Section subAllCloseFrom.
Fixpoint closeFrom (a n : nat) (f : fol.Formula L) {struct n} :
fol.Formula L :=
match n with
| O => f
| S m => fol.forallH L (a + m) (closeFrom a m f)
end.
Opaque le_lt_dec.

Lemma liftCloseFrom :
forall (n : nat) (f : fol.Formula L) (T : fol.System L) (m : nat),
(forall v : nat, In v (freeVarFormula L f) -> v < m) ->
n <= m ->
folProof.SysPrf L T (closeFrom 0 n f) ->
folProof.SysPrf L T
(closeFrom m n
(subAllFormula f
(fun x : nat =>
match le_lt_dec n x with
| left _ => fol.var L x
| right _ => fol.var L (m + x)
end))).