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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.
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Require Import Fcore_digits.
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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
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  | Z0 => F754_zero sx
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  | Zpos m => if Zle_bool e'' (emax - prec) then F754_finite sx m e'' else binary_overflow mode sx
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  | _ => F754_nan (* dummy *)
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  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 ->
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  let z := binary_round_sign mode (Rlt_bool x 0) mx ex lx in
  valid_binary z = true /\
634
  if Rlt_bool (Rabs (round radix2 fexp (round_mode mode) x)) (bpow radix2 emax) then
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    FF2R radix2 z = round radix2 fexp (round_mode mode) x /\
    is_finite_FF z = true
637
  else
638
    z = binary_overflow mode (Rlt_bool x 0).
639
Proof with auto with typeclass_instances.
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intros m x mx ex lx Bx Ex z.
unfold binary_round_sign in z.
revert z.
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rewrite shr_truncate. 2: easy.
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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)).
646
destruct (truncate radix2 fexp (Zpos mx, ex, lx)) as ((m1, e1), l1).
647
rewrite loc_of_shr_record_of_loc, shr_m_shr_record_of_loc.
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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 *)
675
rewrite shr_truncate. 2: apply Zle_refl.
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generalize (truncate_0 radix2 fexp e1 loc_Exact).
destruct (truncate radix2 fexp (Z0, e1, loc_Exact)) as ((m2, e2), l2).
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rewrite shr_m_shr_record_of_loc.
679
intros Hm2.
680
rewrite Hm2.
681
intros z.
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repeat split.
rewrite Rlt_bool_true.
684
repeat split.
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apply sym_eq.
case Rlt_bool ; apply F2R_0.
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rewrite abs_F2R, abs_cond_Zopp, F2R_0.
apply bpow_gt_0.
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(* . 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)).
696
apply canonic_exponent_round...
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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.
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refine (_ (truncate_correct_partial _ _ _ _ _ _ _ Br He)).
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2: now rewrite Hr ; apply F2R_gt_0_compat.
refine (_ (truncate_correct_format radix2 fexp (Zpos m1') e1 _ _ He)).
2: discriminate.
707
rewrite shr_truncate. 2: easy.
708
destruct (truncate radix2 fexp (Zpos m1', e1, loc_Exact)) as ((m2, e2), l2).
709
rewrite shr_m_shr_record_of_loc.
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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.
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rewrite F2R_cond_Zopp, H3, abs_cond_Ropp, abs_F2R.
simpl Zabs.
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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).
730
rewrite Rlt_bool_true.
731
repeat split.
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apply F2R_cond_Zopp.
now apply bounded_lt_emax.
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rewrite (Rlt_bool_false _ (bpow radix2 emax)).
refine (conj _ (refl_equal _)).
unfold binary_overflow.
case overflow_to_inf.
738
apply refl_equal.
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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.
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generalize (prec_gt_0 prec).
clear -Hmax ; zify ; omega.
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change 2%Z with (radix_val radix2).
case_eq (Zpower radix2 prec - 1)%Z.
simpl digits.
752
generalize (Zpower_gt_1 radix2 prec (prec_gt_0 prec)).
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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.
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generalize (Zpower_gt_1 radix2 _ (prec_gt_0 prec)).
770
clear -Hp ; zify ; omega.
771 772
apply Rnot_lt_le.
intros Hx.
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generalize (refl_equal (bounded m2 e2)).
unfold bounded at 2.
rewrite He2.
rewrite Bool.andb_false_r.
777 778
rewrite bounded_canonic_lt_emax with (2 := Hx).
discriminate.
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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.
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apply generic_format_abs...
apply generic_format_round...
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(* . 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.
804

805 806 807 808 809 810 811 812
Definition Bsign x :=
  match x with
  | B754_nan => false
  | B754_zero s => s
  | B754_infinity s => s
  | B754_finite s _ _ _ => s
  end.

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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.

828 829 830 831 832 833 834
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
835 836
    FF2R radix2 z = round radix2 fexp (round_mode m) (x * y) /\
    is_finite_FF z = true
837
  else
838
    z = binary_overflow m (xorb sx sy).
839
Proof.
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intros m sx mx ex Hx sy my ey Hy x y.
unfold x, y.
842
rewrite <- mult_F2R.
843
simpl.
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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).
859
clear x y sx sy Hx Hy H.
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unfold fexp, FLT_exp.
refine (_ (digits_mult_ge radix2 (Zpos mx) (Zpos my) _ _)) ; try discriminate.
862
refine (_ (Zdigits_gt_0 radix2 (Zpos mx) _) (Zdigits_gt_0 radix2 (Zpos my) _)) ; try discriminate.
863
generalize (digits radix2 (Zpos mx)) (digits radix2 (Zpos my)) (digits radix2 (Zpos mx * Zpos my)).
864
clear -Hmax.
865
unfold emin.
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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.
880

881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899
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
900 901
    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)
902
  else
903
    B2FF (Bmult m x y) = binary_overflow m (xorb (Bsign x) (Bsign y)).
904 905
Proof.
intros m [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] ;
906
  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 ] ).
907 908
simpl.
case Bmult_correct_aux.
909 910 911 912
intros H1.
case Rlt_bool.
intros (H2, H3).
split.
913
now rewrite B2R_FF2B.
914 915
now rewrite is_finite_FF2B.
intros H2.
916 917 918
now rewrite B2FF_FF2B.
Qed.

919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942
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.

943
Definition shl mx ex ex' :=
944 945 946 947 948
  match (ex' - ex)%Z with
  | Zneg d => (shift_pos d mx, ex')
  | _ => (mx, ex)
  end.

949 950 951 952 953
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.
954
Proof.
955 956 957
intros mx ex ex'.
unfold shl.
case_eq (ex' - ex)%Z.
958 959 960 961 962 963 964 965
(* d = 0 *)
intros H.
repeat split.
rewrite Zminus_eq with (1 := H).
apply Zle_refl.
(* d > 0 *)
intros d Hd.
repeat split.
966
replace ex' with (ex' - ex + ex)%Z by ring.
967 968 969 970 971 972 973 974 975 976
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.
977
replace (- (ex' - ex))%Z with (ex - ex')%Z by ring.
978 979
apply F2R_change_exp.
apply Zle_0_minus_le.
980
replace (ex - ex')%Z with (- (ex' - ex))%Z by ring.
981 982 983
now rewrite Hd.
apply Zle_refl.
Qed.
984

985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000
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.

1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024
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.
1025

1026
Theorem binary_round_sign_shl_correct :
1027
  forall m sx mx ex,
1028
  let z := binary_round_sign_shl m sx mx ex in
1029 1030
  valid_binary z = true /\
  let x := F2R (Float radix2 (cond_Zopp sx (Zpos mx)) ex) in
1031
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) x)) (bpow radix2 emax) then
1032 1033
    FF2R radix2 z = round radix2 fexp (round_mode m) x /\
    is_finite_FF z = true
1034
  else
1035
    z = binary_overflow m sx.
1036 1037
Proof.
intros m sx mx ex.
1038 1039 1040
unfold binary_round_sign_shl.
generalize (shl_fexp_correct mx ex).
destruct (shl_fexp mx ex) as (mz, ez).
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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.

1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105
Definition binary_normalize mode m e szero :=
  match m with
  | Z0 => B754_zero szero
  | Zpos m => FF2B _ (proj1 (binary_round_sign_shl_correct mode false m e))
  | Zneg m => FF2B _ (proj1 (binary_round_sign_shl_correct mode true m e))
  end.

Theorem binary_normalize_correct :
  forall m mx ex szero,
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (F2R (Float radix2 mx ex)))) (bpow radix2 emax) then
    B2R (binary_normalize m mx ex szero) = round radix2 fexp (round_mode m) (F2R (Float radix2 mx ex)) /\
    is_finite (binary_normalize m mx ex szero) = true
  else
    B2FF (binary_normalize m mx ex szero) = binary_overflow m (Rlt_bool (F2R (Float radix2 mx ex)) 0).
Proof with auto with typeclass_instances.
intros m mx ez szero.
destruct mx as [|mz|mz] ; simpl.
rewrite F2R_0, round_0, Rabs_R0, Rlt_bool_true...
apply bpow_gt_0.
(* . mz > 0 *)
generalize (binary_round_sign_shl_correct m false mz ez).
simpl.
case Rlt_bool_spec.
intros _ (Vz, (Rz, Rz')).
split.
now rewrite B2R_FF2B.
now rewrite is_finite_FF2B.
intros Hz' (Vz, Rz).
rewrite B2FF_FF2B, Rz.
apply f_equal.
apply sym_eq.
apply Rlt_bool_false.
now apply F2R_ge_0_compat.
(* . mz < 0 *)
generalize (binary_round_sign_shl_correct m true mz ez).
simpl.
case Rlt_bool_spec.
intros _ (Vz, (Rz, Rz')).
split.
now rewrite B2R_FF2B.
now rewrite is_finite_FF2B.
intros Hz' (Vz, Rz).
rewrite B2FF_FF2B, Rz.
apply f_equal.
apply sym_eq.
apply Rlt_bool_true.
now apply F2R_lt_0_compat.
Qed.

1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119
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 =>
1120
    let ez := Zmin ex ey in
1121 1122
    binary_normalize m (Zplus (cond_Zopp sx (Zpos (fst (shl mx ex ez)))) (cond_Zopp sy (Zpos (fst (shl my ey ez)))))
      ez (match m with mode_DN => true | _ => false end)
1123 1124
  end.

1125
Theorem Bplus_correct :
1126 1127 1128
  forall m x y,
  is_finite x = true ->
  is_finite y = true ->
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  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x + B2R y))) (bpow radix2 emax) then
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    B2R (Bplus m x y) = round radix2 fexp (round_mode m) (B2R x + B2R y) /\
    is_finite (Bplus m x y) = true
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  else
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    (B2FF (Bplus m x y) = binary_overflow m (Bsign x) /\ Bsign x = Bsign y).
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Proof with auto with typeclass_instances.
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intros m [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] Fx Fy ; try easy.
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(* *)
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rewrite Rplus_0_r, round_0, Rabs_R0, Rlt_bool_true...
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simpl.
case (Bool.eqb sx sy) ; try easy.
now case m.
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apply bpow_gt_0.
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(* *)
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rewrite Rplus_0_l, round_generic, Rlt_bool_true...
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apply B2R_lt_emax.
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apply generic_format_B2R.
(* *)
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rewrite Rplus_0_r, round_generic, Rlt_bool_true...
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apply B2R_lt_emax.
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apply generic_format_B2R.
(* *)
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clear Fx Fy.
simpl.
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set (szero := match m with mode_DN => true | _ => false end).
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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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(* . *)
generalize (binary_normalize_correct m mz ez szero).
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case Rlt_bool_spec.
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easy.
intros Hz' Vz.
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specialize (Sz Hz').
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split.
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rewrite Vz.
now apply f_equal.
apply Sz.
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Qed.
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Definition Bminus m x y := Bplus m x (Bopp y).

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Definition Fdiv_core_binary m1 e1 m2 e2 :=
  let d1 := digits2 m1 in
  let d2 := digits2 m2 in
  let e := (e1 - e2)%Z in
  let (m, e') :=
    match (d2 + prec - d1)%Z with
    | Zpos p => (m1 * Zpower_pos 2 p, e + Zneg p)%Z
    | _ => (m1, e)
    end in
  let '(q, r) :=  Zdiv_eucl m m2 in
  (q, e', new_location m2 r loc_Exact).

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Lemma Bdiv_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 :=
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    let '(mz, ez, lz) := Fdiv_core_binary (Zpos mx) ex (Zpos my) ey in
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    match mz with
    | Zpos mz => binary_round_sign m (xorb sx sy) mz ez lz
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    | _ => F754_nan (* dummy *)
    end in
  valid_binary z = true /\
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (x / y))) (bpow radix2 emax) then
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    FF2R radix2 z = round radix2 fexp (round_mode m) (x / y) /\
    is_finite_FF z = true
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  else
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    z = binary_overflow m (xorb sx sy).
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Proof.
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intros m sx mx ex Hx sy my ey Hy.