(* find, count, has, all. *)
Section SeqFind.
Variable a : pred T.
Fixpoint find s := if s is x :: s' then if a x then 0 else (find s').+1 else 0.
Fixpoint filter s :=
if s is x :: s' then if a x then x :: filter s' else filter s' else [::].
Fixpoint count s := if s is x :: s' then a x + count s' else 0.
Fixpoint has s := if s is x :: s' then a x || has s' else false.
Fixpoint all s := if s is x :: s' then a x && all s' else true.

Lemma count_filter a1 a2 s : count a1 (filter a2 s) = count (predI a1 a2) s.
Fixpoint drop n s {struct s} :=
match s, n with
| _ :: s', n'.+1 => drop n' s'
| _, _ => s
end.