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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) *)
(************************************************************************)
(** * DecimalR
Proofs that conversions between decimal numbers and [R]
are bijections. *)
Require Import Decimal DecimalFacts DecimalPos DecimalZ DecimalQ Rdefinitions.
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) = IRQ (QArith_base.Qmake num den)
end.
Proof.
unfold IQmake_to_decimal.
case (Pos.eq_dec den 1); [now intros->|intro Hden].
assert (Hf : match QArith_base.IQmake_to_decimal num den with
| Some (Decimal i f) => f <> Nil
| _ => True
end).
{ unfold QArith_base.IQmake_to_decimal; simpl.
generalize (Unsigned.nztail_to_uint den).
case Decimal.nztail as [den' e_den'].
case den'; [now simpl|now simpl| |now simpl..]; clear den'; intro den'.
case den'; [ |now simpl..]; clear den'.
case e_den' as [|e_den']; [now simpl; intros H _; apply Hden; injection H|].
intros _.
case Nat.ltb_spec; intro He_den'.
- apply del_head_nonnil.
revert He_den'; case nb_digits as [|n]; [now simpl|].
now intro H; simpl; apply Nat.lt_succ_r, Nat.le_sub_l.
- apply nb_digits_n0.
now rewrite nb_digits_iter_D0, Nat.sub_add. }
replace (match den with 1%positive => _ | _ => _ end)
with (QArith_base.IQmake_to_decimal num den); [|now revert Hden; case den].
generalize (of_IQmake_to_decimal num den).
case QArith_base.IQmake_to_decimal as [d'|]; [|now simpl].
case d' as [i f|]; [|now simpl].
unfold of_decimal; simpl.
injection 1 as H <-.
generalize (f_equal QArith_base.IZ_to_Z H); clear H.
rewrite !IZ_to_Z_IZ_of_Z; injection 1 as <-.
now revert Hf; case f.
Qed.
Lemma of_to (q:IR) : forall d, to_decimal q = Some d -> of_decimal d = q.
Proof.
intro d.
case q as [z|q|r r'|r r']; simpl.
- case z as [z p| |p|p].
+ now simpl.
+ now simpl; injection 1 as <-.
+ simpl; injection 1 as <-.
now unfold of_decimal; simpl; unfold Z.of_uint; rewrite Unsigned.of_to.
+ simpl; injection 1 as <-.
now unfold of_decimal; simpl; unfold Z.of_uint; rewrite Unsigned.of_to.
- case q as [num den].
generalize (of_IQmake_to_decimal num den).
case IQmake_to_decimal as [d'|]; [|now simpl].
case d' as [i f|]; [|now simpl].
now intros H; injection 1 as <-.
- case r as [z|q| |]; [|case q as[num den]|now simpl..];
(case r' as [z'| | |]; [|now simpl..]);
(case z' as [p e| | |]; [|now simpl..]).
+ case (Z.eq_dec p 10); [intros->|intro Hp].
2:{ revert Hp; case p; [now simpl|intro d0..];
(case d0; [intro d1..|]; [now simpl| |now simpl];
case d1; [intro d2..|]; [|now simpl..];
case d2; [intro d3..|]; [now simpl| |now simpl];
now case d3). }
case z as [| |p|p]; [now simpl|..]; injection 1 as <-.
* now unfold of_decimal; simpl; unfold Z.of_uint; rewrite Unsigned.of_to.
* unfold of_decimal; simpl; unfold Z.of_uint; rewrite Unsigned.of_to; simpl.
now rewrite Unsigned.of_to.
* unfold of_decimal; simpl; unfold Z.of_uint; rewrite Unsigned.of_to; simpl.
now rewrite Unsigned.of_to.
+ case (Z.eq_dec p 10); [intros->|intro Hp].
2:{ revert Hp; case p; [now simpl|intro d0..];
(case d0; [intro d1..|]; [now simpl| |now simpl];
case d1; [intro d2..|]; [|now simpl..];
case d2; [intro d3..|]; [now simpl| |now simpl];
now case d3). }
generalize (of_IQmake_to_decimal num den).
case IQmake_to_decimal as [d'|]; [|now simpl].
case d' as [i f|]; [|now simpl].
intros H; injection 1 as <-.
unfold of_decimal; simpl.
change (match f with Nil => _ | _ => _ end) with (of_decimal (Decimal i f)).
rewrite H; clear H.
now unfold Z.of_uint; rewrite Unsigned.of_to.
- case r as [z|q| |]; [|case q as[num den]|now simpl..];
(case r' as [z'| | |]; [|now simpl..]);
(case z' as [p e| | |]; [|now simpl..]).
+ case (Z.eq_dec p 10); [intros->|intro Hp].
2:{ revert Hp; case p; [now simpl|intro d0..];
(case d0; [intro d1..|]; [now simpl| |now simpl];
case d1; [intro d2..|]; [|now simpl..];
case d2; [intro d3..|]; [now simpl| |now simpl];
now case d3). }
case z as [| |p|p]; [now simpl|..]; injection 1 as <-.
* now unfold of_decimal; simpl; unfold Z.of_uint; rewrite Unsigned.of_to.
* unfold of_decimal; simpl; unfold Z.of_uint; rewrite Unsigned.of_to; simpl.
now rewrite Unsigned.of_to.
* unfold of_decimal; simpl; unfold Z.of_uint; rewrite Unsigned.of_to; simpl.
now rewrite Unsigned.of_to.
+ case (Z.eq_dec p 10); [intros->|intro Hp].
2:{ revert Hp; case p; [now simpl|intro d0..];
(case d0; [intro d1..|]; [now simpl| |now simpl];
case d1; [intro d2..|]; [|now simpl..];
case d2; [intro d3..|]; [now simpl| |now simpl];
now case d3). }
generalize (of_IQmake_to_decimal num den).
case IQmake_to_decimal as [d'|]; [|now simpl].
case d' as [i f|]; [|now simpl].
intros H; injection 1 as <-.
unfold of_decimal; simpl.
change (match f with Nil => _ | _ => _ end) with (of_decimal (Decimal i f)).
rewrite H; clear H.
now unfold Z.of_uint; rewrite Unsigned.of_to.
Qed.
Lemma to_of (d:decimal) : to_decimal (of_decimal d) = Some (dnorm d).
Proof.
case d as [i f|i f e].
- unfold of_decimal; simpl.
case (uint_eq_dec f Nil); intro Hf.
+ rewrite Hf; clear f Hf.
unfold to_decimal; simpl.
rewrite IZ_to_Z_IZ_of_Z, DecimalZ.to_of.
case i as [i|i]; [now simpl|]; simpl.
rewrite app_nil_r.
case (uint_eq_dec (nzhead i) Nil); [now intros->|intro Hi].
now rewrite (unorm_nzhead _ Hi); revert Hi; case nzhead.
+ set (r := IRQ _).
set (m := match f with Nil => _ | _ => _ end).
replace m with r; [unfold r|now unfold m; revert Hf; case f].
unfold to_decimal; simpl.
unfold IQmake_to_decimal; simpl.
set (n := Nat.iter _ _ _).
case (Pos.eq_dec n 1); intro Hn.
exfalso; apply Hf.
{ now apply nb_digits_0; revert Hn; unfold n; case nb_digits. }
clear m; set (m := match n with 1%positive | _ => _ end).
replace m with (QArith_base.IQmake_to_decimal (Z.of_int (app_int i f)) n).
2:{ now unfold m; revert Hn; case n. }
unfold QArith_base.IQmake_to_decimal, n; simpl.
rewrite nztail_to_uint_pow10.
clear r; set (r := if _ <? _ then Some (Decimal _ _) else Some _).
clear m; set (m := match nb_digits f with 0 => _ | _ => _ end).
replace m with r; [unfold r|now unfold m; revert Hf; case f].
rewrite DecimalZ.to_of, abs_norm, abs_app_int.
case Nat.ltb_spec; intro Hnf.
* rewrite (del_tail_app_int_exact _ _ Hnf).
rewrite (del_head_app_int_exact _ _ Hnf).
now rewrite (dnorm_i_exact _ _ Hnf).
* rewrite (unorm_app_r _ _ Hnf).
rewrite (iter_D0_unorm _ Hf).
now rewrite dnorm_i_exact'.
- unfold of_decimal; simpl.
rewrite <-(DecimalZ.to_of e).
case (Z.of_int e); clear e; [|intro e..]; simpl.
+ case (uint_eq_dec f Nil); intro Hf.
* rewrite Hf; clear f Hf.
unfold to_decimal; simpl.
rewrite IZ_to_Z_IZ_of_Z, DecimalZ.to_of.
case i as [i|i]; [now simpl|]; simpl.
rewrite app_nil_r.
case (uint_eq_dec (nzhead i) Nil); [now intros->|intro Hi].
now rewrite (unorm_nzhead _ Hi); revert Hi; case nzhead.
* set (r := IRQ _).
set (m := match f with Nil => _ | _ => _ end).
replace m with r; [unfold r|now unfold m; revert Hf; case f].
unfold to_decimal; simpl.
unfold IQmake_to_decimal; simpl.
set (n := Nat.iter _ _ _).
case (Pos.eq_dec n 1); intro Hn.
exfalso; apply Hf.
{ now apply nb_digits_0; revert Hn; unfold n; case nb_digits. }
clear m; set (m := match n with 1%positive | _ => _ end).
replace m with (QArith_base.IQmake_to_decimal (Z.of_int (app_int i f)) n).
2:{ now unfold m; revert Hn; case n. }
unfold QArith_base.IQmake_to_decimal, n; simpl.
rewrite nztail_to_uint_pow10.
clear r; set (r := if _ <? _ then Some (Decimal _ _) else Some _).
clear m; set (m := match nb_digits f with 0 => _ | _ => _ end).
replace m with r; [unfold r|now unfold m; revert Hf; case f].
rewrite DecimalZ.to_of, abs_norm, abs_app_int.
case Nat.ltb_spec; intro Hnf.
-- rewrite (del_tail_app_int_exact _ _ Hnf).
rewrite (del_head_app_int_exact _ _ Hnf).
now rewrite (dnorm_i_exact _ _ Hnf).
-- rewrite (unorm_app_r _ _ Hnf).
rewrite (iter_D0_unorm _ Hf).
now rewrite dnorm_i_exact'.
+ set (i' := match i with Pos _ => _ | _ => _ end).
set (m := match Pos.to_uint e with Nil => _ | _ => _ end).
replace m with (DecimalExp i' f (Pos (Pos.to_uint e))).
2:{ unfold m; generalize (Unsigned.to_uint_nonzero e).
now case Pos.to_uint; [|intro u; case u|..]. }
unfold i'; clear i' m.
case (uint_eq_dec f Nil); intro Hf.
* rewrite Hf; clear f Hf.
unfold to_decimal; simpl.
rewrite IZ_to_Z_IZ_of_Z, DecimalZ.to_of.
case i as [i|i]; [now simpl|]; simpl.
rewrite app_nil_r.
case (uint_eq_dec (nzhead i) Nil); [now intros->|intro Hi].
now rewrite (unorm_nzhead _ Hi); revert Hi; case nzhead.
* set (r := IRQ _).
set (m := match f with Nil => _ | _ => _ end).
replace m with r; [unfold r|now unfold m; revert Hf; case f].
unfold to_decimal; simpl.
unfold IQmake_to_decimal; simpl.
set (n := Nat.iter _ _ _).
case (Pos.eq_dec n 1); intro Hn.
exfalso; apply Hf.
{ now apply nb_digits_0; revert Hn; unfold n; case nb_digits. }
clear m; set (m := match n with 1%positive | _ => _ end).
replace m with (QArith_base.IQmake_to_decimal (Z.of_int (app_int i f)) n).
2:{ now unfold m; revert Hn; case n. }
unfold QArith_base.IQmake_to_decimal, n; simpl.
rewrite nztail_to_uint_pow10.
clear r; set (r := if _ <? _ then Some (Decimal _ _) else Some _).
clear m; set (m := match nb_digits f with 0 => _ | _ => _ end).
replace m with r; [unfold r|now unfold m; revert Hf; case f].
rewrite DecimalZ.to_of, abs_norm, abs_app_int.
case Nat.ltb_spec; intro Hnf.
-- rewrite (del_tail_app_int_exact _ _ Hnf).
rewrite (del_head_app_int_exact _ _ Hnf).
now rewrite (dnorm_i_exact _ _ Hnf).
-- rewrite (unorm_app_r _ _ Hnf).
rewrite (iter_D0_unorm _ Hf).
now rewrite dnorm_i_exact'.
+ case (uint_eq_dec f Nil); intro Hf.
* rewrite Hf; clear f Hf.
unfold to_decimal; simpl.
rewrite IZ_to_Z_IZ_of_Z, DecimalZ.to_of.
case i as [i|i]; [now simpl|]; simpl.
rewrite app_nil_r.
case (uint_eq_dec (nzhead i) Nil); [now intros->|intro Hi].
now rewrite (unorm_nzhead _ Hi); revert Hi; case nzhead.
* set (r := IRQ _).
set (m := match f with Nil => _ | _ => _ end).
replace m with r; [unfold r|now unfold m; revert Hf; case f].
unfold to_decimal; simpl.
unfold IQmake_to_decimal; simpl.
set (n := Nat.iter _ _ _).
case (Pos.eq_dec n 1); intro Hn.
exfalso; apply Hf.
{ now apply nb_digits_0; revert Hn; unfold n; case nb_digits. }
clear m; set (m := match n with 1%positive | _ => _ end).
replace m with (QArith_base.IQmake_to_decimal (Z.of_int (app_int i f)) n).
2:{ now unfold m; revert Hn; case n. }
unfold QArith_base.IQmake_to_decimal, n; simpl.
rewrite nztail_to_uint_pow10.
clear r; set (r := if _ <? _ then Some (Decimal _ _) else Some _).
clear m; set (m := match nb_digits f with 0 => _ | _ => _ end).
replace m with r; [unfold r|now unfold m; revert Hf; case f].
rewrite DecimalZ.to_of, abs_norm, abs_app_int.
case Nat.ltb_spec; intro Hnf.
-- rewrite (del_tail_app_int_exact _ _ Hnf).
rewrite (del_head_app_int_exact _ _ Hnf).
now rewrite (dnorm_i_exact _ _ Hnf).
-- rewrite (unorm_app_r _ _ Hnf).
rewrite (iter_D0_unorm _ Hf).
now rewrite dnorm_i_exact'.
Qed.
(** 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_involutive]. 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.
|
