Section definitions.
Variable radix : Z.
Hypothesis radixMoreThanOne : (1 < radix)%Z.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ radixMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
(* The type float represents the set of numbers who can be written: x = n*b^p with  n and p in Z. (pdic numbers) n = Fnum and p = Fexp 						*)
Record float : Set := Float {Fnum : Z; Fexp : Z}.
intros p q; case p; case q; simpl in |- *; intros;
apply (f_equal2 (A1:=Z) (A2:=Z)); auto.
Qed.
intros x y; case x; case y; intros Fnum2 Fexp2 Fnum1 Fexp1.
case (Z.eq_dec Fnum1 Fnum2); intros H1.
case (Z.eq_dec Fexp1 Fexp2); intros H2.
left; apply floatEq; auto.
right; red in |- *; intros H'; Contradict H2; inversion H'; auto.
right; red in |- *; intros H'; Contradict H1; inversion H'; auto.
Qed.
Definition Fzero (x : Z) := Float 0 x.
Definition is_Fzero (x : float) := Fnum x = 0%Z.
unfold is_Fzero in |- *; intro; CaseEq (Fnum x); intros;
(right; discriminate) || (left; auto).
Qed.
Coercion IZR : Z >-> R.
Coercion INR : nat >-> R.
Coercion Z_of_nat : nat >-> Z.
Definition FtoR (x : float) := (Fnum x * powerRZ (IZR radix) (Fexp x))%R.
Local Coercion FtoR : float >-> R.
intros z; unfold FtoR in |- *; simpl in |- *; auto with real.
Qed.
intros x H; unfold FtoR in |- *.
red in H; rewrite H; simpl in |- *; auto with real.
Qed.
intros x; case x; unfold FtoR in |- *; simpl in |- *.
intros Fnum1 Fexp1 H'; replace 0%R with (Fnum1 * 0)%R;
[ apply Rmult_lt_compat_l | ring ]; auto with real zarith.
Qed.
intros x H'.
case (Rmult_integral _ _ H'); simpl in |- *; auto.
case x; simpl in |- *.
intros Fnum1 Fexp1 H'0; red in |- *; simpl in |- *; auto with real zarith.
apply eq_IZR_R0; auto.
intros H'0; Contradict H'0; apply powerRZ_NOR; auto with real zarith.
Qed.
intros x y H' H'0; Contradict H'.
apply is_Fzero_rep2; auto.
rewrite H'0.
apply is_Fzero_rep1; auto.
Qed.
(* Some inegalities that will be helpful *)
intros x y z H'; apply Rmult_lt_compat_r; auto with real zarith.
Qed.
intros x y z H'; apply Rmult_le_compat_r; auto with real zarith.
Qed.
intros x y z H'; apply Rmult_lt_reg_l with (r := powerRZ radix z);
auto with real zarith.
repeat rewrite (Rmult_comm (powerRZ radix z)); auto.
Qed.

Theorem Rle_monotony_contra_exp :
forall (x y : R) (z : Z),
(x * powerRZ radix z <= y * powerRZ radix z)%R -> (x <= y)%R.