Definition Formula_depth_ind :
  forall P : Formula -> Prop,
  (forall a : Formula, (forall b : Formula, lt_depth b a -> P b) -> P a) ->
  forall a : Formula, P a.

Lemma Formula_depth_ind2 : forall P : Formula -> Prop, (forall t t0 : Term, P (equal t t0)) -> (forall (r : Relations L) (t : Terms (arity L (inl (Functions L) r))), P (atomic r t)) -> (forall f : Formula, P f -> forall f0 : Formula, P f0 -> P (impH f f0)) -> (forall f : Formula, P f -> P (notH f)) -> (forall (v : nat) (a : Formula), (forall b : Formula, lt_depth b (forallH v a) -> P b) -> P (forallH v a)) -> forall f4 : Formula, P f4.