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+(************************************************************************)
+(*         *   The Coq Proof Assistant / The Coq Development Team       *)
+(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
+(* <O___,, *       (see CREDITS file for the list of authors)           *)
+(*   \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)         *)
+(************************************************************************)
+
+(** * 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.