Library Float.MSBProp

Require Export MSB.
Section MSBProp.
Variable b : Fbound.
Variable precision : nat.
Variable radix : Z.

Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
Hypothesis radixMoreThanOne : (1 < radix)%Z.

Let radixMoreThanZERO : (0 < radix)%Z :=
  Zlt_1_O _ (Zlt_le_weak _ _ radixMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
Hypothesis precisionGreaterThanOne : 1 < precision.
Hypothesis pGivesBound : Zpos (vNum b) = Zpower_nat radix precision.

Theorem boundOnePrecision :
  a : float,
 Fbounded b a → (Rabs a < Float 1%nat (precision + Fexp a))%R.
intros a H.
replace (FtoRradix (Float 1%nat (precision + Fexp a))) with
 (FtoRradix (Fshift radix precision (Float 1%nat (precision + Fexp a))));
 [ idtac | apply (FshiftCorrect radix); auto ].
unfold Fshift, FtoRradix, FtoR in |- *; simpl in |- ×.
rewrite <- pGivesBound.
replace (precision + Fexp a - precision)%Z with (Fexp a); [ idtac | ring ].
rewrite Rabs_mult; rewrite (fun x yRabs_pos_eq (powerRZ x y));
 auto with real zarith.
apply Rlt_monotony_exp; auto with float real zarith.
rewrite Faux.Rabsolu_Zabs; auto with float real zarith.
Qed.

Theorem boundNormalMult :
  x y : float,
 Fnormal radix b x
 Fbounded b y
 (Rabs y × Float 1%nat (Fexp x) < radix × (Rabs x × Float 1%nat (Fexp y)))%R.
intros x y H H0.
apply
 Rlt_le_trans
  with (Float (Zpos (vNum b)) (Fexp y) × Float 1%nat (Fexp x))%R.
apply Rmult_lt_compat_r.
unfold FtoRradix in |- *; unfold FtoR in |- *; simpl in |- ×.
replace (1 × powerRZ radix (Fexp x))%R with (powerRZ radix (Fexp x));
 [ idtac | ring ].
apply powerRZ_lt; auto with real arith.
unfold FtoRradix in |- *; apply MaxFloat; auto.
replace (Float (Zpos (vNum b)) (Fexp y) × Float 1%nat (Fexp x))%R with
 (Zpos (vNum b) × Float 1%nat (Fexp x) × Float 1%nat (Fexp y))%R.
replace (radix × (Rabs x × Float 1%nat (Fexp y)))%R with
 (radix × Rabs x × Float 1%nat (Fexp y))%R; [ idtac | ring ].
apply Rmult_le_compat_r.
unfold FtoRradix in |- *; unfold FtoR in |- *; simpl in |- ×.
replace (1 × powerRZ radix (Fexp y))%R with (powerRZ radix (Fexp y));
 [ idtac | ring ].
apply powerRZ_le; auto with real arith.
replace (Zpos (vNum b) × Float 1%nat (Fexp x))%R with
 (FtoRradix (Float (Zpos (vNum b)) (Fexp x))).
rewrite <- (Fabs_correct radix); auto with real zarith.
unfold Fabs, FtoRradix, FtoR in |- ×.
rewrite <- Rmult_assoc.
apply Rle_monotone_exp; auto with real arith.
rewrite <- Rmult_IZR; apply Rle_IZR; simpl in |- ×.
rewrite <- (Zabs_eq radix); auto with zarith; rewrite <- Zabs_Zmult;
 auto with float.
case H; simpl in |- *; auto.
unfold FtoRradix, FtoR in |- *; simpl in |- *; ring.
unfold FtoRradix, FtoR in |- *; simpl in |- *; ring.
Qed.
End MSBProp.