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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * Copyright INRIA, CNRS and contributors *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** * DecimalQ
Proofs that conversions between decimal numbers and [Q]
are bijections. *)
Require Import Decimal DecimalFacts DecimalPos DecimalN DecimalZ QArith.
Lemma of_to (q:IQ) : forall d, to_decimal q = Some d -> of_decimal d = q.
Admitted.
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.
Admitted.
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.
Admitted.
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.
Admitted.
Lemma dnorm_invol d : dnorm (dnorm d) = dnorm d.
Admitted.
Lemma to_of (d:decimal) : to_decimal (of_decimal d) = Some (dnorm d).
Admitted.
(** Some consequences *)
Lemma to_decimal_inj q q' :
to_decimal q <> None -> to_decimal q = to_decimal q' -> q = q'.
Proof.
intros Hnone EQ.
generalize (of_to q) (of_to q').
rewrite <-EQ.
revert Hnone; case to_decimal; [|now simpl].
now intros d _ H1 H2; rewrite <-(H1 d eq_refl), <-(H2 d eq_refl).
Qed.
Lemma to_decimal_surj d : exists q, to_decimal q = Some (dnorm d).
Proof.
exists (of_decimal d). apply to_of.
Qed.
Lemma of_decimal_dnorm d : of_decimal (dnorm d) = of_decimal d.
Proof. now apply to_decimal_inj; rewrite !to_of; [|rewrite dnorm_invol]. Qed.
Lemma of_inj d d' : of_decimal d = of_decimal d' -> dnorm d = dnorm d'.
Proof.
intro H.
apply (@f_equal _ _ (fun x => match x with Some x => x | _ => d end)
(Some (dnorm d)) (Some (dnorm d'))).
now rewrite <- !to_of, H.
Qed.
Lemma of_iff d d' : of_decimal d = of_decimal d' <-> dnorm d = dnorm d'.
Proof.
split. apply of_inj. intros E. rewrite <- of_decimal_dnorm, E.
apply of_decimal_dnorm.
Qed.
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