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1 (**************************************************************************)
2 (*       ___                                                              *)
3 (*      ||M||                                                             *)
4 (*      ||A||       A project by Andrea Asperti                           *)
5 (*      ||T||                                                             *)
6 (*      ||I||       Developers:                                           *)
7 (*      ||T||         The HELM team.                                      *)
8 (*      ||A||         http://helm.cs.unibo.it                             *)
9 (*      \   /                                                             *)
10 (*       \ /        This file is distributed under the terms of the       *)
11 (*        v         GNU General Public License Version 2                  *)
12 (*                                                                        *)
13 (**************************************************************************)
14
15 include "logic/pts.ma".
16
17 ndefinition reflexive1 ≝ λT:Type[1].λP:T → T → CProp[1]. ∀x.P x x.
18 ndefinition symmetric1 ≝ λT:Type[1].λP:T → T → CProp[1]. ∀x,y.P x y → P y x.
19 ndefinition transitive1 ≝ λT:Type[1].λP:T → T → CProp[1]. ∀z,x,y. P x z → P z y → P x y.
20
21 nrecord equivalence_relation1 (A:Type[1]) : Type[2] ≝
22  { eq_rel1:2> A → A → CProp[1];
23    refl1: reflexive1 ? eq_rel1;
24    sym1: symmetric1 ? eq_rel1;
25    trans1: transitive1 ? eq_rel1
26  }.