Section Bool.
Variable A : Type.
Variable f : A -> bool.
(** find whether a boolean function can be satisfied by an elements of the list. *)
Fixpoint existsb (l:list A) : bool :=
match l with
| nil => false
| a::l => f a || existsb l
end.

Lemma existsb_nth : forall l n d, n < length l ->
existsb l = false -> f (nth n l d) = false.