Definition Formula_depth_rec_rec :
  forall P : Formula -> Set,
  (forall a : Formula, (forall b : Formula, lt_depth b a -> P b) -> P a) ->
  forall (n : nat) (b : Formula), depth b <= n -> P b.Definition Formula_depth_rec (P : Formula -> Set)
  (rec : forall a : Formula, (forall b : Formula, lt_depth b a -> P b) -> P a)
  (a : Formula) : P a :=
  Formula_depth_rec_rec P rec (depth a) a (le_n (depth a)).

Lemma Formula_depth_rec_indep : forall (Q P : Formula -> Set) (rec : forall a : Formula, (forall b : Formula, lt_depth b a -> Q b -> P b) -> Q a -> P a), (forall (a : Formula) (z1 z2 : forall b : Formula, lt_depth b a -> Q b -> P b), (forall (b : Formula) (p : lt_depth b a) (q : Q b), z1 b p q = z2 b p q) -> forall q : Q a, rec a z1 q = rec a z2 q) -> forall (a : Formula) (q : Q a), Formula_depth_rec (fun x : Formula => Q x -> P x) rec a q = rec a (fun (b : Formula) _ => Formula_depth_rec (fun x : Formula => Q x -> P x) rec b) q.