(************************************************************************)
(*         *      The Rocq Prover / The Rocq 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)         *)
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(** * DecimalPos  Proofs that conversions between decimal numbers and [positive] are bijections. *)
Module Unsigned.
Local Open Scope N.
(** A direct version of [of_little_uint] *)
Fixpoint of_lu (d:uint) : N :=
match d with
| Nil => 0
| D0 d => 10 * of_lu d
| D1 d => 1 + 10 * of_lu d
| D2 d => 2 + 10 * of_lu d
| D3 d => 3 + 10 * of_lu d
| D4 d => 4 + 10 * of_lu d
| D5 d => 5 + 10 * of_lu d
| D6 d => 6 + 10 * of_lu d
| D7 d => 7 + 10 * of_lu d
| D8 d => 8 + 10 * of_lu d
| D9 d => 9 + 10 * of_lu d
end.
Definition hd d :=
match d with
| Nil => 0
| D0 _ => 0
| D1 _ => 1
| D2 _ => 2
| D3 _ => 3
| D4 _ => 4
| D5 _ => 5
| D6 _ => 6
| D7 _ => 7
| D8 _ => 8
| D9 _ => 9
end.
Definition tl d :=
match d with
| Nil => d
| D0 d | D1 d | D2 d | D3 d | D4 d | D5 d | D6 d | D7 d | D8 d | D9 d => d
end.

(* Specifications: *)
Lemma of_lu_eqn d :
of_lu d = hd d + 10 * (of_lu (tl d)).