Section Formula_Depth_Induction.
Fixpoint depth (A : Formula) : nat :=
match A with
| equal _ _ => 0
| atomic _ _ => 0
| impH A B => S (max (depth A) (depth B))
| notH A => S (depth A)
| forallH _ A => S (depth A)
end.
Definition lt_depth (A B : Formula) : Prop := depth A < depth B.

Lemma depthImp1 : forall A B : Formula, lt_depth A (impH A B).