Require Import Decimal DecimalFacts DecimalPos DecimalN DecimalZ QArith.

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) = 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_decimal' num den :
  match IQmake_to_decimal' num den with
  | None => True
  | Some (DecimalExp _ _ _) => False
  | Some (Decimal i f) => of_decimal (Decimal i f) = IQmake num den
  end.

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

Definition dnorm (d:decimal) : decimal :=
  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
  | Decimal i f => Decimal (norm_i i f) f
  | DecimalExp i f e =>
    match norm e with
    | Pos zero => Decimal (norm_i i f) f
    | e => DecimalExp (norm_i i f) f e
    end
  end.

Lemma dnorm_spec_i d :
  let (i, f) :=
    match d with Decimal i f => (i, f) | DecimalExp i f _ => (i, f) end in
  let i' := match dnorm d with Decimal i _ => i | DecimalExp 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 Decimal _ f => f | DecimalExp _ f _ => f end in
  let f' := match dnorm d with Decimal _ f => f | DecimalExp _ f _ => f end in
  f' = f.

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