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.