Section stratifies.
Variable Q:knot * other -> Prop.
Variable HQ:hereditary ko_age Q.
Fixpoint stratifies (n:nat) : sinv n -> Prop :=
match n as n' return sinv n' -> Prop with
| 0 => fun _ => True
| S n' => fun (p:sinv (S n')) =>
stratifies n' (fst (proj1_sig p)) /\
forall (k:F (sinv n')) (o:other), snd (proj1_sig p) (k,o) <-> Q (existT (F oo sinv) n' k,o)
end.
Definition stratify (n:nat) : { x:sinv n | stratifies n x }.
End stratifies.
Definition unstratify (n:nat) (p:sinv n) : knot * other -> Prop := fun w =>
match w with (existT _ nw w',o) =>
match decompose_nat nw n with
| inleft (existT _ m Hm) => snd (proj1_sig (floor m (S nw) (eq_rect  n _ p (m + S nw) Hm))) (w',o)
| inright H => False
end
end.

Lemma floor_shuffle:
forall (m1 n : nat)
(p1 : sinv (m1 + S n)) (H1 : (m1 + S n) = (S m1 + n)),
floor (S m1) n (eq_rect (m1 + S n) sinv p1 (S m1 + n) H1) = fst (proj1_sig (floor m1 (S n) p1)).