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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).