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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 reflexive2 ≝ λT:Type[2].λP:T → T → CProp[2]. ∀x.P x x.
18 ndefinition symmetric2 ≝ λT:Type[2].λP:T → T → CProp[2]. ∀x,y.P x y → P y x.
19 ndefinition transitive2 ≝ λT:Type[2].λP:T → T → CProp[2]. ∀z,x,y. P x z → P z y → P x y.
20
21 nrecord equivalence_relation2 (A:Type[2]) : Type[3] ≝
22  { eq_rel2:2> A → A → CProp[2];
23    refl2: reflexive2 ? eq_rel2;
24    sym2: symmetric2 ? eq_rel2;
25    trans2: transitive2 ? eq_rel2
26  }.