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(* 02110-1301 USA                                                     *)
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(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                 Coq V6.1                                 *)
(*                               Oct 1st 1996                               *)
(*                                                                          *)
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(*                                 Setoid.v                                 *)
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(*****************************************************************************)
(*          Projet Coq - Calculus of Inductive Constructions V5.10           *)
(*****************************************************************************)
(*                                                                           *)
(*          General Algebra : Setoids                                        *)
(*                                                                           *)
(*          Peter Aczel, Dec. 92                                             *)
(*          rev. Gerard Huet,  Aug. 1993                                     *)
(*                                                                           *)
(*****************************************************************************)
Require Export Relations.
Set Implicit Arguments.
Unset Strict Implicit.
Global Set Asymmetric Patterns.
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.
Inductive Nat : Type :=
| Z : Nat
| Suc : Nat -> Nat.