Inductive full_float :=
  | F754_zero : bool -> full_float
  | F754_infinity : bool -> full_float
  | F754_nan : bool -> positive -> full_float
  | F754_finite : bool -> positive -> Z -> full_float.Inductive full_float :=
  | F754_zero : bool -> full_float
  | F754_infinity : bool -> full_float
  | F754_nan : bool -> positive -> full_float
  | F754_finite : bool -> positive -> Z -> full_float.Inductive full_float :=
  | F754_zero : bool -> full_float
  | F754_infinity : bool -> full_float
  | F754_nan : bool -> positive -> full_float
  | F754_finite : bool -> positive -> Z -> full_float.Inductive full_float :=
  | F754_zero : bool -> full_float
  | F754_infinity : bool -> full_float
  | F754_nan : bool -> positive -> full_float
  | F754_finite : bool -> positive -> Z -> full_float.Inductive full_float :=
  | F754_zero : bool -> full_float
  | F754_infinity : bool -> full_float
  | F754_nan : bool -> positive -> full_float
  | F754_finite : bool -> positive -> Z -> full_float.Definition FF2R beta x :=
  match x with
  | F754_finite s m e => F2R (Float beta (cond_Zopp s (Zpos m)) e)
  | _ => 0%R
  end.Instance fexp_correct : Valid_exp fexp := FLT_exp_valid emin prec.Instance fexp_monotone : Monotone_exp fexp := FLT_exp_monotone emin prec.Definition canonic_mantissa m e :=
  Zeq_bool (fexp (Zpos (digits2_pos m) + e)) e.Definition bounded m e :=
  andb (canonic_mantissa m e) (Zle_bool e (emax - prec)).Inductive binary_float :=
  | B754_zero : bool -> binary_float
  | B754_infinity : bool -> binary_float
  | B754_nan : bool -> nan_pl -> binary_float
  | B754_finite : bool ->
    forall (m : positive) (e : Z), bounded m e = true -> binary_float.Inductive binary_float :=
  | B754_zero : bool -> binary_float
  | B754_infinity : bool -> binary_float
  | B754_nan : bool -> nan_pl -> binary_float
  | B754_finite : bool ->
    forall (m : positive) (e : Z), bounded m e = true -> binary_float.Inductive binary_float :=
  | B754_zero : bool -> binary_float
  | B754_infinity : bool -> binary_float
  | B754_nan : bool -> nan_pl -> binary_float
  | B754_finite : bool ->
    forall (m : positive) (e : Z), bounded m e = true -> binary_float.Inductive binary_float :=
  | B754_zero : bool -> binary_float
  | B754_infinity : bool -> binary_float
  | B754_nan : bool -> nan_pl -> binary_float
  | B754_finite : bool ->
    forall (m : positive) (e : Z), bounded m e = true -> binary_float.Inductive binary_float :=
  | B754_zero : bool -> binary_float
  | B754_infinity : bool -> binary_float
  | B754_nan : bool -> nan_pl -> binary_float
  | B754_finite : bool ->
    forall (m : positive) (e : Z), bounded m e = true -> binary_float.Definition FF2B x :=
  match x as x return valid_binary x = true -> binary_float with
  | F754_finite s m e => B754_finite s m e
  | F754_infinity s => fun _ => B754_infinity s
  | F754_zero s => fun _ => B754_zero s
  | F754_nan b pl => fun H => B754_nan b (exist pl H)
  end.Definition B2FF x :=
  match x with
  | B754_finite s m e _ => F754_finite s m e
  | B754_infinity s => F754_infinity s
  | B754_zero s => F754_zero s
  | B754_nan b (exist pl _) => F754_nan b pl
  end.Definition B2R f :=
  match f with
  | B754_finite s m e _ => F2R (Float radix2 (cond_Zopp s (Zpos m)) e)
  | _ => 0%R
  end.

Theorem B2FF_FF2B : forall x Hx, B2FF (FF2B x Hx) = x.