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 hered (unstratify n p) (unstratify_hered n p).Definition squash (x:nat * F predicate predicate) : knot :=
    match x with (n,f) => existT (F_sinv) n (bimap (unstrat n) (strat n) f) end.Definition unsquash (k:knot) : nat * F predicate predicate :=
    match k with existT _ n f => (n, bimap (strat n) (unstrat n) f) end.Definition knot_level_def (k:knot) : nat :=
    fst (unsquash k).Definition knot_age1_def (k:knot) : option knot :=
    match unsquash k with
    | (O,_) => None
    | (S n,x) => Some (squash (n,x))
    end.Definition knot_unage_def (k:knot) :=
    let (n,k) := unsquash k in squash (S n,k).Program Definition approx (n:nat) (p:predicate) : predicate :=
    fun w => if (le_gt_dec n (knot_level_def (fst w))) then T_bot else proj1_sig p w.

Lemma bimap_bimap : forall A B C X Y Z (s:X->Y) (t:Y->Z) (f:B->C) (g:A->B) x, bimap s f (bimap t g x) = bimap (t oo s) (f oo g) x.