Fixpoint filter1 (A: Type) (pred: A -> bool) (m: t A) {struct m} : t A :=
match m with
| Leaf => Leaf
| Node l o r =>
let o' := match o with None => None | Some x => if pred x then o else None end in
Node' (filter1 pred l) o' (filter1 pred r)
end.

Theorem gfilter1:
forall (A: Type) (pred: A -> bool) (i: elt) (m: t A),
get i (filter1 pred m) =
match get i m with None => None | Some x => if pred x then Some x else None end.