Require drop_props.
Require pc3_props.
Require ty0_defs.
Require ty0_gen.
Require ty0_lift.

   Section ty0_correct. (****************************************************)

(*#* #caption "correctness of types" *)
(*#* #cap #cap c, t1, t2  #alpha t in T3 *)

      Theorem ty0_correct : (g:?; c:?; t1,t2:?)
                            (ty0 g c t1 t2) -> (EX t | (ty0 g c t2 t)).

(*#* #stop file *)

      Intros; XElim H; Intros.
(* case 1 : ty0_conv *)
      XEAuto.
(* case 2 : ty0_sort *)
      XEAuto.
(* case 3 : ty0_abbr *)
      XElim H2; XEAuto.
(* case 4 : ty0_abst *)
      XEAuto.
(* case 5 : ty0_bind *)
      XElim H4; XEAuto.
(* case 6 : ty0_appl *)
      XElim H0; XElim H2; Intros.
      Ty0GenBase; XEAuto.
(* case 7 : ty0_cast *)
      XAuto.
      Qed.

   End ty0_correct.

      Tactic Definition Ty0Correct :=
         Match Context With
            [ _: (ty0 ?1 ?2 ?3 ?4) |- ? ] ->
               LApply (ty0_correct ?1 ?2 ?3 ?4); [ Intros H_x | XAuto ];
               XElim H_x; Intros.

(*#* #start file *)

   Section ty0_shift. (******************************************************)

(*#* #caption "shift lemma for types" *)
(*#* #cap #cap t1, t2 #alpha c in C1, e in C2 *)

      Theorem ty0_shift : (h:?; c,e:?) (drop h (0) c e) ->
                          (g:?; t1,t2:?) (ty0 g c t1 t2) -> (wf0 g e) ->
                          (ty0 g e (app c h t1) (app c h t2)).

(*#* #stop file *)

      XElim h.
(* case 1 : h = 0 *)
      Intros; DropGenBase; Rewrite <- H.
      Repeat Rewrite app_O; XAuto.
(* case 2 : h > 0 *)
      Intros h IHh; XElim c.
(* case 2.1 : CSort *)
      Intros; DropGenBase; Rewrite H.
      Simpl; XAuto.
(* case 2.2 : CTail k *)
      Intros c IHc; Clear IHc; XElim k; Intros; Wf0Ty0.
      DropGenBase; Move H0 after H2; Ty0Correct.
      Simpl; Apply IHh; [ Idtac | EApply ty0_bind | Idtac ]; XEAuto.
      Qed.

   End ty0_shift.

      Hints Resolve ty0_shift : ltlc.

   Section ty0_unique. (*****************************************************)

      Opaque pc3.

(*#* #start file *)

(*#* #caption "uniqueness of types" *)
(*#* #cap #cap c, t1, t2 #alpha u in T *)

      Theorem ty0_unique : (g:?; c:?; u,t1:?) (ty0 g c u t1) ->
                           (t2:?) (ty0 g c u t2) -> (pc3 c t1 t2).

(*#* #stop file *)

      Intros until 1; XElim H; Intros.
(* case 1 : ty0_conv *)
      XEAuto.
(* case 2 : ty0_sort *)
      Ty0GenBase; XAuto.
(* case 3 : ty0_abbr *)
      Ty0GenBase; DropDis; Inversion H4.
      Rewrite H7 in H2; Rewrite H8 in H2; XEAuto.
(* case 4 : ty0_abst *)
      Ty0GenBase; DropDis; Inversion H4.
      Rewrite H7 in H2; Rewrite H8 in H2; XEAuto.
(* case 5 : ty0_bind *)
      Ty0GenBase; XEAuto.
(* case 6 : ty0_appl *)
      Ty0GenBase; EApply pc3_t; [ Idtac | EApply H3 ]; XEAuto.
(* case 7 : ty0_cast *)
      Ty0GenBase; XEAuto.
      Qed.

   End ty0_unique.

      Hints Resolve ty0_unique : ltlc.