(*#* #stop file *)

Require pc3_props.
Require csub0_defs.   

   Section csub0_pc3. (*****************************************************)

      Theorem csub0_pr2: (g:?; c1:?; t1,t2:?) (pr2 c1 t1 t2) ->
                         (c2:?) (csub0 g c1 c2) -> (pr2 c2 t1 t2).
      Intros until 1; XElim H; Intros.
(* case 1: pr2_free *)
      XAuto.
(* case 2: pr2_delta *)
      CSub0Drop; XEAuto.
      Qed.

      Hints Resolve csub0_pr2.

      Opaque pc3.

      Theorem csub0_pc3: (g:?; c1:?; t1,t2:?) (pc3 c1 t1 t2) ->
                         (c2:?) (csub0 g c1 c2) -> (pc3 c2 t1 t2).
      Intros until 1; XElimUsing pc3_ind_left H; XEAuto.
      Qed.

   End csub0_pc3.

      Hints Resolve csub0_pc3 : ltlc.
   
   Section csub0_ty0. (*****************************************************)

      Theorem csub0_ty0: (g:?; c1:?; t1,t2:?) (ty0 g c1 t1 t2) ->
                         (c2:?) (wf0 g c2) -> (csub0 g c1 c2) ->
                         (ty0 g c2 t1 t2).
      Intros until 1; XElim H; Intros.
(* case 1: ty0_conv *)
      EApply ty0_conv; XEAuto.
(* case 2: ty0_sort *)
      XEAuto.
(* case 3: ty0_abbr *)
      CSub0Drop; EApply ty0_abbr; XEAuto.
(* case 4: ty0_abst *)
      CSub0Drop; [ EApply ty0_abst | EApply ty0_abbr ]; XEAuto.
(* case 5: ty0_bind *)
      EApply ty0_bind; XEAuto.
(* case 6: ty0_appl *)
      EApply ty0_appl; XEAuto.
(* case 7: ty0_cast *)
      EApply ty0_cast; XAuto.
      Qed.

      Theorem csub0_ty0_ld: (g:?; c:?; u,v:?) (ty0 g c u v) -> (t1,t2:?)
                            (ty0 g (CTail c (Bind Abst) v) t1 t2) ->
                            (ty0 g (CTail c (Bind Abbr) u) t1 t2).
      Intros; EApply csub0_ty0; XEAuto.
      Qed.

   End csub0_ty0.

      Hints Resolve csub0_ty0 csub0_ty0_ld : ltlc.

      Tactic Definition CSub0Ty0 :=
         Match Context With
            [ _: (ty0 ?1 ?2 ?4 ?); _: (ty0 ?1 ?2 ?3 ?7); _: (pc3 ?2 ?4 ?7);
              H: (ty0 ?1 (CTail ?2 (Bind Abst) ?4) ?5 ?6) |- ? ] ->
               LApply (csub0_ty0_ld ?1 ?2 ?3 ?4); [ Intros H_x | EApply ty0_conv; XEAuto ];
               LApply (H_x ?5 ?6); [ Clear H_x H; Intros | XAuto ].