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 size_take s : size (take n0 s) = if n0 < size s then n0 else size s.