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

(* Specifications: *)
Lemma clos_rt1n_step : forall x y, R x y -> clos_refl_trans_1n R x y.
Lemma clos_rtn1_step : forall x y, R x y -> clos_refl_trans_n1 R x y.
Lemma clos_rt1n_rt : forall x y,
clos_refl_trans_1n R x y -> clos_refl_trans R x y.
Lemma clos_rt_rt1n : forall x y,
clos_refl_trans R x y -> clos_refl_trans_1n R x y.
Lemma clos_rt_rt1n_iff : forall x y,
clos_refl_trans R x y <-> clos_refl_trans_1n R x y.