Definition reflexive := forall x, R x x.Definition irreflexive := forall x, R x x = false.Definition left_transitive := forall x y, R x y -> R x =1 R y.Definition right_transitive := forall x y, R x y -> R^~ x =1 R^~ y.

(* Specifications: *)
Lemma sym_left_transitive : left_transitive.
Lemma sym_right_transitive : right_transitive.