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.

Lemma prod_delta_star w q1 q2 : @delta_star prod (q1, q2) w = (delta_star q1 w, delta_star q2 w).