Require Import Decimal DecimalFacts DecimalPos DecimalZ DecimalQ Rdefinitions.

Lemma of_IQmake_to_decimal num den :
  match IQmake_to_decimal num den with
  | None => True
  | Some (DecimalExp _ _ _) => False
  | Some (Decimal i f) =>
    of_decimal (Decimal i f) = IRQ (QArith_base.Qmake num den)
  end.

Lemma of_to (q:IR) : forall d, to_decimal q = Some d -> of_decimal d = q.

Lemma to_of (d:decimal) : to_decimal (of_decimal d) = Some (dnorm d).

Lemma to_decimal_inj q q' :
  to_decimal q <> None -> to_decimal q = to_decimal q' -> q = q'.

Lemma to_decimal_surj d : exists q, to_decimal q = Some (dnorm d).

Lemma of_decimal_dnorm d : of_decimal (dnorm d) = of_decimal d.

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

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