The Coq Proof Assistant

# Library Coq.Numbers.DecimalR

# DecimalR

Proofs that conversions between decimal numbers and R are bijections.
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).
Some consequences
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'.