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 *)
(** Direct reflexive-transitive closure is equivalent to transitivity by left-step extension *)
(** Direct reflexive-transitive closure is equivalent to transitivity by right-step extension *)

Lemma clos_rt_rtn1_iff : forall x y,
clos_refl_trans R x y <-> clos_refl_trans_n1 R x y.