Section ContainsAutomation.
End ContainsAutomation.
End FOL.
Definition tmap {S1 S2 : Signature} (f : @form S1 -> @form S2) (T : @theory S1) : @theory S2 :=
fun phi => exists psi, T psi /\ f psi = phi.
Hint Constructors vec_in : core.
Infix "⊏" := contains_L (at level 20).
Infix "⊑" := subset_T (at level 20).
Infix "∈" := contains (at level 70).
Infix "⋄" := extend (at level 20).
Hint Resolve contains_nil contains_cons contains_cons2 contains_app : contains_theory.
Hint Resolve contains_extend1 contains_extend2 contains_extend3 : contains_theory.
Ltac use_theory A := exists A; split; [eauto 15 with contains_theory|].
(* **** Enumerability *)
Fixpoint vecs_from X (A : list X) (n : nat) : list (vector X n) :=
match n with
| 0 => [Vector.nil]
| S n => [ cons x v | (x,  v) ∈ (A × vecs_from A n) ]
end.

Lemma vecs_from_correct X (A : list X) (n : nat) (v : vector X n) :
(forall x, vec_in x v -> x el A) <-> v el vecs_from A n.