Require Import Decimal DecimalFacts.
Require Import Hexadecimal HexadecimalFacts HexadecimalPos HexadecimalZ.
Require Import HexadecimalQ Rdefinitions.

Lemma of_IQmake_to_hexadecimal num den :
  match IQmake_to_hexadecimal num den with
  | None => True
  | Some (HexadecimalExp _ _ _) => False
  | Some (Hexadecimal i f) =>
    of_hexadecimal (Hexadecimal i f) = IRQ (QArith_base.Qmake num den)
  end.

Lemma of_to (q:IR) : forall d, to_hexadecimal q = Some d -> of_hexadecimal d = q.

Lemma to_of (d:hexadecimal) : to_hexadecimal (of_hexadecimal d) = Some (dnorm d).

Lemma to_hexadecimal_inj q q' :
  to_hexadecimal q <> None -> to_hexadecimal q = to_hexadecimal q' -> q = q'.

Lemma to_hexadecimal_surj d : exists q, to_hexadecimal q = Some (dnorm d).

Lemma of_hexadecimal_dnorm d : of_hexadecimal (dnorm d) = of_hexadecimal d.

Lemma of_inj d d' : of_hexadecimal d = of_hexadecimal d' -> dnorm d = dnorm d'.

Lemma of_iff d d' : of_hexadecimal d = of_hexadecimal d' <-> dnorm d = dnorm d'.