Module TF := TF'.
(* Put the discrete pointed order on rhs *)
Inductive leT : T -> T -> Prop :=
| leT_refl : forall t, leT t t
| leT_bot: forall t, leT T_bot t.
Fixpoint sinv (n: nat) : Type :=
match n with
| O => unit
| S n => prodT (sinv n) ((F (sinv n) * other) -> T)
end.
Fixpoint floor (m:nat) (n:nat) (p:sinv (m+n)) : sinv n :=
match m as m' return forall (p : sinv (m'+n)), sinv n with
| O => fun p => p
| S m' => fun p => floor m' n (fst p)
end p.
Definition knot := { n:nat & F (sinv n) }.
Definition predicate := knot * other -> T.
Fixpoint stratify (n:nat) (Q:predicate) {struct n} : sinv n :=
match n as n' return sinv n' with
| O => tt
| S n' => ( stratify n' Q, fun v => Q (existT (F oo sinv) n' (fst v),snd v) )
end.
Definition unstratify (n:nat) (p:sinv n) : predicate := fun w =>
match w with (existT _ nw w',e) =>
match decompose_nat nw n with
| inleft (existT _ m Hm) => snd (floor m (S nw) (eq_rect  n _ p (m + S nw) Hm)) (w', e)
| inright H => T_bot
end
end.
Definition proof_irr_nat := eq_proofs_unicity dec_eq_nat.
Arguments proof_irr_nat [x] [y] _ _.

Lemma stratify_unstratify : forall n,
stratify n oo unstratify n = id (sinv n).