- bug fix in notation precedences

[helm.git] / matita / matita / lib / basics / relations.ma

diff --git a/matita/matita/lib/basics/relations.ma b/matita/matita/lib/basics/relations.ma
index afb5d5aae05515a1c9801c45abb9064f2a55a91f..ed5130955886e557a1e11b8c83b09f7ad649cb48 100644 (file)
--- a/matita/matita/lib/basics/relations.ma
+++ b/matita/matita/lib/basics/relations.ma
@@ -23,6 +23,9 @@ definition relation : Type[0] → Type[0]
 definition relation2 : Type[0] → Type[0] → Type[0]
 ≝ λA,B.A→B→Prop.
 
+definition relation3 : Type[0] → Type[0] → Type[0] → Type[0]
+≝ λA,B,C.A→B→C→Prop.
+
 definition reflexive: ∀A.∀R :relation A.Prop
 ≝ λA.λR.∀x:A.R x x.
 
@@ -45,6 +48,14 @@ definition tight_apart: ∀A.∀eq,ap:relation A.Prop
 definition antisymmetric: ∀A.∀R:relation A.Prop
 ≝ λA.λR.∀x,y:A. R x y → ¬(R y x).
 
+(* Reflexive closure ************)
+
+definition RC: ∀A:Type[0]. relation A → relation A ≝
+               λA,R,x,y. R … x y ∨ x = y.
+
+lemma RC_reflexive: ∀A,R. reflexive A (RC … R).
+/2 width=1/ qed.
+
 (********** operations **********)
 definition Rcomp ≝ λA.λR1,R2:relation A.λa1,a2.
   ∃am.R1 a1 am ∧ R2 am a2.
@@ -161,5 +172,12 @@ interpretation "functional extentional equality"
 definition bi_relation: Type[0] → Type[0] → Type[0]
 ≝ λA,B.A→B→A→B→Prop.
 
-definition bi_reflexive: ∀A,B. ∀R :bi_relation A B. Prop
+definition bi_reflexive: ∀A,B. ∀R:bi_relation A B. Prop
 ≝ λA,B,R. ∀x,y. R x y x y.
+
+definition bi_symmetric: ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R.
+                         ∀a1,a2,b1,b2. R a2 b2 a1 b1 → R a1 b1 a2 b2.
+
+definition bi_transitive: ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R.
+                          ∀a1,a,b1,b. R a1 b1 a b →
+                          ∀a2,b2. R a b a2 b2 → R a1 b1 a2 b2.