Definition strat (n:nat) (p:predicate) : sinv n :=
    proj1_sig (stratify (proj1_sig p) (proj2_sig p) n).Definition unstrat (n:nat) (p:sinv n) : predicate :=
    exist (hereditary ko_age) (unstratify n p) (unstratify_hered n p).Definition squash (x:nat * F predicate) : knot :=
    match x with (n,f) => existT (F oo sinv) n (fmap F (strat n) f) end.Definition unsquash (k:knot) : nat * F predicate :=
    match k with existT _ n f => (n, fmap F (unstrat n) f) end.Definition level (x:knot) : nat := fst (unsquash x).Program Definition approx (n:nat) (p:predicate) : predicate :=
     fun w => level (fst w) < n /\ p w.

Lemma knot_age_age1 : forall k k', k_age1 k = Some k' <-> match unsquash k with | (O,_) => None | (S n,x) => Some (squash (n,x)) end = Some k'.