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.

(* Specifications: *)
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)).
Lemma unstratify_hered : forall n p, hered (unstratify n p).
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)).
Lemma stratifies_unstratify_more : forall (n m1 m2:nat) (p1:sinv (m1+n)) (p2:sinv (m2+n)), floor m1 n p1 = floor m2 n p2 -> (stratifies (unstratify (m1+n) p1) n (floor m1 n p1) -> stratifies (unstratify (m2+n) p2) n (floor m2 n p2)).
Lemma stratify_unstratify : forall n p H, proj1_sig (stratify (unstratify n p) H n) = p.
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)).
Lemma unstratify_hered : forall n p, hered (unstratify n p).
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)).
Lemma stratifies_unstratify_more : forall (n m1 m2:nat) (p1:sinv (m1+n)) (p2:sinv (m2+n)), floor m1 n p1 = floor m2 n p2 -> (stratifies (unstratify (m1+n) p1) n (floor m1 n p1) -> stratifies (unstratify (m2+n) p2) n (floor m2 n p2)).
Lemma stratify_unstratify : forall n p H, proj1_sig (stratify (unstratify n p) H n) = p.