Section reduce_strictly_pos.
Variable E : Set.
Variable f : E -> Z.
Hypothesis positive_function : forall a : E, (f a >= 0)%Z.
Hypothesis eq_E_dec : eq_dec E.

(* Specifications: *)
Lemma reduce_in_queue_strictly_positive :
forall (x : E) (q : queue E),
(f x > 0)%Z -> In_queue E x q -> (reduce E f q > 0)%Z.