The Coq Proof Assistant

# Library Coq.Numbers.HexadecimalR

# HexadecimalR

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