Notation "'[tele' x .. z ]" :=
  (TeleS (fun x => .. (TeleS (fun z => TeleO)) ..))
  (x binder, z binder, format "[tele  '[hv' x  ..  z ']' ]").
Notation "'[tele' ]" := (TeleO)
  (format "[tele ]").

Definition test {TT : tele} (t : TT → Prop) : Prop :=
  ∀.. x, t x ∧ t x.
Notation "'[TEST'  x .. z ,  P ']'" :=
  (test (TT:=[tele x .. z])
        (tele_app (λ x, .. (λ z, P) ..)))
  (x binder, z binder).
(* Error: x should only be used in the recursive part of a pattern. *)

Notation "'[TEST'  x .. z ,  P ']'" :=
  (test (TT:=(TeleS (fun x => .. (TeleS (fun z => TeleO)) ..)))
        (tele_app (λ x, .. (λ z, P) ..)))
  (x binder, z binder).

From Coq Require Import Utf8.
Set Default Proof Using "Type".

Local Set Universe Polymorphism.

(** Telescopes *)
Inductive tele : Type :=
  | TeleO : tele
  | TeleS {X} (binder : X → tele) : tele.

Arguments TeleS {_} _.

(** The telescope version of Coq's function type *)
Fixpoint tele_fun (TT : tele) (T : Type) : Type :=
  match TT with
  | TeleO => T
  | TeleS b => ∀ x, tele_fun (b x) T
  end.

Notation "TT -t> A" :=
  (tele_fun TT A) (at level 99, A at level 200, right associativity).

(** A sigma-like type for an "element" of a telescope, i.e. the data it
  takes to get a [T] from a [TT -t> T]. *)
Inductive tele_arg : tele → Type :=
| TargO : tele_arg TeleO
(* the [x] is the only relevant data here *)
| TargS {X} {binder} (x : X) : tele_arg (binder x) → tele_arg (TeleS binder).

Definition tele_app {TT : tele} {T} (f : TT -t> T) : tele_arg TT → T :=
  λ a, (fix rec {TT} (a : tele_arg TT) : (TT -t> T) → T :=
     match a in tele_arg TT return (TT -t> T) → T with
     | TargO => λ t : T, t
     | TargS x a => λ f, rec a (f x)
     end) TT a f.
Arguments tele_app {!_ _} _ !_ /.

Coercion tele_arg : tele >-> Sortclass.
Coercion tele_app : tele_fun >-> Funclass.

(** The actual testcase *)
Notation "'[tele' x .. z ]" :=
  (TeleS (fun x => .. (TeleS (fun z => TeleO)) ..))
  (x binder, z binder, format "[tele  '[hv' x  ..  z ']' ]").
Notation "'[tele' ]" := (TeleO)
  (format "[tele ]").

Definition test {TT : tele} (t : TT → Prop) : Prop.
Admitted.
Notation "'[TEST'  x .. z ,  P ']'" :=
  (test (TT:=[tele x .. z])
        (tele_app (λ x, .. (λ z, P) ..)))
  (x binder, z binder).