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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** * DecimalZ
Proofs that conversions between decimal numbers and [Z]
are bijections. *)
Require Import Decimal DecimalFacts DecimalPos DecimalN ZArith.
Lemma of_to (z:Z) : Z.of_int (Z.to_int z) = z.
Proof.
destruct z; simpl.
- trivial.
- unfold Z.of_uint. rewrite DecimalPos.Unsigned.of_to. now destruct p.
- unfold Z.of_uint. rewrite DecimalPos.Unsigned.of_to. destruct p; auto.
Qed.
Lemma to_of (d:int) : Z.to_int (Z.of_int d) = norm d.
Proof.
destruct d; simpl; unfold Z.to_int, Z.of_uint.
- rewrite <- (DecimalN.Unsigned.to_of d). unfold N.of_uint.
now destruct (Pos.of_uint d).
- destruct (Pos.of_uint d) eqn:Hd; simpl; f_equal.
+ generalize (DecimalPos.Unsigned.to_of d). rewrite Hd. simpl.
intros H. symmetry in H. apply unorm_0 in H. now rewrite H.
+ assert (Hp := DecimalPos.Unsigned.to_of d). rewrite Hd in Hp. simpl in *.
rewrite Hp. unfold unorm in *.
destruct (nzhead d); trivial.
generalize (DecimalPos.Unsigned.of_to p). now rewrite Hp.
Qed.
(** Some consequences *)
Lemma to_int_inj n n' : Z.to_int n = Z.to_int n' -> n = n'.
Proof.
intro EQ.
now rewrite <- (of_to n), <- (of_to n'), EQ.
Qed.
Lemma to_int_surj d : exists n, Z.to_int n = norm d.
Proof.
exists (Z.of_int d). apply to_of.
Qed.
Lemma of_int_norm d : Z.of_int (norm d) = Z.of_int d.
Proof.
unfold Z.of_int, Z.of_uint.
destruct d.
- simpl. now rewrite DecimalPos.Unsigned.of_uint_norm.
- simpl. destruct (nzhead d) eqn:H;
[ induction d; simpl; auto; discriminate |
destruct (nzhead_nonzero _ _ H) | .. ];
f_equal; f_equal; apply DecimalPos.Unsigned.of_iff;
unfold unorm; now rewrite H.
Qed.
Lemma of_inj d d' :
Z.of_int d = Z.of_int d' -> norm d = norm d'.
Proof.
intros. rewrite <- !to_of. now f_equal.
Qed.
Lemma of_iff d d' : Z.of_int d = Z.of_int d' <-> norm d = norm d'.
Proof.
split. apply of_inj. intros E. rewrite <- of_int_norm, E.
apply of_int_norm.
Qed.
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