->
  forall x f : R,
  Rnd_UP_pt F x f -> Rnd_DN_pt F (-x) (-f).forall F : R -> Prop,
(forall x : R, F x -> F (- x)) ->
forall x f : R,
Rnd_UP_pt F x f -> Rnd_DN_pt F (- x) (- f)
Proof.forall F : R -> Prop,
(forall x : R, F x -> F (- x)) ->
forall x f : R,
Rnd_UP_pt F x f -> Rnd_DN_pt F (- x) (- f)
intros F HF x f H.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_UP_pt F x f
Rnd_DN_pt F (- x) (- f)
repeat split.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_UP_pt F x f
F (- f)
F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_UP_pt F x f
- f <= - x
F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_UP_pt F x f
forall g : R, F g -> g <= - x -> g <= - f
apply HF.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_UP_pt F x f
F f
F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_UP_pt F x f
- f <= - x
F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_UP_pt F x f
forall g : R, F g -> g <= - x -> g <= - f
apply H.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_UP_pt F x f
- f <= - x
F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_UP_pt F x f
forall g : R, F g -> g <= - x -> g <= - f
apply Ropp_le_contravar.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_UP_pt F x f
x <= f
F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_UP_pt F x f
forall g : R, F g -> g <= - x -> g <= - f
apply H.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_UP_pt F x f
forall g : R, F g -> g <= - x -> g <= - f
intros g Hg.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_UP_pt F x f
g:R
Hg:F g
g <= - x -> g <= - f
rewrite <- (Ropp_involutive g).F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_UP_pt F x f
g:R
Hg:F g
- - g <= - x -> - - g <= - f
intros Hxg.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_UP_pt F x f
g:R
Hg:F g
Hxg:- - g <= - x
- - g <= - f
apply Ropp_le_contravar.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_UP_pt F x f
g:R
Hg:F g
Hxg:- - g <= - x
f <= - g
apply H.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_UP_pt F x f
g:R
Hg:F g
Hxg:- - g <= - x
F (- g)
F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_UP_pt F x f
g:R
Hg:F g
Hxg:- - g <= - x
x <= - g
now apply HF.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H:Rnd_UP_pt F x f
g:R
Hg:F g
Hxg:- - g <= - x
x <= - g
now apply Ropp_le_cancel.
Qed.

Theorem Rnd_DN_opp :
  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.forall F : R -> Prop,
(forall x : R, F x -> F (- x)) ->
forall rnd1 rnd2 : R -> R,
Rnd_DN F rnd1 ->
Rnd_UP F rnd2 -> forall x : R, rnd1 (- x) = - rnd2 x
Proof.forall F : R -> Prop,
(forall x : R, F x -> F (- x)) ->
forall rnd1 rnd2 : R -> R,
Rnd_DN F rnd1 ->
Rnd_UP F rnd2 -> forall x : R, rnd1 (- x) = - rnd2 x
intros F HF rnd1 rnd2 H1 H2 x.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
rnd1, rnd2:R -> R
H1:Rnd_DN F rnd1
H2:Rnd_UP F rnd2
x:R
rnd1 (- x) = - rnd2 x
rewrite <- (Ropp_involutive (rnd1 (-x))).F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
rnd1, rnd2:R -> R
H1:Rnd_DN F rnd1
H2:Rnd_UP F rnd2
x:R
- - rnd1 (- x) = - rnd2 x
apply f_equal.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
rnd1, rnd2:R -> R
H1:Rnd_DN F rnd1
H2:Rnd_UP F rnd2
x:R
- rnd1 (- x) = rnd2 x
apply (Rnd_UP_unique F (fun x => - rnd1 (-x))) ; trivial.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
rnd1, rnd2:R -> R
H1:Rnd_DN F rnd1
H2:Rnd_UP F rnd2
x:R
Rnd_UP F (fun x0 : R => - rnd1 (- x0))
intros y.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
rnd1, rnd2:R -> R
H1:Rnd_DN F rnd1
H2:Rnd_UP F rnd2
x, y:R
Rnd_UP_pt F y (- rnd1 (- y))
pattern y at 1 ; rewrite <- Ropp_involutive.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
rnd1, rnd2:R -> R
H1:Rnd_DN F rnd1
H2:Rnd_UP F rnd2
x, y:R
Rnd_UP_pt F (- - y) (- rnd1 (- y))
apply Rnd_UP_pt_opp.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
rnd1, rnd2:R -> R
H1:Rnd_DN F rnd1
H2:Rnd_UP F rnd2
x, y:R
forall x0 : R, F x0 -> F (- x0)
F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
rnd1, rnd2:R -> R
H1:Rnd_DN F rnd1
H2:Rnd_UP F rnd2
x, y:R
Rnd_DN_pt F (- y) (rnd1 (- y))
apply HF.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
rnd1, rnd2:R -> R
H1:Rnd_DN F rnd1
H2:Rnd_UP F rnd2
x, y:R
Rnd_DN_pt F (- y) (rnd1 (- y))
apply H1.
Qed.

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

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

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

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

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

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

Theorem Rnd_ZR_abs :
  forall (F : R -> Prop) (rnd: R-> R),
  Rnd_ZR F rnd ->
  forall x : R,  (Rabs (rnd x) <= Rabs x)%R.forall (F : R -> Prop) (rnd : R -> R),
Rnd_ZR F rnd -> forall x : R, Rabs (rnd x) <= Rabs x
Proof.forall (F : R -> Prop) (rnd : R -> R),
Rnd_ZR F rnd -> forall x : R, Rabs (rnd x) <= Rabs x
intros F rnd H x.F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
Rabs (rnd x) <= Rabs x
assert (F 0%R).F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
F 0
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
Rabs (rnd x) <= Rabs x
replace 0%R with (rnd 0%R).F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
F (rnd 0)
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
rnd 0 = 0
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
Rabs (rnd x) <= Rabs x
eapply H.F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
0 <= 0
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
rnd 0 = 0
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
Rabs (rnd x) <= Rabs x
apply Rle_refl.F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
rnd 0 = 0
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
Rabs (rnd x) <= Rabs x
destruct (H 0%R) as (H1, H2).F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H1:0 <= 0 -> Rnd_DN_pt F 0 (rnd 0)
H2:0 <= 0 -> Rnd_UP_pt F 0 (rnd 0)
rnd 0 = 0
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
Rabs (rnd x) <= Rabs x
apply Rle_antisym.F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H1:0 <= 0 -> Rnd_DN_pt F 0 (rnd 0)
H2:0 <= 0 -> Rnd_UP_pt F 0 (rnd 0)
rnd 0 <= 0
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H1:0 <= 0 -> Rnd_DN_pt F 0 (rnd 0)
H2:0 <= 0 -> Rnd_UP_pt F 0 (rnd 0)
0 <= rnd 0
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
Rabs (rnd x) <= Rabs x
apply H1.F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H1:0 <= 0 -> Rnd_DN_pt F 0 (rnd 0)
H2:0 <= 0 -> Rnd_UP_pt F 0 (rnd 0)
0 <= 0
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H1:0 <= 0 -> Rnd_DN_pt F 0 (rnd 0)
H2:0 <= 0 -> Rnd_UP_pt F 0 (rnd 0)
0 <= rnd 0
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
Rabs (rnd x) <= Rabs x
apply Rle_refl.F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H1:0 <= 0 -> Rnd_DN_pt F 0 (rnd 0)
H2:0 <= 0 -> Rnd_UP_pt F 0 (rnd 0)
0 <= rnd 0
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
Rabs (rnd x) <= Rabs x
apply H2.F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H1:0 <= 0 -> Rnd_DN_pt F 0 (rnd 0)
H2:0 <= 0 -> Rnd_UP_pt F 0 (rnd 0)
0 <= 0
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
Rabs (rnd x) <= Rabs x
apply Rle_refl.F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
Rabs (rnd x) <= Rabs x
(* . *)
destruct (Rle_or_lt 0 x).F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
H1:0 <= x
Rabs (rnd x) <= Rabs x
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
H1:x < 0
Rabs (rnd x) <= Rabs x
(* positive *)
rewrite Rabs_pos_eq with (1 := H1).F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
H1:0 <= x
Rabs (rnd x) <= x
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
H1:x < 0
Rabs (rnd x) <= Rabs x
rewrite Rabs_pos_eq.F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
H1:0 <= x
rnd x <= x
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
H1:0 <= x
0 <= rnd x
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
H1:x < 0
Rabs (rnd x) <= Rabs x
now apply (proj1 (H x)).F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
H1:0 <= x
0 <= rnd x
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
H1:x < 0
Rabs (rnd x) <= Rabs x
now apply (proj1 (H x)).F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
H1:x < 0
Rabs (rnd x) <= Rabs x
(* negative *)
apply Rlt_le in H1.F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
H1:x <= 0
Rabs (rnd x) <= Rabs x
rewrite Rabs_left1 with (1 := H1).F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
H1:x <= 0
Rabs (rnd x) <= - x
rewrite Rabs_left1.F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
H1:x <= 0
- rnd x <= - x
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
H1:x <= 0
rnd x <= 0
apply Ropp_le_contravar.F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
H1:x <= 0
x <= rnd x
F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
H1:x <= 0
rnd x <= 0
now apply (proj2 (H x)).F:R -> Prop
rnd:R -> R
H:Rnd_ZR F rnd
x:R
H0:F 0
H1:x <= 0
rnd x <= 0
now apply (proj2 (H x)).
Qed.

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

Theorem Rnd_N_pt_DN_or_UP :
  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.forall (F : R -> Prop) (x f : R),
Rnd_N_pt F x f -> Rnd_DN_pt F x f \/ Rnd_UP_pt F x f
Proof.forall (F : R -> Prop) (x f : R),
Rnd_N_pt F x f -> Rnd_DN_pt F x f \/ Rnd_UP_pt F x f
intros F x f (Hf1,Hf2).F:R -> Prop
x, f:R
Hf1:F f
Hf2:forall g : R, F g -> Rabs (f - x) <= Rabs (g - x)
Rnd_DN_pt F x f \/ Rnd_UP_pt F x f
destruct (Rle_or_lt x f) as [Hxf|Hxf].F:R -> Prop
x, f:R
Hf1:F f
Hf2:forall g : R, F g -> Rabs (f - x) <= Rabs (g - x)
Hxf:x <= f
Rnd_DN_pt F x f \/ Rnd_UP_pt F x f
F:R -> Prop
x, f:R
Hf1:F f
Hf2:forall g : R, F g -> Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
Rnd_DN_pt F x f \/ Rnd_UP_pt F x f
(* . *)
right.F:R -> Prop
x, f:R
Hf1:F f
Hf2:forall g : R, F g -> Rabs (f - x) <= Rabs (g - x)
Hxf:x <= f
Rnd_UP_pt F x f
F:R -> Prop
x, f:R
Hf1:F f
Hf2:forall g : R, F g -> Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
Rnd_DN_pt F x f \/ Rnd_UP_pt F x f
repeat split ; try assumption.F:R -> Prop
x, f:R
Hf1:F f
Hf2:forall g : R, F g -> Rabs (f - x) <= Rabs (g - x)
Hxf:x <= f
forall g : R, F g -> x <= g -> f <= g
F:R -> Prop
x, f:R
Hf1:F f
Hf2:forall g : R, F g -> Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
Rnd_DN_pt F x f \/ Rnd_UP_pt F x f
intros g Hg Hxg.F:R -> Prop
x, f:R
Hf1:F f
Hf2:forall g0 : R, F g0 -> Rabs (f - x) <= Rabs (g0 - x)
Hxf:x <= f
g:R
Hg:F g
Hxg:x <= g
f <= g
F:R -> Prop
x, f:R
Hf1:F f
Hf2:forall g : R, F g -> Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
Rnd_DN_pt F x f \/ Rnd_UP_pt F x f
specialize (Hf2 g Hg).F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:Rabs (f - x) <= Rabs (g - x)
Hxf:x <= f
Hg:F g
Hxg:x <= g
f <= g
F:R -> Prop
x, f:R
Hf1:F f
Hf2:forall g : R, F g -> Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
Rnd_DN_pt F x f \/ Rnd_UP_pt F x f
rewrite 2!Rabs_pos_eq in Hf2.F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:f - x <= g - x
Hxf:x <= f
Hg:F g
Hxg:x <= g
f <= g
F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:f - x <= Rabs (g - x)
Hxf:x <= f
Hg:F g
Hxg:x <= g
0 <= g - x
F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:Rabs (f - x) <= Rabs (g - x)
Hxf:x <= f
Hg:F g
Hxg:x <= g
0 <= f - x
F:R -> Prop
x, f:R
Hf1:F f
Hf2:forall g : R, F g -> Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
Rnd_DN_pt F x f \/ Rnd_UP_pt F x f
now apply Rplus_le_reg_r with (-x)%R.F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:f - x <= Rabs (g - x)
Hxf:x <= f
Hg:F g
Hxg:x <= g
0 <= g - x
F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:Rabs (f - x) <= Rabs (g - x)
Hxf:x <= f
Hg:F g
Hxg:x <= g
0 <= f - x
F:R -> Prop
x, f:R
Hf1:F f
Hf2:forall g : R, F g -> Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
Rnd_DN_pt F x f \/ Rnd_UP_pt F x f
now apply Rle_0_minus.F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:Rabs (f - x) <= Rabs (g - x)
Hxf:x <= f
Hg:F g
Hxg:x <= g
0 <= f - x
F:R -> Prop
x, f:R
Hf1:F f
Hf2:forall g : R, F g -> Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
Rnd_DN_pt F x f \/ Rnd_UP_pt F x f
now apply Rle_0_minus.F:R -> Prop
x, f:R
Hf1:F f
Hf2:forall g : R, F g -> Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
Rnd_DN_pt F x f \/ Rnd_UP_pt F x f
(* . *)
left.F:R -> Prop
x, f:R
Hf1:F f
Hf2:forall g : R, F g -> Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
Rnd_DN_pt F x f
repeat split ; try assumption.F:R -> Prop
x, f:R
Hf1:F f
Hf2:forall g : R, F g -> Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
f <= x
F:R -> Prop
x, f:R
Hf1:F f
Hf2:forall g : R, F g -> Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
forall g : R, F g -> g <= x -> g <= f
now apply Rlt_le.F:R -> Prop
x, f:R
Hf1:F f
Hf2:forall g : R, F g -> Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
forall g : R, F g -> g <= x -> g <= f
intros g Hg Hxg.F:R -> Prop
x, f:R
Hf1:F f
Hf2:forall g0 : R, F g0 -> Rabs (f - x) <= Rabs (g0 - x)
Hxf:f < x
g:R
Hg:F g
Hxg:g <= x
g <= f
specialize (Hf2 g Hg).F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
Hg:F g
Hxg:g <= x
g <= f
rewrite 2!Rabs_left1 in Hf2.F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:- (f - x) <= - (g - x)
Hxf:f < x
Hg:F g
Hxg:g <= x
g <= f
F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:- (f - x) <= Rabs (g - x)
Hxf:f < x
Hg:F g
Hxg:g <= x
g - x <= 0
F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
Hg:F g
Hxg:g <= x
f - x <= 0
generalize (Ropp_le_cancel _ _ Hf2).F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:- (f - x) <= - (g - x)
Hxf:f < x
Hg:F g
Hxg:g <= x
g - x <= f - x -> g <= f
F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:- (f - x) <= Rabs (g - x)
Hxf:f < x
Hg:F g
Hxg:g <= x
g - x <= 0
F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
Hg:F g
Hxg:g <= x
f - x <= 0
intros H.F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:- (f - x) <= - (g - x)
Hxf:f < x
Hg:F g
Hxg:g <= x
H:g - x <= f - x
g <= f
F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:- (f - x) <= Rabs (g - x)
Hxf:f < x
Hg:F g
Hxg:g <= x
g - x <= 0
F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
Hg:F g
Hxg:g <= x
f - x <= 0
now apply Rplus_le_reg_r with (-x)%R.F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:- (f - x) <= Rabs (g - x)
Hxf:f < x
Hg:F g
Hxg:g <= x
g - x <= 0
F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
Hg:F g
Hxg:g <= x
f - x <= 0
now apply Rle_minus.F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
Hg:F g
Hxg:g <= x
f - x <= 0
apply Rlt_le.F:R -> Prop
x, f:R
Hf1:F f
g:R
Hf2:Rabs (f - x) <= Rabs (g - x)
Hxf:f < x
Hg:F g
Hxg:g <= x
f - x < 0
now apply Rlt_minus.
Qed.

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

Theorem Rnd_N_pt_opp_inv :
  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.forall F : R -> Prop,
(forall x : R, F x -> F (- x)) ->
forall x f : R,
Rnd_N_pt F (- x) (- f) -> Rnd_N_pt F x f
Proof.forall F : R -> Prop,
(forall x : R, F x -> F (- x)) ->
forall x f : R,
Rnd_N_pt F (- x) (- f) -> Rnd_N_pt F x f
intros F HF x f (H1,H2).F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H1:F (- f)
H2:forall g : R,
F g -> Rabs (- f - - x) <= Rabs (g - - x)
Rnd_N_pt F x f
rewrite <- (Ropp_involutive f).F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H1:F (- f)
H2:forall g : R,
F g -> Rabs (- f - - x) <= Rabs (g - - x)
Rnd_N_pt F x (- - f)
repeat split.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H1:F (- f)
H2:forall g : R,
F g -> Rabs (- f - - x) <= Rabs (g - - x)
F (- - f)
F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H1:F (- f)
H2:forall g : R,
F g -> Rabs (- f - - x) <= Rabs (g - - x)
forall g : R, F g -> Rabs (- - f - x) <= Rabs (g - x)
apply HF.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H1:F (- f)
H2:forall g : R,
F g -> Rabs (- f - - x) <= Rabs (g - - x)
F (- f)
F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H1:F (- f)
H2:forall g : R,
F g -> Rabs (- f - - x) <= Rabs (g - - x)
forall g : R, F g -> Rabs (- - f - x) <= Rabs (g - x)
apply H1.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H1:F (- f)
H2:forall g : R,
F g -> Rabs (- f - - x) <= Rabs (g - - x)
forall g : R, F g -> Rabs (- - f - x) <= Rabs (g - x)
intros g H3.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H1:F (- f)
H2:forall g0 : R,
F g0 -> Rabs (- f - - x) <= Rabs (g0 - - x)
g:R
H3:F g
Rabs (- - f - x) <= Rabs (g - x)
rewrite Ropp_involutive.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H1:F (- f)
H2:forall g0 : R,
F g0 -> Rabs (- f - - x) <= Rabs (g0 - - x)
g:R
H3:F g
Rabs (f - x) <= Rabs (g - x)
replace (f - x)%R with (-(-f - -x))%R by ring.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H1:F (- f)
H2:forall g0 : R,
F g0 -> Rabs (- f - - x) <= Rabs (g0 - - x)
g:R
H3:F g
Rabs (- (- f - - x)) <= Rabs (g - x)
replace (g - x)%R with (-(-g - -x))%R by ring.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H1:F (- f)
H2:forall g0 : R,
F g0 -> Rabs (- f - - x) <= Rabs (g0 - - x)
g:R
H3:F g
Rabs (- (- f - - x)) <= Rabs (- (- g - - x))
rewrite 2!Rabs_Ropp.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H1:F (- f)
H2:forall g0 : R,
F g0 -> Rabs (- f - - x) <= Rabs (g0 - - x)
g:R
H3:F g
Rabs (- f - - x) <= Rabs (- g - - x)
apply H2.F:R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
H1:F (- f)
H2:forall g0 : R,
F g0 -> Rabs (- f - - x) <= Rabs (g0 - - x)
g:R
H3:F g
F (- g)
now apply HF.
Qed.

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.forall (F : R -> Prop) (x y f g : R),
Rnd_N_pt F x f -> Rnd_N_pt F y g -> x < y -> f <= g
Proof.forall (F : R -> Prop) (x y f g : R),
Rnd_N_pt F x f -> Rnd_N_pt F y g -> x < y -> f <= g
intros F x y f g (Hf,Hx) (Hg,Hy) Hxy.F:R -> Prop
x, y, f, g:R
Hf:F f
Hx:forall g0 : R,
F g0 -> Rabs (f - x) <= Rabs (g0 - x)
Hg:F g
Hy:forall g0 : R,
F g0 -> Rabs (g - y) <= Rabs (g0 - y)
Hxy:x < y
f <= g
apply Rnot_lt_le.F:R -> Prop
x, y, f, g:R
Hf:F f
Hx:forall g0 : R,
F g0 -> Rabs (f - x) <= Rabs (g0 - x)
Hg:F g
Hy:forall g0 : R,
F g0 -> Rabs (g - y) <= Rabs (g0 - y)
Hxy:x < y
~ g < f
intros Hgf.F:R -> Prop
x, y, f, g:R
Hf:F f
Hx:forall g0 : R,
F g0 -> Rabs (f - x) <= Rabs (g0 - x)
Hg:F g
Hy:forall g0 : R,
F g0 -> Rabs (g - y) <= Rabs (g0 - y)
Hxy:x < y
Hgf:g < f
False
assert (Hfgx := Hx g Hg).F:R -> Prop
x, y, f, g:R
Hf:F f
Hx:forall g0 : R,
F g0 -> Rabs (f - x) <= Rabs (g0 - x)
Hg:F g
Hy:forall g0 : R,
F g0 -> Rabs (g - y) <= Rabs (g0 - y)
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
False
assert (Hgfy := Hy f Hf).F:R -> Prop
x, y, f, g:R
Hf:F f
Hx:forall g0 : R,
F g0 -> Rabs (f - x) <= Rabs (g0 - x)
Hg:F g
Hy:forall g0 : R,
F g0 -> Rabs (g - y) <= Rabs (g0 - y)
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
False
clear F Hf Hg Hx Hy.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
False
destruct (Rle_or_lt x g) as [Hxg|Hgx].x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hxg:x <= g
False
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
False
(* x <= g < f *)
apply Rle_not_lt with (1 := Hfgx).x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hxg:x <= g
Rabs (g - x) < Rabs (f - x)
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
False
rewrite 2!Rabs_pos_eq.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hxg:x <= g
g - x < f - x
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hxg:x <= g
0 <= f - x
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hxg:x <= g
0 <= g - x
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
False
now apply Rplus_lt_compat_r.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hxg:x <= g
0 <= f - x
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hxg:x <= g
0 <= g - x
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
False
apply Rle_0_minus.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hxg:x <= g
x <= f
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hxg:x <= g
0 <= g - x
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
False
apply Rlt_le.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hxg:x <= g
x < f
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hxg:x <= g
0 <= g - x
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
False
now apply Rle_lt_trans with (1 := Hxg).x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hxg:x <= g
0 <= g - x
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
False
now apply Rle_0_minus.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
False
assert (Hgy := Rlt_trans _ _ _ Hgx Hxy).x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
False
destruct (Rle_or_lt f y) as [Hfy|Hyf].x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hfy:f <= y
False
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
False
(* g < f <= y *)
apply Rle_not_lt with (1 := Hgfy).x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hfy:f <= y
Rabs (f - y) < Rabs (g - y)
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
False
rewrite (Rabs_left (g - y)).x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hfy:f <= y
Rabs (f - y) < - (g - y)
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hfy:f <= y
g - y < 0
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
False
2: now apply Rlt_minus.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hfy:f <= y
Rabs (f - y) < - (g - y)
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
False
rewrite Rabs_left1.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hfy:f <= y
- (f - y) < - (g - y)
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hfy:f <= y
f - y <= 0
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
False
apply Ropp_lt_contravar.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hfy:f <= y
g - y < f - y
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hfy:f <= y
f - y <= 0
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
False
now apply Rplus_lt_compat_r.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hfy:f <= y
f - y <= 0
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
False
now apply Rle_minus.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
False
(* g < x < y < f *)
rewrite Rabs_left, Rabs_pos_eq, Ropp_minus_distr in Hgfy.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
False
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:- (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - y
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - y < 0
rewrite Rabs_pos_eq, Rabs_left, Ropp_minus_distr in Hfgx.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:f - x <= x - g
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
False
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:f - x <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - x < 0
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - x
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:- (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - y
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - y < 0
apply Rle_not_lt with (1 := Rplus_le_compat _ _ _ _ Hfgx Hgfy).x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:f - x <= x - g
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
x - g + (f - y) < f - x + (y - g)
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:f - x <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - x < 0
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - x
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:- (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - y
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - y < 0
apply Rminus_lt.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:f - x <= x - g
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
x - g + (f - y) - (f - x + (y - g)) < 0
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:f - x <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - x < 0
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - x
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:- (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - y
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - y < 0
ring_simplify.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:f - x <= x - g
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
2 * x - 2 * y < 0
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:f - x <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - x < 0
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - x
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:- (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - y
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - y < 0
apply Rlt_minus.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:f - x <= x - g
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
2 * x < 2 * y
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:f - x <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - x < 0
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - x
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:- (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - y
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - y < 0
apply Rmult_lt_compat_l.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:f - x <= x - g
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 < 2
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:f - x <= x - g
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
x < y
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:f - x <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - x < 0
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - x
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:- (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - y
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - y < 0
now apply IZR_lt.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:f - x <= x - g
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
x < y
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:f - x <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - x < 0
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - x
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:- (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - y
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - y < 0
exact Hxy.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:f - x <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - x < 0
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - x
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:- (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - y
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - y < 0
now apply Rlt_minus.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - x
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:- (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - y
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - y < 0
apply Rle_0_minus.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
x <= f
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:- (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - y
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - y < 0
apply Rlt_le.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:y - g <= f - y
Hgx:g < x
Hgy:g < y
Hyf:y < f
x < f
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:- (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - y
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - y < 0
now apply Rlt_trans with (1 := Hxy).x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:- (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
0 <= f - y
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - y < 0
apply Rle_0_minus.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:- (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
y <= f
x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - y < 0
now apply Rlt_le.x, y, f, g:R
Hxy:x < y
Hgf:g < f
Hfgx:Rabs (f - x) <= Rabs (g - x)
Hgfy:Rabs (g - y) <= Rabs (f - y)
Hgx:g < x
Hgy:g < y
Hyf:y < f
g - y < 0
now apply Rlt_minus.
Qed.

Theorem Rnd_N_pt_unique :
  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.forall (F : R -> Prop) (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.forall (F : R -> Prop) (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
intros F x d u f1 f2 Hd Hu Hdu.F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
assert (forall f1 f2, Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 < f2 -> False).F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
forall f0 f3 : R,
Rnd_N_pt F x f0 -> Rnd_N_pt F x f3 -> f0 < f3 -> False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
clear f1 f2.F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
forall f1 f2 : R,
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 < f2 -> False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2 intros f1 f2 Hf1 Hf2 H12.F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
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].F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hd2:Rnd_DN_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
apply Rlt_not_eq with (1 := H12).F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hd2:Rnd_DN_pt F x f2
f1 = f2
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
now apply Rnd_DN_pt_unique with (1 := Hd1).F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
apply Hdu.F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
x - d = u - x
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
rewrite Rnd_DN_pt_unique with (1 := Hd) (2 := Hd1).F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
x - f1 = u - x
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
rewrite Rnd_UP_pt_unique with (1 := Hu) (2 := Hu2).F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
x - f1 = f2 - x
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
rewrite <- (Rabs_pos_eq (x - f1)).F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
Rabs (x - f1) = f2 - x
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
0 <= x - f1
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
rewrite <- (Rabs_pos_eq (f2 - x)).F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
Rabs (x - f1) = Rabs (f2 - x)
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
0 <= f2 - x
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
0 <= x - f1
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
rewrite Rabs_minus_sym.F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
Rabs (f1 - x) = Rabs (f2 - x)
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
0 <= f2 - x
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
0 <= x - f1
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
apply Rle_antisym.F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
Rabs (f1 - x) <= Rabs (f2 - x)
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
Rabs (f2 - x) <= Rabs (f1 - x)
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
0 <= f2 - x
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
0 <= x - f1
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
apply Hf1.F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
F f2
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
Rabs (f2 - x) <= Rabs (f1 - x)
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
0 <= f2 - x
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
0 <= x - f1
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2 apply Hf2.F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
Rabs (f2 - x) <= Rabs (f1 - x)
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
0 <= f2 - x
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
0 <= x - f1
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
apply Hf2.F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
F f1
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
0 <= f2 - x
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
0 <= x - f1
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2 apply Hf1.F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
0 <= f2 - x
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
0 <= x - f1
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
apply Rle_0_minus.F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
x <= f2
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
0 <= x - f1
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
apply Hu2.F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
0 <= x - f1
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
apply Rle_0_minus.F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hd1:Rnd_DN_pt F x f1
Hu2:Rnd_UP_pt F x f2
f1 <= x
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
apply Hd1.F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
False
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
apply Rlt_not_le with (1 := H12).F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
f2 <= f1
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
apply Rle_trans with x.F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
f2 <= x
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
x <= f1
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
apply Hd2.F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hd2:Rnd_DN_pt F x f2
x <= f1
F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
apply Hu1.F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
False
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
apply Rgt_not_eq with (1 := H12).F:R -> Prop
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
f1, f2:R
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
H12:f1 < f2
Hu1:Rnd_UP_pt F x f1
Hu2:Rnd_UP_pt F x f2
f2 = f1
F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
now apply Rnd_UP_pt_unique with (1 := Hu2).F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Rnd_N_pt F x f1 -> Rnd_N_pt F x f2 -> f1 = f2
intros Hf1 Hf2.F:R -> Prop
x, d, u, f1, f2:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hdu:x - d <> u - x
H:forall f0 f3 : R,
Rnd_N_pt F x f0 ->
Rnd_N_pt F x f3 -> f0 < f3 -> False
Hf1:Rnd_N_pt F x f1
Hf2:Rnd_N_pt F x f2
f1 = f2
now apply Rle_antisym ; apply Rnot_lt_le ; refine (H _ _ _ _).
Qed.

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

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

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

Theorem Rnd_N_pt_ge_0 :
  forall F : R -> Prop, F 0 ->
  forall x f, 0 <= x ->
  Rnd_N_pt F x f ->
  0 <= f.forall F : R -> Prop,
F 0 ->
forall x f : R, 0 <= x -> Rnd_N_pt F x f -> 0 <= f
Proof.forall F : R -> Prop,
F 0 ->
forall x f : R, 0 <= x -> Rnd_N_pt F x f -> 0 <= f
intros F HF x f [Hx|Hx] Hxf.F:R -> Prop
HF:F 0
x, f:R
Hx:0 < x
Hxf:Rnd_N_pt F x f
0 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 = x
Hxf:Rnd_N_pt F x f
0 <= f
eapply Rnd_N_pt_monotone ; try eassumption.F:R -> Prop
HF:F 0
x, f:R
Hx:0 < x
Hxf:Rnd_N_pt F x f
Rnd_N_pt F 0 0
F:R -> Prop
HF:F 0
x, f:R
Hx:0 = x
Hxf:Rnd_N_pt F x f
0 <= f
now apply Rnd_N_pt_0.F:R -> Prop
HF:F 0
x, f:R
Hx:0 = x
Hxf:Rnd_N_pt F x f
0 <= f
right.F:R -> Prop
HF:F 0
x, f:R
Hx:0 = x
Hxf:Rnd_N_pt F x f
0 = f
apply sym_eq.F:R -> Prop
HF:F 0
x, f:R
Hx:0 = x
Hxf:Rnd_N_pt F x f
f = 0
apply Rnd_N_pt_idempotent with F.F:R -> Prop
HF:F 0
x, f:R
Hx:0 = x
Hxf:Rnd_N_pt F x f
Rnd_N_pt F 0 f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 = x
Hxf:Rnd_N_pt F x f
F 0
now rewrite Hx.F:R -> Prop
HF:F 0
x, f:R
Hx:0 = x
Hxf:Rnd_N_pt F x f
F 0
exact HF.
Qed.

Theorem Rnd_N_pt_le_0 :
  forall F : R -> Prop, F 0 ->
  forall x f, x <= 0 ->
  Rnd_N_pt F x f ->
  f <= 0.forall F : R -> Prop,
F 0 ->
forall x f : R, x <= 0 -> Rnd_N_pt F x f -> f <= 0
Proof.forall F : R -> Prop,
F 0 ->
forall x f : R, x <= 0 -> Rnd_N_pt F x f -> f <= 0
intros F HF x f [Hx|Hx] Hxf.F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Hxf:Rnd_N_pt F x f
f <= 0
F:R -> Prop
HF:F 0
x, f:R
Hx:x = 0
Hxf:Rnd_N_pt F x f
f <= 0
eapply Rnd_N_pt_monotone ; try eassumption.F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Hxf:Rnd_N_pt F x f
Rnd_N_pt F 0 0
F:R -> Prop
HF:F 0
x, f:R
Hx:x = 0
Hxf:Rnd_N_pt F x f
f <= 0
now apply Rnd_N_pt_0.F:R -> Prop
HF:F 0
x, f:R
Hx:x = 0
Hxf:Rnd_N_pt F x f
f <= 0
right.F:R -> Prop
HF:F 0
x, f:R
Hx:x = 0
Hxf:Rnd_N_pt F x f
f = 0
apply Rnd_N_pt_idempotent with F.F:R -> Prop
HF:F 0
x, f:R
Hx:x = 0
Hxf:Rnd_N_pt F x f
Rnd_N_pt F 0 f
F:R -> Prop
HF:F 0
x, f:R
Hx:x = 0
Hxf:Rnd_N_pt F x f
F 0
now rewrite <- Hx.F:R -> Prop
HF:F 0
x, f:R
Hx:x = 0
Hxf:Rnd_N_pt F x f
F 0
exact HF.
Qed.

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).forall F : R -> Prop,
F 0 ->
(forall x : R, F x -> F (- x)) ->
forall x f : R,
Rnd_N_pt F x f -> Rnd_N_pt F (Rabs x) (Rabs f)
Proof.forall F : R -> Prop,
F 0 ->
(forall x : R, F x -> F (- x)) ->
forall x f : R,
Rnd_N_pt F x f -> Rnd_N_pt F (Rabs x) (Rabs f)
intros F HF0 HF x f Hxf.F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Rnd_N_pt F (Rabs x) (Rabs f)
unfold Rabs at 1.F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Rnd_N_pt F (if Rcase_abs x then - x else x) (Rabs f)
destruct (Rcase_abs x) as [Hx|Hx].F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x < 0
Rnd_N_pt F (- x) (Rabs f)
F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x >= 0
Rnd_N_pt F x (Rabs f)
rewrite Rabs_left1.F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x < 0
Rnd_N_pt F (- x) (- f)
F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x < 0
f <= 0
F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x >= 0
Rnd_N_pt F x (Rabs f)
apply Rnd_N_pt_opp_inv.F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x < 0
forall x0 : R, F x0 -> F (- x0)
F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x < 0
Rnd_N_pt F (- - x) (- - f)
F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x < 0
f <= 0
F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x >= 0
Rnd_N_pt F x (Rabs f)
exact HF.F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x < 0
Rnd_N_pt F (- - x) (- - f)
F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x < 0
f <= 0
F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x >= 0
Rnd_N_pt F x (Rabs f)
now rewrite 2!Ropp_involutive.F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x < 0
f <= 0
F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x >= 0
Rnd_N_pt F x (Rabs f)
apply Rnd_N_pt_le_0 with (3 := Hxf).F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x < 0
F 0
F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x < 0
x <= 0
F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x >= 0
Rnd_N_pt F x (Rabs f)
exact HF0.F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x < 0
x <= 0
F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x >= 0
Rnd_N_pt F x (Rabs f)
now apply Rlt_le.F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x >= 0
Rnd_N_pt F x (Rabs f)
rewrite Rabs_pos_eq.F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x >= 0
Rnd_N_pt F x f
F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x >= 0
0 <= f
exact Hxf.F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x >= 0
0 <= f
apply Rnd_N_pt_ge_0 with (3 := Hxf).F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x >= 0
F 0
F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x >= 0
0 <= x
exact HF0.F:R -> Prop
HF0:F 0
HF:forall x0 : R, F x0 -> F (- x0)
x, f:R
Hxf:Rnd_N_pt F x f
Hx:x >= 0
0 <= x
now apply Rge_le.
Qed.

Theorem Rnd_N_pt_DN_UP :
  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.forall (F : R -> Prop) (x d u f : R),
F f ->
Rnd_DN_pt F x d ->
Rnd_UP_pt F x u ->
Rabs (f - x) <= x - d ->
Rabs (f - x) <= u - x -> Rnd_N_pt F x f
Proof.forall (F : R -> Prop) (x d u f : R),
F f ->
Rnd_DN_pt F x d ->
Rnd_UP_pt F x u ->
Rabs (f - x) <= x - d ->
Rabs (f - x) <= u - x -> Rnd_N_pt F x f
intros F x d u f Hf Hxd Hxu Hd Hu.F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
Rnd_N_pt F x f
split.F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
F f
F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
forall g : R, F g -> Rabs (f - x) <= Rabs (g - x)
exact Hf.F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
forall g : R, F g -> Rabs (f - x) <= Rabs (g - x)
intros g Hg.F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Rabs (f - x) <= Rabs (g - x)
destruct (Rnd_DN_UP_pt_split F x d u Hxd Hxu g Hg) as [Hgd|Hgu].F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgd:g <= d
Rabs (f - x) <= Rabs (g - x)
F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgu:u <= g
Rabs (f - x) <= Rabs (g - x)
(* g <= d *)
apply Rle_trans with (1 := Hd).F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgd:g <= d
x - d <= Rabs (g - x)
F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgu:u <= g
Rabs (f - x) <= Rabs (g - x)
rewrite Rabs_left1.F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgd:g <= d
x - d <= - (g - x)
F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgd:g <= d
g - x <= 0
F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgu:u <= g
Rabs (f - x) <= Rabs (g - x)
rewrite Ropp_minus_distr.F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgd:g <= d
x - d <= x - g
F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgd:g <= d
g - x <= 0
F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgu:u <= g
Rabs (f - x) <= Rabs (g - x)
apply Rplus_le_compat_l.F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgd:g <= d
- d <= - g
F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgd:g <= d
g - x <= 0
F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgu:u <= g
Rabs (f - x) <= Rabs (g - x)
now apply Ropp_le_contravar.F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgd:g <= d
g - x <= 0
F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgu:u <= g
Rabs (f - x) <= Rabs (g - x)
apply Rle_minus.F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgd:g <= d
g <= x
F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgu:u <= g
Rabs (f - x) <= Rabs (g - x)
apply Rle_trans with (1 := Hgd).F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgd:g <= d
d <= x
F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgu:u <= g
Rabs (f - x) <= Rabs (g - x)
apply Hxd.F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgu:u <= g
Rabs (f - x) <= Rabs (g - x)
(* u <= g *)
apply Rle_trans with (1 := Hu).F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgu:u <= g
u - x <= Rabs (g - x)
rewrite Rabs_pos_eq.F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgu:u <= g
u - x <= g - x
F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgu:u <= g
0 <= g - x
now apply Rplus_le_compat_r.F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgu:u <= g
0 <= g - x
apply Rle_0_minus.F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgu:u <= g
x <= g
apply Rle_trans with (2 := Hgu).F:R -> Prop
x, d, u, f:R
Hf:F f
Hxd:Rnd_DN_pt F x d
Hxu:Rnd_UP_pt F x u
Hd:Rabs (f - x) <= x - d
Hu:Rabs (f - x) <= u - x
g:R
Hg:F g
Hgu:u <= g
x <= u
apply Hxu.
Qed.

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

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

Definition Rnd_NG_pt_unique_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.

Theorem Rnd_NG_pt_unique :
  forall (F : R -> Prop) (P : R -> R -> Prop),
  Rnd_NG_pt_unique_prop F P ->
  forall x f1 f2 : R,
  Rnd_NG_pt F P x f1 -> Rnd_NG_pt F P x f2 ->
  f1 = f2.forall (F : R -> Prop) (P : R -> R -> Prop),
Rnd_NG_pt_unique_prop F P ->
forall x f1 f2 : R,
Rnd_NG_pt F P x f1 -> Rnd_NG_pt F P x f2 -> f1 = f2
Proof.forall (F : R -> Prop) (P : R -> R -> Prop),
Rnd_NG_pt_unique_prop F P ->
forall x f1 f2 : R,
Rnd_NG_pt F P x f1 -> Rnd_NG_pt F P x f2 -> f1 = f2
intros F P HP x f1 f2 (H1a,H1b) (H2a,H2b).F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1 \/ (forall f0 : R, Rnd_N_pt F x f0 -> f0 = f1)
H2a:Rnd_N_pt F x f2
H2b:P x f2 \/ (forall f0 : R, Rnd_N_pt F x f0 -> f0 = f2)
f1 = f2
destruct H1b as [H1b|H1b].F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:P x f2 \/ (forall f0 : R, Rnd_N_pt F x f0 -> f0 = f2)
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f1
H2a:Rnd_N_pt F x f2
H2b:P x f2 \/ (forall f0 : R, Rnd_N_pt F x f0 -> f0 = f2)
f1 = f2
destruct H2b as [H2b|H2b].F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:P x f2
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f2
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f1
H2a:Rnd_N_pt F x f2
H2b:P x f2 \/ (forall f0 : R, Rnd_N_pt F x f0 -> f0 = f2)
f1 = f2
destruct (Rnd_N_pt_DN_or_UP _ _ _ H1a) as [H1c|H1c] ;
  destruct (Rnd_N_pt_DN_or_UP _ _ _ H2a) as [H2c|H2c].F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:P x f2
H1c:Rnd_DN_pt F x f1
H2c:Rnd_DN_pt F x f2
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:P x f2
H1c:Rnd_DN_pt F x f1
H2c:Rnd_UP_pt F x f2
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:P x f2
H1c:Rnd_UP_pt F x f1
H2c:Rnd_DN_pt F x f2
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:P x f2
H1c:Rnd_UP_pt F x f1
H2c:Rnd_UP_pt F x f2
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f2
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f1
H2a:Rnd_N_pt F x f2
H2b:P x f2 \/ (forall f0 : R, Rnd_N_pt F x f0 -> f0 = f2)
f1 = f2
eapply Rnd_DN_pt_unique ; eassumption.F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:P x f2
H1c:Rnd_DN_pt F x f1
H2c:Rnd_UP_pt F x f2
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:P x f2
H1c:Rnd_UP_pt F x f1
H2c:Rnd_DN_pt F x f2
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:P x f2
H1c:Rnd_UP_pt F x f1
H2c:Rnd_UP_pt F x f2
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f2
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f1
H2a:Rnd_N_pt F x f2
H2b:P x f2 \/ (forall f0 : R, Rnd_N_pt F x f0 -> f0 = f2)
f1 = f2
now apply (HP x f1 f2).F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:P x f2
H1c:Rnd_UP_pt F x f1
H2c:Rnd_DN_pt F x f2
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:P x f2
H1c:Rnd_UP_pt F x f1
H2c:Rnd_UP_pt F x f2
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f2
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f1
H2a:Rnd_N_pt F x f2
H2b:P x f2 \/ (forall f0 : R, Rnd_N_pt F x f0 -> f0 = f2)
f1 = f2
apply sym_eq.F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:P x f2
H1c:Rnd_UP_pt F x f1
H2c:Rnd_DN_pt F x f2
f2 = f1
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:P x f2
H1c:Rnd_UP_pt F x f1
H2c:Rnd_UP_pt F x f2
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f2
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f1
H2a:Rnd_N_pt F x f2
H2b:P x f2 \/ (forall f0 : R, Rnd_N_pt F x f0 -> f0 = f2)
f1 = f2
now apply (HP x f2 f1 H2c H2a H1c H1a).F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:P x f2
H1c:Rnd_UP_pt F x f1
H2c:Rnd_UP_pt F x f2
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f2
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f1
H2a:Rnd_N_pt F x f2
H2b:P x f2 \/ (forall f0 : R, Rnd_N_pt F x f0 -> f0 = f2)
f1 = f2
eapply Rnd_UP_pt_unique ; eassumption.F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:P x f1
H2a:Rnd_N_pt F x f2
H2b:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f2
f1 = f2
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f1
H2a:Rnd_N_pt F x f2
H2b:P x f2 \/ (forall f0 : R, Rnd_N_pt F x f0 -> f0 = f2)
f1 = f2
now apply H2b.F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f1
H2a:Rnd_N_pt F x f2
H2b:P x f2 \/ (forall f0 : R, Rnd_N_pt F x f0 -> f0 = f2)
f1 = f2
apply sym_eq.F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, f1, f2:R
H1a:Rnd_N_pt F x f1
H1b:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f1
H2a:Rnd_N_pt F x f2
H2b:P x f2 \/ (forall f0 : R, Rnd_N_pt F x f0 -> f0 = f2)
f2 = f1
now apply H1b.
Qed.

Theorem Rnd_NG_pt_monotone :
  forall (F : R -> Prop) (P : R -> R -> Prop),
  Rnd_NG_pt_unique_prop F P ->
  round_pred_monotone (Rnd_NG_pt F P).forall (F : R -> Prop) (P : R -> R -> Prop),
Rnd_NG_pt_unique_prop F P ->
round_pred_monotone (Rnd_NG_pt F P)
Proof.forall (F : R -> Prop) (P : R -> R -> Prop),
Rnd_NG_pt_unique_prop F P ->
round_pred_monotone (Rnd_NG_pt F P)
intros F P HP x y f g (Hf,Hx) (Hg,Hy) [Hxy|Hxy].F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, y, f, g:R
Hf:Rnd_N_pt F x f
Hx:P x f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Hg:Rnd_N_pt F y g
Hy:P y g \/
(forall f2 : R, Rnd_N_pt F y f2 -> f2 = g)
Hxy:x < y
f <= g
F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, y, f, g:R
Hf:Rnd_N_pt F x f
Hx:P x f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Hg:Rnd_N_pt F y g
Hy:P y g \/
(forall f2 : R, Rnd_N_pt F y f2 -> f2 = g)
Hxy:x = y
f <= g
now apply Rnd_N_pt_monotone with F x y.F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, y, f, g:R
Hf:Rnd_N_pt F x f
Hx:P x f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Hg:Rnd_N_pt F y g
Hy:P y g \/
(forall f2 : R, Rnd_N_pt F y f2 -> f2 = g)
Hxy:x = y
f <= g
apply Req_le.F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, y, f, g:R
Hf:Rnd_N_pt F x f
Hx:P x f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Hg:Rnd_N_pt F y g
Hy:P y g \/
(forall f2 : R, Rnd_N_pt F y f2 -> f2 = g)
Hxy:x = y
f = g
rewrite <- Hxy in Hg, Hy.F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
x, y, f, g:R
Hf:Rnd_N_pt F x f
Hx:P x f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Hg:Rnd_N_pt F x g
Hy:P x g \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = g)
Hxy:x = y
f = g
eapply Rnd_NG_pt_unique ; try split ; eassumption.
Qed.

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

Theorem Rnd_NG_pt_opp_inv :
  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.forall (F : R -> Prop) (P : R -> R -> Prop),
(forall x : R, F x -> F (- x)) ->
(forall x f : R, 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.forall (F : R -> Prop) (P : R -> R -> Prop),
(forall x : R, F x -> F (- x)) ->
(forall x f : R, P x f -> P (- x) (- f)) ->
forall x f : R,
Rnd_NG_pt F P (- x) (- f) -> Rnd_NG_pt F P x f
intros F P HF HP x f (H1,H2).F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:P (- x) (- f) \/
(forall f2 : R, Rnd_N_pt F (- x) f2 -> f2 = - f)
Rnd_NG_pt F P x f
split.F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:P (- x) (- f) \/
(forall f2 : R, Rnd_N_pt F (- x) f2 -> f2 = - f)
Rnd_N_pt F x f
F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:P (- x) (- f) \/
(forall f2 : R, Rnd_N_pt F (- x) f2 -> f2 = - f)
P x f \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
now apply Rnd_N_pt_opp_inv.F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:P (- x) (- f) \/
(forall f2 : R, Rnd_N_pt F (- x) f2 -> f2 = - f)
P x f \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
destruct H2 as [H2|H2].F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:P (- x) (- f)
P x f \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:forall f2 : R, Rnd_N_pt F (- x) f2 -> f2 = - f
P x f \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
left.F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:P (- x) (- f)
P x f
F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:forall f2 : R, Rnd_N_pt F (- x) f2 -> f2 = - f
P x f \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
rewrite <- (Ropp_involutive x), <- (Ropp_involutive f).F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:P (- x) (- f)
P (- - x) (- - f)
F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:forall f2 : R, Rnd_N_pt F (- x) f2 -> f2 = - f
P x f \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
now apply HP.F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:forall f2 : R, Rnd_N_pt F (- x) f2 -> f2 = - f
P x f \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
right.F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:forall f2 : R, Rnd_N_pt F (- x) f2 -> f2 = - f
forall f2 : R, Rnd_N_pt F x f2 -> f2 = f
intros f2 Hxf2.F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:forall f0 : R, Rnd_N_pt F (- x) f0 -> f0 = - f
f2:R
Hxf2:Rnd_N_pt F x f2
f2 = f
rewrite <- (Ropp_involutive f).F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:forall f0 : R, Rnd_N_pt F (- x) f0 -> f0 = - f
f2:R
Hxf2:Rnd_N_pt F x f2
f2 = - - f
rewrite <- H2 with (-f2).F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:forall f0 : R, Rnd_N_pt F (- x) f0 -> f0 = - f
f2:R
Hxf2:Rnd_N_pt F x f2
f2 = - - f2
F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:forall f0 : R, Rnd_N_pt F (- x) f0 -> f0 = - f
f2:R
Hxf2:Rnd_N_pt F x f2
Rnd_N_pt F (- x) (- f2)
apply sym_eq.F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:forall f0 : R, Rnd_N_pt F (- x) f0 -> f0 = - f
f2:R
Hxf2:Rnd_N_pt F x f2
- - f2 = f2
F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:forall f0 : R, Rnd_N_pt F (- x) f0 -> f0 = - f
f2:R
Hxf2:Rnd_N_pt F x f2
Rnd_N_pt F (- x) (- f2)
apply Ropp_involutive.F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:forall f0 : R, Rnd_N_pt F (- x) f0 -> f0 = - f
f2:R
Hxf2:Rnd_N_pt F x f2
Rnd_N_pt F (- x) (- f2)
apply Rnd_N_pt_opp_inv.F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:forall f0 : R, Rnd_N_pt F (- x) f0 -> f0 = - f
f2:R
Hxf2:Rnd_N_pt F x f2
forall x0 : R, F x0 -> F (- x0)
F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:forall f0 : R, Rnd_N_pt F (- x) f0 -> f0 = - f
f2:R
Hxf2:Rnd_N_pt F x f2
Rnd_N_pt F (- - x) (- - f2)
exact HF.F:R -> Prop
P:R -> R -> Prop
HF:forall x0 : R, F x0 -> F (- x0)
HP:forall x0 f0 : R, P x0 f0 -> P (- x0) (- f0)
x, f:R
H1:Rnd_N_pt F (- x) (- f)
H2:forall f0 : R, Rnd_N_pt F (- x) f0 -> f0 = - f
f2:R
Hxf2:Rnd_N_pt F x f2
Rnd_N_pt F (- - x) (- - f2)
now rewrite 2!Ropp_involutive.
Qed.

Theorem Rnd_NG_unique :
  forall (F : R -> Prop) (P : R -> R -> Prop),
  Rnd_NG_pt_unique_prop F P ->
  forall rnd1 rnd2 : R -> R,
  Rnd_NG F P rnd1 -> Rnd_NG F P rnd2 ->
  forall x, rnd1 x = rnd2 x.forall (F : R -> Prop) (P : R -> R -> Prop),
Rnd_NG_pt_unique_prop F P ->
forall rnd1 rnd2 : R -> R,
Rnd_NG F P rnd1 ->
Rnd_NG F P rnd2 -> forall x : R, rnd1 x = rnd2 x
Proof.forall (F : R -> Prop) (P : R -> R -> Prop),
Rnd_NG_pt_unique_prop F P ->
forall rnd1 rnd2 : R -> R,
Rnd_NG F P rnd1 ->
Rnd_NG F P rnd2 -> forall x : R, rnd1 x = rnd2 x
intros F P HP rnd1 rnd2 H1 H2 x.F:R -> Prop
P:R -> R -> Prop
HP:Rnd_NG_pt_unique_prop F P
rnd1, rnd2:R -> R
H1:Rnd_NG F P rnd1
H2:Rnd_NG F P rnd2
x:R
rnd1 x = rnd2 x
now apply Rnd_NG_pt_unique 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.forall F : R -> Prop,
F 0 ->
forall x f : R,
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
Proof.forall F : R -> Prop,
F 0 ->
forall x f : R,
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
intros F HF x f.F:R -> Prop
HF:F 0
x, f:R
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
destruct (Rle_or_lt 0 x) as [Hx|Hx].F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
(* *)
split ; intros (H1, H2).F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
(* . *)
assert (Hf := Rnd_N_pt_ge_0 F HF x f Hx H1).F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
split.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
Rnd_N_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
exact H1.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
destruct (Rnd_N_pt_DN_or_UP _ _ _ H1) as [H3|H3].F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_DN_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
(* . . *)
right.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_DN_pt F x f
forall f2 : R, Rnd_N_pt F x f2 -> f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
intros f2 Hxf2.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R,
Rnd_N_pt F x f0 -> Rabs f0 <= Rabs f
Hf:0 <= f
H3:Rnd_DN_pt F x f
f2:R
Hxf2:Rnd_N_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
specialize (H2 _ Hxf2).F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
destruct (Rnd_N_pt_DN_or_UP _ _ _ Hxf2) as [H4|H4].F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
eapply Rnd_DN_pt_unique ; eassumption.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
apply Rle_antisym.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
rewrite Rabs_pos_eq with (1 := Hf) in H2.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= f
Hf:0 <= f
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
rewrite Rabs_pos_eq in H2.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:f2 <= f
Hf:0 <= f
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= f
Hf:0 <= f
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
0 <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
exact H2.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= f
Hf:0 <= f
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
0 <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
now apply Rnd_N_pt_ge_0 with F x.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
apply Rle_trans with x.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f <= x
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
x <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
apply H3.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
x <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
apply H4.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
(* . . *)
left.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
rewrite Rabs_pos_eq with (1 := Hf).F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs x <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
rewrite Rabs_pos_eq with (1 := Hx).F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:0 <= f
H3:Rnd_UP_pt F x f
x <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
apply H3.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
(* . *)
split.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
forall f2 : R, Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
exact H1.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
forall f2 : R, Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
intros f2 Hxf2.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f0 : R, Rnd_N_pt F x f0 -> f0 = f)
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
destruct H2 as [H2|H2].F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
assert (Hf := Rnd_N_pt_ge_0 F HF x f Hx H1).F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
assert (Hf2 := Rnd_N_pt_ge_0 F HF x f2 Hx Hxf2).F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
rewrite 2!Rabs_pos_eq ; trivial.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
rewrite 2!Rabs_pos_eq in H2 ; trivial.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:x <= f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
destruct (Rnd_N_pt_DN_or_UP _ _ _ Hxf2) as [H3|H3].F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:x <= f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
H3:Rnd_DN_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:x <= f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
H3:Rnd_UP_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
apply Rle_trans with (2 := H2).F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:x <= f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
H3:Rnd_DN_pt F x f2
f2 <= x
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:x <= f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
H3:Rnd_UP_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
apply H3.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:x <= f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
H3:Rnd_UP_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
apply H3.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:x <= f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
H3:Rnd_UP_pt F x f2
F f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:x <= f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
H3:Rnd_UP_pt F x f2
x <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
apply H1.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:x <= f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
H3:Rnd_UP_pt F x f2
x <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
apply H2.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
rewrite (H2 _ Hxf2).F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
apply Rle_refl.F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
(* *)
assert (Hx' := Rlt_le _ _ Hx).F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Hx':x <= 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
clear Hx.F:R -> Prop
HF:F 0
x, f:R
Hx':x <= 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f rename Hx' into Hx.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
Rnd_NA_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
split ; intros (H1, H2).F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
(* . *)
assert (Hf := Rnd_N_pt_le_0 F HF x f Hx H1).F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:f <= 0
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
split.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:f <= 0
Rnd_N_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:f <= 0
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
exact H1.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:f <= 0
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
destruct (Rnd_N_pt_DN_or_UP _ _ _ H1) as [H3|H3].F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_DN_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
(* . . *)
left.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_DN_pt F x f
Rabs x <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
rewrite Rabs_left1 with (1 := Hf).F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_DN_pt F x f
Rabs x <= - f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
rewrite Rabs_left1 with (1 := Hx).F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_DN_pt F x f
- x <= - f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
apply Ropp_le_contravar.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_DN_pt F x f
f <= x
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
apply H3.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
(* . . *)
right.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
forall f2 : R, Rnd_N_pt F x f2 -> f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
intros f2 Hxf2.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R,
Rnd_N_pt F x f0 -> Rabs f0 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
f2:R
Hxf2:Rnd_N_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
specialize (H2 _ Hxf2).F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
destruct (Rnd_N_pt_DN_or_UP _ _ _ Hxf2) as [H4|H4].F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
apply Rle_antisym.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
apply Rle_trans with x.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f2 <= x
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
x <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
apply H4.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
x <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
apply H3.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
rewrite Rabs_left1 with (1 := Hf) in H2.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= - f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
rewrite Rabs_left1 in H2.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:- f2 <= - f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= - f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f2 <= 0
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
now apply Ropp_le_cancel.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= - f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f2 <= 0
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
now apply Rnd_N_pt_le_0 with F x.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f2 <= Rabs f
Hf:f <= 0
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
eapply Rnd_UP_pt_unique ; eassumption.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_NA_pt F x f
(* . *)
split.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
forall f2 : R, Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
exact H1.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
forall f2 : R, Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
intros f2 Hxf2.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f \/
(forall f0 : R, Rnd_N_pt F x f0 -> f0 = f)
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
destruct H2 as [H2|H2].F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
assert (Hf := Rnd_N_pt_le_0 F HF x f Hx H1).F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
assert (Hf2 := Rnd_N_pt_le_0 F HF x f2 Hx Hxf2).F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
Rabs f2 <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
rewrite 2!Rabs_left1 ; trivial.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs x <= Rabs f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
- f2 <= - f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
rewrite 2!Rabs_left1 in H2 ; trivial.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:- x <= - f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
- f2 <= - f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
apply Ropp_le_contravar.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:- x <= - f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
apply Ropp_le_cancel in H2.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
destruct (Rnd_N_pt_DN_or_UP _ _ _ Hxf2) as [H3|H3].F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
H3:Rnd_DN_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
H3:Rnd_UP_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
apply H3.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
H3:Rnd_DN_pt F x f2
F f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
H3:Rnd_DN_pt F x f2
f <= x
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
H3:Rnd_UP_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
apply H1.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
H3:Rnd_DN_pt F x f2
f <= x
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
H3:Rnd_UP_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
apply H2.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
H3:Rnd_UP_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
apply Rle_trans with (1 := H2).F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
H3:Rnd_UP_pt F x f2
x <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
apply H3.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f2 <= Rabs f
rewrite (H2 _ Hxf2).F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f
apply Rle_refl.
Qed.

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

Theorem Rnd_NA_pt_unique :
  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.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.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
intros F HF x f1 f2 H1 H2.F:R -> Prop
HF:F 0
x, f1, f2:R
H1:Rnd_NA_pt F x f1
H2:Rnd_NA_pt F x f2
f1 = f2
apply (Rnd_NG_pt_unique F _ (Rnd_NA_pt_unique_prop F HF) x).F:R -> Prop
HF:F 0
x, f1, f2:R
H1:Rnd_NA_pt F x f1
H2:Rnd_NA_pt F x f2
Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b) x f1
F:R -> Prop
HF:F 0
x, f1, f2:R
H1:Rnd_NA_pt F x f1
H2:Rnd_NA_pt F x f2
Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b) x f2
now apply -> Rnd_NA_NG_pt.F:R -> Prop
HF:F 0
x, f1, f2:R
H1:Rnd_NA_pt F x f1
H2:Rnd_NA_pt F x f2
Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b) x f2
now apply -> Rnd_NA_NG_pt.
Qed.

Theorem Rnd_NA_pt_N :
  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.forall F : R -> Prop,
F 0 ->
forall x f : R,
Rnd_N_pt F x f -> Rabs x <= Rabs f -> Rnd_NA_pt F x f
Proof.forall F : R -> Prop,
F 0 ->
forall x f : R,
Rnd_N_pt F x f -> Rabs x <= Rabs f -> Rnd_NA_pt F x f
intros F HF x f Rxf Hxf.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
Rnd_NA_pt F x f
split.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
Rnd_N_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
forall f2 : R, Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
apply Rxf.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
forall f2 : R, Rnd_N_pt F x f2 -> Rabs f2 <= Rabs f
intros g Rxg.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
Rabs g <= Rabs f
destruct (Rabs_eq_Rabs (f - x) (g - x)) as [H|H].F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
Rabs (f - x) = Rabs (g - x)
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = g - x
Rabs g <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
Rabs g <= Rabs f
apply Rle_antisym.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
Rabs (f - x) <= Rabs (g - x)
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
Rabs (g - x) <= Rabs (f - x)
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = g - x
Rabs g <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
Rabs g <= Rabs f
apply Rxf.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
F g
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
Rabs (g - x) <= Rabs (f - x)
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = g - x
Rabs g <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
Rabs g <= Rabs f
apply Rxg.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
Rabs (g - x) <= Rabs (f - x)
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = g - x
Rabs g <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
Rabs g <= Rabs f
apply Rxg.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
F f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = g - x
Rabs g <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
Rabs g <= Rabs f
apply Rxf.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = g - x
Rabs g <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
Rabs g <= Rabs f
(* *)
replace g with f.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = g - x
Rabs f <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = g - x
f = g
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
Rabs g <= Rabs f
apply Rle_refl.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = g - x
f = g
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
Rabs g <= Rabs f
apply Rplus_eq_reg_r with (1 := H).F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
Rabs g <= Rabs f
(* *)
assert (g = 2 * x - f)%R.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
g = 2 * x - f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Rabs g <= Rabs f
replace (2 * x - f)%R with (x - (f - x))%R by ring.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
g = x - (f - x)
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Rabs g <= Rabs f
rewrite H.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
g = x - - (g - x)
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Rabs g <= Rabs f
ring.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Rabs g <= Rabs f
destruct (Rle_lt_dec 0 x) as [Hx|Hx].F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:0 <= x
Rabs g <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs g <= Rabs f
(* . *)
revert Hxf.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:0 <= x
Rabs x <= Rabs f -> Rabs g <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs g <= Rabs f
rewrite Rabs_pos_eq with (1 := Hx).F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:0 <= x
x <= Rabs f -> Rabs g <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs g <= Rabs f
rewrite 2!Rabs_pos_eq ; try ( apply (Rnd_N_pt_ge_0 F HF x) ; assumption ).F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:0 <= x
x <= f -> g <= f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs g <= Rabs f
intros Hxf.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:0 <= x
Hxf:x <= f
g <= f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs g <= Rabs f
rewrite H0.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:0 <= x
Hxf:x <= f
2 * x - f <= f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs g <= Rabs f
apply Rplus_le_reg_r with f.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:0 <= x
Hxf:x <= f
2 * x - f + f <= f + f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs g <= Rabs f
ring_simplify.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:0 <= x
Hxf:x <= f
2 * x <= 2 * f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs g <= Rabs f
apply Rmult_le_compat_l with (2 := Hxf).F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:0 <= x
Hxf:x <= f
0 <= 2
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs g <= Rabs f
now apply IZR_le.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs x <= Rabs f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs g <= Rabs f
(* . *)
revert Hxf.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs x <= Rabs f -> Rabs g <= Rabs f
apply Rlt_le in Hx.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
Rabs x <= Rabs f -> Rabs g <= Rabs f
rewrite Rabs_left1 with (1 := Hx).F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
- x <= Rabs f -> Rabs g <= Rabs f
rewrite 2!Rabs_left1 ; try ( apply (Rnd_N_pt_le_0 F HF x) ; assumption ).F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
- x <= - f -> - g <= - f
intros Hxf.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
Hxf:- x <= - f
- g <= - f
rewrite H0.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
Hxf:- x <= - f
- (2 * x - f) <= - f
apply Ropp_le_contravar.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
Hxf:- x <= - f
f <= 2 * x - f
apply Rplus_le_reg_r with f.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
Hxf:- x <= - f
f + f <= 2 * x - f + f
ring_simplify.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
Hxf:- x <= - f
2 * f <= 2 * x
apply Rmult_le_compat_l.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
Hxf:- x <= - f
0 <= 2
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
Hxf:- x <= - f
f <= x
now apply IZR_le.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
Hxf:- x <= - f
f <= x
now apply Ropp_le_cancel.
Qed.

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

Theorem Rnd_NA_pt_monotone :
  forall F : R -> Prop,
  F 0 ->
  round_pred_monotone (Rnd_NA_pt F).forall F : R -> Prop,
F 0 -> round_pred_monotone (Rnd_NA_pt F)
Proof.forall F : R -> Prop,
F 0 -> round_pred_monotone (Rnd_NA_pt F)
intros F HF x y f g Hxf Hyg Hxy.F:R -> Prop
HF:F 0
x, y, f, g:R
Hxf:Rnd_NA_pt F x f
Hyg:Rnd_NA_pt F y g
Hxy:x <= y
f <= g
apply (Rnd_NG_pt_monotone F _ (Rnd_NA_pt_unique_prop F HF) x y).F:R -> Prop
HF:F 0
x, y, f, g:R
Hxf:Rnd_NA_pt F x f
Hyg:Rnd_NA_pt F y g
Hxy:x <= y
Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b) x f
F:R -> Prop
HF:F 0
x, y, f, g:R
Hxf:Rnd_NA_pt F x f
Hyg:Rnd_NA_pt F y g
Hxy:x <= y
Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b) y g
F:R -> Prop
HF:F 0
x, y, f, g:R
Hxf:Rnd_NA_pt F x f
Hyg:Rnd_NA_pt F y g
Hxy:x <= y
x <= y
now apply -> Rnd_NA_NG_pt.F:R -> Prop
HF:F 0
x, y, f, g:R
Hxf:Rnd_NA_pt F x f
Hyg:Rnd_NA_pt F y g
Hxy:x <= y
Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b) y g
F:R -> Prop
HF:F 0
x, y, f, g:R
Hxf:Rnd_NA_pt F x f
Hyg:Rnd_NA_pt F y g
Hxy:x <= y
x <= y
now apply -> Rnd_NA_NG_pt.F:R -> Prop
HF:F 0
x, y, f, g:R
Hxf:Rnd_NA_pt F x f
Hyg:Rnd_NA_pt F y g
Hxy:x <= y
x <= y
exact Hxy.
Qed.

Theorem Rnd_NA_pt_refl :
  forall F : R -> Prop,
  forall x : R, F x ->
  Rnd_NA_pt F x x.forall (F : R -> Prop) (x : R), F x -> Rnd_NA_pt F x x
Proof.forall (F : R -> Prop) (x : R), F x -> Rnd_NA_pt F x x
intros F x Hx.F:R -> Prop
x:R
Hx:F x
Rnd_NA_pt F x x
split.F:R -> Prop
x:R
Hx:F x
Rnd_N_pt F x x
F:R -> Prop
x:R
Hx:F x
forall f2 : R, Rnd_N_pt F x f2 -> Rabs f2 <= Rabs x
now apply Rnd_N_pt_refl.F:R -> Prop
x:R
Hx:F x
forall f2 : R, Rnd_N_pt F x f2 -> Rabs f2 <= Rabs x
intros f Hxf.F:R -> Prop
x:R
Hx:F x
f:R
Hxf:Rnd_N_pt F x f
Rabs f <= Rabs x
apply Req_le.F:R -> Prop
x:R
Hx:F x
f:R
Hxf:Rnd_N_pt F x f
Rabs f = Rabs x
apply f_equal.F:R -> Prop
x:R
Hx:F x
f:R
Hxf:Rnd_N_pt F x f
f = x
now apply Rnd_N_pt_idempotent with (1 := Hxf).
Qed.

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

Theorem Rnd_N0_NG_pt :
  forall F : R -> Prop,
  F 0 ->
  forall x f,
  Rnd_N0_pt F x f <-> Rnd_NG_pt F (fun x f => Rabs f <= Rabs x) x f.forall F : R -> Prop,
F 0 ->
forall x f : R,
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
Proof.forall F : R -> Prop,
F 0 ->
forall x f : R,
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
intros F HF x f.F:R -> Prop
HF:F 0
x, f:R
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
destruct (Rle_or_lt 0 x) as [Hx|Hx].F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
(* *)
split ; intros (H1, H2).F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
(* . *)
assert (Hf := Rnd_N_pt_ge_0 F HF x f Hx H1).F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:0 <= f
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
split.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:0 <= f
Rnd_N_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:0 <= f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
exact H1.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:0 <= f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
destruct (Rnd_N_pt_DN_or_UP _ _ _ H1) as [H3|H3].F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_DN_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
(* . . *)
left.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_DN_pt F x f
Rabs f <= Rabs x
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
rewrite Rabs_pos_eq with (1 := Hf).F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_DN_pt F x f
f <= Rabs x
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
rewrite Rabs_pos_eq with (1 := Hx).F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_DN_pt F x f
f <= x
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
apply H3.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
(* . . *)
right.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
forall f2 : R, Rnd_N_pt F x f2 -> f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
intros f2 Hxf2.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R,
Rnd_N_pt F x f0 -> Rabs f <= Rabs f0
Hf:0 <= f
H3:Rnd_UP_pt F x f
f2:R
Hxf2:Rnd_N_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
specialize (H2 _ Hxf2).F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
destruct (Rnd_N_pt_DN_or_UP _ _ _ Hxf2) as [H4|H4].F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
apply Rle_antisym.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
apply Rle_trans with x.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f2 <= x
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
x <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
apply H4.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
x <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
apply H3.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
rewrite Rabs_pos_eq with (1 := Hf) in H2.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
rewrite Rabs_pos_eq in H2.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:f <= f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
0 <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
exact H2.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
0 <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
now apply Rnd_N_pt_ge_0 with F x.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:0 <= f
H3:Rnd_UP_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
eapply Rnd_UP_pt_unique ; eassumption.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
(* . *)
split.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
forall f2 : R, Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
exact H1.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
forall f2 : R, Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
intros f2 Hxf2.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f0 : R, Rnd_N_pt F x f0 -> f0 = f)
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
destruct H2 as [H2|H2].F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
assert (Hf := Rnd_N_pt_ge_0 F HF x f Hx H1).F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
assert (Hf2 := Rnd_N_pt_ge_0 F HF x f2 Hx Hxf2).F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
rewrite 2!Rabs_pos_eq ; trivial.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
rewrite 2!Rabs_pos_eq in H2 ; trivial.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
destruct (Rnd_N_pt_DN_or_UP _ _ _ Hxf2) as [H3|H3].F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
H3:Rnd_DN_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
H3:Rnd_UP_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
apply H3.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
H3:Rnd_DN_pt F x f2
F f
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
H3:Rnd_DN_pt F x f2
f <= x
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
H3:Rnd_UP_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
apply H1.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
H3:Rnd_DN_pt F x f2
f <= x
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
H3:Rnd_UP_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
apply H2.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
H3:Rnd_UP_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
apply Rle_trans with (1 := H2).F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:f <= x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:0 <= f
Hf2:0 <= f2
H3:Rnd_UP_pt F x f2
x <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
apply H3.F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
rewrite (H2 _ Hxf2).F:R -> Prop
HF:F 0
x, f:R
Hx:0 <= x
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
apply Rle_refl.F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
(* *)
assert (Hx' := Rlt_le _ _ Hx).F:R -> Prop
HF:F 0
x, f:R
Hx:x < 0
Hx':x <= 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
clear Hx.F:R -> Prop
HF:F 0
x, f:R
Hx':x <= 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f rename Hx' into Hx.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
Rnd_N0_pt F x f <->
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
split ; intros (H1, H2).F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
(* . *)
assert (Hf := Rnd_N_pt_le_0 F HF x f Hx H1).F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
split.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
Rnd_N_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
exact H1.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
destruct (Rnd_N_pt_DN_or_UP _ _ _ H1) as [H3|H3].F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_DN_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
(* . . *)
right.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_DN_pt F x f
forall f2 : R, Rnd_N_pt F x f2 -> f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
intros f2 Hxf2.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R,
Rnd_N_pt F x f0 -> Rabs f <= Rabs f0
Hf:f <= 0
H3:Rnd_DN_pt F x f
f2:R
Hxf2:Rnd_N_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
specialize (H2 _ Hxf2).F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
destruct (Rnd_N_pt_DN_or_UP _ _ _ Hxf2) as [H4|H4].F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_DN_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
eapply Rnd_DN_pt_unique ; eassumption.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 = f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
apply Rle_antisym.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
2: apply Rle_trans with x.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f <= x
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
x <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
2: apply H3.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
x <= f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
2: apply H4.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
rewrite Rabs_left1 with (1 := Hf) in H2.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:- f <= Rabs f2
Hf:f <= 0
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
rewrite Rabs_left1 in H2.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:- f <= - f2
Hf:f <= 0
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:- f <= Rabs f2
Hf:f <= 0
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 <= 0
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
now apply Ropp_le_cancel.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
f2:R
H2:- f <= Rabs f2
Hf:f <= 0
H3:Rnd_DN_pt F x f
Hxf2:Rnd_N_pt F x f2
H4:Rnd_UP_pt F x f2
f2 <= 0
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
now apply Rnd_N_pt_le_0 with F x.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
(* . . *)
left.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_UP_pt F x f
Rabs f <= Rabs x
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
rewrite Rabs_left1 with (1 := Hf).F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_UP_pt F x f
- f <= Rabs x
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
rewrite Rabs_left1 with (1 := Hx).F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_UP_pt F x f
- f <= - x
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
apply Ropp_le_contravar.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f2 : R,
Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
Hf:f <= 0
H3:Rnd_UP_pt F x f
x <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
apply H3.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N0_pt F x f
(* . *)
split.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
Rnd_N_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
forall f2 : R, Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
exact H1.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f2 : R, Rnd_N_pt F x f2 -> f2 = f)
forall f2 : R, Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
intros f2 Hxf2.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x \/
(forall f0 : R, Rnd_N_pt F x f0 -> f0 = f)
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
destruct H2 as [H2|H2].F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
assert (Hf := Rnd_N_pt_le_0 F HF x f Hx H1).F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
assert (Hf2 := Rnd_N_pt_le_0 F HF x f2 Hx Hxf2).F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
Rabs f <= Rabs f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
rewrite 2!Rabs_left1 ; trivial.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:Rabs f <= Rabs x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
- f <= - f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
rewrite 2!Rabs_left1 in H2 ; trivial.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:- f <= - x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
- f <= - f2
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
apply Ropp_le_contravar.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:- f <= - x
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
apply Ropp_le_cancel in H2.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:x <= f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
destruct (Rnd_N_pt_DN_or_UP _ _ _ Hxf2) as [H3|H3].F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:x <= f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
H3:Rnd_DN_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:x <= f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
H3:Rnd_UP_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
2: apply H3.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:x <= f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
H3:Rnd_DN_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:x <= f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
H3:Rnd_UP_pt F x f2
F f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:x <= f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
H3:Rnd_UP_pt F x f2
x <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
2: apply H1.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:x <= f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
H3:Rnd_DN_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:x <= f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
H3:Rnd_UP_pt F x f2
x <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
2: apply H2.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:x <= f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
H3:Rnd_DN_pt F x f2
f2 <= f
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
apply Rle_trans with (2 := H2).F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:x <= f
f2:R
Hxf2:Rnd_N_pt F x f2
Hf:f <= 0
Hf2:f2 <= 0
H3:Rnd_DN_pt F x f2
f2 <= x
F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
apply H3.F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f2
rewrite (H2 _ Hxf2).F:R -> Prop
HF:F 0
x, f:R
Hx:x <= 0
H1:Rnd_N_pt F x f
H2:forall f0 : R, Rnd_N_pt F x f0 -> f0 = f
f2:R
Hxf2:Rnd_N_pt F x f2
Rabs f <= Rabs f
apply Rle_refl.
Qed.

Lemma Rnd_N0_pt_unique_prop :
  forall F : R -> Prop,
  F 0 ->
  Rnd_NG_pt_unique_prop F (fun x f => Rabs f <= Rabs x).forall F : R -> Prop,
F 0 ->
Rnd_NG_pt_unique_prop F
  (fun x f : R => Rabs f <= Rabs x)
Proof.forall F : R -> Prop,
F 0 ->
Rnd_NG_pt_unique_prop F
  (fun x f : R => Rabs f <= Rabs x)
intros F HF x d u Hxd1 Hxd2 Hxu1 Hxu2 Hd Hu.F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
d = u
apply Rle_antisym.F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
d <= u
F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
u <= d
apply Rle_trans with x.F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
d <= x
F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
x <= u
F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
u <= d
apply Hxd1.F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
x <= u
F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
u <= d
apply Hxu1.F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
u <= d
destruct (Rle_or_lt 0 x) as [Hx|Hx].F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
Hx:0 <= x
u <= d
F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
Hx:x < 0
u <= d
apply Hxd1.F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
Hx:0 <= x
F u
F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
Hx:0 <= x
u <= x
F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
Hx:x < 0
u <= d
apply Hxu1.F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
Hx:0 <= x
u <= x
F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
Hx:x < 0
u <= d
rewrite Rabs_pos_eq with (1 := Hx) in Hu.F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= x
Hx:0 <= x
u <= x
F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
Hx:x < 0
u <= d
rewrite Rabs_pos_eq in Hu.F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:u <= x
Hx:0 <= x
u <= x
F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= x
Hx:0 <= x
0 <= u
F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
Hx:x < 0
u <= d
exact Hu.F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= x
Hx:0 <= x
0 <= u
F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
Hx:x < 0
u <= d
apply Rle_trans with (1:=Hx).F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= x
Hx:0 <= x
x <= u
F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
Hx:x < 0
u <= d
apply Hxu1.F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
Hx:x < 0
u <= d
(* *)
apply Hxu1.F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
Hx:x < 0
F d
F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
Hx:x < 0
x <= d
apply Hxd1.F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= Rabs x
Hu:Rabs u <= Rabs x
Hx:x < 0
x <= d
rewrite Rabs_left with (1 := Hx) in Hd.F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= - x
Hu:Rabs u <= Rabs x
Hx:x < 0
x <= d
rewrite Rabs_left1 in Hd.F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:- d <= - x
Hu:Rabs u <= Rabs x
Hx:x < 0
x <= d
F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= - x
Hu:Rabs u <= Rabs x
Hx:x < 0
d <= 0
now apply Ropp_le_cancel.F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= - x
Hu:Rabs u <= Rabs x
Hx:x < 0
d <= 0
apply Rlt_le, Rle_lt_trans with (2:=Hx).F:R -> Prop
HF:F 0
x, d, u:R
Hxd1:Rnd_DN_pt F x d
Hxd2:Rnd_N_pt F x d
Hxu1:Rnd_UP_pt F x u
Hxu2:Rnd_N_pt F x u
Hd:Rabs d <= - x
Hu:Rabs u <= Rabs x
Hx:x < 0
d <= x
apply Hxd1.
Qed.

Theorem Rnd_N0_pt_unique :
  forall F : R -> Prop,
  F 0 ->
  forall x f1 f2 : R,
  Rnd_N0_pt F x f1 -> Rnd_N0_pt F x f2 ->
  f1 = f2.forall F : R -> Prop,
F 0 ->
forall x f1 f2 : R,
Rnd_N0_pt F x f1 -> Rnd_N0_pt F x f2 -> f1 = f2
Proof.forall F : R -> Prop,
F 0 ->
forall x f1 f2 : R,
Rnd_N0_pt F x f1 -> Rnd_N0_pt F x f2 -> f1 = f2
intros F HF x f1 f2 H1 H2.F:R -> Prop
HF:F 0
x, f1, f2:R
H1:Rnd_N0_pt F x f1
H2:Rnd_N0_pt F x f2
f1 = f2
apply (Rnd_NG_pt_unique F _ (Rnd_N0_pt_unique_prop F HF) x).F:R -> Prop
HF:F 0
x, f1, f2:R
H1:Rnd_N0_pt F x f1
H2:Rnd_N0_pt F x f2
Rnd_NG_pt F (fun x0 f : R => Rabs f <= Rabs x0) x f1
F:R -> Prop
HF:F 0
x, f1, f2:R
H1:Rnd_N0_pt F x f1
H2:Rnd_N0_pt F x f2
Rnd_NG_pt F (fun x0 f : R => Rabs f <= Rabs x0) x f2
now apply -> Rnd_N0_NG_pt.F:R -> Prop
HF:F 0
x, f1, f2:R
H1:Rnd_N0_pt F x f1
H2:Rnd_N0_pt F x f2
Rnd_NG_pt F (fun x0 f : R => Rabs f <= Rabs x0) x f2
now apply -> Rnd_N0_NG_pt.
Qed.

Theorem Rnd_N0_pt_N :
  forall F : R -> Prop,
  F 0 ->
  forall x f : R,
  Rnd_N_pt F x f ->
  (Rabs f <= Rabs x)%R ->
  Rnd_N0_pt F x f.forall F : R -> Prop,
F 0 ->
forall x f : R,
Rnd_N_pt F x f -> Rabs f <= Rabs x -> Rnd_N0_pt F x f
Proof.forall F : R -> Prop,
F 0 ->
forall x f : R,
Rnd_N_pt F x f -> Rabs f <= Rabs x -> Rnd_N0_pt F x f
intros F HF x f Rxf Hxf.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
Rnd_N0_pt F x f
split.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
Rnd_N_pt F x f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
forall f2 : R, Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
apply Rxf.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
forall f2 : R, Rnd_N_pt F x f2 -> Rabs f <= Rabs f2
intros g Rxg.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
Rabs f <= Rabs g
destruct (Rabs_eq_Rabs (f - x) (g - x)) as [H|H].F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
Rabs (f - x) = Rabs (g - x)
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = g - x
Rabs f <= Rabs g
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
Rabs f <= Rabs g
apply Rle_antisym.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
Rabs (f - x) <= Rabs (g - x)
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
Rabs (g - x) <= Rabs (f - x)
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = g - x
Rabs f <= Rabs g
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
Rabs f <= Rabs g
apply Rxf.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
F g
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
Rabs (g - x) <= Rabs (f - x)
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = g - x
Rabs f <= Rabs g
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
Rabs f <= Rabs g
apply Rxg.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
Rabs (g - x) <= Rabs (f - x)
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = g - x
Rabs f <= Rabs g
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
Rabs f <= Rabs g
apply Rxg.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
F f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = g - x
Rabs f <= Rabs g
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
Rabs f <= Rabs g
apply Rxf.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = g - x
Rabs f <= Rabs g
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
Rabs f <= Rabs g
(* *)
replace g with f.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = g - x
Rabs f <= Rabs f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = g - x
f = g
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
Rabs f <= Rabs g
apply Rle_refl.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = g - x
f = g
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
Rabs f <= Rabs g
apply Rplus_eq_reg_r with (1 := H).F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
Rabs f <= Rabs g
(* *)
assert (g = 2 * x - f)%R.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
g = 2 * x - f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Rabs f <= Rabs g
replace (2 * x - f)%R with (x - (f - x))%R by ring.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
g = x - (f - x)
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Rabs f <= Rabs g
rewrite H.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
g = x - - (g - x)
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Rabs f <= Rabs g
ring.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Rabs f <= Rabs g
destruct (Rle_lt_dec 0 x) as [Hx|Hx].F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:0 <= x
Rabs f <= Rabs g
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs f <= Rabs g
(* . *)
revert Hxf.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:0 <= x
Rabs f <= Rabs x -> Rabs f <= Rabs g
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs f <= Rabs g
rewrite Rabs_pos_eq with (1 := Hx).F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:0 <= x
Rabs f <= x -> Rabs f <= Rabs g
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs f <= Rabs g
rewrite 2!Rabs_pos_eq ; try ( apply (Rnd_N_pt_ge_0 F HF x) ; assumption ).F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:0 <= x
f <= x -> f <= g
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs f <= Rabs g
intros Hxf.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:0 <= x
Hxf:f <= x
f <= g
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs f <= Rabs g
rewrite H0.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:0 <= x
Hxf:f <= x
f <= 2 * x - f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs f <= Rabs g
apply Rplus_le_reg_r with f.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:0 <= x
Hxf:f <= x
f + f <= 2 * x - f + f
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs f <= Rabs g
ring_simplify.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:0 <= x
Hxf:f <= x
2 * f <= 2 * x
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs f <= Rabs g
apply Rmult_le_compat_l with (2 := Hxf).F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:0 <= x
Hxf:f <= x
0 <= 2
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs f <= Rabs g
now apply IZR_le.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
Hxf:Rabs f <= Rabs x
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs f <= Rabs g
(* . *)
revert Hxf.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x < 0
Rabs f <= Rabs x -> Rabs f <= Rabs g
apply Rlt_le in Hx.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
Rabs f <= Rabs x -> Rabs f <= Rabs g
rewrite Rabs_left1 with (1 := Hx).F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
Rabs f <= - x -> Rabs f <= Rabs g
rewrite 2!Rabs_left1 ; try ( apply (Rnd_N_pt_le_0 F HF x) ; assumption ).F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
- f <= - x -> - f <= - g
intros Hxf.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
Hxf:- f <= - x
- f <= - g
rewrite H0.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
Hxf:- f <= - x
- f <= - (2 * x - f)
apply Ropp_le_contravar.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
Hxf:- f <= - x
2 * x - f <= f
apply Rplus_le_reg_r with f.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
Hxf:- f <= - x
2 * x - f + f <= f + f
ring_simplify.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
Hxf:- f <= - x
2 * x <= 2 * f
apply Rmult_le_compat_l.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
Hxf:- f <= - x
0 <= 2
F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
Hxf:- f <= - x
x <= f
now apply IZR_le.F:R -> Prop
HF:F 0
x, f:R
Rxf:Rnd_N_pt F x f
g:R
Rxg:Rnd_N_pt F x g
H:f - x = - (g - x)
H0:g = 2 * x - f
Hx:x <= 0
Hxf:- f <= - x
x <= f
now apply Ropp_le_cancel.
Qed.

Theorem Rnd_N0_unique :
  forall (F : R -> Prop),
  F 0 ->
  forall rnd1 rnd2 : R -> R,
  Rnd_N0 F rnd1 -> Rnd_N0 F rnd2 ->
  forall x, rnd1 x = rnd2 x.forall F : R -> Prop,
F 0 ->
forall rnd1 rnd2 : R -> R,
Rnd_N0 F rnd1 ->
Rnd_N0 F rnd2 -> forall x : R, rnd1 x = rnd2 x
Proof.forall F : R -> Prop,
F 0 ->
forall rnd1 rnd2 : R -> R,
Rnd_N0 F rnd1 ->
Rnd_N0 F rnd2 -> forall x : R, rnd1 x = rnd2 x
intros F HF rnd1 rnd2 H1 H2 x.F:R -> Prop
HF:F 0
rnd1, rnd2:R -> R
H1:Rnd_N0 F rnd1
H2:Rnd_N0 F rnd2
x:R
rnd1 x = rnd2 x
now apply Rnd_N0_pt_unique with F x.
Qed.

Theorem Rnd_N0_pt_monotone :
  forall F : R -> Prop,
  F 0 ->
  round_pred_monotone (Rnd_N0_pt F).forall F : R -> Prop,
F 0 -> round_pred_monotone (Rnd_N0_pt F)
Proof.forall F : R -> Prop,
F 0 -> round_pred_monotone (Rnd_N0_pt F)
intros F HF x y f g Hxf Hyg Hxy.F:R -> Prop
HF:F 0
x, y, f, g:R
Hxf:Rnd_N0_pt F x f
Hyg:Rnd_N0_pt F y g
Hxy:x <= y
f <= g
apply (Rnd_NG_pt_monotone F _ (Rnd_N0_pt_unique_prop F HF) x y).F:R -> Prop
HF:F 0
x, y, f, g:R
Hxf:Rnd_N0_pt F x f
Hyg:Rnd_N0_pt F y g
Hxy:x <= y
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
F:R -> Prop
HF:F 0
x, y, f, g:R
Hxf:Rnd_N0_pt F x f
Hyg:Rnd_N0_pt F y g
Hxy:x <= y
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) y g
F:R -> Prop
HF:F 0
x, y, f, g:R
Hxf:Rnd_N0_pt F x f
Hyg:Rnd_N0_pt F y g
Hxy:x <= y
x <= y
now apply -> Rnd_N0_NG_pt.F:R -> Prop
HF:F 0
x, y, f, g:R
Hxf:Rnd_N0_pt F x f
Hyg:Rnd_N0_pt F y g
Hxy:x <= y
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) y g
F:R -> Prop
HF:F 0
x, y, f, g:R
Hxf:Rnd_N0_pt F x f
Hyg:Rnd_N0_pt F y g
Hxy:x <= y
x <= y
now apply -> Rnd_N0_NG_pt.F:R -> Prop
HF:F 0
x, y, f, g:R
Hxf:Rnd_N0_pt F x f
Hyg:Rnd_N0_pt F y g
Hxy:x <= y
x <= y
exact Hxy.
Qed.

Theorem Rnd_N0_pt_refl :
  forall F : R -> Prop,
  forall x : R, F x ->
  Rnd_N0_pt F x x.forall (F : R -> Prop) (x : R), F x -> Rnd_N0_pt F x x
Proof.forall (F : R -> Prop) (x : R), F x -> Rnd_N0_pt F x x
intros F x Hx.F:R -> Prop
x:R
Hx:F x
Rnd_N0_pt F x x
split.F:R -> Prop
x:R
Hx:F x
Rnd_N_pt F x x
F:R -> Prop
x:R
Hx:F x
forall f2 : R, Rnd_N_pt F x f2 -> Rabs x <= Rabs f2
now apply Rnd_N_pt_refl.F:R -> Prop
x:R
Hx:F x
forall f2 : R, Rnd_N_pt F x f2 -> Rabs x <= Rabs f2
intros f Hxf.F:R -> Prop
x:R
Hx:F x
f:R
Hxf:Rnd_N_pt F x f
Rabs x <= Rabs f
apply Req_le.F:R -> Prop
x:R
Hx:F x
f:R
Hxf:Rnd_N_pt F x f
Rabs x = Rabs f
apply f_equal.F:R -> Prop
x:R
Hx:F x
f:R
Hxf:Rnd_N_pt F x f
x = f
now apply sym_eq, Rnd_N_pt_idempotent with (1 := Hxf).
Qed.

Theorem Rnd_N0_pt_idempotent :
  forall F : R -> Prop,
  forall x f : R,
  Rnd_N0_pt F x f -> F x ->
  f = x.forall (F : R -> Prop) (x f : R),
Rnd_N0_pt F x f -> F x -> f = x
Proof.forall (F : R -> Prop) (x f : R),
Rnd_N0_pt F x f -> F x -> f = x
intros F x f (Hf,_) Hx.F:R -> Prop
x, f:R
Hf:Rnd_N_pt F x f
Hx:F x
f = x
now apply Rnd_N_pt_idempotent with F.
Qed.




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

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

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

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

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.forall (F1 F2 : R -> Prop) (a b : R),
F1 a ->
(forall x : R, a <= x <= b -> F1 x <-> F2 x) ->
forall x f : R,
a <= x <= b -> Rnd_DN_pt F1 x f -> Rnd_DN_pt F2 x f
Proof.forall (F1 F2 : R -> Prop) (a b : R),
F1 a ->
(forall x : R, a <= x <= b -> F1 x <-> F2 x) ->
forall x f : R,
a <= x <= b -> Rnd_DN_pt F1 x f -> Rnd_DN_pt F2 x f
intros F1 F2 a b Ha HF x f Hx (H1, (H2, H3)).F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
Rnd_DN_pt F2 x f
split.F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
F2 f
F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
f <= x /\ (forall g : R, F2 g -> g <= x -> g <= f)
apply -> HF.F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
F1 f
F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
a <= f <= b
F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
f <= x /\ (forall g : R, F2 g -> g <= x -> g <= f)
exact H1.F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
a <= f <= b
F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
f <= x /\ (forall g : R, F2 g -> g <= x -> g <= f)
split.F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
a <= f
F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
f <= b
F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
f <= x /\ (forall g : R, F2 g -> g <= x -> g <= f)
now apply H3.F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
f <= b
F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
f <= x /\ (forall g : R, F2 g -> g <= x -> g <= f)
now apply Rle_trans with (1 := H2).F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
f <= x /\ (forall g : R, F2 g -> g <= x -> g <= f)
split.F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
f <= x
F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
forall g : R, F2 g -> g <= x -> g <= f
exact H2.F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
forall g : R, F2 g -> g <= x -> g <= f
intros k Hk Hl.F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
k <= f
destruct (Rlt_or_le k a) as [H|H].F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:k < a
k <= f
F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:a <= k
k <= f
apply Rlt_le.F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:k < a
k < f
F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:a <= k
k <= f
apply Rlt_le_trans with (1 := H).F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:k < a
a <= f
F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:a <= k
k <= f
now apply H3.F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:a <= k
k <= f
apply H3.F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:a <= k
F1 k
F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:a <= k
k <= x
apply <- HF.F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:a <= k
F2 k
F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:a <= k
a <= k <= b
F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:a <= k
k <= x
exact Hk.F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:a <= k
a <= k <= b
F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:a <= k
k <= x
split.F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:a <= k
a <= k
F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:a <= k
k <= b
F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:a <= k
k <= x
exact H.F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:a <= k
k <= b
F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:a <= k
k <= x
now apply Rle_trans with (1 := Hl).F1, F2:R -> Prop
a, b:R
Ha:F1 a
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
k:R
Hk:F2 k
Hl:k <= x
H:a <= k
k <= x
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.forall (F1 F2 : R -> Prop) (a b : R),
F1 b ->
(forall x : R, a <= x <= b -> F1 x <-> F2 x) ->
forall x f : R,
a <= x <= b -> Rnd_UP_pt F1 x f -> Rnd_UP_pt F2 x f
Proof.forall (F1 F2 : R -> Prop) (a b : R),
F1 b ->
(forall x : R, a <= x <= b -> F1 x <-> F2 x) ->
forall x f : R,
a <= x <= b -> Rnd_UP_pt F1 x f -> Rnd_UP_pt F2 x f
intros F1 F2 a b Hb HF x f Hx (H1, (H2, H3)).F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
Rnd_UP_pt F2 x f
split.F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
F2 f
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
x <= f /\ (forall g : R, F2 g -> x <= g -> f <= g)
apply -> HF.F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
F1 f
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
a <= f <= b
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
x <= f /\ (forall g : R, F2 g -> x <= g -> f <= g)
exact H1.F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
a <= f <= b
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
x <= f /\ (forall g : R, F2 g -> x <= g -> f <= g)
split.F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
a <= f
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
f <= b
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
x <= f /\ (forall g : R, F2 g -> x <= g -> f <= g)
now apply Rle_trans with (2 := H2).F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
f <= b
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
x <= f /\ (forall g : R, F2 g -> x <= g -> f <= g)
now apply H3.F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
x <= f /\ (forall g : R, F2 g -> x <= g -> f <= g)
split.F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
x <= f
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
forall g : R, F2 g -> x <= g -> f <= g
exact H2.F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
forall g : R, F2 g -> x <= g -> f <= g
intros k Hk Hl.F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
f <= k
destruct (Rle_or_lt k b) as [H|H].F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:k <= b
f <= k
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:b < k
f <= k
apply H3.F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:k <= b
F1 k
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:k <= b
x <= k
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:b < k
f <= k
apply <- HF.F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:k <= b
F2 k
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:k <= b
a <= k <= b
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:k <= b
x <= k
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:b < k
f <= k
exact Hk.F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:k <= b
a <= k <= b
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:k <= b
x <= k
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:b < k
f <= k
split.F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:k <= b
a <= k
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:k <= b
k <= b
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:k <= b
x <= k
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:b < k
f <= k
now apply Rle_trans with (2 := Hl).F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:k <= b
k <= b
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:k <= b
x <= k
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:b < k
f <= k
exact H.F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:k <= b
x <= k
F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:b < k
f <= k
exact Hl.F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:b < k
f <= k
apply Rlt_le.F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:b < k
f < k
apply Rle_lt_trans with (2 := H).F1, F2:R -> Prop
a, b:R
Hb:F1 b
HF:forall x0 : R, a <= x0 <= b -> F1 x0 <-> F2 x0
x, f:R
Hx:a <= x <= b
H1:F1 f
H2:x <= f
H3:forall g : R, F1 g -> x <= g -> f <= g
k:R
Hk:F2 k
Hl:x <= k
H:b < k
f <= b
now apply H3.
Qed.

ensures a real number can always be rounded

Inductive satisfies_any (F : R -> Prop) :=
  Satisfies_any :
    F 0 -> ( forall x : R, F x -> F (-x) ) ->
    round_pred_total (Rnd_DN_pt F) -> satisfies_any F.

Theorem satisfies_any_eq :
  forall F1 F2 : R -> Prop,
  ( forall x, F1 x <-> F2 x ) ->
  satisfies_any F1 ->
  satisfies_any F2.forall F1 F2 : R -> Prop,
(forall x : R, F1 x <-> F2 x) ->
satisfies_any F1 -> satisfies_any F2
Proof.forall F1 F2 : R -> Prop,
(forall x : R, F1 x <-> F2 x) ->
satisfies_any F1 -> satisfies_any F2
intros F1 F2 Heq (Hzero, Hsym, Hrnd).F1, F2:R -> Prop
Heq:forall x : R, F1 x <-> F2 x
Hzero:F1 0
Hsym:forall x : R, F1 x -> F1 (- x)
Hrnd:round_pred_total (Rnd_DN_pt F1)
satisfies_any F2
split.F1, F2:R -> Prop
Heq:forall x : R, F1 x <-> F2 x
Hzero:F1 0
Hsym:forall x : R, F1 x -> F1 (- x)
Hrnd:round_pred_total (Rnd_DN_pt F1)
F2 0
F1, F2:R -> Prop
Heq:forall x : R, F1 x <-> F2 x
Hzero:F1 0
Hsym:forall x : R, F1 x -> F1 (- x)
Hrnd:round_pred_total (Rnd_DN_pt F1)
forall x : R, F2 x -> F2 (- x)
F1, F2:R -> Prop
Heq:forall x : R, F1 x <-> F2 x
Hzero:F1 0
Hsym:forall x : R, F1 x -> F1 (- x)
Hrnd:round_pred_total (Rnd_DN_pt F1)
round_pred_total (Rnd_DN_pt F2)
now apply -> Heq.F1, F2:R -> Prop
Heq:forall x : R, F1 x <-> F2 x
Hzero:F1 0
Hsym:forall x : R, F1 x -> F1 (- x)
Hrnd:round_pred_total (Rnd_DN_pt F1)
forall x : R, F2 x -> F2 (- x)
F1, F2:R -> Prop
Heq:forall x : R, F1 x <-> F2 x
Hzero:F1 0
Hsym:forall x : R, F1 x -> F1 (- x)
Hrnd:round_pred_total (Rnd_DN_pt F1)
round_pred_total (Rnd_DN_pt F2)
intros x Hx.F1, F2:R -> Prop
Heq:forall x0 : R, F1 x0 <-> F2 x0
Hzero:F1 0
Hsym:forall x0 : R, F1 x0 -> F1 (- x0)
Hrnd:round_pred_total (Rnd_DN_pt F1)
x:R
Hx:F2 x
F2 (- x)
F1, F2:R -> Prop
Heq:forall x : R, F1 x <-> F2 x
Hzero:F1 0
Hsym:forall x : R, F1 x -> F1 (- x)
Hrnd:round_pred_total (Rnd_DN_pt F1)
round_pred_total (Rnd_DN_pt F2)
apply -> Heq.F1, F2:R -> Prop
Heq:forall x0 : R, F1 x0 <-> F2 x0
Hzero:F1 0
Hsym:forall x0 : R, F1 x0 -> F1 (- x0)
Hrnd:round_pred_total (Rnd_DN_pt F1)
x:R
Hx:F2 x
F1 (- x)
F1, F2:R -> Prop
Heq:forall x : R, F1 x <-> F2 x
Hzero:F1 0
Hsym:forall x : R, F1 x -> F1 (- x)
Hrnd:round_pred_total (Rnd_DN_pt F1)
round_pred_total (Rnd_DN_pt F2)
apply Hsym.F1, F2:R -> Prop
Heq:forall x0 : R, F1 x0 <-> F2 x0
Hzero:F1 0
Hsym:forall x0 : R, F1 x0 -> F1 (- x0)
Hrnd:round_pred_total (Rnd_DN_pt F1)
x:R
Hx:F2 x
F1 x
F1, F2:R -> Prop
Heq:forall x : R, F1 x <-> F2 x
Hzero:F1 0
Hsym:forall x : R, F1 x -> F1 (- x)
Hrnd:round_pred_total (Rnd_DN_pt F1)
round_pred_total (Rnd_DN_pt F2)
now apply <- Heq.F1, F2:R -> Prop
Heq:forall x : R, F1 x <-> F2 x
Hzero:F1 0
Hsym:forall x : R, F1 x -> F1 (- x)
Hrnd:round_pred_total (Rnd_DN_pt F1)
round_pred_total (Rnd_DN_pt F2)
intros x.F1, F2:R -> Prop
Heq:forall x0 : R, F1 x0 <-> F2 x0
Hzero:F1 0
Hsym:forall x0 : R, F1 x0 -> F1 (- x0)
Hrnd:round_pred_total (Rnd_DN_pt F1)
x:R
exists f : R, Rnd_DN_pt F2 x f
destruct (Hrnd x) as (f, (H1, (H2, H3))).F1, F2:R -> Prop
Heq:forall x0 : R, F1 x0 <-> F2 x0
Hzero:F1 0
Hsym:forall x0 : R, F1 x0 -> F1 (- x0)
Hrnd:round_pred_total (Rnd_DN_pt F1)
x, f:R
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
exists f0 : R, Rnd_DN_pt F2 x f0
exists f.F1, F2:R -> Prop
Heq:forall x0 : R, F1 x0 <-> F2 x0
Hzero:F1 0
Hsym:forall x0 : R, F1 x0 -> F1 (- x0)
Hrnd:round_pred_total (Rnd_DN_pt F1)
x, f:R
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
Rnd_DN_pt F2 x f
split.F1, F2:R -> Prop
Heq:forall x0 : R, F1 x0 <-> F2 x0
Hzero:F1 0
Hsym:forall x0 : R, F1 x0 -> F1 (- x0)
Hrnd:round_pred_total (Rnd_DN_pt F1)
x, f:R
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
F2 f
F1, F2:R -> Prop
Heq:forall x0 : R, F1 x0 <-> F2 x0
Hzero:F1 0
Hsym:forall x0 : R, F1 x0 -> F1 (- x0)
Hrnd:round_pred_total (Rnd_DN_pt F1)
x, f:R
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
f <= x /\ (forall g : R, F2 g -> g <= x -> g <= f)
now apply -> Heq.F1, F2:R -> Prop
Heq:forall x0 : R, F1 x0 <-> F2 x0
Hzero:F1 0
Hsym:forall x0 : R, F1 x0 -> F1 (- x0)
Hrnd:round_pred_total (Rnd_DN_pt F1)
x, f:R
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
f <= x /\ (forall g : R, F2 g -> g <= x -> g <= f)
split.F1, F2:R -> Prop
Heq:forall x0 : R, F1 x0 <-> F2 x0
Hzero:F1 0
Hsym:forall x0 : R, F1 x0 -> F1 (- x0)
Hrnd:round_pred_total (Rnd_DN_pt F1)
x, f:R
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
f <= x
F1, F2:R -> Prop
Heq:forall x0 : R, F1 x0 <-> F2 x0
Hzero:F1 0
Hsym:forall x0 : R, F1 x0 -> F1 (- x0)
Hrnd:round_pred_total (Rnd_DN_pt F1)
x, f:R
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
forall g : R, F2 g -> g <= x -> g <= f
exact H2.F1, F2:R -> Prop
Heq:forall x0 : R, F1 x0 <-> F2 x0
Hzero:F1 0
Hsym:forall x0 : R, F1 x0 -> F1 (- x0)
Hrnd:round_pred_total (Rnd_DN_pt F1)
x, f:R
H1:F1 f
H2:f <= x
H3:forall g : R, F1 g -> g <= x -> g <= f
forall g : R, F2 g -> g <= x -> g <= f
intros g Hg Hgx.F1, F2:R -> Prop
Heq:forall x0 : R, F1 x0 <-> F2 x0
Hzero:F1 0
Hsym:forall x0 : R, F1 x0 -> F1 (- x0)
Hrnd:round_pred_total (Rnd_DN_pt F1)
x, f:R
H1:F1 f
H2:f <= x
H3:forall g0 : R, F1 g0 -> g0 <= x -> g0 <= f
g:R
Hg:F2 g
Hgx:g <= x
g <= f
apply H3.F1, F2:R -> Prop
Heq:forall x0 : R, F1 x0 <-> F2 x0
Hzero:F1 0
Hsym:forall x0 : R, F1 x0 -> F1 (- x0)
Hrnd:round_pred_total (Rnd_DN_pt F1)
x, f:R
H1:F1 f
H2:f <= x
H3:forall g0 : R, F1 g0 -> g0 <= x -> g0 <= f
g:R
Hg:F2 g
Hgx:g <= x
F1 g
F1, F2:R -> Prop
Heq:forall x0 : R, F1 x0 <-> F2 x0
Hzero:F1 0
Hsym:forall x0 : R, F1 x0 -> F1 (- x0)
Hrnd:round_pred_total (Rnd_DN_pt F1)
x, f:R
H1:F1 f
H2:f <= x
H3:forall g0 : R, F1 g0 -> g0 <= x -> g0 <= f
g:R
Hg:F2 g
Hgx:g <= x
g <= x
now apply <- Heq.F1, F2:R -> Prop
Heq:forall x0 : R, F1 x0 <-> F2 x0
Hzero:F1 0
Hsym:forall x0 : R, F1 x0 -> F1 (- x0)
Hrnd:round_pred_total (Rnd_DN_pt F1)
x, f:R
H1:F1 f
H2:f <= x
H3:forall g0 : R, F1 g0 -> g0 <= x -> g0 <= f
g:R
Hg:F2 g
Hgx:g <= x
g <= x
exact Hgx.
Qed.

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

Theorem satisfies_any_imp_UP :
  forall F : R -> Prop,
  satisfies_any F ->
  round_pred (Rnd_UP_pt F).forall F : R -> Prop,
satisfies_any F -> round_pred (Rnd_UP_pt F)
Proof.forall F : R -> Prop,
satisfies_any F -> round_pred (Rnd_UP_pt F)
intros F Hany.F:R -> Prop
Hany:satisfies_any F
round_pred (Rnd_UP_pt F)
split.F:R -> Prop
Hany:satisfies_any F
round_pred_total (Rnd_UP_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_UP_pt F)
intros x.F:R -> Prop
Hany:satisfies_any F
x:R
exists f : R, Rnd_UP_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_UP_pt F)
destruct (proj1 (satisfies_any_imp_DN F Hany) (-x)) as (f, Hf).F:R -> Prop
Hany:satisfies_any F
x, f:R
Hf:Rnd_DN_pt F (- x) f
exists f0 : R, Rnd_UP_pt F x f0
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_UP_pt F)
exists (-f).F:R -> Prop
Hany:satisfies_any F
x, f:R
Hf:Rnd_DN_pt F (- x) f
Rnd_UP_pt F x (- f)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_UP_pt F)
rewrite <- (Ropp_involutive x).F:R -> Prop
Hany:satisfies_any F
x, f:R
Hf:Rnd_DN_pt F (- x) f
Rnd_UP_pt F (- - x) (- f)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_UP_pt F)
apply Rnd_UP_pt_opp.F:R -> Prop
Hany:satisfies_any F
x, f:R
Hf:Rnd_DN_pt F (- x) f
forall x0 : R, F x0 -> F (- x0)
F:R -> Prop
Hany:satisfies_any F
x, f:R
Hf:Rnd_DN_pt F (- x) f
Rnd_DN_pt F (- x) f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_UP_pt F)
apply Hany.F:R -> Prop
Hany:satisfies_any F
x, f:R
Hf:Rnd_DN_pt F (- x) f
Rnd_DN_pt F (- x) f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_UP_pt F)
exact Hf.F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_UP_pt F)
apply Rnd_UP_pt_monotone.
Qed.

Theorem satisfies_any_imp_ZR :
  forall F : R -> Prop,
  satisfies_any F ->
  round_pred (Rnd_ZR_pt F).forall F : R -> Prop,
satisfies_any F -> round_pred (Rnd_ZR_pt F)
Proof.forall F : R -> Prop,
satisfies_any F -> round_pred (Rnd_ZR_pt F)
intros F Hany.F:R -> Prop
Hany:satisfies_any F
round_pred (Rnd_ZR_pt F)
split.F:R -> Prop
Hany:satisfies_any F
round_pred_total (Rnd_ZR_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
intros x.F:R -> Prop
Hany:satisfies_any F
x:R
exists f : R, Rnd_ZR_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
destruct (Rle_or_lt 0 x) as [Hx|Hx].F:R -> Prop
Hany:satisfies_any F
x:R
Hx:0 <= x
exists f : R, Rnd_ZR_pt F x f
F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
exists f : R, Rnd_ZR_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
(* positive *)
destruct (proj1 (satisfies_any_imp_DN F Hany) x) as (f, Hf).F:R -> Prop
Hany:satisfies_any F
x:R
Hx:0 <= x
f:R
Hf:Rnd_DN_pt F x f
exists f0 : R, Rnd_ZR_pt F x f0
F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
exists f : R, Rnd_ZR_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
exists f.F:R -> Prop
Hany:satisfies_any F
x:R
Hx:0 <= x
f:R
Hf:Rnd_DN_pt F x f
Rnd_ZR_pt F x f
F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
exists f : R, Rnd_ZR_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
split.F:R -> Prop
Hany:satisfies_any F
x:R
Hx:0 <= x
f:R
Hf:Rnd_DN_pt F x f
0 <= x -> Rnd_DN_pt F x f
F:R -> Prop
Hany:satisfies_any F
x:R
Hx:0 <= x
f:R
Hf:Rnd_DN_pt F x f
x <= 0 -> Rnd_UP_pt F x f
F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
exists f : R, Rnd_ZR_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
now intros _.F:R -> Prop
Hany:satisfies_any F
x:R
Hx:0 <= x
f:R
Hf:Rnd_DN_pt F x f
x <= 0 -> Rnd_UP_pt F x f
F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
exists f : R, Rnd_ZR_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
intros Hx'.F:R -> Prop
Hany:satisfies_any F
x:R
Hx:0 <= x
f:R
Hf:Rnd_DN_pt F x f
Hx':x <= 0
Rnd_UP_pt F x f
F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
exists f : R, Rnd_ZR_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
(* zero *)
assert (x = 0).F:R -> Prop
Hany:satisfies_any F
x:R
Hx:0 <= x
f:R
Hf:Rnd_DN_pt F x f
Hx':x <= 0
x = 0
F:R -> Prop
Hany:satisfies_any F
x:R
Hx:0 <= x
f:R
Hf:Rnd_DN_pt F x f
Hx':x <= 0
H:x = 0
Rnd_UP_pt F x f
F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
exists f : R, Rnd_ZR_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
now apply Rle_antisym.F:R -> Prop
Hany:satisfies_any F
x:R
Hx:0 <= x
f:R
Hf:Rnd_DN_pt F x f
Hx':x <= 0
H:x = 0
Rnd_UP_pt F x f
F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
exists f : R, Rnd_ZR_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
rewrite H in Hf |- *.F:R -> Prop
Hany:satisfies_any F
x:R
Hx:0 <= x
f:R
Hf:Rnd_DN_pt F 0 f
Hx':x <= 0
H:x = 0
Rnd_UP_pt F 0 f
F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
exists f : R, Rnd_ZR_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
clear H Hx Hx'.F:R -> Prop
Hany:satisfies_any F
x, f:R
Hf:Rnd_DN_pt F 0 f
Rnd_UP_pt F 0 f
F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
exists f : R, Rnd_ZR_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
rewrite Rnd_DN_pt_idempotent with (1 := Hf).F:R -> Prop
Hany:satisfies_any F
x, f:R
Hf:Rnd_DN_pt F 0 f
Rnd_UP_pt F 0 0
F:R -> Prop
Hany:satisfies_any F
x, f:R
Hf:Rnd_DN_pt F 0 f
F 0
F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
exists f : R, Rnd_ZR_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
apply Rnd_UP_pt_refl.F:R -> Prop
Hany:satisfies_any F
x, f:R
Hf:Rnd_DN_pt F 0 f
F 0
F:R -> Prop
Hany:satisfies_any F
x, f:R
Hf:Rnd_DN_pt F 0 f
F 0
F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
exists f : R, Rnd_ZR_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
apply Hany.F:R -> Prop
Hany:satisfies_any F
x, f:R
Hf:Rnd_DN_pt F 0 f
F 0
F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
exists f : R, Rnd_ZR_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
apply Hany.F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
exists f : R, Rnd_ZR_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
(* negative *)
destruct (proj1 (satisfies_any_imp_UP F Hany) x) as (f, Hf).F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
f:R
Hf:Rnd_UP_pt F x f
exists f0 : R, Rnd_ZR_pt F x f0
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
exists f.F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
f:R
Hf:Rnd_UP_pt F x f
Rnd_ZR_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
split.F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
f:R
Hf:Rnd_UP_pt F x f
0 <= x -> Rnd_DN_pt F x f
F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
f:R
Hf:Rnd_UP_pt F x f
x <= 0 -> Rnd_UP_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
intros Hx'.F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
f:R
Hf:Rnd_UP_pt F x f
Hx':0 <= x
Rnd_DN_pt F x f
F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
f:R
Hf:Rnd_UP_pt F x f
x <= 0 -> Rnd_UP_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
elim (Rlt_irrefl 0).F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
f:R
Hf:Rnd_UP_pt F x f
Hx':0 <= x
0 < 0
F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
f:R
Hf:Rnd_UP_pt F x f
x <= 0 -> Rnd_UP_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
now apply Rle_lt_trans with x.F:R -> Prop
Hany:satisfies_any F
x:R
Hx:x < 0
f:R
Hf:Rnd_UP_pt F x f
x <= 0 -> Rnd_UP_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
now intros _.F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_ZR_pt F)
(* . *)
apply Rnd_ZR_pt_monotone.F:R -> Prop
Hany:satisfies_any F
F 0
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 ->
  round_pred_total (Rnd_NG_pt F P).forall (F : R -> Prop) (P : R -> R -> Prop),
satisfies_any F ->
NG_existence_prop F P ->
round_pred_total (Rnd_NG_pt F P)
Proof.forall (F : R -> Prop) (P : R -> R -> Prop),
satisfies_any F ->
NG_existence_prop F P ->
round_pred_total (Rnd_NG_pt F P)
intros F P Hany HP x.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x:R
exists f : R, Rnd_NG_pt F P x f
destruct (proj1 (satisfies_any_imp_DN F Hany) x) as (d, Hd).F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
exists f : R, Rnd_NG_pt F P x f
destruct (proj1 (satisfies_any_imp_UP F Hany) x) as (u, Hu).F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
exists f : R, Rnd_NG_pt F P x f
destruct (total_order_T (Rabs (u - x)) (Rabs (d - x))) as [[H|H]|H].F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
(* |up(x) - x| < |dn(x) - x| *)
exists u.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
Rnd_NG_pt F P x u
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
split.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
Rnd_N_pt F x u
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
(* - . *)
split.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
F u
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
forall g : R, F g -> Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Hu.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
forall g : R, F g -> Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
intros g Hg.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
destruct (Rle_or_lt x g) as [Hxg|Hxg].F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g < x
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
rewrite 2!Rabs_pos_eq.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
u - x <= g - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
0 <= g - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
0 <= u - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g < x
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Rplus_le_compat_r.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
u <= g
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
0 <= g - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
0 <= u - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g < x
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
now apply Hu.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
0 <= g - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
0 <= u - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g < x
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
now apply Rle_0_minus.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
0 <= u - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g < x
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Rle_0_minus.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
x <= u
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g < x
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Hu.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g < x
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Rlt_le in Hxg.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Rlt_le.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
Rabs (u - x) < Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Rlt_le_trans with (1 := H).F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
do 2 rewrite <- (Rabs_minus_sym x).F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
Rabs (x - d) <= Rabs (x - g)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
rewrite 2!Rabs_pos_eq.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
x - d <= x - g
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
0 <= x - g
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
0 <= x - d
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Rplus_le_compat_l.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
- d <= - g
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
0 <= x - g
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
0 <= x - d
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Ropp_le_contravar.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
g <= d
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
0 <= x - g
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
0 <= x - d
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
now apply Hd.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
0 <= x - g
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
0 <= x - d
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
now apply Rle_0_minus.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
0 <= x - d
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Rle_0_minus.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
d <= x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Hd.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
(* - . *)
right.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
forall f2 : R, Rnd_N_pt F x f2 -> f2 = u
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
intros f Hf.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
f:R
Hf:Rnd_N_pt F x f
f = u
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
destruct (Rnd_N_pt_DN_or_UP_eq F x _ _ _ Hd Hu Hf) as [K|K] ; rewrite K.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
f:R
Hf:Rnd_N_pt F x f
K:f = d
d = u
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
f:R
Hf:Rnd_N_pt F x f
K:f = u
u = u
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
elim Rlt_not_le with (1 := H).F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
f:R
Hf:Rnd_N_pt F x f
K:f = d
Rabs (d - x) <= Rabs (u - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
f:R
Hf:Rnd_N_pt F x f
K:f = u
u = u
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
rewrite <- K.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
f:R
Hf:Rnd_N_pt F x f
K:f = d
Rabs (f - x) <= Rabs (u - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
f:R
Hf:Rnd_N_pt F x f
K:f = u
u = u
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Hf.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
f:R
Hf:Rnd_N_pt F x f
K:f = d
F u
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
f:R
Hf:Rnd_N_pt F x f
K:f = u
u = u
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Hu.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) < Rabs (d - x)
f:R
Hf:Rnd_N_pt F x f
K:f = u
u = u
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply refl_equal.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
(* |up(x) - x| = |dn(x) - x| *)
destruct (Req_dec x d) as [He|Hne].F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
He:x = d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
(* - x = d = u *)
exists x.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
He:x = d
Rnd_NG_pt F P x x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
split.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
He:x = d
Rnd_N_pt F x x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
He:x = d
P x x \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Rnd_N_pt_refl.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
He:x = d
F x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
He:x = d
P x x \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
rewrite He.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
He:x = d
F d
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
He:x = d
P x x \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Hd.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
He:x = d
P x x \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
right.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
He:x = d
forall f2 : R, Rnd_N_pt F x f2 -> f2 = x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
intros.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
He:x = d
f2:R
H0:Rnd_N_pt F x f2
f2 = x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Rnd_N_pt_idempotent with (1 := H0).F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
He:x = d
f2:R
H0:Rnd_N_pt F x f2
F x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
rewrite He.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
He:x = d
f2:R
H0:Rnd_N_pt F x f2
F d
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Hd.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
assert (Hf : ~F x).F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
~ F x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
intros Hf.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:F x
False
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Hne.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:F x
x = d
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply sym_eq.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:F x
d = x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
now apply Rnd_DN_pt_idempotent with (1 := Hd).F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
destruct (HP x _ _ Hf Hd Hu) as [H'|H'].F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
(* - u >> d *)
exists u.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
Rnd_NG_pt F P x u
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
split.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
Rnd_N_pt F x u
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
split.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
F u
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
forall g : R, F g -> Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Hu.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
forall g : R, F g -> Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
intros g Hg.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
destruct (Rle_or_lt x g) as [Hxg|Hxg].F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:x <= g
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g < x
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
rewrite 2!Rabs_pos_eq.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:x <= g
u - x <= g - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:x <= g
0 <= g - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:x <= g
0 <= u - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g < x
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Rplus_le_compat_r.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:x <= g
u <= g
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:x <= g
0 <= g - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:x <= g
0 <= u - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g < x
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
now apply Hu.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:x <= g
0 <= g - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:x <= g
0 <= u - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g < x
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
now apply Rle_0_minus.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:x <= g
0 <= u - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g < x
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Rle_0_minus.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:x <= g
x <= u
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g < x
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Hu.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g < x
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Rlt_le in Hxg.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g <= x
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
rewrite H.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g <= x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
rewrite 2!Rabs_left1.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g <= x
- (d - x) <= - (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g <= x
g - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g <= x
d - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Ropp_le_contravar.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g <= x
g - x <= d - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g <= x
g - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g <= x
d - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Rplus_le_compat_r.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g <= x
g <= d
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g <= x
g - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g <= x
d - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
now apply Hd.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g <= x
g - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g <= x
d - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
now apply Rle_minus.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g <= x
d - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Rle_minus.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
g:R
Hg:F g
Hxg:g <= x
d <= x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Hd.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x u
P x u \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = u)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
now left.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
exists f : R, Rnd_NG_pt F P x f
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
(* - d >> u *)
exists d.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
Rnd_NG_pt F P x d
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
split.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
Rnd_N_pt F x d
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
split.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
F d
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
forall g : R, F g -> Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Hd.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
forall g : R, F g -> Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
intros g Hg.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
destruct (Rle_or_lt x g) as [Hxg|Hxg].F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:x <= g
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g < x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
rewrite <- H.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:x <= g
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g < x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
rewrite 2!Rabs_pos_eq.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:x <= g
u - x <= g - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:x <= g
0 <= g - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:x <= g
0 <= u - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g < x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Rplus_le_compat_r.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:x <= g
u <= g
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:x <= g
0 <= g - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:x <= g
0 <= u - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g < x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
now apply Hu.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:x <= g
0 <= g - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:x <= g
0 <= u - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g < x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
now apply Rle_0_minus.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:x <= g
0 <= u - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g < x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Rle_0_minus.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:x <= g
x <= u
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g < x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Hu.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g < x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Rlt_le in Hxg.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g <= x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
rewrite 2!Rabs_left1.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g <= x
- (d - x) <= - (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g <= x
g - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g <= x
d - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Ropp_le_contravar.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g <= x
g - x <= d - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g <= x
g - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g <= x
d - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Rplus_le_compat_r.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g <= x
g <= d
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g <= x
g - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g <= x
d - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
now apply Hd.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g <= x
g - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g <= x
d - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
now apply Rle_minus.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g <= x
d - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Rle_minus.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
g:R
Hg:F g
Hxg:g <= x
d <= x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
apply Hd.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) = Rabs (d - x)
Hne:x <> d
Hf:~ F x
H':P x d
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
now left.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
exists f : R, Rnd_NG_pt F P x f
(* |up(x) - x| > |dn(x) - x| *)
exists d.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
Rnd_NG_pt F P x d
split.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
Rnd_N_pt F x d
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
split.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
F d
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
forall g : R, F g -> Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
apply Hd.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
forall g : R, F g -> Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
intros g Hg.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
destruct (Rle_or_lt x g) as [Hxg|Hxg].F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g < x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
apply Rlt_le.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
Rabs (d - x) < Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g < x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
apply Rlt_le_trans with (1 := H).F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
Rabs (u - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g < x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
rewrite 2!Rabs_pos_eq.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
u - x <= g - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
0 <= g - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
0 <= u - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g < x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
apply Rplus_le_compat_r.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
u <= g
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
0 <= g - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
0 <= u - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g < x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
now apply Hu.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
0 <= g - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
0 <= u - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g < x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
now apply Rle_0_minus.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
0 <= u - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g < x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
apply Rle_0_minus.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:x <= g
x <= u
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g < x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
apply Hu.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g < x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
apply Rlt_le in Hxg.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
Rabs (d - x) <= Rabs (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
rewrite 2!Rabs_left1.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
- (d - x) <= - (g - x)
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
g - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
d - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
apply Ropp_le_contravar.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
g - x <= d - x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
g - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
d - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
apply Rplus_le_compat_r.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
g <= d
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
g - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
d - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
now apply Hd.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
g - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
d - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
now apply Rle_minus.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
d - x <= 0
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
apply Rle_minus.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
g:R
Hg:F g
Hxg:g <= x
d <= x
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
apply Hd.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
P x d \/ (forall f2 : R, Rnd_N_pt F x f2 -> f2 = d)
right.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
forall f2 : R, Rnd_N_pt F x f2 -> f2 = d
intros f Hf.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
f:R
Hf:Rnd_N_pt F x f
f = d
destruct (Rnd_N_pt_DN_or_UP_eq F x _ _ _ Hd Hu Hf) as [K|K] ; rewrite K.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
f:R
Hf:Rnd_N_pt F x f
K:f = d
d = d
F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
f:R
Hf:Rnd_N_pt F x f
K:f = u
u = d
apply refl_equal.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
f:R
Hf:Rnd_N_pt F x f
K:f = u
u = d
elim Rlt_not_le with (1 := H).F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
f:R
Hf:Rnd_N_pt F x f
K:f = u
Rabs (u - x) <= Rabs (d - x)
rewrite <- K.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
f:R
Hf:Rnd_N_pt F x f
K:f = u
Rabs (f - x) <= Rabs (d - x)
apply Hf.F:R -> Prop
P:R -> R -> Prop
Hany:satisfies_any F
HP:NG_existence_prop F P
x, d:R
Hd:Rnd_DN_pt F x d
u:R
Hu:Rnd_UP_pt F x u
H:Rabs (u - x) > Rabs (d - x)
f:R
Hf:Rnd_N_pt F x f
K:f = u
F d
apply Hd.
Qed.

Theorem satisfies_any_imp_NA :
  forall F : R -> Prop,
  satisfies_any F ->
  round_pred (Rnd_NA_pt F).forall F : R -> Prop,
satisfies_any F -> round_pred (Rnd_NA_pt F)
Proof.forall F : R -> Prop,
satisfies_any F -> round_pred (Rnd_NA_pt F)
intros F Hany.F:R -> Prop
Hany:satisfies_any F
round_pred (Rnd_NA_pt F)
split.F:R -> Prop
Hany:satisfies_any F
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
assert (H : round_pred_total (Rnd_NG_pt F (fun a b => (Rabs a <= Rabs b)%R))).F:R -> Prop
Hany:satisfies_any F
round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
apply satisfies_any_imp_NG.F:R -> Prop
Hany:satisfies_any F
satisfies_any F
F:R -> Prop
Hany:satisfies_any F
NG_existence_prop F (fun a b : R => Rabs a <= Rabs b)
F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
apply Hany.F:R -> Prop
Hany:satisfies_any F
NG_existence_prop F (fun a b : R => Rabs a <= Rabs b)
F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
intros x d u Hf Hd Hu.F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Rabs x <= Rabs u \/ Rabs x <= Rabs d
F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
destruct (Rle_lt_dec 0 x) as [Hx|Hx].F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:0 <= x
Rabs x <= Rabs u \/ Rabs x <= Rabs d
F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
Rabs x <= Rabs u \/ Rabs x <= Rabs d
F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
left.F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:0 <= x
Rabs x <= Rabs u
F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
Rabs x <= Rabs u \/ Rabs x <= Rabs d
F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
rewrite Rabs_pos_eq with (1 := Hx).F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:0 <= x
x <= Rabs u
F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
Rabs x <= Rabs u \/ Rabs x <= Rabs d
F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
rewrite Rabs_pos_eq.F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:0 <= x
x <= u
F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:0 <= x
0 <= u
F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
Rabs x <= Rabs u \/ Rabs x <= Rabs d
F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
apply Hu.F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:0 <= x
0 <= u
F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
Rabs x <= Rabs u \/ Rabs x <= Rabs d
F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
apply Rle_trans with (1 := Hx).F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:0 <= x
x <= u
F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
Rabs x <= Rabs u \/ Rabs x <= Rabs d
F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
apply Hu.F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
Rabs x <= Rabs u \/ Rabs x <= Rabs d
F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
right.F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
Rabs x <= Rabs d
F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
rewrite Rabs_left with (1 := Hx).F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
- x <= Rabs d
F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
rewrite Rabs_left1.F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
- x <= - d
F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
d <= 0
F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
apply Ropp_le_contravar.F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
d <= x
F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
d <= 0
F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
apply Hd.F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
d <= 0
F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
apply Rlt_le in Hx.F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x <= 0
d <= 0
F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
apply Rle_trans with (2 := Hx).F:R -> Prop
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x <= 0
d <= x
F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
apply Hd.F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
round_pred_total (Rnd_NA_pt F)
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
intros x.F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
x:R
exists f : R, Rnd_NA_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
destruct (H x) as (f, Hf).F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
x, f:R
Hf:Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b) x f
exists f0 : R, Rnd_NA_pt F x f0
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
exists f.F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
x, f:R
Hf:Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b) x f
Rnd_NA_pt F x f
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
apply <- Rnd_NA_NG_pt.F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
x, f:R
Hf:Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b) x f
Rnd_NG_pt F (fun x0 f0 : R => Rabs x0 <= Rabs f0) x f
F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
x, f:R
Hf:Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b) x f
F 0
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
apply Hf.F:R -> Prop
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b))
x, f:R
Hf:Rnd_NG_pt F (fun a b : R => Rabs a <= Rabs b) x f
F 0
F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
apply Hany.F:R -> Prop
Hany:satisfies_any F
round_pred_monotone (Rnd_NA_pt F)
apply Rnd_NA_pt_monotone.F:R -> Prop
Hany:satisfies_any F
F 0
apply Hany.
Qed.

Theorem satisfies_any_imp_N0 :
  forall F : R -> Prop,
  F 0 -> satisfies_any F ->
  round_pred (Rnd_N0_pt F).forall F : R -> Prop,
F 0 -> satisfies_any F -> round_pred (Rnd_N0_pt F)
Proof.forall F : R -> Prop,
F 0 -> satisfies_any F -> round_pred (Rnd_N0_pt F)
intros F HF0 Hany.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred (Rnd_N0_pt F)
split.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_total (Rnd_N0_pt F)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
assert (H : round_pred_total (Rnd_NG_pt F (fun a b => (Rabs b <= Rabs a)%R))).F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
round_pred_total (Rnd_N0_pt F)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
apply satisfies_any_imp_NG.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
satisfies_any F
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
NG_existence_prop F (fun a b : R => Rabs b <= Rabs a)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
round_pred_total (Rnd_N0_pt F)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
apply Hany.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
NG_existence_prop F (fun a b : R => Rabs b <= Rabs a)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
round_pred_total (Rnd_N0_pt F)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
intros x d u Hf Hd Hu.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Rabs u <= Rabs x \/ Rabs d <= Rabs x
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
round_pred_total (Rnd_N0_pt F)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
destruct (Rle_lt_dec 0 x) as [Hx|Hx].F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:0 <= x
Rabs u <= Rabs x \/ Rabs d <= Rabs x
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
Rabs u <= Rabs x \/ Rabs d <= Rabs x
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
round_pred_total (Rnd_N0_pt F)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
right.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:0 <= x
Rabs d <= Rabs x
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
Rabs u <= Rabs x \/ Rabs d <= Rabs x
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
round_pred_total (Rnd_N0_pt F)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
rewrite Rabs_pos_eq with (1 := Hx).F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:0 <= x
Rabs d <= x
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
Rabs u <= Rabs x \/ Rabs d <= Rabs x
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
round_pred_total (Rnd_N0_pt F)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
rewrite Rabs_pos_eq.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:0 <= x
d <= x
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:0 <= x
0 <= d
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
Rabs u <= Rabs x \/ Rabs d <= Rabs x
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
round_pred_total (Rnd_N0_pt F)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
apply Hd.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:0 <= x
0 <= d
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
Rabs u <= Rabs x \/ Rabs d <= Rabs x
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
round_pred_total (Rnd_N0_pt F)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
apply Hd; try easy.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
Rabs u <= Rabs x \/ Rabs d <= Rabs x
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
round_pred_total (Rnd_N0_pt F)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
left.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
Rabs u <= Rabs x
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
round_pred_total (Rnd_N0_pt F)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
rewrite Rabs_left with (1 := Hx).F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
Rabs u <= - x
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
round_pred_total (Rnd_N0_pt F)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
rewrite Rabs_left1.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
- u <= - x
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
u <= 0
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
round_pred_total (Rnd_N0_pt F)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
apply Ropp_le_contravar.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
x <= u
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
u <= 0
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
round_pred_total (Rnd_N0_pt F)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
apply Hu.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
u <= 0
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
round_pred_total (Rnd_N0_pt F)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
apply Hu; try easy.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
x, d, u:R
Hf:~ F x
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
Hx:x < 0
x <= 0
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
round_pred_total (Rnd_N0_pt F)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
now apply Rlt_le.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
round_pred_total (Rnd_N0_pt F)
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
intros x.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
x:R
exists f : R, Rnd_N0_pt F x f
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
destruct (H x) as (f, Hf).F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
x, f:R
Hf:Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a) x f
exists f0 : R, Rnd_N0_pt F x f0
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
exists f.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
x, f:R
Hf:Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a) x f
Rnd_N0_pt F x f
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
apply <- Rnd_N0_NG_pt.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
x, f:R
Hf:Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a) x f
Rnd_NG_pt F (fun x0 f0 : R => Rabs f0 <= Rabs x0) x f
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
x, f:R
Hf:Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a) x f
F 0
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
apply Hf.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
H:round_pred_total
  (Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a))
x, f:R
Hf:Rnd_NG_pt F (fun a b : R => Rabs b <= Rabs a) x f
F 0
F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
apply HF0.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
round_pred_monotone (Rnd_N0_pt F)
apply Rnd_N0_pt_monotone.F:R -> Prop
HF0:F 0
Hany:satisfies_any F
F 0
apply HF0.
Qed.

End RND_prop.