Require Import Decimal DecimalFacts DecimalPos DecimalN DecimalZ.
Require Import Hexadecimal HexadecimalFacts HexadecimalPos HexadecimalN HexadecimalZ QArith.

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.

Definition dnorm (d:hexadecimal) : hexadecimal :=
  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).

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'.