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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * Copyright INRIA, CNRS and contributors *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** * HexadecimalQ
Proofs that conversions between hexadecimal numbers and [Q]
are bijections. *)
Require Import Decimal DecimalFacts DecimalPos DecimalN DecimalZ.
Require Import Hexadecimal HexadecimalFacts HexadecimalPos HexadecimalN HexadecimalZ QArith.
Lemma of_to (q:IQ) : forall d, to_hexadecimal q = Some d -> of_hexadecimal d = q.
Admitted.
(* normalize without fractional part, for instance norme 0x1.2p-1 is 0x12e-5 *)
Definition hnorme (d:hexadecimal) : hexadecimal :=
let '(i, f, e) :=
match d with
| Hexadecimal i f => (i, f, Decimal.Pos Decimal.Nil)
| HexadecimalExp i f e => (i, f, e)
end in
let i := norm (app_int i f) in
let e := (Z.of_int e - 4 * Z.of_nat (nb_digits f))%Z in
match e with
| Z0 => Hexadecimal i Nil
| Zpos e => Hexadecimal (Pos.iter double i e) Nil
| Zneg _ => HexadecimalExp i Nil (Decimal.norm (Z.to_int e))
end.
Lemma hnorme_spec d :
match hnorme d with
| Hexadecimal i Nil => i = norm i
| HexadecimalExp i Nil e =>
i = norm i /\ e = Decimal.norm e /\ e <> Decimal.Pos Decimal.zero
| _ => False
end.
Admitted.
Lemma hnorme_invol d : hnorme (hnorme d) = hnorme d.
Admitted.
Lemma to_of (d:hexadecimal) :
to_hexadecimal (of_hexadecimal d) = Some (hnorme d).
Admitted.
(** Some consequences *)
Lemma to_hexadecimal_inj q q' :
to_hexadecimal q <> None -> to_hexadecimal q = to_hexadecimal q' -> q = q'.
Admitted.
Lemma to_hexadecimal_surj d : exists q, to_hexadecimal q = Some (hnorme d).
Admitted.
Lemma of_hexadecimal_hnorme d : of_hexadecimal (hnorme d) = of_hexadecimal d.
Admitted.
Lemma of_inj d d' :
of_hexadecimal d = of_hexadecimal d' -> hnorme d = hnorme d'.
Admitted.
Lemma of_iff d d' :
of_hexadecimal d = of_hexadecimal d' <-> hnorme d = hnorme d'.
Admitted.
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