begin div algo

1 files changed, 3 insertions, 52 deletions

diff --git a/kvx/ExtFloats.v b/kvx/ExtFloats.v
index ea663735..332d3e3d 100644
--- a/kvx/ExtFloats.v
+++ b/kvx/ExtFloats.v
@@ -15,7 +15,7 @@
 
 From Flocq Require Import Core Digits Operations Round Bracket Sterbenz
                           Binary Round_odd.
-Require Import Floats Integers ZArith IEEE754_extra Zdiv Psatz.
+Require Import Floats Integers ZArith.
 
 Module ExtFloat.
 (** TODO check with the actual KVX;
@@ -32,6 +32,8 @@ Definition max (x : float) (y : float) : float :=
   | Some Eq | Some Gt => x
   | Some Lt | None => y
   end.
+
+Definition one := Float.of_int (Int.repr (1%Z)).
 End ExtFloat.
 
 Module ExtFloat32.
@@ -53,54 +55,3 @@ Definition one := Float32.of_int (Int.repr (1%Z)).
 Definition inv (x : float32) : float32 :=
   Float32.div one x.
 End ExtFloat32.
-
-
-Definition div_approx_reals (a b : Z) (x : R) :=
-    let q:=ZnearestE x in
-    let r:=a-q*b in
-    if r <? 0
-    then q-1
-    else q.
-
-Lemma floor_ball1:
-  forall x : R, forall y : Z,
-    (Rabs (x - IZR y) < 1)%R -> Zfloor x = (y-1)%Z \/ Zfloor x = y.
-Proof.
-  intros x y BALL.
-  apply Rabs_lt_inv in BALL.
-  case (Rcompare_spec x (IZR y)); intro CMP.
-  - left. apply Zfloor_imp.
-    ring_simplify (y-1+1).
-    rewrite minus_IZR. lra.
-  - subst.
-    rewrite Zfloor_IZR. right. reflexivity.
-  - right. apply Zfloor_imp.
-    rewrite plus_IZR. lra.
-Qed.
-
-Theorem div_approx_reals_correct:
-  forall a b : Z, forall x : R,
-    b > 0 ->
-    (Rabs (x - IZR a/ IZR b) < 1/2)%R ->
-    div_approx_reals a b x = (a/b)%Z.
-Proof.                    
-  intros a b x bPOS GAP.
-  assert (0 < IZR b)%R by (apply IZR_lt ; lia).
-  unfold div_approx_reals.
-  pose proof (Znearest_imp2 (fun x => negb (Z.even x)) x) as NEAR.
-  assert (Rabs (IZR (ZnearestE x) - IZR a/ IZR b) < 1)%R as BALL.
-  { pose proof (Rabs_triang (IZR (ZnearestE x) - x)
-                            (x - IZR a/ IZR b)) as TRI.
-    ring_simplify (IZR (ZnearestE x) - x + (x - IZR a / IZR b))%R in TRI.
-    lra.
-  }
-  clear GAP NEAR.
-  rewrite Rabs_minus_sym in BALL.
-  pose proof (floor_ball1 _ _ BALL) as FLOOR.
-  clear BALL.
-  rewrite Zfloor_div in FLOOR by lia.
-  pose proof (Z_div_mod_eq_full a b) as DIV_MOD.
-  assert (0 < b) as bPOS' by lia.
-  pose proof (Z.mod_pos_bound a b bPOS') as MOD_BOUNDS.
-  case Z.ltb_spec; intro; destruct FLOOR; lia.
-Qed.