Section Equivalences.
(** *** Equivalences between the different definition of the reflexive, symmetric, transitive closures *)
(** *** Contributed by P. Castéran *)
(** Direct transitive closure vs left-step extension *)
(** Direct transitive closure vs right-step extension *)

Lemma clos_trans_tn1_iff : forall x y,
clos_trans R x y <-> clos_trans_n1 R x y.