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.