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_predUI a1 a2 s : count (predU a1 a2) s + count (predI a1 a2) s = count a1 s + count a2 s.