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.

(* Specifications: *)
Theorem FF2R_B2FF : forall x, FF2R radix2 (B2FF x) = B2R x.
Theorem B2FF_FF2B : forall x Hx, B2FF (FF2B x Hx) = x.
Theorem valid_binary_B2FF : forall x, valid_binary (B2FF x) = true.
Theorem FF2B_B2FF : forall x H, FF2B (B2FF x) H = x.
Theorem FF2B_B2FF_valid : forall x, FF2B (B2FF x) (valid_binary_B2FF x) = x.
Theorem B2R_FF2B : forall x Hx, B2R (FF2B x Hx) = FF2R radix2 x.
Theorem match_FF2B : forall {T} fz fi fn ff x Hx, match FF2B x Hx return T with | B754_zero sx => fz sx | B754_infinity sx => fi sx | B754_nan b (exist p _) => fn b p | B754_finite sx mx ex _ => ff sx mx ex end = match x with | F754_zero sx => fz sx | F754_infinity sx => fi sx | F754_nan b p => fn b p | F754_finite sx mx ex => ff sx mx ex end.
Theorem canonic_canonic_mantissa : forall (sx : bool) mx ex, canonic_mantissa mx ex = true -> canonic radix2 fexp (Float radix2 (cond_Zopp sx (Zpos mx)) ex).
Theorem generic_format_B2R : forall x, generic_format radix2 fexp (B2R x).
Theorem FLT_format_B2R : forall x, FLT_format radix2 emin prec (B2R x).
Theorem B2FF_inj : forall x y : binary_float, B2FF x = B2FF y -> x = y.