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

(* Specifications: *)
Lemma clos_t1n_trans : forall x y, clos_trans_1n R x y -> clos_trans R x y.
Lemma clos_trans_t1n : forall x y, clos_trans R x y -> clos_trans_1n R x y.
Lemma clos_trans_t1n_iff : forall x y,
clos_trans R x y <-> clos_trans_1n R x y.