(** A constructive proof that any non empty decidable subset of natural numbers has a least element *)
Set Implicit Arguments.
Definition has_unique_least_element (A:Type) (R:A->A->Prop) (P:A->Prop) :=
exists! x, P x /\ forall x', P x' -> R x x'.

(* Specifications: *)
Lemma dec_inh_nat_subset_has_unique_least_element :
forall P:nat->Prop, (forall n, P n \/ ~ P n) ->
(exists n, P n) -> has_unique_least_element le P.