(* This program is free software; you can redistribute it and/or      *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Union.
Require Export Singleton.
Require Export Diff.
Require Export Classical_Prop.
Section fparts_in_def.
Variable E : Setoid.
(* Add_part. *)
Definition add_part (A : part_set E) (x : E) := union A (single x).

(* Specifications: *)
Lemma add_part_comp :
forall (A A' : part_set E) (x x' : E),
Equal A A' -> Equal x x' -> Equal (add_part A x) (add_part A' x').