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_cat s1 s2 : count (s1 ++ s2) = count s1 + count s2.