Definition dec_type := sigT (fun P: Z -> Prop => forall z: Z, {P z} + {~ P z}).Definition dec_False: dec_type :=
  existT (fun P: Z -> Prop => forall z: Z, {P z} + {~ P z}) (fun z => False) (fun z => right (fun H => H)).Definition dec_True: dec_type :=
  existT (fun P: Z -> Prop => forall z: Z, {P z} + {~ P z}) (fun z => True) (fun z => left I).Fixpoint cons_in_list {A} (a: A) (al' al: list A) (H: forall x, In x al' -> In x al) (bl: list {x:A| In x al'}) : list {x: A | In x al} :=
  match bl with
  | nil => nil
  | exist x i :: bl0 =>exist _ x (H x i)  :: cons_in_list a al' al H bl0
  end.Fixpoint make_in_list {A} (al: list A) : list {x: A | In x al} := 
  match al as ax return (al = ax -> list {x : A | In x ax}) with
           | nil => fun _ => nil
           | a::al' => fun H: al = a::al' =>
                      exist _ a (or_introl eq_refl) ::
                         eq_rect al (fun l : list A => list {x : A | In x l}) 
                          (cons_in_list a al' al (fun (x : A) (H0 : In x al') =>
                                 eq_ind_r (fun al0 : list A => In x al0) (in_cons _ _ _ H0) H)
                               (make_in_list al'))
                        (a :: al') H
           end (eq_refl _).

Lemma in_make_in_list: forall {A} (a: A) (al: list A) H, In (exist (fun x => In x al) a H) (make_in_list al).