Proofs that conversions between hexadecimal numbers and Q are bijections.

Require Import Decimal DecimalFacts DecimalPos DecimalN DecimalZ.

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) = IQmake (IZ_of_Z num) den
end.

Lemma IZ_of_Z_IZ_to_Z z z' : IZ_to_Z z = Some z' -> IZ_of_Z z' = z.

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) = IQmake num den
end.

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

let norm_i i f :=
match i with
| Pos i => Pos (unorm i)
| Neg i => match nzhead (app i f) with Nil => Pos zero | _ => Neg (unorm i) end
end in
match d with
| Hexadecimal i f => Hexadecimal (norm_i i f) f
| HexadecimalExp i f e =>
match Decimal.norm e with
| Decimal.Pos Decimal.zero => Hexadecimal (norm_i i f) f
| e => HexadecimalExp (norm_i i f) f e
end
end.

Lemma dnorm_spec_i d :
let (i, f) :=
match d with Hexadecimal i f => (i, f) | HexadecimalExp i f _ => (i, f) end in
let i' := match dnorm d with Hexadecimal i _ => i | HexadecimalExp i _ _ => i end in
match i with
| Pos i => i' = Pos (unorm i)
| Neg i =>
(i' = Neg (unorm i) /\ (nzhead i <> Nil \/ nzhead f <> Nil))
\/ (i' = Pos zero /\ (nzhead i = Nil /\ nzhead f = Nil))
end.

Lemma dnorm_spec_f d :
let f := match d with Hexadecimal _ f => f | HexadecimalExp _ f _ => f end in
let f' := match dnorm d with Hexadecimal _ f => f | HexadecimalExp _ f _ => f end in
f' = f.

Lemma dnorm_spec_e d :
match d, dnorm d with
| Hexadecimal _ _, Hexadecimal _ _ => True
| HexadecimalExp _ _ e, Hexadecimal _ _ =>
Decimal.norm e = Decimal.Pos Decimal.zero
| HexadecimalExp _ _ e, HexadecimalExp _ _ e' =>
e' = Decimal.norm e /\ e' <> Decimal.Pos Decimal.zero
| Hexadecimal _ _, HexadecimalExp _ _ _ => False
end.

Lemma dnorm_involutive d : dnorm (dnorm d) = dnorm d.

Lemma IZ_to_Z_IZ_of_Z z : IZ_to_Z (IZ_of_Z z) = Some z.

Lemma dnorm_i_exact i f :
(nb_digits f < nb_digits (unorm (app (abs i) f)))%nat ->
match i with
| Pos i => Pos (unorm i)
| Neg i =>
match nzhead (app i f) with
| Nil => Pos zero
| _ => Neg (unorm i)
end
end = norm i.

Lemma dnorm_i_exact' i f :
(nb_digits (unorm (app (abs i) f)) <= nb_digits f)%nat ->
match i with
| Pos i => Pos (unorm i)
| Neg i =>
match nzhead (app i f) with
| Nil => Pos zero
| _ => Neg (unorm i)
end
end =
match norm (app_int i f) with
| Pos _ => Pos zero
| Neg _ => Neg zero
end.

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

Some consequences