-definition Transitive: ∀A. ∀R: relation A. Prop ≝ λA,R.
- ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2.
+definition Transitive (A) (R:relation A): Prop ≝
+ ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2.
+
+definition left_cancellable (A) (R:relation A): Prop ≝
+ ∀a0,a1. R a0 a1 → ∀a2. R a0 a2 → R a1 a2.
+
+definition right_cancellable (A) (R:relation A): Prop ≝
+ ∀a1,a0. R a1 a0 → ∀a2. R a2 a0 → R a1 a2.
+
+definition pw_confluent2 (A) (R1,R2:relation A): predicate A ≝
+ λa0.
+ ∀a1. R1 a0 a1 → ∀a2. R2 a0 a2 →
+ ∃∃a. R2 a1 a & R1 a2 a.