unfold explicit_noetherian in |- *; unfold universal in |- *;
unfold hereditary in |- *; unfold adjoint in |- *;
unfold sub in |- *; unfold a_set in |- *; intros.
pattern x in |- *; apply H; exact H0.
Qed.

Lemma noetherian_induction :
forall (A : Set) (R : A -> A -> Prop),
explicit_noetherian _ R ->
forall (x : A) (P : A -> Prop),
(forall y : A, (forall z : A, explicit_rel_plus _ R y z -> P z) -> P y) ->
P x.