+lemma ex2_1_commute: ∀A0. ∀P0,P1:A0→Prop. (∃∃x0. P0 x0 & P1 x0) → ∃∃x0. P1 x0 & P0 x0.
+#A0 #P0 #P1 * /2 width=3/
+qed.
+
+(* relations *)
+
+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.
+
+(* Note: confluent1 would be redundant if \Pi-reduction where enabled *)
+definition confluent: ∀A. predicate (relation A) ≝ λA,R.
+ ∀a0. confluent1 … R a0.
+
+(* arithmetics *)
+