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(* coqc version 8.8.0 (May 2018) compiled on May 2 2018 16:49:46 with OCaml 4.02.3
   coqtop version 8.8.0 (May 2018) *)

(* This was triggering a "Not_found" at the time of printing/showing the goal *)

Notation "t $ r" := (t r)
  (at level 65, right associativity, only parsing).

Inductive tele : Type :=
  | TeleO : tele
  | TeleS {X} (binder : X -> tele) : tele.

Fixpoint tele_fun (TT : tele) (T : Type) : Type :=
  match TT with
  | TeleO => T
  | TeleS b => forall x, tele_fun (b x) T
  end.

Inductive tele_arg : tele -> Type :=
| TargO : tele_arg TeleO
| TargS {X} {binder} (x : X) : tele_arg (binder x) -> tele_arg (TeleS binder).

Axiom tele_app : forall {TT : tele} {T} (f : tele_fun TT T), tele_arg TT -> T.

Coercion tele_arg : tele >-> Sortclass.

Inductive val :=
  | LitV
  | PairV (v1 v2 : val)
  | InjLV (v : val)
  | InjRV (v : val).
Axiom coPset : Set.
Axiom atomic_update : forall {PROP : Type} {TA TB : tele}, coPset -> coPset -> (TA -> PROP) -> (TA -> TB -> PROP) -> (TA -> TB -> PROP) -> PROP.
Notation "'AU' '<<' ∀ x1 .. xn , α '>>' @ Eo , Ei '<<' β , 'COMM' Φ '>>'" :=
  (atomic_update (TA:=TeleS (fun x1 => .. (TeleS (fun xn => TeleO)) .. ))
                 (TB:=TeleO)
                 Eo Ei
                 (tele_app (TT:=TeleS (fun x1 => .. (TeleS (fun xn => TeleO)) .. )) $
                       fun x1 => .. (fun xn => α) ..)
                 (tele_app (TT:=TeleS (fun x1 => .. (TeleS (fun xn => TeleO)) .. )) $
                       fun x1 => .. (fun xn => tele_app (TT:=TeleO) β) .. )
                 (tele_app (TT:=TeleS (fun x1 => .. (TeleS (fun xn => TeleO)) .. )) $
                       fun x1 => .. (fun xn => tele_app (TT:=TeleO) Φ) .. )
  )
  (at level 20, Eo, Ei, α, β, Φ at level 200, x1 binder, xn binder,
   format "'[   ' 'AU'  '<<'  ∀  x1  ..  xn ,  α  '>>'  '/' @  Eo ,  Ei  '/' '[   ' '<<'  β ,  '/' COMM  Φ  '>>' ']' ']'") : bi_scope.

Axiom ident : Set.
Inductive env (A : Type) : Type :=  Enil : env A | Esnoc : env A -> ident -> A -> env A.
Record envs (PROP : Type) : Type
  := Envs { env_spatial : env PROP }.
Axiom positive : Set.
Axiom Qp : Set.
Axiom one : positive.
Goal   forall (T : Type) (T0 : forall _ : T, Type) (P : Set)
    (u : T) (γs : P) (Q : T0 u) (Φ o : forall _ : val, T0 u)
    (stack_content0 : forall (_ : P) (_ : list val), T0 u)
    (c c0 : coPset) (l : forall (A : Type) (_ : list A), list A)
    (e0 : forall (_ : env (T0 u)) (_ : positive), envs (T0 u))
    (i0 : ident) (o1 : forall (_ : Qp) (_ : val), T0 u)
    (b0 : forall _ : env (T0 u), T0 u) (P0 : forall _ : T0 u, Type)
    (u0 : forall (_ : T0 u) (_ : T0 u), T0 u),
  P0
    (@atomic_update (T0 u)
       (@TeleS val (fun _ : val => @TeleS Qp (fun _ : Qp => TeleO))) TeleO c c0
       (@tele_app (@TeleS val (fun _ : val => @TeleS Qp (fun _ : Qp => TeleO)))
          (T0 u) (fun (v : val) (q : Qp) => o1 q v))
       (@tele_app (@TeleS val (fun _ : val => @TeleS Qp (fun _ : Qp => TeleO)))
          (forall _ : tele_arg TeleO, T0 u)
          (fun (v : val) (q : Qp) => @tele_app TeleO (T0 u) (o1 q v)))
       (@tele_app (@TeleS val (fun _ : val => @TeleS Qp (fun _ : Qp => TeleO)))
          (forall _ : tele_arg TeleO, T0 u)
          (fun (x : val) (_ : Qp) =>
           @tele_app TeleO (T0 u)
             (u0
                (b0
                   match
                     e0
                       (@Esnoc (T0 u) (@Enil (T0 u)) i0
                          (@atomic_update (T0 u)
                             (@TeleS (list val) (fun _ : list val => TeleO)) TeleO
                             c c0
                             (@tele_app
                                (@TeleS (list val) (fun _ : list val => TeleO))
                                (T0 u) (fun l0 : list val => stack_content0 γs l0))
                             (@tele_app
                                (@TeleS (list val) (fun _ : list val => TeleO))
                                (forall _ : tele_arg TeleO, T0 u)
                                (fun l0 : list val =>
                                 @tele_app TeleO (T0 u)
                                   (stack_content0 γs (l val l0))))
                             (@tele_app
                                (@TeleS (list val) (fun _ : list val => TeleO))
                                (forall _ : tele_arg TeleO, T0 u)
                                (fun x1 : list val =>
                                 @tele_app TeleO (T0 u)
                                   (u0 Q
                                      (Φ
                                         match x1 return val with
                                         | nil => InjLV LitV
                                         | cons v _ => InjRV v
                                         end)))))) one
                     return (env (T0 u))
                   with
                   | Envs _ env_spatial0 => env_spatial0
                   end) (o x)))))
.
  Show.
Abort.
