1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "logic/pts.ma".
17 definition reflexive1 ≝ λT:Type[1].λP:T → T → CProp[1]. ∀x.P x x.
18 definition symmetric1 ≝ λT:Type[1].λP:T → T → CProp[1]. ∀x,y.P x y → P y x.
19 definition transitive1 ≝ λT:Type[1].λP:T → T → CProp[1]. ∀z,x,y. P x z → P z y → P x y.
21 record equivalence_relation1 (A:Type[1]) : Type[2] ≝
22 { eq_rel1:2> A → A → CProp[1];
23 refl1: <A href="cic:/matita/ng/properties/relations1/reflexive1.def(1)">reflexive1</A> ? eq_rel1;
24 sym1: <A href="cic:/matita/ng/properties/relations1/symmetric1.def(1)">symmetric1</A> ? eq_rel1;
25 trans1: <A href="cic:/matita/ng/properties/relations1/transitive1.def(1)">transitive1</A> ? eq_rel1