(* 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 Parts.
(** Title "Singletons." *)
Section Single1.
Variable E : Setoid.
Definition single : E -> part_set E.
intros x.
apply (Build_Predicate (Pred_fun:=fun y : E => Equal y x)).
red in |- *.
intros x0 y H' H'0; try assumption.
apply Trans with x0; auto with algebra.
Defined.

(* Specifications: *)
Lemma in_single : forall x : E, in_part x (single x).
simpl in |- *; auto with algebra.