Definition procedure_family(A : nat -> Type) := forall n, A n.Definition efficiency_predicate := forall (A : nat -> Type), procedure_family A -> Prop.

Theorem expected_poly_time_fst_f : forall (A B C : nat -> Type)(f : forall n, A n -> (B n * C n)), efficient _ f -> efficient _ (fun n (a : A n) => fst (f n a)).