intros; change (Pterms P wt a) in |- *; elim a; simpl in |- *; auto.
Qed.
(* Principe d induction pour les sub_explicits *)
Goal (sub_explicits -> Prop) -> forall b : wsort, TS b -> Prop.
intros P b; elim b.
exact P.
exact (fun x : TS wt => True).
Defined Psubst.

(* Specifications: *)
Lemma sub_explicits_ind :
forall P : sub_explicits -> Prop,
P id ->
P shift ->
(forall s : sub_explicits, P s -> forall a : terms, P (cons a s)) ->
(forall s t : sub_explicits, P s -> P t -> P (comp s t)) ->
(forall s : sub_explicits, P s -> P (lift s)) ->
(forall n : nat, P (meta_x n)) -> forall s : sub_explicits, P s.