(*#* #stop file *)

Require lift_gen.
Require lift_props.
Require subst0_gen.
Require pr0_defs.
Require pr0_lift.

   Section pr0_gen_abbr. (***************************************************)

      Theorem pr0_gen_abbr : (u1,t1,x:?) (pr0 (TTail (Bind Abbr) u1 t1) x) ->
                             (EX u2 t2 | x = (TTail (Bind Abbr) u2 t2) &
                                         (pr0 u1 u2) &
                                         (pr0 t1 t2) \/
                                         (EX y | (pr0 t1 y) & (subst0 (0) u2 y t2))
                             ) \/
                             (pr0 t1 (lift (1) (0) x)).
      Intros.
      Inversion H; Clear H; XDEAuto 6.
      Qed.

   End pr0_gen_abbr.

   Section pr0_gen_void. (***************************************************)

      Theorem pr0_gen_void : (u1,t1,x:?) (pr0 (TTail (Bind Void) u1 t1) x) ->
                             (EX u2 t2 | x = (TTail (Bind Void) u2 t2) &
                                         (pr0 u1 u2) & (pr0 t1 t2)
                             ) \/
                             (pr0 t1 (lift (1) (0) x)).
      Intros.
      Inversion H; Clear H; XEAuto.
      Qed.

   End pr0_gen_void.

   Section pr0_gen_lift. (***************************************************)

      Tactic Definition IH :=
         Match Context With
            | [ H: (_:?; _:?) ?0 = (lift ? ? ?) -> ?;
                H0: ?0 = (lift ? ?2 ?3) |- ? ] ->
               LApply (H ?3 ?2); [ Clear H H0; Intros H_x | XAuto ];
               XElim H_x; Intro; Intros H_x; Intro;
               Try Rewrite H_x; Try Rewrite H_x in H3; Clear H_x.

(*#* #start file *)

(*#* #caption "generation lemma for lift" *)
(*#* #cap #alpha t1 in U1, t2 in U2, x in T, d in i *)

      Theorem pr0_gen_lift : (t1,x:?; h,d:?) (pr0 (lift h d t1) x) ->
                             (EX t2 | x = (lift h d t2) & (pr0 t1 t2)).

(*#* #stop file *)

      Intros until 1; InsertEq H '(lift h d t1);
      UnIntro H d; UnIntro H t1; XElim H; Clear y x; Intros;
      Rename x into t3; Rename x0 into d.
(* case 1 : pr0_r *)
      XEAuto.
(* case 2 : pr0_c *)
      NewInduction k; LiftGen; Rewrite H3; Clear H3 t0;
      IH; IH; XEAuto.
(* case 3 : pr0_beta *)
      LiftGen; Rewrite H3; Clear H3 t0.
      LiftGen; Rewrite H3; Clear H3 H5 x1 k.
      IH; IH; XEAuto.
(* case 4 : pr0_gamma *)
      LiftGen; Rewrite H6; Clear H6 t0.
      LiftGen; Rewrite H6; Clear H6 x1.
      IH; IH; IH.
      Rewrite <- lift_d; [ Simpl | XAuto ].
      Rewrite <- lift_flat; XEAuto.
(* case 5 : pr0_delta *)
      LiftGen; Rewrite H4; Clear H4 t0.
      IH; IH; Arith3In H3 d; Subst0Gen.
      Rewrite H3; XEAuto.
(* case 6 : pr0_zeta *)
      LiftGen; Rewrite H2; Clear H2 t0.
      Arith7In H4 d; LiftGen; Rewrite H2; Clear H2 x1.
      IH; XEAuto.
(* case 7 : pr0_zeta *)
      LiftGen; Rewrite H1; Clear H1 t0.
      IH; XEAuto.
      Qed.

   End pr0_gen_lift.

      Tactic Definition Pr0Gen :=
         Match Context With
            | [ H: (pr0 (TTail (Bind Abbr) ?1 ?2) ?3) |- ? ] ->
               LApply (pr0_gen_abbr ?1 ?2 ?3); [ Clear H; Intros H | XAuto ];
               XElim H;
               [ Intros H; XElim H; Do 4 Intro; Intros H_x;
                 XElim H_x; [ Intros | Intros H_x; XElim H_x; Intros ]
               | Intros ]
            | [ H: (pr0 (TTail (Bind Void) ?1 ?2) ?3) |- ? ] ->
               LApply (pr0_gen_void ?1 ?2 ?3); [ Clear H; Intros H | XAuto ];
               XElim H; [ Intros H; XElim H; Intros | Intros ]
            | [ H: (pr0 (lift ?0 ?1 ?2) ?3) |- ? ] ->
               LApply (pr0_gen_lift ?2 ?3 ?0 ?1); [ Clear H; Intros H | XAuto ];
               XElim H; Intros
            | _ -> Pr0GenBase.