Section FOL.
Context {Sigma : Signature}.
(* **** Subformula *)
Inductive sf : form -> form -> Prop :=
| SImplL phi psi : sf phi (phi --> psi)
| SImplR phi psi : sf psi (phi --> psi)
| SAll phi t : sf (phi [t .: ids]) (∀ phi).
Hint Constructors sf : core.
(* **** Formula size *)
Fixpoint size (phi : form ) :=
match phi with
| Pred _ _ => 0
| Fal => 0
| Impl phi psi => S (size phi + size psi)
| All phi => S (size phi)
end.

Lemma subst_size rho phi :
size (subst_form rho phi) = size phi.