matita/library/higher_order_defs/relations.ma

   1 (**************************************************************************)
   2 (*       ___                                                                *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
   8 (*      ||A||       E.Tassi, S.Zacchiroli                                 *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU Lesser General Public License Version 2.1         *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 set "baseuri" "cic:/matita/higher_order_defs/relations/".
  16
  17 include "logic/connectives.ma".
  18
  19 definition reflexive: \forall A:Type.\forall R:A \to A \to Prop.Prop
  20 \def
  21 \lambda A.\lambda R.\forall x:A.R x x.
  22
  23 definition symmetric: \forall A:Type.\forall R:A \to A \to Prop.Prop
  24 \def
  25 \lambda A.\lambda R.\forall x,y:A.R x y \to R y x.
  26
  27 definition transitive: \forall A:Type.\forall R:A \to A \to Prop.Prop
  28 \def
  29 \lambda A.\lambda R.\forall x,y,z:A.R x y \to R y z \to R x z.
  30
  31 definition irreflexive: \forall A:Type.\forall R:A \to A \to Prop.Prop
  32 \def
  33 \lambda A.\lambda R.\forall x:A.\lnot (R x x).