Definition chain (S: Ensemble T) : Prop :=
  forall x y:T, In S x -> In S y -> (R x y \/ R y x).Definition maximal (x:T) : Prop :=
  forall y:T, R x y -> x = y.Inductive tower : Ensemble T :=
  | sup_intro: forall (S: Ensemble T), Included S tower ->
      forall c:chain S,
      In tower (proj1_sig (chain_sup S c))
  | inflation_intro: forall x:T, In tower x ->
      In tower (proj1_sig (inflation x)).Inductive tower : Ensemble T :=
  | sup_intro: forall (S: Ensemble T), Included S tower ->
      forall c:chain S,
      In tower (proj1_sig (chain_sup S c))
  | inflation_intro: forall x:T, In tower x ->
      In tower (proj1_sig (inflation x)).Inductive tower : Ensemble T :=
  | sup_intro: forall (S: Ensemble T), Included S tower ->
      forall c:chain S,
      In tower (proj1_sig (chain_sup S c))
  | inflation_intro: forall x:T, In tower x ->
      In tower (proj1_sig (inflation x)).

(* Specifications: *)
Lemma tower_is_chain: chain tower.
Lemma ZL': False.