Hint Resolve trans_incl.
Hint Resolve comp_incl.
Hint Resolve union_incl.
Hint Resolve star_incl.
Hint Resolve plus_incl.
Hint Resolve trans_eeq.
Hint Resolve comp_eeq.
Hint Resolve union_eeq.
Hint Resolve star_eeq.
Hint Resolve plus_eeq.
Section InclEeq.
Variables X Y: Type.
Variables S R T: relation2 X Y.

(* Specifications: *)
Lemma incl_refl: incl R R.
Proof fun x y H => H.
Lemma incl_trans: incl R S -> incl S T -> incl R T.
Lemma star_S: forall x y z, star R x y -> R y z -> star R x z.
Lemma star_trans: forall x y z, star R x y -> star R y z -> star R x z.
Lemma plus_star: forall x y, plus R x y -> star R x y.
Lemma plus_star_plus: forall y x z, plus R x y -> star R y z -> plus R x z.
Lemma star_plus_plus: forall y x z, star R x y -> plus R y z -> plus R x z.
Lemma plus_trans: forall y x z, plus R x y -> plus R y z -> plus R x z.
Lemma plus1: forall x y, R x y -> plus R x y.
Lemma plus_S: forall y x z, star R x y -> R y z -> plus R x z.
Lemma S_plus: forall y x z, R x y -> star R y z -> plus R x z.
End star.