Proof using.
unfold in_output_list.
simpl. intuition.
find_inversion. auto.
Qed.

Lemma assoc_Some_In :
forall K V (K_eq_dec : forall k k' : K, {k = k'} + {k <> k'}) k v l,
assoc (V:=V) K_eq_dec l k = Some v ->
In (k, v) l.