(* 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                                                     *)
Require Export Relations.
Set Implicit Arguments.
Unset Strict Implicit.
(* evite le clash d'univers entre le setoide des foncteur et SET *)
(***  SETOID  ***)
Structure > Setoid' : Type :=
{Carrier' :> Type;
Equal' : Relation Carrier';
Prf_equiv' :> Equivalence Equal'}.
Infix "=_S'" := Equal' (at level 70).
(*** rewrite rules ***)

(* Specifications: *)
Lemma Refl' : forall (S : Setoid') (x : S), x =_S' x.
Lemma Sym' : forall (S : Setoid') (x y : S), x =_S' y -> y =_S' x.
Lemma Trans' :
forall (S : Setoid') (x y z : S), x =_S' y -> y =_S' z -> x =_S' z.