X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fbasics%2Frelations.ma;h=8b6a79be5deec90a9f7820287174045f277bb533;hb=0460fd3dc2909efe0baa6592281d0cf0527165ff;hp=a1bae198b707723b66f5ba16db61ac1cac58deab;hpb=24fc5e5485245fe879e17d46176530b688930b3b;p=helm.git

diff --git a/matita/matita/lib/basics/relations.ma b/matita/matita/lib/basics/relations.ma
index a1bae198b..8b6a79be5 100644
--- a/matita/matita/lib/basics/relations.ma
+++ b/matita/matita/lib/basics/relations.ma
@@ -11,10 +11,18 @@
 
 include "basics/logic.ma".
 
+(********** predicates *********)
+
+definition predicate: Type[0] → Type[0]
+≝ λA.A→Prop.
+
 (********** relations **********)
 definition relation : Type[0] → Type[0]
 ≝ λA.A→A→Prop. 
 
+definition relation2 : Type[0] → Type[0] → Type[0]
+≝ λA,B.A→B→Prop.
+
 definition reflexive: ∀A.∀R :relation A.Prop
 ≝ λA.λR.∀x:A.R x x.
 
@@ -37,7 +45,44 @@ 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).
 
-(********** functions **********)
+(********** operations **********)
+definition Rcomp ≝ λA.λR1,R2:relation A.λa1,a2.
+  ∃am.R1 a1 am ∧ R2 am a2.
+interpretation "relation composition" 'compose R1 R2 = (Rcomp ? R1 R2).
+
+definition Runion ≝ λA.λR1,R2:relation A.λa,b. R1 a b ∨ R2 a b.
+interpretation "union of relations" 'union R1 R2 = (Runion ? R1 R2).
+    
+definition Rintersection ≝ λA.λR1,R2:relation A.λa,b.R1 a b ∧ R2 a b.
+interpretation "interesecion of relations" 'intersects R1 R2 = (Rintersection ? R1 R2).
+
+definition inv ≝ λA.λR:relation A.λa,b.R b a.
+
+(*********** sub relation ***********)
+definition subR ≝ λA.λR,S:relation A.(∀a,b. R a b → S a b).
+interpretation "relation inclusion" 'subseteq R S = (subR ? R S).
+
+lemma sub_comp_l:  ∀A.∀R,R1,R2:relation A.
+  R1 ⊆ R2 → R1 ∘ R ⊆ R2 ∘ R.
+#A #R #R1 #R2 #Hsub #a #b * #c * /4/
+qed.
+
+lemma sub_comp_r:  ∀A.∀R,R1,R2:relation A.
+  R1 ⊆ R2 → R ∘ R1 ⊆ R ∘ R2.
+#A #R #R1 #R2 #Hsub #a #b * #c * /4/
+qed.
+
+lemma sub_assoc_l: ∀A.∀R1,R2,R3:relation A.
+  R1 ∘ (R2 ∘ R3) ⊆ (R1 ∘ R2) ∘ R3.
+#A #R1 #R2 #R3 #a #b * #c * #Hac * #d * /5/
+qed.
+
+lemma sub_assoc_r: ∀A.∀R1,R2,R3:relation A.
+  (R1 ∘ R2) ∘ R3 ⊆ R1 ∘ (R2 ∘ R3).
+#A #R1 #R2 #R3 #a #b * #c * * #d * /5 width=5/ 
+qed.
+
+(************* functions ************)
 
 definition compose ≝
   λA,B,C:Type[0].λf:B→C.λg:A→B.λx:A.f (g x).