(*#* #stop file *)

Require lift_gen.
Require drop_defs.

(*#* #caption "main properties of drop" #clauses *)

   Section confluence. (*****************************************************)

      Tactic Definition IH :=
         Match Context With
            [ H1: (drop ?1 ?2 c ?3); H2: (drop ?1 ?2 c ?4) |- ? ] ->
               LApply (H ?4 ?2 ?1); [ Clear H H2; Intros H | XAuto ];
               LApply (H ?3); [ Clear H H1; Intros | XAuto ].

(*#* #caption "confluence, first case" *)
(*#* #cap #alpha c in C, x1 in C1, x2 in C2, d in i *)

      Theorem drop_mono : (c,x1:?; d,h:?) (drop h d c x1) ->
                          (x2:?) (drop h d c x2) -> x1 = x2.
      XElim c.
(* case 1: CSort *)
      Intros; Repeat DropGenBase; Rewrite H0; XAuto.
(* case 2: CTail k *)
      XElim d.
(* case 2.1: d = 0 *)
      XElim h; Intros; Repeat DropGenBase; Try Rewrite <- H0; XEAuto.
(* case 2.2: d > 0 *)
      Intros; Repeat DropGenBase; Rewrite H1; Rewrite H2; Rewrite H5 in H3;
      LiftGen; IH; XAuto.
      Qed.

(*#* #caption "confluence, second case" *)
(*#* #cap #alpha c in C1, c0 in E1, e in C2, e0 in E2, u in V1, v in V2, i in k, d in i *)

      Theorem drop_conf_lt: (b:?; i:?; u:?; c0,c:?)
                            (drop i (0) c (CTail c0 (Bind b) u)) ->
                            (e:?; h,d:?) (drop h (S (plus i d)) c e) ->
                            (EX v e0 | u = (lift h d v) &
                                       (drop i (0) e (CTail e0 (Bind b) v)) &
                                       (drop h d c0 e0)
                            ).
      XElim i.
(* case 1 : i = 0 *)
      Intros until 1.
      DropGenBase.
      Rewrite H in H0; Clear H.
      Inversion H0; XEAuto.
(* case 2 : i > 0 *)
      Intros i; XElim c.
(* case 2.1 : CSort *)
      Intros; Inversion H0.
(* case 2.2 : CTail k *)
      XElim k; Intros; Repeat DropGenBase; Rewrite H2; Clear H2 H3 e t.
(* case 2.2.1 : Bind *)
      LApply (H u c0 c); [ Clear H H0 H1; Intros H | XAuto ].
      LApply (H x0 h d); [ Clear H H9; Intros H | XAuto ].
      XElim H; XEAuto.
(* case 2.2.2 : Flat *)
      LApply H0; [ Clear H H0 H1; Intros H | XAuto ].
      LApply (H x0 h d); [ Clear H H9; Intros H | XAuto ].
      XElim H; XEAuto.
      Qed.

(*#* #caption "confluence, third case" *)
(*#* #cap #alpha c in C, a in C1, e in C2, i in k, d in i *)

      Theorem drop_conf_ge: (i:?; a,c:?) (drop i (0) c a) ->
                            (e:?; h,d:?) (drop h d c e) -> (le (plus d h) i) ->
                            (drop (minus i h) (0) e a).
      XElim i.
(* case 1 : i = 0 *)
      Intros until 1.
      DropGenBase; Rewrite H in H0; Clear H c.
      Inversion H1; Rewrite H2; Simpl; Clear H1.
      PlusO; Rewrite H in H0; Rewrite H1 in H0; Clear H H1 d h.
      DropGenBase; Rewrite <- H; XAuto.
(* case 2 : i > 0 *)
      Intros i; XElim c.
(* case 2.1 : CSort *)
      Intros; Repeat DropGenBase; Rewrite H1; Rewrite H0; XAuto.
(* case 2.2 : CTail k *)
      XElim k; Intros; DropGenBase;
      ( NewInduction d;
      [ NewInduction h; DropGenBase;
        [ Rewrite <- H2; Simpl; XAuto | Clear IHh ]
      | DropGenBase; Rewrite H2; Clear IHd H2 H4 e t ] ).
(* case 2.2.1 : Bind, d = 0, h > 0 *)
      LApply (H a c); [ Clear H H0 H1; Intros H | XAuto ].
      LApply (H e h (0)); XAuto.
(* case 2.2.2 : Bind, d > 0 *)
      LApply (H a c); [ Clear H H0 H1; Intros H | XAuto ].
      LApply (H x0 h d); [ Clear H H4; Intros H | XAuto ].
      LApply H; [ Clear H; Simpl in H3; Intros H | XAuto ].
      Rewrite <- minus_Sn_m; XEAuto.
(* case 2.2.3 : Flat, d = 0, h > 0 *)
      LApply H0; [ Clear H H0 H1; Intros H | XAuto ].
      LApply (H e (S h) (0)); XAuto.
(* case 2.2.4 : Flat, d > 0 *)
      LApply H0; [ Clear H H0 H1; Intros H | XAuto ].
      LApply (H x0 h (S d)); [ Clear H H4; Intros H | XAuto ].
      LApply H; [ Clear H; Simpl in H3; Intros H | XAuto ].
      Rewrite <- minus_Sn_m in H; [ Idtac | XEAuto ].
      Rewrite <- minus_Sn_m; XEAuto.
      Qed.

   End confluence.

   Section transitivity. (***************************************************)

(*#* #caption "transitivity, first case" *)
(*#* #cap #alpha c1 in C1, c2 in C2, e1 in D1, e2 in D2, d in i, i in k *)

      Theorem drop_trans_le : (i,d:?) (le i d) ->
                              (c1,c2:?; h:?) (drop h d c1 c2) ->
                              (e2:?) (drop i (0) c2 e2) ->
                              (EX e1 | (drop i (0) c1 e1) & (drop h (minus d i) e1 e2)).
      XElim i.
(* case 1 : i = 0 *)
      Intros.
      DropGenBase; Rewrite H1 in H0.
      Rewrite <- minus_n_O; XEAuto.
(* case 2 : i > 0 *)
      Intros i IHi; XElim d.
(* case 2.1 : d = 0 *)
      Intros; Inversion H.
(* case 2.2 : d > 0 *)
      Intros d IHd; XElim c1.
(* case 2.2.1 : CSort *)
      Intros.
      DropGenBase; Rewrite H0 in H1.
      DropGenBase; Rewrite H1; XEAuto.
(* case 2.2.2 : CTail k *)
      Intros c1 IHc; XElim k; Intros;
      DropGenBase; Rewrite H0 in H1; Rewrite H2; Clear IHd H0 H2 c2 t;
      DropGenBase.
(* case 2.2.2.1 : Bind *)
      LApply (IHi d); [ Clear IHi; Intros IHi | XAuto ].
      LApply (IHi c1 x0 h); [ Clear IHi H8; Intros IHi | XAuto ].
      LApply (IHi e2); [ Clear IHi H0; Intros IHi | XAuto ].
      XElim IHi; XEAuto.
(* case 2.2.2.2 : Flat *)
      LApply (IHc x0 h); [ Clear IHc H8; Intros IHc | XAuto ].
      LApply (IHc e2); [ Clear IHc H0; Intros IHc | XAuto ].
      XElim IHc; XEAuto.
      Qed.

(*#* #caption "transitivity, second case" *)
(*#* #cap #alpha c1 in C1, c2 in C, e2 in C2, d in i, i in k *)

      Theorem drop_trans_ge : (i:?; c1,c2:?; d,h:?) (drop h d c1 c2) ->
                              (e2:?) (drop i (0) c2 e2) -> (le d i) ->
                              (drop (plus i h) (0) c1 e2).
      XElim i.
(* case 1: i = 0 *)
      Intros.
      DropGenBase; Rewrite <- H0.
      Inversion H1; Rewrite H2 in H; XAuto.
(* case 2 : i > 0 *)
      Intros i IHi; XElim c1; Simpl.
(* case 2.1: CSort *)
      Intros.
      DropGenBase; Rewrite H in H0.
      DropGenBase; Rewrite H0; XAuto.
(* case 2.2: CTail *)
      Intros c1 IHc; XElim d.
(* case 2.2.1: d = 0 *)
      XElim h; Intros.
(* case 2.2.1.1: h = 0 *)
      DropGenBase; Rewrite <- H in H0;
      DropGenBase; Rewrite <- plus_n_O; XAuto.
(* case 2.2.1.2: h > 0 *)
      DropGenBase; Rewrite <- plus_n_Sm.
      Apply drop_drop; RRw; XEAuto. (**) (* explicit constructor *)
(* case 2.2.2: d > 0 *)
      Intros d IHd; Intros.
      DropGenBase; Rewrite H in IHd; Rewrite H in H0; Rewrite H2 in IHd; Rewrite H2; Clear IHd H H2 c2 t;
      DropGenBase; Apply drop_drop; NewInduction k; Simpl; XEAuto. (**) (* explicit constructor *)
      Qed.

   End transitivity.

      Tactic Definition DropDis :=
         Match Context With
            [ H1: (drop ?1 ?2 ?3 ?4); H2: (drop ?1 ?2 ?3 ?5) |- ? ] ->
               LApply (drop_mono ?3 ?5 ?2 ?1); [ Intros H_x | XAuto ];
               LApply (H_x ?4); [ Clear H_x H1; Intros H1; Rewrite H1 in H2 | XAuto ]
            | [ H1: (drop ?0 (0) ?1 (CTail ?2 (Bind ?3) ?4));
                H2: (drop ?5 (S (plus ?0 ?6)) ?1 ?7) |- ? ] ->
               LApply (drop_conf_lt ?3 ?0 ?4 ?2 ?1); [ Clear H1; Intros H1 | XAuto ];
               LApply (H1 ?7 ?5 ?6); [ Clear H1 H2; Intros H1 | XAuto ];
               XElim H1; Intros
            | [ _: (drop ?0 (0) ?1 ?2); _: (drop ?5 (0) ?1 ?7);
                _: (lt ?5 ?0) |- ? ] ->
               LApply (drop_conf_ge ?0 ?2 ?1); [ Intros H_x | XAuto ];
               LApply (H_x ?7 ?5 (0)); [ Clear H_x; Intros H_x | XAuto ];
               Simpl in H_x; LApply H_x; [ Clear H_x; Intros | XAuto ]
            | [ _: (drop ?1 (0) ?2 (CTail ?3 (Bind ?) ?));
                _: (drop (1) ?1 ?2 ?4) |- ? ] ->
               LApply (drop_conf_ge (S ?1) ?3 ?2); [ Intros H_x | XEAuto ];
               LApply (H_x ?4 (1) ?1); [ Clear H_x; Intros H_x | XAuto ];
               LApply H_x; [ Clear H_x; Intros | Rewrite plus_sym; XAuto ]; (
               Match Context With
                  [ H: (drop (minus (S ?1) (1)) (0) ?4 ?3) |- ? ] ->
                    Simpl in H; Rewrite <- minus_n_O in H )
            | [ H0: (drop ?0 (0) ?1 ?2); H2: (lt ?6 ?0);
                H1: (drop (1) ?6 ?1 ?7) |- ? ] ->
               LApply (drop_conf_ge ?0 ?2 ?1); [ Intros H_x | XAuto ];
               LApply (H_x ?7 (1) ?6); [ Clear H_x; Intros H_x | XAuto ];
               LApply H_x; [ Clear H_x; Intros | Rewrite plus_sym; XAuto ]
            | [ H0: (drop ?0 (0) ?1 ?2);
                H1: (drop ?5 ?6 ?1 ?7) |- ? ] ->
               LApply (drop_conf_ge ?0 ?2 ?1); [ Intros H_x | XAuto ];
               LApply (H_x ?7 ?5 ?6); [ Clear H_x; Intros H_x | XAuto ];
               LApply H_x; [ Clear H_x; Intros | XAuto ]
            | [ H0: (lt ?1 ?2);
                H1: (drop ?3 ?2 ?4 ?5); H2: (drop ?1 (0) ?5 ?6) |- ? ] ->
               LApply (drop_trans_le ?1 ?2); [ Intros H_x | XAuto ];
               LApply (H_x ?4 ?5 ?3); [ Clear H_x H1; Intros H_x | XAuto ];
               LApply (H_x ?6); [ Clear H_x H2; Intros H_x | XAuto ];
               XElim H_x; Intros
            | [ H0: (le ?1 ?2);
                H1: (drop ?3 ?1 ?4 ?5); H2: (drop ?2 (0) ?5 ?6) |- ? ] ->
               LApply (drop_trans_ge ?2 ?4 ?5 ?1 ?3); [ Clear H1; Intros H1 | XAuto ];
               LApply (H1 ?6); [ Clear H1 H2; Intros H1 | XAuto ];
               LApply H1; [ Clear H1; Intros | XAuto ].

(*#* #single *)