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

(* Specifications: *)
Lemma clos_rstn1_rst : forall x y,
clos_refl_sym_trans_n1 R x y -> clos_refl_sym_trans R x y.
Lemma clos_rstn1_trans : forall x y z, clos_refl_sym_trans_n1 R x y ->
clos_refl_sym_trans_n1 R y z -> clos_refl_sym_trans_n1 R x z.
Lemma clos_rstn1_sym : forall x y, clos_refl_sym_trans_n1 R x y ->
clos_refl_sym_trans_n1 R y x.
Lemma clos_rst_rstn1 : forall x y,
clos_refl_sym_trans R x y -> clos_refl_sym_trans_n1 R x y.
Lemma clos_rst_rstn1_iff : forall x y,
clos_refl_sym_trans R x y <-> clos_refl_sym_trans_n1 R x y.