definition antisymmetric: ∀A.∀R:relation A.Prop
≝ λA.λR.∀x,y:A. R x y → ¬(R y x).
+definition singlevalued: ∀A,B. predicate (relation2 A B) ≝ λA,B,R.
+ ∀a,b1. R a b1 → ∀b2. R a b2 → b1 = b2.
+
+definition confluent1: ∀A. relation A → predicate A ≝ λA,R,a0.
+ ∀a1. R a0 a1 → ∀a2. R a0 a2 →
+ ∃∃a. R a1 a & R a2 a.
+
+definition confluent: ∀A. predicate (relation A) ≝ λA,R.
+ ∀a0. confluent1 … R a0.
+
+(* 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.
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.