Section Product_automaton.
Variable op: Prop -> Prop -> Prop.
Variable op_dec: forall P Q, dec P -> dec Q -> dec (op P Q).
Definition prod_delta: A (x) A' -> Sig -> A (x) A' :=
fun P x  => match P with (q1,q2) => (delta_S q1 x, delta_S q2 x) end.
Definition prod_pred := (fun P => match P with (q1,q2) => op (@F A q1) (@F A' q2) end).
Global Instance prod_pred_dec P: dec (prod_pred P).
Definition prod_F  := DecPred prod_pred.
Definition prod := DFA (s,s) prod_F prod_delta.
End Product_automaton.
Arguments prod op {op_dec}.
Definition intersect := prod and.
Definition U :=  prod or.
Definition diff := prod (fun P Q => P /\ ~ Q).
Definition lang_incl := forall w, in_lang A w -> in_lang A' w.

Lemma lang_incl_iff : lang_incl <-> empty diff.