bir georganization, most of the structures done

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

diff --git a/helm/software/matita/dama/constructive_higher_order_relations.ma b/helm/software/matita/dama/constructive_higher_order_relations.ma
index b66ba684309c47c2cb606f2607e4db506b374f31..789c7c9c50df894f8c4cb5270278f7009d6afa49 100644 (file)
--- a/helm/software/matita/dama/constructive_higher_order_relations.ma
+++ b/helm/software/matita/dama/constructive_higher_order_relations.ma
@@ -29,3 +29,22 @@ definition symmetric ≝
 
 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.
+
+  
\ No newline at end of file