matitaB/matita/nlibrary/properties/relations.ma

   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 definition reflexive ≝ λT:Type[0].λP:T → T → CProp[0]. ∀x.P x x.
  18 definition symmetric ≝ λT:Type[0].λP:T → T → CProp[0]. ∀x,y.P x y → P y x.
  19 definition transitive ≝ λT:Type[0].λP:T → T → CProp[0]. ∀z,x,y. P x z → P z y → P x y.
  20
  21 record equivalence_relation (A:Type[0]) : Type[1] ≝
  22  { eq_rel:2> A → A → CProp[0];
  23    refl: <A href="cic:/matita/ng/properties/relations/reflexive.def(1)">reflexive</A> ? eq_rel;
  24    sym: <A href="cic:/matita/ng/properties/relations/symmetric.def(1)">symmetric</A> ? eq_rel;
  25    trans: <A href="cic:/matita/ng/properties/relations/transitive.def(1)">transitive</A> ? eq_rel
  26  }.