Definition lift_chain := Chain lift_chainP.Definition image_chain := Chain image_chainP.Notation "[ f '^^' s 'by' M ]" := (@image_chain _ _ s f M)
  (at level 0, format "[ f  '^^'  s  'by'  M ]") : form_scope.

(* Specifications: *)
Lemma id_mono : monotone (@id T).
Lemma const_mono : monotone (fun x : T1 => y).
Lemma const_chainP : chain_axiom (Pred1 y).