Section FMinMax.
Variable b : Fbound.
Variable radix : Z.
Variable precision : nat.
Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
Hypothesis radixMoreThanOne : (1 < radix)%Z.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ radixMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
Hypothesis precisionNotZero : precision <> 0.
Hypothesis pGivesBound : Zpos (vNum b) = Zpower_nat radix precision.
(* a function that returns a boundd greater than a given nat *)
Definition boundNat (n : nat) := Float 1%nat (digit radix n).
intros n; unfold FtoRradix, FtoR, boundNat in |- *; simpl in |- *.
rewrite Rmult_1_l.
rewrite <- Zpower_nat_Z_powerRZ; auto with real zarith.
rewrite INR_IZR_INZ; auto with real zarith.
apply Rle_lt_trans with (Z.abs n); [rewrite (Z.abs_eq (Z_of_nat n))|idtac];auto with real zarith.
Qed.
intros n; repeat split; unfold boundNat in |- *; simpl in |- *;
auto with zarith.
apply vNumbMoreThanOne with (radix := radix) (precision := precision);
auto with zarith.
apply Z.le_trans with 0%Z;[case (dExp b)|idtac]; auto with zarith.
Qed.
(* A function that returns a bounded greater than a given r *)
Definition boundR (r : R) := boundNat (Z.abs_nat (up (Rabs r))).
intros r; case (Rle_or_lt r 0); intros H'.
apply Rle_lt_trans with (1 := H').
unfold boundR, boundNat, FtoRradix, FtoR in |- *; simpl in |- *;
auto with real.
rewrite Rmult_1_l; auto with real zarith.
apply Rlt_trans with (2 := boundNatCorrect (Z.abs_nat (up (Rabs r)))).
replace (Rabs r) with r; auto with real.
apply Rlt_le_trans with (r2 := IZR (up r)); auto with real zarith.
case (archimed r); auto.
rewrite INR_IZR_INZ; auto with real zarith.
unfold Rabs in |- *; case (Rcase_abs r); auto with real.
intros H'0; Contradict H'0; auto with real.
Qed.
intros R; unfold boundR in |- *.
rewrite Rabs_Ropp; auto.
Qed.
intros r; case (Rle_or_lt r 0); intros H'.
rewrite boundRrOpp.
pattern r at 2 in |- *; rewrite <- (Ropp_involutive r).
unfold FtoRradix in |- *; rewrite Fopp_correct.
apply Ropp_lt_contravar; apply boundRCorrect1; auto.
apply Rle_lt_trans with 0%R; auto.
replace 0%R with (-0)%R; auto with real.
unfold FtoRradix in |- *; rewrite Fopp_correct.
apply Ropp_le_contravar.
unfold boundR, boundNat, FtoRradix, FtoR in |- *; simpl in |- *;
auto with real zarith.
rewrite Rmult_1_l; apply Rlt_le; auto with real zarith arith.
Qed.
(* A function that returns a list containing all the bounded smaller than a given real *)
Definition mBFloat (p : R) :=
map (fun p : Z * Z => Float (fst p) (snd p))
(mProd Z Z (Z * Z)
(mZlist (- pPred (vNum b)) (pPred (vNum b)))
(mZlist (- dExp b) (Fexp (boundR p)))).

Theorem mBFadic_correct1 :
forall (r : R) (q : float),
~ is_Fzero q ->
(Fopp (boundR r) < q)%R ->
(q < boundR r)%R -> Fbounded b q -> In q (mBFloat r).