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[helm.git] / matita / contribs / dama / dama / constructive_higher_order_relations.ma

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+
+
+include "constructive_connectives.ma".
+include "higher_order_defs/relations.ma".
+
+definition cotransitive ≝
+ λC:Type.λlt:C→C→Type.∀x,y,z:C. lt x y → lt x z ∨ lt z y. 
+
+definition coreflexive ≝ λC:Type.λlt:C→C→Type. ∀x:C. ¬ (lt x x).
+
+definition antisymmetric ≝
+ λC:Type.λle:C→C→Type.λeq:C→C→Type.∀x,y:C.le x y → le y x → eq x y.
+
+definition symmetric ≝
+ λC:Type.λle:C→C→Type.∀x,y:C.le x y → le y x.
+
+definition transitive ≝
+ λC:Type.λle:C→C→Type.∀x,y,z:C.le x y → le y z → le x z.
+
+definition associative ≝
+ λC:Type.λop:C→C→C.λeq:C→C→Type.∀x,y,z. eq (op x (op y z)) (op (op x y) z).
+
+definition commutative ≝
+ λC:Type.λop:C→C→C.λeq:C→C→Type.∀x,y. eq (op x y) (op y x).
+
+alias id "antisymmetric" = "cic:/matita/higher_order_defs/relations/antisymmetric.con".
+theorem antisimmetric_to_cotransitive_to_transitive:
+ ∀C:Type.∀le:C→C→Prop. antisymmetric ? le → cotransitive ? le → transitive ? le.  
+intros (T f Af cT); unfold transitive; intros (x y z fxy fyz);
+lapply (cT ??z fxy) as H; cases H; [assumption] cases (Af ? ? fyz H1);
+qed.
+
+lemma Or_symmetric: symmetric ? Or.
+unfold; intros (x y H); cases H; [right|left] assumption;
+qed.
+
+