Proof using.
intros. induction l; simpl in *; intuition.
find_rewrite. break_match; intuition.
repeat find_higher_order_rewrite. auto.
Qed.

(* Specifications: *)
Lemma argmax_None :
forall A (f : A -> nat) l,
argmax f l = None ->
l = [].
Lemma argmin_None :
forall A (f : A -> nat) l,
argmin f l = None ->
l = [].