(* *********************************************************************)
(*                                                                     *)
(*              The Compcert verified compiler                         *)
(*                                                                     *)
(*          Xavier Leroy, INRIA Paris-Rocquencourt                     *)
(*                                                                     *)
(*  Copyright Institut National de Recherche en Informatique et en     *)
(*  Automatique.  All rights reserved.  This file is distributed       *)
(*  under the terms of the GNU General Public License as published by  *)
(*  the Free Software Foundation, either version 2 of the License, or  *)
(*  (at your option) any later version.  This file is also distributed *)
(*  under the terms of the INRIA Non-Commercial License Agreement.     *)
(*                                                                     *)
(* *********************************************************************)

(* * Axiomatization of floating-point numbers. *)

(* * In contrast with what we do with machine integers, we do not bother
  to formalize precisely IEEE floating-point arithmetic.  Instead, we
  simply axiomatize a type [float] for IEEE double-precision floats
  and the associated operations.  *)

include "utilities/Coqlib.ma".
include "common/Integers.ma".

axiom float: Type[0].

(*Module Float.*)

axiom Fzero: float.
axiom Fone: float.

axiom Fneg: float → float.
axiom Fabs: float → float.
axiom singleoffloat: float → float.
axiom intoffloat: float → int.
axiom intuoffloat: float → int.
axiom floatofint: int → float.
axiom floatofintu: int → float.

axiom Fadd: float → float → float.
axiom Fsub: float → float → float.
axiom Fmul: float → float → float.
axiom Fdiv: float → float → float.

axiom Fcmp: comparison → float → float → bool.

axiom eq_dec: ∀f1,f2: float. (f1 = f2) + (f1 ≠ f2).

(* * Below are the only properties of floating-point arithmetic that we
  rely on in the compiler proof. *)

axiom addf_commut: ∀f1,f2. Fadd f1 f2 = Fadd f2 f1.

axiom subf_addf_opp: ∀f1,f2. Fsub f1 f2 = Fadd f1 (Fneg f2).

axiom singleoffloat_idem:
  ∀f. singleoffloat (singleoffloat f) = singleoffloat f.

axiom Fcmp_ne_eq:
  ∀ f1,f2. Fcmp Cne f1 f2 = ¬(Fcmp Ceq f1 f2).
axiom Fcmp_le_lt_eq:
  ∀ f1,f2. Fcmp Cle f1 f2 = (Fcmp Clt f1 f2 ∨ Fcmp Ceq f1 f2).
axiom Fcmp_ge_gt_eq:
  ∀f1,f2. Fcmp Cge f1 f2 = (Fcmp Cgt f1 f2 ∨ Fcmp Ceq f1 f2).

axiom Feq_zero_true: Fcmp Ceq Fzero Fzero = true.
axiom Feq_zero_false: ∀f. f ≠ Fzero → Fcmp Ceq f Fzero = false.

(*End Float.*)