Mise à jour terminée. Pour connaître les apports de la version 13.8.4 par rapport à notre ancienne version vous pouvez lire les "Release Notes" suivantes :
https://about.gitlab.com/releases/2021/02/11/security-release-gitlab-13-8-4-released/
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Fappli_IEEE.v 55.3 KB
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(**
This file is part of the Flocq formalization of floating-point
arithmetic in Coq: http://flocq.gforge.inria.fr/

Copyright (C) 2010 Sylvie Boldo
#<br />#
Copyright (C) 2010 Guillaume Melquiond

This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 3 of the License, or (at your option) any later version.

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

(** * IEEE-754 arithmetic *)
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Require Import Fcore.
Require Import Fcalc_digits.
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Require Import Fcalc_round.
Require Import Fcalc_bracket.
Require Import Fcalc_ops.
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Require Import Fcalc_div.
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Require Import Fcalc_sqrt.
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Require Import Fprop_relative.
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Section AnyRadix.

Inductive full_float :=
  | F754_zero : bool -> full_float
  | F754_infinity : bool -> full_float
  | F754_nan : full_float
  | F754_finite : bool -> positive -> Z -> full_float.

Definition FF2R r x :=
  match x with
  | F754_finite s m e => F2R (Float r (cond_Zopp s (Zpos m)) e)
  | _ => R0
  end.

End AnyRadix.

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Section Binary.

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Variable prec emax : Z.
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Context (prec_gt_0_ : Prec_gt_0 prec).
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Hypothesis Hmax : (prec < emax)%Z.
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Let emin := (3 - emax - prec)%Z.
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Let fexp := FLT_exp emin prec.
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Instance fexp_correct : Valid_exp fexp := FLT_exp_valid emin prec.
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Instance fexp_monotone : Monotone_exp fexp := FLT_exp_monotone emin prec.
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Definition bounded_prec m e :=
  Zeq_bool (fexp (Z_of_nat (S (digits2_Pnat m)) + e)) e.

Definition bounded m e :=
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  andb (bounded_prec m e) (Zle_bool e (emax - prec)).
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Definition valid_binary x :=
  match x with
  | F754_finite _ m e => bounded m e
  | _ => true
  end.

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Inductive binary_float :=
  | B754_zero : bool -> binary_float
  | B754_infinity : bool -> binary_float
  | B754_nan : binary_float
  | B754_finite : bool ->
    forall (m : positive) (e : Z), bounded m e = true -> binary_float.

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Definition FF2B x :=
  match x as x return valid_binary x = true -> binary_float with
  | F754_finite s m e => B754_finite s m e
  | F754_infinity s => fun _ => B754_infinity s
  | F754_zero s => fun _ => B754_zero s
  | F754_nan => fun _ => B754_nan
  end.

Definition B2FF x :=
  match x with
  | B754_finite s m e _ => F754_finite s m e
  | B754_infinity s => F754_infinity s
  | B754_zero s => F754_zero s
  | B754_nan => F754_nan
  end.

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Definition radix2 := Build_radix 2 (refl_equal true).

Definition B2R f :=
  match f with
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  | B754_finite s m e _ => F2R (Float radix2 (cond_Zopp s (Zpos m)) e)
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  | _ => R0
  end.

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Theorem FF2R_B2FF :
  forall x,
  FF2R radix2 (B2FF x) = B2R x.
Proof.
now intros [sx|sx| |sx mx ex Hx].
Qed.

Theorem B2FF_FF2B :
  forall x Hx,
  B2FF (FF2B x Hx) = x.
Proof.
now intros [sx|sx| |sx mx ex] Hx.
Qed.

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Theorem valid_binary_B2FF :
  forall x,
  valid_binary (B2FF x) = true.
Proof.
now intros [sx|sx| |sx mx ex Hx].
Qed.

Theorem FF2B_B2FF :
  forall x H,
  FF2B (B2FF x) H = x.
Proof.
intros [sx|sx| |sx mx ex Hx] H ; try easy.
apply f_equal.
apply eqbool_irrelevance.
Qed.

Theorem FF2B_B2FF_valid :
  forall x,
  FF2B (B2FF x) (valid_binary_B2FF x) = x.
Proof.
intros x.
apply FF2B_B2FF.
Qed.

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Theorem B2R_FF2B :
  forall x Hx,
  B2R (FF2B x Hx) = FF2R radix2 x.
Proof.
now intros [sx|sx| |sx mx ex] Hx.
Qed.

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Theorem match_FF2B :
  forall {T} fz fi fn ff x Hx,
  match FF2B x Hx return T with
  | B754_zero sx => fz sx
  | B754_infinity sx => fi sx
  | B754_nan => fn
  | B754_finite sx mx ex _ => ff sx mx ex
  end =
  match x with
  | F754_zero sx => fz sx
  | F754_infinity sx => fi sx
  | F754_nan => fn
  | F754_finite sx mx ex => ff sx mx ex
  end.
Proof.
now intros T fz fi fn ff [sx|sx| |sx mx ex] Hx.
Qed.

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Theorem canonic_bounded_prec :
  forall (sx : bool) mx ex,
  bounded_prec mx ex = true ->
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  canonic radix2 fexp (Float radix2 (cond_Zopp sx (Zpos mx)) ex).
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Proof.
intros sx mx ex H.
assert (Hx := Zeq_bool_eq _ _ H). clear H.
apply sym_eq.
simpl.
pattern ex at 2 ; rewrite <- Hx.
apply (f_equal fexp).
rewrite ln_beta_F2R_digits.
rewrite <- digits_abs.
rewrite Z_of_nat_S_digits2_Pnat.
now case sx.
now case sx.
Qed.

Theorem generic_format_B2R :
  forall x,
  generic_format radix2 fexp (B2R x).
Proof.
intros [sx|sx| |sx mx ex Hx] ; try apply generic_format_0.
simpl.
apply generic_format_canonic.
apply canonic_bounded_prec.
now destruct (andb_prop _ _ Hx) as (H, _).
Qed.

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Theorem binary_unicity :
  forall x y : binary_float,
  B2FF x = B2FF y ->
  x = y.
Proof.
intros [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] ; try easy.
(* *)
intros H.
now inversion H.
(* *)
intros H.
now inversion H.
(* *)
intros H.
inversion H.
clear H.
revert Hx.
rewrite H2, H3.
intros Hx.
apply f_equal.
apply eqbool_irrelevance.
Qed.

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Definition is_finite_strict f :=
  match f with
  | B754_finite _ _ _ _ => true
  | _ => false
  end.

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Theorem finite_binary_unicity :
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  forall x y : binary_float,
  is_finite_strict x = true ->
  is_finite_strict y = true ->
  B2R x = B2R y ->
  x = y.
Proof.
intros [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] ; try easy.
simpl.
intros _ _ Heq.
assert (Hs: sx = sy).
(* *)
revert Heq. clear.
case sx ; case sy ; try easy ;
  intros Heq ; apply False_ind ; revert Heq.
apply Rlt_not_eq.
apply Rlt_trans with R0.
now apply F2R_lt_0_compat.
now apply F2R_gt_0_compat.
apply Rgt_not_eq.
apply Rgt_trans with R0.
now apply F2R_gt_0_compat.
now apply F2R_lt_0_compat.
assert (mx = my /\ ex = ey).
(* *)
refine (_ (canonic_unicity _ fexp _ _ _ _ Heq)).
rewrite Hs.
now case sy ; intro H ; injection H ; split.
apply canonic_bounded_prec.
exact (proj1 (andb_prop _ _ Hx)).
apply canonic_bounded_prec.
exact (proj1 (andb_prop _ _ Hy)).
(* *)
revert Hx.
rewrite Hs, (proj1 H), (proj2 H).
intros Hx.
apply f_equal.
apply eqbool_irrelevance.
Qed.

Definition is_finite f :=
  match f with
  | B754_finite _ _ _ _ => true
  | B754_zero _ => true
  | _ => false
  end.

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Definition is_finite_FF f :=
  match f with
  | F754_finite _ _ _ => true
  | F754_zero _ => true
  | _ => false
  end.

Theorem is_finite_FF2B :
  forall x Hx,
  is_finite (FF2B x Hx) = is_finite_FF x.
Proof.
now intros [| | |].
Qed.

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Theorem is_finite_FF_B2FF :
  forall x,
  is_finite_FF (B2FF x) = is_finite x.
Proof.
now intros [| | |].
Qed.

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Definition Bopp x :=
  match x with
  | B754_nan => x
  | B754_infinity sx => B754_infinity (negb sx)
  | B754_finite sx mx ex Hx => B754_finite (negb sx) mx ex Hx
  | B754_zero sx => B754_zero (negb sx)
  end.

Theorem Bopp_involutive :
  forall x, Bopp (Bopp x) = x.
Proof.
now intros [sx|sx| |sx mx ex Hx] ; simpl ; try rewrite Bool.negb_involutive.
Qed.

Theorem B2R_Bopp :
  forall x,
  B2R (Bopp x) = (- B2R x)%R.
Proof.
intros [sx|sx| |sx mx ex Hx] ; apply sym_eq ; try apply Ropp_0.
simpl.
rewrite opp_F2R.
now case sx.
Qed.

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Theorem bounded_lt_emax :
  forall mx ex,
  bounded mx ex = true ->
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  (F2R (Float radix2 (Zpos mx) ex) < bpow radix2 emax)%R.
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Proof.
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intros mx ex Hx.
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destruct (andb_prop _ _ Hx) as (H1,H2).
generalize (Zeq_bool_eq _ _ H1). clear H1. intro H1.
generalize (Zle_bool_imp_le _ _ H2). clear H2. intro H2.
generalize (ln_beta_F2R_digits radix2 (Zpos mx) ex).
destruct (ln_beta radix2 (F2R (Float radix2 (Zpos mx) ex))) as (e',Ex).
unfold ln_beta_val.
intros H.
apply Rlt_le_trans with (bpow radix2 e').
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change (Zpos mx) with (Zabs (Zpos mx)).
rewrite <- abs_F2R.
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apply Ex.
apply Rgt_not_eq.
now apply F2R_gt_0_compat.
apply bpow_le.
rewrite H. 2: discriminate.
revert H1. clear -H2.
rewrite Z_of_nat_S_digits2_Pnat.
change Fcalc_digits.radix2 with radix2.
unfold fexp, FLT_exp.
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generalize (digits radix2 (Zpos mx)).
intros ; zify ; subst.
clear -H H2. clearbody emin.
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omega.
Qed.

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Theorem B2R_lt_emax :
  forall x,
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  (Rabs (B2R x) < bpow radix2 emax)%R.
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Proof.
intros [sx|sx| |sx mx ex Hx] ; simpl ; try ( rewrite Rabs_R0 ; apply bpow_gt_0 ).
rewrite abs_F2R, abs_cond_Zopp.
now apply bounded_lt_emax.
Qed.

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Theorem bounded_canonic_lt_emax :
  forall mx ex,
  canonic radix2 fexp (Float radix2 (Zpos mx) ex) ->
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  (F2R (Float radix2 (Zpos mx) ex) < bpow radix2 emax)%R ->
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  bounded mx ex = true.
Proof.
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intros mx ex Cx Bx.
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apply andb_true_intro.
split.
unfold bounded_prec.
unfold canonic, Fexp in Cx.
rewrite Cx at 2.
rewrite Z_of_nat_S_digits2_Pnat.
change Fcalc_digits.radix2 with radix2.
unfold canonic_exponent.
rewrite ln_beta_F2R_digits. 2: discriminate.
now apply -> Zeq_is_eq_bool.
apply Zle_bool_true.
unfold canonic, Fexp in Cx.
rewrite Cx.
unfold canonic_exponent, fexp, FLT_exp.
destruct (ln_beta radix2 (F2R (Float radix2 (Zpos mx) ex))) as (e',Ex). simpl.
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apply Zmax_lub.
cut (e' - 1 < emax)%Z. clear ; omega.
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apply lt_bpow with radix2.
apply Rle_lt_trans with (2 := Bx).
change (Zpos mx) with (Zabs (Zpos mx)).
rewrite <- abs_F2R.
apply Ex.
apply Rgt_not_eq.
now apply F2R_gt_0_compat.
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unfold emin.
generalize (prec_gt_0 prec).
clear -Hmax ; omega.
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Qed.

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Record shr_record := { shr_m : Z ; shr_r : bool ; shr_s : bool }.

Definition shr_1 mrs :=
  let '(Build_shr_record m r s) := mrs in
  let s := orb r s in
  match m with
  | Z0 => Build_shr_record Z0 false s
  | Zpos xH => Build_shr_record Z0 true s
  | Zpos (xO p) => Build_shr_record (Zpos p) false s
  | Zpos (xI p) => Build_shr_record (Zpos p) true s
  | Zneg xH => Build_shr_record Z0 true s
  | Zneg (xO p) => Build_shr_record (Zneg p) false s
  | Zneg (xI p) => Build_shr_record (Zneg p) true s
  end.

Definition loc_of_shr_record mrs :=
  match mrs with
  | Build_shr_record _ false false => loc_Exact
  | Build_shr_record _ false true => loc_Inexact Lt
  | Build_shr_record _ true false => loc_Inexact Eq
  | Build_shr_record _ true true => loc_Inexact Gt
  end.

Definition shr_record_of_loc m l :=
  match l with
  | loc_Exact => Build_shr_record m false false
  | loc_Inexact Lt => Build_shr_record m false true
  | loc_Inexact Eq => Build_shr_record m true false
  | loc_Inexact Gt => Build_shr_record m true true
  end.

Theorem shr_m_shr_record_of_loc :
  forall m l,
  shr_m (shr_record_of_loc m l) = m.
Proof.
now intros m [|[| |]].
Qed.

Theorem loc_of_shr_record_of_loc :
  forall m l,
  loc_of_shr_record (shr_record_of_loc m l) = l.
Proof.
now intros m [|[| |]].
Qed.

Definition shr mrs e n :=
  match n with
  | Zpos p => (iter_pos p _ shr_1 mrs, (e + n)%Z)
  | _ => (mrs, e)
  end.

Theorem inbetween_shr_1 :
  forall x mrs e,
  (0 <= shr_m mrs)%Z ->
  inbetween_float radix2 (shr_m mrs) e x (loc_of_shr_record mrs) ->
  inbetween_float radix2 (shr_m (shr_1 mrs)) (e + 1) x (loc_of_shr_record (shr_1 mrs)).
Proof.
intros x mrs e Hm Hl.
refine (_ (new_location_even_correct (F2R (Float radix2 (shr_m (shr_1 mrs)) (e + 1))) (bpow radix2 e) 2 _ _ _ x (if shr_r (shr_1 mrs) then 1 else 0) (loc_of_shr_record mrs) _ _)) ; try easy.
2: apply bpow_gt_0.
2: now case (shr_r (shr_1 mrs)) ; split.
change (Z2R 2) with (bpow radix2 1).
rewrite <- bpow_plus.
rewrite (Zplus_comm 1), <- (F2R_bpow radix2 (e + 1)).
unfold inbetween_float, F2R. simpl.
rewrite Z2R_plus, Rmult_plus_distr_r.
replace (new_location_even 2 (if shr_r (shr_1 mrs) then 1%Z else 0%Z) (loc_of_shr_record mrs)) with (loc_of_shr_record (shr_1 mrs)).
easy.
clear -Hm.
destruct mrs as (m, r, s).
now destruct m as [|[m|m|]|m] ; try (now elim Hm) ; destruct r as [|] ; destruct s as [|].
rewrite (F2R_change_exp radix2 e).
2: apply Zle_succ.
unfold F2R. simpl.
rewrite <- 2!Rmult_plus_distr_r, <- 2!Z2R_plus.
rewrite Zplus_assoc.
replace (shr_m (shr_1 mrs) * 2 ^ (e + 1 - e) + (if shr_r (shr_1 mrs) then 1%Z else 0%Z))%Z with (shr_m mrs).
exact Hl.
ring_simplify (e + 1 - e)%Z.
change (2^1)%Z with 2%Z.
rewrite Zmult_comm.
clear -Hm.
destruct mrs as (m, r, s).
now destruct m as [|[m|m|]|m] ; try (now elim Hm) ; destruct r as [|] ; destruct s as [|].
Qed.

Theorem inbetween_shr :
  forall x m e l n,
  (0 <= m)%Z ->
  inbetween_float radix2 m e x l ->
  let '(mrs, e') := shr (shr_record_of_loc m l) e n in
  inbetween_float radix2 (shr_m mrs) e' x (loc_of_shr_record mrs).
Proof.
intros x m e l n Hm Hl.
destruct n as [|n|n].
now destruct l as [|[| |]].
2: now destruct l as [|[| |]].
unfold shr.
rewrite iter_nat_of_P.
rewrite Zpos_eq_Z_of_nat_o_nat_of_P.
induction (nat_of_P n).
simpl.
rewrite Zplus_0_r.
now destruct l as [|[| |]].
simpl iter_nat.
rewrite inj_S.
unfold Zsucc.
rewrite  Zplus_assoc.
revert IHn0.
apply inbetween_shr_1.
clear -Hm.
induction n0.
now destruct l as [|[| |]].
simpl.
revert IHn0.
generalize (iter_nat n0 shr_record shr_1 (shr_record_of_loc m l)).
clear.
intros (m, r, s) Hm.
now destruct m as [|[m|m|]|m] ; try (now elim Hm) ; destruct r as [|] ; destruct s as [|].
Qed.

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Definition digits2 m :=
  match m with Z0 => m | Zpos p => Z_of_nat (S (digits2_Pnat p)) | Zneg p => Z_of_nat (S (digits2_Pnat p)) end.

Theorem digits2_digits :
  forall m,
  digits2 m = digits radix2 m.
Proof.
unfold digits2.
intros [|m|m] ; try apply Z_of_nat_S_digits2_Pnat.
easy.
Qed.

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Definition shr_fexp m e l :=
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  shr (shr_record_of_loc m l) e (fexp (digits2 m + e) - e).
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Theorem shr_truncate :
  forall m e l,
  (0 <= m)%Z ->
  shr_fexp m e l =
  let '(m', e', l') := truncate radix2 fexp (m, e, l) in (shr_record_of_loc m' l', e').
Proof.
intros m e l Hm.
case_eq (truncate radix2 fexp (m, e, l)).
intros (m', e') l'.
unfold shr_fexp.
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rewrite digits2_digits.
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case_eq (fexp (digits radix2 m + e) - e)%Z.
(* *)
intros He.
unfold truncate.
rewrite He.
simpl.
intros H.
now inversion H.
(* *)
intros p Hp.
assert (He: (e <= fexp (digits radix2 m + e))%Z).
clear -Hp ; zify ; omega.
destruct (inbetween_float_ex radix2 m e l) as (x, Hx).
generalize (inbetween_shr x m e l (fexp (digits radix2 m + e) - e) Hm Hx).
assert (Hx0 : (0 <= x)%R).
apply Rle_trans with (F2R (Float radix2 m e)).
now apply F2R_ge_0_compat.
exact (proj1 (inbetween_float_bounds _ _ _ _ _ Hx)).
case_eq (shr (shr_record_of_loc m l) e (fexp (digits radix2 m + e) - e)).
intros mrs e'' H3 H4 H1.
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generalize (truncate_correct radix2 _ x m e l Hx0 Hx (or_introl _ He)).
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rewrite H1.
intros (H2,_).
rewrite <- Hp, H3.
assert (e'' = e').
change (snd (mrs, e'') = snd (fst (m',e',l'))).
rewrite <- H1, <- H3.
unfold truncate.
now rewrite Hp.
rewrite H in H4 |- *.
apply (f_equal (fun v => (v, _))).
destruct (inbetween_float_unique _ _ _ _ _ _ _ H2 H4) as (H5, H6).
rewrite H5, H6.
case mrs.
now intros m0 [|] [|].
(* *)
intros p Hp.
unfold truncate.
rewrite Hp.
simpl.
intros H.
now inversion H.
Qed.

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Inductive mode := mode_NE | mode_ZR | mode_DN | mode_UP | mode_NA.

Definition round_mode m :=
  match m with
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  | mode_NE => ZnearestE
  | mode_ZR => Ztrunc
  | mode_DN => Zfloor
  | mode_UP => Zceil
  | mode_NA => ZnearestA
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  end.
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Definition choice_mode m sx mx lx :=
  match m with
  | mode_NE => cond_incr (round_N (negb (Zeven mx)) lx) mx
  | mode_ZR => mx
  | mode_DN => cond_incr (round_sign_DN sx lx) mx
  | mode_UP => cond_incr (round_sign_UP sx lx) mx
  | mode_NA => cond_incr (round_N true lx) mx
  end.

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Global Instance valid_rnd_round_mode : forall m, Valid_rnd (round_mode m).
Proof.
destruct m ; unfold round_mode ; auto with typeclass_instances.
Qed.

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Definition overflow_to_inf m s :=
  match m with
  | mode_NE => true
  | mode_NA => true
  | mode_ZR => false
  | mode_UP => negb s
  | mode_DN => s
  end.

Definition binary_overflow m s :=
  if overflow_to_inf m s then F754_infinity s
  else F754_finite s (match (Zpower 2 prec - 1)%Z with Zpos p => p | _ => xH end) (emax - prec).

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Definition binary_round_sign mode sx mx ex lx :=
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  let '(mrs', e') := shr_fexp (Zpos mx) ex lx in
  let '(mrs'', e'') := shr_fexp (choice_mode mode sx (shr_m mrs') (loc_of_shr_record mrs')) e' loc_Exact in
  match shr_m mrs'' with
622
  | Z0 => F754_zero sx
623
  | Zpos m => if Zle_bool e'' (emax - prec) then F754_finite sx m e'' else binary_overflow mode sx
624
  | _ => F754_nan (* dummy *)
625 626 627 628 629 630
  end.

Theorem binary_round_sign_correct :
  forall mode x mx ex lx,
  inbetween_float radix2 (Zpos mx) ex (Rabs x) lx ->
  (ex <= fexp (digits radix2 (Zpos mx) + ex))%Z ->
631 632
  let z := binary_round_sign mode (Rlt_bool x 0) mx ex lx in
  valid_binary z = true /\
633
  if Rlt_bool (Rabs (round radix2 fexp (round_mode mode) x)) (bpow radix2 emax) then
634 635
    FF2R radix2 z = round radix2 fexp (round_mode mode) x /\
    is_finite_FF z = true
636
  else
637
    z = binary_overflow mode (Rlt_bool x 0).
638
Proof with auto with typeclass_instances.
639 640 641
intros m x mx ex lx Bx Ex z.
unfold binary_round_sign in z.
revert z.
642
rewrite shr_truncate. 2: easy.
643 644
refine (_ (round_trunc_sign_any_correct _ _ (round_mode m) (choice_mode m) _ x (Zpos mx) ex lx Bx (or_introl _ Ex))).
refine (_ (truncate_correct_partial _ _ _ _ _ _ _ Bx Ex)).
645
destruct (truncate radix2 fexp (Zpos mx, ex, lx)) as ((m1, e1), l1).
646
rewrite loc_of_shr_record_of_loc, shr_m_shr_record_of_loc.
647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673
set (m1' := choice_mode m (Rlt_bool x 0) m1 l1).
intros (H1a,H1b) H1c.
rewrite H1c.
assert (Hm: (m1 <= m1')%Z).
(* . *)
unfold m1', choice_mode, cond_incr.
case m ;
  try apply Zle_refl ;
  match goal with |- (m1 <= if ?b then _ else _)%Z =>
    case b ; [ apply Zle_succ | apply Zle_refl ] end.
assert (Hr: Rabs (round radix2 fexp (round_mode m) x) = F2R (Float radix2 m1' e1)).
(* . *)
rewrite <- (Zabs_eq m1').
replace (Zabs m1') with (Zabs (cond_Zopp (Rlt_bool x 0) m1')).
rewrite <- abs_F2R.
now apply f_equal.
apply abs_cond_Zopp.
apply Zle_trans with (2 := Hm).
apply Zlt_succ_le.
apply F2R_gt_0_reg with radix2 e1.
apply Rle_lt_trans with (1 := Rabs_pos x).
exact (proj2 (inbetween_float_bounds _ _ _ _ _ H1a)).
(* . *)
assert (Br: inbetween_float radix2 m1' e1 (Rabs (round radix2 fexp (round_mode m) x)) loc_Exact).
now apply inbetween_Exact.
destruct m1' as [|m1'|m1'].
(* . m1' = 0 *)
674
rewrite shr_truncate. 2: apply Zle_refl.
675 676
generalize (truncate_0 radix2 fexp e1 loc_Exact).
destruct (truncate radix2 fexp (Z0, e1, loc_Exact)) as ((m2, e2), l2).
677
rewrite shr_m_shr_record_of_loc.
678
intros Hm2.
679
rewrite Hm2.
680
intros z.
681 682
repeat split.
rewrite Rlt_bool_true.
683
repeat split.
684 685
apply sym_eq.
case Rlt_bool ; apply F2R_0.
686 687
rewrite abs_F2R, abs_cond_Zopp, F2R_0.
apply bpow_gt_0.
688 689 690 691 692 693 694
(* . 0 < m1' *)
assert (He: (e1 <= fexp (digits radix2 (Zpos m1') + e1))%Z).
rewrite <- ln_beta_F2R_digits, <- Hr, ln_beta_abs.
2: discriminate.
rewrite H1b.
rewrite canonic_exponent_abs.
fold (canonic_exponent radix2 fexp (round radix2 fexp (round_mode m) x)).
695
apply canonic_exponent_round...
696 697 698 699 700 701
rewrite H1c.
case (Rlt_bool x 0).
apply Rlt_not_eq.
now apply F2R_lt_0_compat.
apply Rgt_not_eq.
now apply F2R_gt_0_compat.
702
refine (_ (truncate_correct_partial _ _ _ _ _ _ _ Br He)).
703 704 705
2: now rewrite Hr ; apply F2R_gt_0_compat.
refine (_ (truncate_correct_format radix2 fexp (Zpos m1') e1 _ _ He)).
2: discriminate.
706
rewrite shr_truncate. 2: easy.
707
destruct (truncate radix2 fexp (Zpos m1', e1, loc_Exact)) as ((m2, e2), l2).
708
rewrite shr_m_shr_record_of_loc.
709 710 711 712 713
intros (H3,H4) (H2,_).
destruct m2 as [|m2|m2].
elim Rgt_not_eq with (2 := H3).
rewrite F2R_0.
now apply F2R_gt_0_compat.
714 715
rewrite F2R_cond_Zopp, H3, abs_cond_Ropp, abs_F2R.
simpl Zabs.
716 717 718 719 720 721 722 723 724 725 726 727 728
case_eq (Zle_bool e2 (emax - prec)) ; intros He2.
assert (bounded m2 e2 = true).
apply andb_true_intro.
split.
unfold bounded_prec.
apply Zeq_bool_true.
rewrite Z_of_nat_S_digits2_Pnat.
rewrite <- ln_beta_F2R_digits.
apply sym_eq.
now rewrite H3 in H4.
discriminate.
exact He2.
apply (conj H).
729
rewrite Rlt_bool_true.
730
repeat split.
731 732
apply F2R_cond_Zopp.
now apply bounded_lt_emax.
733 734 735 736
rewrite (Rlt_bool_false _ (bpow radix2 emax)).
refine (conj _ (refl_equal _)).
unfold binary_overflow.
case overflow_to_inf.
737
apply refl_equal.
738 739 740 741 742 743 744 745
unfold valid_binary, bounded.
rewrite Zle_bool_refl.
rewrite Bool.andb_true_r.
apply Zeq_bool_true.
rewrite Z_of_nat_S_digits2_Pnat.
change Fcalc_digits.radix2 with radix2.
replace (digits radix2 (Zpos (match (Zpower 2 prec - 1)%Z with Zpos p => p | _ => xH end))) with prec.
unfold fexp, FLT_exp, emin.
746 747
generalize (prec_gt_0 prec).
clear -Hmax ; zify ; omega.
748 749 750
change 2%Z with (radix_val radix2).
case_eq (Zpower radix2 prec - 1)%Z.
simpl digits.
751
generalize (Zpower_gt_1 radix2 prec (prec_gt_0 prec)).
752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767
clear ; omega.
intros p Hp.
apply Zle_antisym.
cut (prec - 1 < digits radix2 (Zpos p))%Z. clear ; omega.
apply digits_gt_Zpower.
simpl Zabs. rewrite <- Hp.
cut (Zpower radix2 (prec - 1) < Zpower radix2 prec)%Z. clear ; omega.
apply lt_Z2R.
rewrite 2!Z2R_Zpower. 2: now apply Zlt_le_weak.
apply bpow_lt.
apply Zlt_pred.
now apply Zlt_0_le_0_pred.
apply digits_le_Zpower.
simpl Zabs. rewrite <- Hp.
apply Zlt_pred.
intros p Hp.
768
generalize (Zpower_gt_1 radix2 _ (prec_gt_0 prec)).
769
clear -Hp ; zify ; omega.
770 771
apply Rnot_lt_le.
intros Hx.
772 773 774 775
generalize (refl_equal (bounded m2 e2)).
unfold bounded at 2.
rewrite He2.
rewrite Bool.andb_false_r.
776 777
rewrite bounded_canonic_lt_emax with (2 := Hx).
discriminate.
778 779 780 781 782 783 784
unfold canonic.
now rewrite <- H3.
elim Rgt_not_eq with (2 := H3).
apply Rlt_trans with R0.
now apply F2R_lt_0_compat.
now apply F2R_gt_0_compat.
rewrite <- Hr.
785 786
apply generic_format_abs...
apply generic_format_round...
787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802
(* . not m1' < 0 *)
elim Rgt_not_eq with (2 := Hr).
apply Rlt_le_trans with R0.
now apply F2R_lt_0_compat.
apply Rabs_pos.
(* *)
apply Rlt_le_trans with (2 := proj1 (inbetween_float_bounds _ _ _ _ _ Bx)).
now apply F2R_gt_0_compat.
(* all the modes are valid *)
clear. case m.
exact inbetween_int_NE_sign.
exact inbetween_int_ZR_sign.
exact inbetween_int_DN_sign.
exact inbetween_int_UP_sign.
exact inbetween_int_NA_sign.
Qed.
803

804 805 806 807 808 809 810 811
Definition Bsign x :=
  match x with
  | B754_nan => false
  | B754_zero s => s
  | B754_infinity s => s
  | B754_finite s _ _ _ => s
  end.

812 813 814 815 816 817 818 819 820 821 822 823 824 825 826
Definition Bsign_FF x :=
  match x with
  | F754_nan => false
  | F754_zero s => s
  | F754_infinity s => s
  | F754_finite s _ _ => s
  end.

Theorem Bsign_FF2B :
  forall x H,
  Bsign (FF2B x H) = Bsign_FF x.
Proof.
now intros [sx|sx| |sx mx ex] H.
Qed.

827 828 829 830 831 832 833
Lemma Bmult_correct_aux :
  forall m sx mx ex (Hx : bounded mx ex = true) sy my ey (Hy : bounded my ey = true),
  let x := F2R (Float radix2 (cond_Zopp sx (Zpos mx)) ex) in
  let y := F2R (Float radix2 (cond_Zopp sy (Zpos my)) ey) in
  let z := binary_round_sign m (xorb sx sy) (mx * my) (ex + ey) loc_Exact in
  valid_binary z = true /\
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (x * y))) (bpow radix2 emax) then
834 835
    FF2R radix2 z = round radix2 fexp (round_mode m) (x * y) /\
    is_finite_FF z = true
836
  else
837
    z = binary_overflow m (xorb sx sy).
838
Proof.
839 840
intros m sx mx ex Hx sy my ey Hy x y.
unfold x, y.
841
rewrite <- mult_F2R.
842
simpl.
843 844 845 846 847 848 849 850 851 852 853 854 855 856 857
replace (xorb sx sy) with (Rlt_bool (F2R (Float radix2 (cond_Zopp sx (Zpos mx) * cond_Zopp sy (Zpos my)) (ex + ey))) 0).
apply binary_round_sign_correct.
constructor.
rewrite abs_F2R.
apply F2R_eq_compat.
rewrite Zabs_Zmult.
now rewrite 2!abs_cond_Zopp.
(* *)
change (Zpos (mx * my)) with (Zpos mx * Zpos my)%Z.
assert (forall m e, bounded m e = true -> fexp (digits radix2 (Zpos m) + e) = e)%Z.
clear. intros m e Hb.
destruct (andb_prop _ _ Hb) as (H,_).
apply Zeq_bool_eq.
now rewrite <- Z_of_nat_S_digits2_Pnat.
generalize (H _ _ Hx) (H _ _ Hy).
858
clear x y sx sy Hx Hy H.
859 860 861 862
unfold fexp, FLT_exp.
refine (_ (digits_mult_ge radix2 (Zpos mx) (Zpos my) _ _)) ; try discriminate.
refine (_ (digits_gt_0 radix2 (Zpos mx) _) (digits_gt_0 radix2 (Zpos my) _)) ; try discriminate.
generalize (digits radix2 (Zpos mx)) (digits radix2 (Zpos my)) (digits radix2 (Zpos mx * Zpos my)).
863
clear -Hmax.
864
unfold emin.
865 866 867 868 869 870 871 872 873 874 875 876 877 878
intros dx dy dxy Hx Hy Hxy.
zify ; intros ; subst.
omega.
(* *)
case sx ; case sy.
apply Rlt_bool_false.
now apply F2R_ge_0_compat.
apply Rlt_bool_true.
now apply F2R_lt_0_compat.
apply Rlt_bool_true.
now apply F2R_lt_0_compat.
apply Rlt_bool_false.
now apply F2R_ge_0_compat.
Qed.
879

880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898
Definition Bmult m x y :=
  match x, y with
  | B754_nan, _ => x
  | _, B754_nan => y
  | B754_infinity sx, B754_infinity sy => B754_infinity (xorb sx sy)
  | B754_infinity sx, B754_finite sy _ _ _ => B754_infinity (xorb sx sy)
  | B754_finite sx _ _ _, B754_infinity sy => B754_infinity (xorb sx sy)
  | B754_infinity _, B754_zero _ => B754_nan
  | B754_zero _, B754_infinity _ => B754_nan
  | B754_finite sx _ _ _, B754_zero sy => B754_zero (xorb sx sy)
  | B754_zero sx, B754_finite sy _ _ _ => B754_zero (xorb sx sy)
  | B754_zero sx, B754_zero sy => B754_zero (xorb sx sy)
  | B754_finite sx mx ex Hx, B754_finite sy my ey Hy =>
    FF2B _ (proj1 (Bmult_correct_aux m sx mx ex Hx sy my ey Hy))
  end.

Theorem Bmult_correct :
  forall m x y,
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x * B2R y))) (bpow radix2 emax) then
899 900
    B2R (Bmult m x y) = round radix2 fexp (round_mode m) (B2R x * B2R y) /\
    is_finite (Bmult m x y) = andb (is_finite x) (is_finite y)
901
  else
902
    B2FF (Bmult m x y) = binary_overflow m (xorb (Bsign x) (Bsign y)).
903 904
Proof.
intros m [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] ;
905
  try ( rewrite ?Rmult_0_r, ?Rmult_0_l, round_0, Rabs_R0, Rlt_bool_true ; [ split ; apply refl_equal | apply bpow_gt_0 | auto with typeclass_instances ] ).
906 907
simpl.
case Bmult_correct_aux.
908 909 910 911
intros H1.
case Rlt_bool.
intros (H2, H3).
split.
912
now rewrite B2R_FF2B.
913 914
now rewrite is_finite_FF2B.
intros H2.
915 916 917
now rewrite B2FF_FF2B.
Qed.

918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941
Definition Bmult_FF m x y :=
  match x, y with
  | F754_nan, _ => x
  | _, F754_nan => y
  | F754_infinity sx, F754_infinity sy => F754_infinity (xorb sx sy)
  | F754_infinity sx, F754_finite sy _ _ => F754_infinity (xorb sx sy)
  | F754_finite sx _ _, F754_infinity sy => F754_infinity (xorb sx sy)
  | F754_infinity _, F754_zero _ => F754_nan
  | F754_zero _, F754_infinity _ => F754_nan
  | F754_finite sx _ _, F754_zero sy => F754_zero (xorb sx sy)
  | F754_zero sx, F754_finite sy _ _ => F754_zero (xorb sx sy)
  | F754_zero sx, F754_zero sy => F754_zero (xorb sx sy)
  | F754_finite sx mx ex, F754_finite sy my ey =>
    binary_round_sign m (xorb sx sy) (mx * my) (ex + ey) loc_Exact
  end.

Theorem B2FF_Bmult :
  forall m x y,
  B2FF (Bmult m x y) = Bmult_FF m (B2FF x) (B2FF y).
Proof.
intros m [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] ; try easy.
apply B2FF_FF2B.
Qed.

942
Definition shl mx ex ex' :=
943 944 945 946 947
  match (ex' - ex)%Z with
  | Zneg d => (shift_pos d mx, ex')
  | _ => (mx, ex)
  end.

948 949 950 951 952
Theorem shl_correct :
  forall mx ex ex',
  let (mx', ex'') := shl mx ex ex' in
  F2R (Float radix2 (Zpos mx) ex) = F2R (Float radix2 (Zpos mx') ex'') /\
  (ex'' <= ex')%Z.
953
Proof.
954 955 956
intros mx ex ex'.
unfold shl.
case_eq (ex' - ex)%Z.
957 958 959 960 961 962 963 964
(* d = 0 *)
intros H.
repeat split.
rewrite Zminus_eq with (1 := H).
apply Zle_refl.
(* d > 0 *)
intros d Hd.
repeat split.
965
replace ex' with (ex' - ex + ex)%Z by ring.
966 967 968 969 970 971 972 973 974 975
rewrite Hd.
pattern ex at 1 ; rewrite <- Zplus_0_l.
now apply Zplus_le_compat_r.
(* d < 0 *)
intros d Hd.
rewrite shift_pos_correct, Zmult_comm.
change (Zpower_pos 2 d) with (Zpower radix2 (Zpos d)).
change (Zpos d) with (Zopp (Zneg d)).
rewrite <- Hd.
split.
976
replace (- (ex' - ex))%Z with (ex - ex')%Z by ring.
977 978
apply F2R_change_exp.
apply Zle_0_minus_le.
979
replace (ex - ex')%Z with (- (ex' - ex))%Z by ring.
980 981 982
now rewrite Hd.
apply Zle_refl.
Qed.
983

984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999
Theorem snd_shl :
  forall mx ex ex',
  (ex' <= ex)%Z ->
  snd (shl mx ex ex') = ex'.
Proof.
intros mx ex ex' He.
unfold shl.
case_eq (ex' - ex)%Z ; simpl.
intros H.
now rewrite Zminus_eq with (1 := H).
intros p.
clear -He ; zify ; omega.
intros.
apply refl_equal.
Qed.

1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023
Definition shl_fexp mx ex :=
  shl mx ex (fexp (Z_of_nat (S (digits2_Pnat mx)) + ex)).

Theorem shl_fexp_correct :
  forall mx ex,
  let (mx', ex') := shl_fexp mx ex in
  F2R (Float radix2 (Zpos mx) ex) = F2R (Float radix2 (Zpos mx') ex') /\
  (ex' <= fexp (digits radix2 (Zpos mx') + ex'))%Z.
Proof.
intros mx ex.
unfold shl_fexp.
generalize (shl_correct mx ex (fexp (Z_of_nat (S (digits2_Pnat mx)) + ex))).
rewrite Z_of_nat_S_digits2_Pnat.
case shl.
intros mx' ex' (H1, H2).
split.
exact H1.
rewrite <- ln_beta_F2R_digits. 2: easy.
rewrite <- H1.
now rewrite ln_beta_F2R_digits.
Qed.

Definition binary_round_sign_shl m sx mx ex :=
  let '(mz, ez) := shl_fexp mx ex in binary_round_sign m sx mz ez loc_Exact.
1024

1025
Theorem binary_round_sign_shl_correct :
1026
  forall m sx mx ex,
1027
  let z := binary_round_sign_shl m sx mx ex in
1028 1029
  valid_binary z = true /\
  let x := F2R (Float radix2 (cond_Zopp sx (Zpos mx)) ex) in
1030
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) x)) (bpow radix2 emax) then
1031 1032
    FF2R radix2 z = round radix2 fexp (round_mode m) x /\
    is_finite_FF z = true
1033
  else
1034
    z = binary_overflow m sx.
1035 1036
Proof.
intros m sx mx ex.
1037 1038 1039
unfold binary_round_sign_shl.
generalize (shl_fexp_correct mx ex).
destruct (shl_fexp mx ex) as (mz, ez).
1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055
intros (H1, H2).
set (x := F2R (Float radix2 (cond_Zopp sx (Zpos mx)) ex)).
replace sx with (Rlt_bool x 0).
apply binary_round_sign_correct.
constructor.
unfold x.
now rewrite abs_F2R, abs_cond_Zopp.
exact H2.
unfold x.
case sx.
apply Rlt_bool_true.
now apply F2R_lt_0_compat.
apply Rlt_bool_false.
now apply F2R_ge_0_compat.
Qed.

1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069
Definition Bplus m x y :=
  match x, y with
  | B754_nan, _ => x
  | _, B754_nan => y
  | B754_infinity sx, B754_infinity sy =>
    if Bool.eqb sx sy then x else B754_nan
  | B754_infinity _, _ => x
  | _, B754_infinity _ => y
  | B754_zero sx, B754_zero sy =>
    if Bool.eqb sx sy then x else
    match m with mode_DN => B754_zero true | _ => B754_zero false end
  | B754_zero _, _ => y
  | _, B754_zero _ => x
  | B754_finite sx mx ex Hx, B754_finite sy my ey Hy =>
1070 1071 1072
    let ez := Zmin ex ey in
    match Zplus (cond_Zopp sx (Zpos (fst (shl mx ex ez)))) (cond_Zopp sy (Zpos (fst (shl my ey ez)))) with
    | Z0 =>
1073
      match m with mode_DN => B754_zero true | _ => B754_zero false end
1074
    | Zpos mz =>
1075
      FF2B (binary_round_sign_shl m false mz ez) (proj1 (binary_round_sign_shl_correct _ _ _ _))
1076
    | Zneg mz =>
1077
      FF2B (binary_round_sign_shl m true mz ez) (proj1 (binary_round_sign_shl_correct _ _ _ _))
1078
    end
1079 1080
  end.

1081
Theorem Bplus_correct :
1082 1083 1084
  forall m x y,
  is_finite x = true ->
  is_finite y = true ->
1085
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x + B2R y))) (bpow radix2 emax) then
1086 1087
    B2R (Bplus m x y) = round radix2 fexp (round_mode m) (B2R x + B2R y) /\
    is_finite (Bplus m x y) = true
1088
  else
1089
    (B2FF (Bplus m x y) = binary_overflow m (Bsign x) /\ Bsign x = Bsign y).
1090
Proof with auto with typeclass_instances.
1091
intros m [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] Fx Fy ; try easy.
1092
(* *)
1093
rewrite Rplus_0_r, round_0, Rabs_R0, Rlt_bool_true...
1094 1095 1096
simpl.
case (Bool.eqb sx sy) ; try easy.
now case m.
1097
apply bpow_gt_0.
1098
(* *)
1099
rewrite Rplus_0_l, round_generic, Rlt_bool_true...
1100
apply B2R_lt_emax.
1101 1102
apply generic_format_B2R.
(* *)
1103
rewrite Rplus_0_r, round_generic, Rlt_bool_true...
1104
apply B2R_lt_emax.
1105 1106
apply generic_format_B2R.
(* *)
1107 1108
clear Fx Fy.
simpl.
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set (ez := Zmin ex ey).
set (mz := (cond_Zopp sx (Zpos (fst (shl mx ex ez))) + cond_Zopp sy (Zpos (fst (shl my ey ez))))%Z).
assert (Hp: (F2R (Float radix2 (cond_Zopp sx (Zpos mx)) ex) +
  F2R (Float radix2 (cond_Zopp sy (Zpos my)) ey))%R = F2R (Float radix2 mz ez)).
rewrite 2!F2R_cond_Zopp.
generalize (shl_correct mx ex ez).
generalize (shl_correct my ey ez).
generalize (snd_shl mx ex ez (Zle_min_l ex ey)).
generalize (snd_shl my ey ez (Zle_min_r ex ey)).
destruct (shl mx ex ez) as (mx', ex').
destruct (shl my ey ez) as (my', ey').
simpl.
intros H1 H2.
rewrite H1, H2.
clear H1 H2.
intros (H1, _) (H2, _).
rewrite H1, H2.
clear H1 H2.
rewrite <- 2!F2R_cond_Zopp.
unfold F2R. simpl.
now rewrite <- Rmult_plus_distr_r, <- Z2R_plus.
rewrite Hp.
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assert (Sz: (bpow radix2 emax <= Rabs (round radix2 fexp (round_mode m) (F2R (Float radix2 mz ez))))%R -> sx = Rlt_bool (F2R (Float radix2 mz ez)) 0 /\ sx = sy).
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(* . *)
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rewrite <- Hp.
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intros Bz.
destruct (Bool.bool_dec sx sy) as [Hs|Hs].
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(* .. *)
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refine (conj _ Hs).
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rewrite Hs.
apply sym_eq.
case sy.
apply Rlt_bool_true.
rewrite <- (Rplus_0_r 0).
apply Rplus_lt_compat.
now apply F2R_lt_0_compat.
now apply F2R_lt_0_compat.
apply Rlt_bool_false.
rewrite <- (Rplus_0_r 0).
apply Rplus_le_compat.
now apply F2R_ge_0_compat.
now apply F2R_ge_0_compat.
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(* .. *)
elim Rle_not_lt with (1 := Bz).
generalize (bounded_lt_emax _ _ Hx) (bounded_lt_emax _ _ Hy) (andb_prop _ _ Hx) (andb_prop _ _ Hy).
intros Bx By (Hx',_) (Hy',_).
generalize (canonic_bounded_prec sx _ _ Hx') (canonic_bounded_prec sy _ _ Hy').
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clear -Bx By Hs.
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intros Cx Cy.
destruct sx.
(* ... *)
destruct sy.
now elim Hs.
clear Hs.
apply Rabs_lt.
split.
apply Rlt_le_trans with (F2R (Float radix2 (cond_Zopp true (Zpos mx)) ex)).
rewrite <- opp_F2R.
now apply Ropp_lt_contravar.
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apply round_monotone_l...
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now apply generic_format_canonic.
pattern (F2R (Float radix2 (cond_Zopp true (Zpos mx)) ex)) at 1 ; rewrite <- Rplus_0_r.
apply Rplus_le_compat_l.
now apply F2R_ge_0_compat.
apply Rle_lt_trans with (2 := By).
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apply round_monotone_r...
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now apply generic_format_canonic.
rewrite <- (Rplus_0_l (F2R (Float radix2 (Zpos my) ey))).
apply Rplus_le_compat_r.
now apply F2R_le_0_compat.
(* ... *)
destruct sy.
2: now elim Hs.
clear Hs.
apply Rabs_lt.
split.
apply Rlt_le_trans with (F2R (Float radix2 (cond_Zopp true (Zpos my)) ey)).
rewrite <- opp_F2R.
now apply Ropp_lt_contravar.
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apply round_monotone_l...
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now apply generic_format_canonic.
pattern (F2R (Float radix2 (cond_Zopp true (Zpos my)) ey)) at 1 ; rewrite <- Rplus_0_l.
apply Rplus_le_compat_r.
now apply F2R_ge_0_compat.
apply Rle_lt_trans with (2 := Bx).
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apply round_monotone_r...
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now apply generic_format_canonic.
rewrite <- (Rplus_0_r (F2R (Float radix2 (Zpos mx) ex))).
apply Rplus_le_compat_l.
now apply F2R_le_0_compat.
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destruct mz as [|mz|mz].
(* . mz = 0 *)
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rewrite F2R_0, round_0, Rabs_R0, Rlt_bool_true...
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now case m.
apply bpow_gt_0.
(* . mz > 0 *)
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generalize (binary_round_sign_shl_correct m false mz ez).
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simpl.
case Rlt_bool_spec.
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intros _ (Vz, (Rz, Rz')).
split.
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now rewrite B2R_FF2B.
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now rewrite is_finite_FF2B.
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intros Hz' (Vz, Rz).
specialize (Sz Hz').
refine (conj _ (proj2 Sz)).
rewrite B2FF_FF2B.
rewrite (proj1 Sz).
rewrite Rlt_bool_false.
exact Rz.
now apply F2R_ge_0_compat.
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(* . mz < 0 *)
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generalize (binary_round_sign_shl_correct m true mz ez).
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simpl.
case Rlt_bool_spec.
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intros _ (Vz, (Rz, Rz')).
split.
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now rewrite B2R_FF2B.
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now rewrite is_finite_FF2B.
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intros Hz' (Vz, Rz).
specialize (Sz Hz').
refine (conj _ (proj2 Sz)).
rewrite B2FF_FF2B.
rewrit