Section stratifies.
Variable Q:knot * other -> T.
Variable HQ:hered 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_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 -> T := 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 => T_bot
end
end.

Lemma unstratify_Q : forall n (p:sinv n) Q,
stratifies Q n p ->
forall (k:knot) (o:other),
projT1 k < n ->
(unstratify n p (k,o) = Q (k,o)).