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 takel_cat s1 : n0 <= size s1 -> forall s2, take n0 (s1 ++ s2) = take n0 s1.