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[helm.git] / matita / library / higher_order_defs / relations.ma

diff --git a/matita/library/higher_order_defs/relations.ma b/matita/library/higher_order_defs/relations.ma
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+(**************************************************************************)
+(*       ___                                                               *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
+(*      ||A||       E.Tassi, S.Zacchiroli                                 *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU Lesser General Public License Version 2.1         *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/higher_order_defs/relations/".
+
+include "logic/connectives.ma".
+
+definition relation : Type \to Type
+\def \lambda A:Type.A \to A \to Prop. 
+
+definition reflexive: \forall A:Type.\forall R :relation A.Prop
+\def 
+\lambda A.\lambda R.\forall x:A.R x x.
+
+definition symmetric: \forall A:Type.\forall R: relation A.Prop
+\def 
+\lambda A.\lambda R.\forall x,y:A.R x y \to R y x.
+
+definition transitive: \forall A:Type.\forall R:relation A.Prop
+\def 
+\lambda A.\lambda R.\forall x,y,z:A.R x y \to R y z \to R x z.
+
+definition irreflexive: \forall A:Type.\forall R:relation A.Prop
+\def 
+\lambda A.\lambda R.\forall x:A.\lnot (R x x).
+
+definition cotransitive: \forall A:Type.\forall R:relation A.Prop
+\def
+\lambda A.\lambda R.\forall x,y:A.R x y \to \forall z:A. R x z \lor R z y.
+
+definition tight_apart: \forall A:Type.\forall eq,ap:relation A.Prop
+\def 
+\lambda A.\lambda eq,ap.\forall x,y:A. (\not (ap x y) \to eq x y) \land
+(eq x y \to \not (ap x y)).
+
+definition antisymmetric: \forall A:Type.\forall R:relation A.Prop
+\def 
+\lambda A.\lambda R.\forall x,y:A. R x y \to \not (R y x).
+
+