(*#* #stop file *)

Require Export drop_defs.
Require Export pr0_defs.

      Inductive pr2 [c:C; t1,t2:T] : Prop :=
(* structural rules *)
         | pr2_pr0   : (pr0 t1 t2) -> (pr2 c t1 t2)
(* axiom rules *)
         | pr2_delta : (d:?; u:?; i:?)
                       (drop i (0) c (CTail d (Bind Abbr) u)) ->
                       (subst0 i u t1 t2) -> (pr2 c t1 t2).

      Hint pr2 : ltlc := Constructors pr2.

   Section pr2_gen_base. (***************************************************)

      Theorem pr2_gen_sort : (c:?; x:?; n:?) (pr2 c (TSort n) x) ->
                             x = (TSort n).
      Intros; Inversion H;
      Try Subst0GenBase; XEAuto.
      Qed.

      Theorem pr2_gen_bref : (c:?; x:?; n:?) (pr2 c (TBRef n) x) ->
                             x = (TBRef n) \/
                             (EX d u | (drop n (0) c (CTail d (Bind Abbr) u)) &
                                        x = (lift (S n) (0) u)
                             ).
      Intros; Inversion H;
      Try Subst0GenBase; Try Rewrite <- H1 in H0; XEAuto.
      Qed.

      Theorem pr2_gen_abst : (c:?; u1,t1,x:?)
                             (pr2 c (TTail (Bind Abst) u1 t1) x) ->
                             (EX u2 t2 | x = (TTail (Bind Abst) u2 t2) &
                                         (pr2 c u1 u2) & (b:?; u:?)
                                         (pr2 (CTail c (Bind b) u) t1 t2)
                             ).
      Intros; Inversion H;
      Try Pr0GenBase; Try Subst0GenBase; XDEAuto 6.
      Qed.

      Theorem pr2_gen_appl : (c:?; u1,t1,x:?)
                             (pr2 c (TTail (Flat Appl) u1 t1) x) -> (OR
                             (EX u2 t2 | x = (TTail (Flat Appl) u2 t2) &
                                         (pr2 c u1 u2) & (pr2 c t1 t2)
                             ) |
                             (EX y1 z1 u2 t2 | t1 = (TTail (Bind Abst) y1 z1) &
                                               x = (TTail (Bind Abbr) u2 t2) &
                                               (pr0 u1 u2) & (pr0 z1 t2)
                             ) |
                             (EX b y1 z1 u2 v2 t2 |
                                ~b=Abst &
                                t1 = (TTail (Bind b) y1 z1) &
                                x = (TTail (Bind b) v2 (TTail (Flat Appl) (lift (1) (0) u2) t2)) &
                                (pr0 u1 u2) & (pr0 y1 v2) & (pr0 z1 t2))
                             ).
      Intros; Inversion H;
      Try Pr0GenBase; Try Subst0GenBase; XEAuto.
      Qed.

      Theorem pr2_gen_cast : (c:?; u1,t1,x:?)
                             (pr2 c (TTail (Flat Cast) u1 t1) x) ->
                             (EX u2 t2 | x = (TTail (Flat Cast) u2 t2) &
                                         (pr2 c u1 u2) & (pr2 c t1 t2)
                             ) \/
                            (pr0 t1 x).
      Intros; Inversion H;
      Try Pr0GenBase; Try Subst0GenBase; XEAuto.
      Qed.

   End pr2_gen_base.

      Tactic Definition Pr2GenBase :=
         Match Context With
            | [ H: (pr2 ?1 (TTail (Bind Abst) ?2 ?3) ?4) |- ? ] ->
               LApply (pr2_gen_abst ?1 ?2 ?3 ?4); [ Clear H; Intros H | XAuto ];
               XElim H; Intros.

   Section pr2_props. (******************************************************)

      Theorem pr2_thin_dx : (c:?; t1,t2:?) (pr2 c t1 t2) ->
                            (u:?; f:?) (pr2 c (TTail (Flat f) u t1)
                                              (TTail (Flat f) u t2)).
      Intros; Inversion H; XEAuto.
      Qed.

      Theorem pr2_tail_1 : (c:?; u1,u2:?) (pr2 c u1 u2) ->
                           (k:?; t:?) (pr2 c (TTail k u1 t) (TTail k u2 t)).
      Intros; Inversion H; XEAuto.
      Qed.

      Theorem pr2_tail_2 : (c:?; u,t1,t2:?; k:?) (pr2 (CTail c k u) t1 t2) ->
                           (pr2 c (TTail k u t1) (TTail k u t2)).
      XElim k; Intros;
      (Inversion H; [ XAuto | Clear H ];
      (NewInduction i; DropGenBase; [ Inversion H; XEAuto | XEAuto ])).
      Qed.

      Hints Resolve pr2_tail_2 : ltlc.

      Theorem pr2_shift : (i:?; c,e:?) (drop i (0) c e) ->
                          (t1,t2:?) (pr2 c t1 t2) ->
                          (pr2 e (app c i t1) (app c i t2)).
      XElim i.
(* case 1 : i = 0 *)
      Intros.
      DropGenBase; Rewrite H in H0.
      Repeat Rewrite app_O; XAuto.
(* case 2 : i > 0 *)
      XElim c.
(* case 2.1 : CSort *)
      Intros; DropGenBase; Rewrite H0; XAuto.
(* case 2.2 : CTail *)
      XElim k; Intros; Simpl; DropGenBase; XAuto.
      Qed.

   End pr2_props.

      Hints Resolve pr2_thin_dx pr2_tail_1 pr2_tail_2 pr2_shift : ltlc.