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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)         *)
+(************************************************************************)
+
+(** * DecimalN
+
+    Proofs that conversions between decimal numbers and [N]
+    are bijections *)
+
+Require Import Decimal DecimalFacts DecimalPos PArith NArith.
+
+Module Unsigned.
+
+Lemma of_to (n:N) : N.of_uint (N.to_uint n) = n.
+Proof.
+ destruct n.
+ - reflexivity.
+ - apply DecimalPos.Unsigned.of_to.
+Qed.
+
+Lemma to_of (d:uint) : N.to_uint (N.of_uint d) = unorm d.
+Proof.
+ exact (DecimalPos.Unsigned.to_of d).
+Qed.
+
+Lemma to_uint_inj n n' : N.to_uint n = N.to_uint n' -> n = n'.
+Proof.
+ intros E. now rewrite <- (of_to n), <- (of_to n'), E.
+Qed.
+
+Lemma to_uint_surj d : exists p, N.to_uint p = unorm d.
+Proof.
+ exists (N.of_uint d). apply to_of.
+Qed.
+
+Lemma of_uint_norm d : N.of_uint (unorm d) = N.of_uint d.
+Proof.
+ now induction d.
+Qed.
+
+Lemma of_inj d d' :
+ N.of_uint d = N.of_uint d' -> unorm d = unorm d'.
+Proof.
+ intros. rewrite <- !to_of. now f_equal.
+Qed.
+
+Lemma of_iff d d' : N.of_uint d = N.of_uint d' <-> unorm d = unorm d'.
+Proof.
+ split. apply of_inj. intros E. rewrite <- of_uint_norm, E.
+ apply of_uint_norm.
+Qed.
+
+End Unsigned.
+
+(** Conversion from/to signed decimal numbers *)
+
+Module Signed.
+
+Lemma of_to (n:N) : N.of_int (N.to_int n) = Some n.
+Proof.
+ unfold N.to_int, N.of_int, norm. f_equal.
+ rewrite Unsigned.of_uint_norm. apply Unsigned.of_to.
+Qed.
+
+Lemma to_of (d:int)(n:N) : N.of_int d = Some n -> N.to_int n = norm d.
+Proof.
+ unfold N.of_int.
+ destruct (norm d) eqn:Hd; intros [= <-].
+ unfold N.to_int. rewrite Unsigned.to_of. f_equal.
+ revert Hd; destruct d; simpl.
+ - intros [= <-]. apply unorm_invol.
+ - destruct (nzhead d); now intros [= <-].
+Qed.
+
+Lemma to_int_inj n n' : N.to_int n = N.to_int n' -> n = n'.
+Proof.
+ intro E.
+ assert (E' : Some n = Some n').
+ { now rewrite <- (of_to n), <- (of_to n'), E. }
+ now injection E'.
+Qed.
+
+Lemma to_int_pos_surj d : exists n, N.to_int n = norm (Pos d).
+Proof.
+ exists (N.of_uint d). unfold N.to_int. now rewrite Unsigned.to_of.
+Qed.
+
+Lemma of_int_norm d : N.of_int (norm d) = N.of_int d.
+Proof.
+ unfold N.of_int. now rewrite norm_invol.
+Qed.
+
+Lemma of_inj_pos d d' :
+ N.of_int (Pos d) = N.of_int (Pos d') -> unorm d = unorm d'.
+Proof.
+ unfold N.of_int. simpl. intros [= H]. apply Unsigned.of_inj.
+ change Pos.of_uint with N.of_uint in H.
+ now rewrite <- Unsigned.of_uint_norm, H, Unsigned.of_uint_norm.
+Qed.
+
+End Signed.