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(** * Derivability of choice for decidable relations with well-ordered codomain *)
(** Countable codomains, such as [nat], can be equipped with a well-order, which implies the existence of a least element on inhabited decidable subsets. As a consequence, the relational form of the axiom of choice is derivable on [nat] for decidable relations.  We show instead that functional relation reification and the functional form of the axiom of choice are equivalent on decidable relations with [nat] as codomain. *)

Lemma classical_denumerable_description_imp_fun_choice :
forall A:Type,
FunctionalRelReification_on A nat ->
forall R:A->nat->Prop,
(forall x y, decidable (R x y)) -> FunctionalChoice_on_rel R.