Definition FtoR (x : float) := (Fnum x * powerRZ (IZR radix) (Fexp x))%R.

(* Specifications: *)
Theorem FzeroisReallyZero : forall z : Z, Fzero z = 0%R :>R.
Theorem is_Fzero_rep1 : forall x : float, is_Fzero x -> x = 0%R :>R.
Theorem LtFnumZERO : forall x : float, (0 < Fnum x)%Z -> (0 < x)%R.
Theorem is_Fzero_rep2 : forall x : float, x = 0%R :>R -> is_Fzero x.
Theorem NisFzeroComp : forall x y : float, ~ is_Fzero x -> x = y :>R -> ~ is_Fzero y.
Theorem Rlt_monotony_exp : forall (x y : R) (z : Z), (x < y)%R -> (x * powerRZ radix z < y * powerRZ radix z)%R.
Theorem Rle_monotone_exp : forall (x y : R) (z : Z), (x <= y)%R -> (x * powerRZ radix z <= y * powerRZ radix z)%R.
Theorem Rlt_monotony_contra_exp : forall (x y : R) (z : Z), (x * powerRZ radix z < y * powerRZ radix z)%R -> (x < y)%R.
Theorem Rle_monotony_contra_exp : forall (x y : R) (z : Z), (x * powerRZ radix z <= y * powerRZ radix z)%R -> (x <= y)%R.
Theorem FtoREqInv1 : forall p q : float, ~ is_Fzero p -> p = q :>R -> Fnum p = Fnum q -> p = q.
Theorem FtoREqInv2 : forall p q : float, p = q :>R -> Fexp p = Fexp q -> p = q.
Theorem Rlt_Float_Zlt : forall p q r : Z, (Float p r < Float q r)%R -> (p < q)%Z.
Theorem Rle_Float_Zle : forall p q r : Z, (Float p r <= Float q r)%R -> (p <= q)%Z.
Theorem oneExp_le : forall x y : Z, (x <= y)%Z -> (Float 1%nat x <= Float 1%nat y)%R.
Theorem oneExp_lt : forall x y : Z, (x < y)%Z -> (Float 1%nat x < Float 1%nat y)%R.
Theorem oneExp_Zlt : forall x y : Z, (Float 1%nat x < Float 1%nat y)%R -> (x < y)%Z.
Theorem oneExp_Zle : forall x y : Z, (Float 1%nat x <= Float 1%nat y)%R -> (x <= y)%Z.