Definition FinContainer : Traversal.
exists finContainer finContainer_traverse.
apply finContainer_free_applicative.
apply finContainer_traverse_identity.
apply finContainer_traverse_compose.
Defined.
End FinContainer.
(*============================================================================*)
(*The main result                                                             *)
(*============================================================================*)

(* Specifications: *)
Theorem Traversal_is_finite_container :
forall (T:Traversal), exists (spec : FinContainerSpec)
(f : forall a, T a -> FinContainer spec a)
(g : forall a, FinContainer spec a -> T a),
(forall a, (forall x, g a (f a x) = x) /\ forall y, f a (g a y) = y) /\
(forall a b (F : Applicative) (h: a -> F b) (x : T a),
map (f b) (traverse h x) = traverse h (f a x)).