Inductive step : term -> term -> Prop :=
| stepApp  s t     : app (lambda s) (lambda t) ≻ subst s 0 (lambda t)
| stepAppR s t  t' : t ≻ t' -> app s t ≻ app s t'
| stepAppL s s' t  : s ≻ s' -> app s t ≻ app s' t
where "s '≻' t" := (step s t).Inductive step : term -> term -> Prop :=
| stepApp  s t     : app (lambda s) (lambda t) ≻ subst s 0 (lambda t)
| stepAppR s t  t' : t ≻ t' -> app s t ≻ app s t'
| stepAppL s s' t  : s ≻ s' -> app s t ≻ app s' t
where "s '≻' t" := (step s t).Inductive step : term -> term -> Prop :=
| stepApp  s t     : app (lambda s) (lambda t) ≻ subst s 0 (lambda t)
| stepAppR s t  t' : t ≻ t' -> app s t ≻ app s t'
| stepAppL s s' t  : s ≻ s' -> app s t ≻ app s' t
where "s '≻' t" := (step s t).Inductive step : term -> term -> Prop :=
| stepApp  s t     : app (lambda s) (lambda t) ≻ subst s 0 (lambda t)
| stepAppR s t  t' : t ≻ t' -> app s t ≻ app s t'
| stepAppL s s' t  : s ≻ s' -> app s t ≻ app s' t
where "s '≻' t" := (step s t).Inductive step : term -> term -> Prop :=
| stepApp  s t     : app (lambda s) (lambda t) ≻ subst s 0 (lambda t)
| stepAppR s t  t' : t ≻ t' -> app s t ≻ app s t'
| stepAppL s s' t  : s ≻ s' -> app s t ≻ app s' t
where "s '≻' t" := (step s t).Ltac inv_step :=
  match goal with
  | H : step (app _ _) _ |- _ => inv H
  | H : step (lambda _) _ |- _ => inv H
  | H : step (var _) _ |- _ => inv H
  end.(** ** Reflexive-transitive closure and step-indexed closure *)

Inductive star : term -> term -> Prop :=
  star_refl s : star s s
| star_step s u t : step s u -> star u t -> star s t.(** ** Reflexive-transitive closure and step-indexed closure *)

Inductive star : term -> term -> Prop :=
  star_refl s : star s s
| star_step s u t : step s u -> star u t -> star s t.(** ** Reflexive-transitive closure and step-indexed closure *)

Inductive star : term -> term -> Prop :=
  star_refl s : star s s
| star_step s u t : step s u -> star u t -> star s t.Inductive stepn : nat -> term -> term -> Prop :=
  stepn_refl s : stepn 0 s s
| stepn_step s u t n : step s u -> stepn n u t -> stepn (S n) s t.Inductive stepn : nat -> term -> term -> Prop :=
  stepn_refl s : stepn 0 s s
| stepn_step s u t n : step s u -> stepn n u t -> stepn (S n) s t.Inductive stepn : nat -> term -> term -> Prop :=
  stepn_refl s : stepn 0 s s
| stepn_step s u t n : step s u -> stepn n u t -> stepn (S n) s t.Instance star_trans : Transitive star.Instance star_PreOrder : PreOrder star.

Lemma rho_closed s : closed s -> closed (rho s).