(*#* #stop file *)

Require lift_gen.
Require lift_props.
Require subst1_defs.
Require subst1_lift.
Require subst1_confluence.
Require drop_props.
Require csubst1_defs.
Require pc3_gen.
Require pc3_gen_context.
Require ty0_defs.
Require ty0_lift.

(* NOTE: these break the recursion between ty0_sred_cpr0_pr0 and ty0_gen_lift *)

   Section ty0_gen_cabbr. (**************************************************)

      Tactic Definition IH d a0 a :=
         Match Context With
            [ H: (e:?; u:?; d:?) ? -> (a0:?) ? -> (a:?) ? -> ? -> ? |- ? ] ->
               LApply (H e u0 d); [ Clear H; Intros H | XAuto ];
               LApply (H a0); [ Clear H; Intros H | XAuto ];
               LApply (H a); [ Clear H; Intros H | XEAuto ];
               LApply H; [ Clear H; Intros H | XAuto ];
               XElim H; Intros.

(* NOTE: This can be generalized removing the last three premises *)
      Theorem ty0_gen_cabbr: (g:?; c:?; t1,t2:?) (ty0 g c t1 t2) ->
                             (e:?; u:?; d:?) (drop d (0) c (CTail e (Bind Abbr) u)) ->
                             (a0:?) (csubst1 d u c a0) ->
                             (a:?) (wf0 g a) -> (drop (1) d a0 a) ->
                             (EX y1 y2 | (subst1 d u t1 (lift (1) d y1)) &
                                         (subst1 d u t2 (lift (1) d y2)) &
                                         (ty0 g a y1 y2)
                             ).
      Intros until 1; XElim H; Intros.
(* case 1 : ty0_conv *)
      Repeat IH d a0 a; EApply ex3_2_intro;
      [ XEAuto | XEAuto | EApply ty0_conv; Try EApply pc3_gen_cabbr; XEAuto ].
(* case 2 : ty0_sort *)
      EApply ex3_2_intro; Try Rewrite lift_sort; XAuto.
(* case 3 : ty0_abbr *)
      Apply (lt_eq_gt_e n d0); Intros; Clear c t1 t2.
(* case 3.1 : n < d0 *)
      Clear H1; DropDis; Rewrite minus_x_Sy in H1; [ DropGenBase | XAuto ].
      CSubst1Drop; Rewrite minus_x_Sy in H0; [ Idtac | XAuto ].
      CSubst1GenBase; Rewrite H0 in H8; Clear H0 x; Simpl in H9.
      Rewrite (lt_plus_minus n d0) in H6; [ Idtac | XAuto ].
      DropDis; Rewrite H0 in H9; Clear H0 x0.
      IH '(minus d0 (S n)) x1 x3.
      Subst1Confluence; Rewrite H0 in H11; Clear H0 x0.
      Pattern 3 d0; Rewrite (le_plus_minus_sym (S n) d0); [ Idtac | XAuto ].
      Pattern 4 d0; Rewrite (le_plus_minus (S n) d0); [ Idtac | XAuto ].
      EApply ex3_2_intro;
      [ Rewrite lift_lref_lt | Rewrite lift_d | EApply ty0_abbr ]; XEAuto.
(* case 3.2 : n = d0 *)
      Rewrite H7; Rewrite H7 in H0; Clear H2 H7 n.
      DropDis; Inversion H0; Rewrite H8 in H4; Clear H0 H7 H8 e u0.
      CSubst1Drop; DropDis.
      EApply ex3_2_intro;
      [ EApply subst1_single; Rewrite lift_free; Simpl; XEAuto
      | Rewrite lift_free; Simpl; XEAuto
      | XEAuto ].
(* case 3.3 : n > d0 *)
      Clear H2 H3 e; CSubst1Drop; DropDis.
      Pattern 1 n; Rewrite (lt_plus_minus (0) n); [ Idtac | XEAuto ].
      Arith4c '(0) '(minus n (1)).
      EApply ex3_2_intro;
      [ Rewrite lift_lref_ge
      | Rewrite lift_free; Simpl
      | Pattern 2 n; Rewrite (minus_x_SO n)
      ]; XEAuto.
(* case 4 : ty0_abst *)
      Apply (lt_eq_gt_e n d0); Intros; Clear c t1 t2.
(* case 4.1 : n < d0 *)
      Clear H1; DropDis; Rewrite minus_x_Sy in H1; [ DropGenBase | XAuto ].
      CSubst1Drop; Rewrite minus_x_Sy in H0; [ Idtac | XAuto ].
      CSubst1GenBase; Rewrite H0 in H8; Clear H0 x; Simpl in H9.
      Rewrite (lt_plus_minus n d0) in H6; [ Idtac | XAuto ].
      DropDis; Rewrite H0 in H9; Clear H0 x0.
      IH '(minus d0 (S n)) x1 x3.
      Subst1Confluence; Rewrite H0 in H11; Clear H0 x0.
      Pattern 3 d0; Rewrite (le_plus_minus_sym (S n) d0); [ Idtac | XAuto ].
      Pattern 4 d0; Rewrite (le_plus_minus (S n) d0); [ Idtac | XAuto ].
      EApply ex3_2_intro;
      [ Rewrite lift_lref_lt | Rewrite lift_d | EApply ty0_abst ]; XEAuto.
(* case 4.2 : n = d0 *)
      Rewrite H7; Rewrite H7 in H0; DropDis; Inversion H0.
(* case 4.3 : n > d0 *)
      Clear H2 H3 e; CSubst1Drop; DropDis.
      Pattern 1 n; Rewrite (lt_plus_minus (0) n); [ Idtac | XEAuto ].
      Arith4c '(0) '(minus n (1)).
      EApply ex3_2_intro;
      [ Rewrite lift_lref_ge
      | Rewrite lift_free; Simpl
      | Pattern 2 n; Rewrite (minus_x_SO n)
      ]; XEAuto.
(* case 5 : ty0_bind *)
      IH d a0 a; Clear H H1 H3 c t1 t2.
      IH '(S d) '(CTail a0 (Bind b) (lift (1) d x0)) '(CTail a (Bind b) x0).
      IH '(S d) '(CTail a0 (Bind b) (lift (1) d x0)) '(CTail a (Bind b) x0).
      Subst1Confluence; Rewrite H4 in H11; Clear H4 x5.
      EApply ex3_2_intro; Try Rewrite lift_bind; XEAuto.
(* case 6 : ty0_appl *)
      Repeat IH d a0 a; Clear H H1 c t1 t2.
      Subst1GenBase; SymEqual; LiftGenBase; Rewrite H in H8; Rewrite H11 in H1; Rewrite H12 in H7; Clear H H11 H12 x1 x4 x5.
      Subst1Confluence; Rewrite H in H8; Clear H x6.
      EApply ex3_2_intro; Try Rewrite lift_flat;
      [ Idtac | EApply subst1_tail; [ Idtac | Rewrite lift_bind ] | Idtac ]; XEAuto.
(* case 7 : ty0_cast *)
      Rename u into u0; Repeat IH d a0 a; Clear H H1 c t1 t2.
      Subst1Confluence; Rewrite H in H10; Clear H x3.
      EApply ex3_2_intro; [ Rewrite lift_flat | Idtac | Idtac ]; XEAuto.
      Qed.

   End ty0_gen_cabbr.

   Section ty0_gen_cvoid. (**************************************************)

      Tactic Definition IH d a :=
         Match Context With
            [ H: (e:?; u:?; d:?) ? -> (a:?) ? -> ? -> ? |- ? ] ->
               LApply (H e u0 d); [ Clear H; Intros H | XAuto ];
               LApply (H a); [ Clear H; Intros H | XEAuto ];
               LApply H; [ Clear H; Intros H | XAuto ];
               XElim H; Intros.

(* NOTE: This can be generalized removing the last two premises *)
      Theorem ty0_gen_cvoid: (g:?; c:?; t1,t2:?) (ty0 g c t1 t2) ->
                             (e:?; u:?; d:?) (drop d (0) c (CTail e (Bind Void) u)) ->
                             (a:?) (wf0 g a) -> (drop (1) d c a) ->
                             (EX y1 y2 | t1 = (lift (1) d y1) &
                                         t2 = (lift (1) d y2) &
                                         (ty0 g a y1 y2)
                             ).
      Intros until 1; XElim H; Intros.
(* case 1 : ty0_conv *)
      Repeat IH d a; Rewrite H0 in H3; Rewrite H7 in H3; Pc3Gen; XEAuto.
(* case 2 : ty0_sort *)
      EApply ex3_2_intro; Try Rewrite lift_sort; XEAuto.
(* case 3 : ty0_abbr *)
      Apply (lt_eq_gt_e n d0); Intros.
(* case 3.1 : n < d0 *)
      DropDis; Rewrite minus_x_Sy in H7; [ DropGenBase | XAuto ].
      Rewrite (lt_plus_minus n d0) in H5; [ Idtac | XAuto ].
      DropDis; Rewrite H0 in H2; Clear H0 H1 u.
      IH '(minus d0 (S n)) x1; Rewrite H1; Clear H1 t.
      LiftGen; Rewrite <- H0 in H2; Clear H0 x2.
      Rewrite <- lift_d; [ Idtac | XAuto ].
      Rewrite <- le_plus_minus; [ Idtac | XAuto ].
      EApply ex3_2_intro; [ Rewrite lift_lref_lt | Idtac | EApply ty0_abbr ]; XEAuto.
(* case 3.2 : n = d0 *)
      Rewrite H6 in H0; DropDis; Inversion H0.
(* case 3.3 : n > d0 *)
      Clear H2 H3 c e t1 t2 u0; DropDis.
      Pattern 1 n; Rewrite (lt_plus_minus (0) n); [ Idtac | XEAuto ].
      Arith4c '(0) '(minus n (1)).
      EApply ex3_2_intro;
      [ Rewrite lift_lref_ge
      | Rewrite lift_free; Simpl
      | Pattern 2 n; Rewrite (minus_x_SO n)
      ]; XEAuto.
(* case 4 : ty0_abst *)
      Apply (lt_eq_gt_e n d0); Intros.
(* case 4.1 : n < d0 *)
      DropDis; Rewrite minus_x_Sy in H7; [ DropGenBase | XAuto ].
      Rewrite (lt_plus_minus n d0) in H5; [ Idtac | XAuto ].
      DropDis; Rewrite H0; Rewrite H0 in H2; Clear H0 H1 u.
      IH '(minus d0 (S n)) x1; Clear H1 t.
      LiftGen; Rewrite <- H0 in H2; Clear H0 x2.
      Rewrite <- lift_d; [ Idtac | XAuto ].
      Rewrite <- le_plus_minus; [ Idtac | XAuto ].
      EApply ex3_2_intro; [ Rewrite lift_lref_lt | Idtac | EApply ty0_abst ]; XEAuto.
(* case 4.2 : n = d0 *)
      Rewrite H6 in H0; DropDis; Inversion H0.
(* case 4.3 : n > d0 *)
      Clear H2 H3 c e t1 t2 u0; DropDis.
      Pattern 1 n; Rewrite (lt_plus_minus (0) n); [ Idtac | XEAuto ].
      Arith4c '(0) '(minus n (1)).
      EApply ex3_2_intro;
      [ Rewrite lift_lref_ge
      | Rewrite lift_free; [ Simpl | Simpl | Idtac ]
      | Pattern 2 n; Rewrite (minus_x_SO n)
      ]; XEAuto.
(* case 5 : ty0_bind *)
      IH d a; Rewrite H0; Rewrite H0 in H2; Rewrite H0 in H4; Clear H H0 H1 H3 H8 u t.
      IH '(S d) '(CTail a (Bind b) x0); Rewrite H; Rewrite H in H2; Clear H H0 t3 t4.
      IH '(S d) '(CTail a (Bind b) x0); Rewrite H; Clear H t0.
      LiftGen; Rewrite <- H in H2; Clear H x5.
      LiftTailRwBack; XEAuto.
(* case 6 : ty0_appl *)
      IH d a; Rewrite H2; Clear H H1 H2 v.
      LiftGenBase; Rewrite H in H7; Rewrite H1; Rewrite H1 in H0; Rewrite H2; Clear H H1 H2 u t x1.
      IH d a; Rewrite H; Clear H w.
      LiftGen; Rewrite <- H in H1; Clear H x4.
      LiftTailRwBack; XEAuto.
(* case 7 : ty0_cast *)
      Rename u into u0.
      IH d a; Rewrite H2 in H0; Rewrite H2; Clear H H1 H2 H6 t3 t4.
      IH d a; Rewrite H; Clear H t0.
      LiftGen; Rewrite <- H in H1; Clear H x3.
      LiftTailRwBack; XEAuto.
      Qed.

   End ty0_gen_cvoid.

      Tactic Definition Ty0GenContext :=
         Match Context With
            | [ H: (ty0 ?1 (CTail ?2 (Bind Abbr) ?3) ?4 ?5) |- ? ] ->
               LApply (ty0_gen_cabbr ?1 (CTail ?2 (Bind Abbr) ?3) ?4 ?5); [ Clear H; Intros H | XAuto ];
               LApply (H ?2 ?3 (0)); [ Clear H; Intros H | XAuto ];
               LApply (H (CTail ?2 (Bind Abbr) ?3)); [ Clear H; Intros H | XAuto ];
               LApply (H ?2); [ Clear H; Intros H | XAuto ];
               LApply H; [ Clear H; Intros H | XAuto ];
               XElim H; Intros
            | [ H: (ty0 ?1 (CTail ?2 (Bind Void) ?3) ?4 ?5) |- ? ] ->
               LApply (ty0_gen_cvoid ?1 (CTail ?2 (Bind Void) ?3) ?4 ?5); [ Clear H; Intros H | XAuto ];
               LApply (H ?2 ?3 (0)); [ Clear H; Intros H | XAuto ];
               LApply (H ?2); [ Clear H; Intros H | XAuto ];
               LApply H; [ Clear H; Intros H | XAuto ];
               XElim H; Intros
            | _ -> Ty0GenBase.

(*#* #start file *)

(*#* #single *)