Definition connect_sym := symmetric connect.

(* Specifications: *)
Lemma same_connect : left_transitive connect.
Lemma same_connect_r : right_transitive connect.
Lemma same_connect1 x y : e x y -> connect x =1 connect y.
Lemma same_connect1r x y : e x y -> connect^~ x =1 connect^~ y.
Lemma rootP x y : reflect (root x = root y) (connect x y).
Lemma root_root x : root (root x) = root x.
Lemma roots_root x : roots (root x).