Require Export contexts_defs.
Require Export drop_defs.
Require Export pr0_defs.

(*#* #caption "current axioms for the relation $\\CprZ{}{}$",
   "reflexivity", "compatibility" 
*)
(*#* #cap #cap c, c1, c2 #alpha u1 in V1, u2 in V2, k in z *)

      Inductive cpr0 : C -> C -> Prop :=
         | cpr0_refl : (c:?) (cpr0 c c)
         | cpr0_comp : (c1,c2:?) (cpr0 c1 c2) -> (u1,u2:?) (pr0 u1 u2) ->
                       (k:?) (cpr0 (CTail c1 k u1) (CTail c2 k u2)).

(*#* #stop file *)

      Hint cpr0 : ltlc := Constructors cpr0.

   Section cpr0_drop. (******************************************************)

      Theorem cpr0_drop : (c1,c2:?) (cpr0 c1 c2) -> (h:?; e1:?; u1:?; k:?)
                          (drop h (0) c1 (CTail e1 k u1)) ->
                          (EX e2 u2 | (drop h (0) c2 (CTail e2 k u2)) &
                                      (cpr0 e1 e2) & (pr0 u1 u2)
                          ).
      Intros until 1; XElim H.
(* case 1 : cpr0_refl *)
      XEAuto.
(* case 2 : cpr0_comp *)
      XElim h.
(* case 2.1 : h = 0 *)
      Intros; DropGenBase.
      Inversion H2; Rewrite H6 in H1; Rewrite H4 in H; XEAuto.
(* case 2.2 : h > 0 *)
      XElim k; Intros; DropGenBase.
(* case 2.2.1 : Bind *)
      LApply (H0 n e1 u0 k); [ Clear H0 H3; Intros H0 | XAuto ].
      XElim H0; XEAuto.
(* case 2.2.2 : Flat *)
      LApply (H0 (S n) e1 u0 k); [ Clear H0 H3; Intros H0 | XAuto ].
      XElim H0; XEAuto.
      Qed.

      Theorem cpr0_drop_back : (c1,c2:?) (cpr0 c2 c1) -> (h:?; e1:?; u1:?; k:?)
                               (drop h (0) c1 (CTail e1 k u1)) ->
                               (EX e2 u2 | (drop h (0) c2 (CTail e2 k u2)) &
                                           (cpr0 e2 e1) & (pr0 u2 u1)
                               ).
      Intros until 1; XElim H.
(* case 1 : cpr0_refl *)
      XEAuto.
(* case 2 : cpr0_comp *)
      XElim h.
(* case 2.1 : h = 0 *)
      Intros; DropGenBase.
      Inversion H2; Rewrite H6 in H1; Rewrite H4 in H; XEAuto.
(* case 2.2 : h > 0 *)
      XElim k; Intros; DropGenBase.
(* case 2.2.1 : Bind *)
      LApply (H0 n e1 u0 k); [ Clear H0 H3; Intros H0 | XAuto ].
      XElim H0; XEAuto.
(* case 2.2.2 : Flat *)
      LApply (H0 (S n) e1 u0 k); [ Clear H0 H3; Intros H0 | XAuto ].
      XElim H0; XEAuto.
      Qed.

   End cpr0_drop.

      Tactic Definition Cpr0Drop :=
         Match Context With
            | [ _: (drop ?1 (0) ?2 (CTail ?3 ?4 ?5));
                _: (cpr0 ?2 ?6) |- ? ] ->
               LApply (cpr0_drop ?2 ?6); [ Intros H_x | XAuto ];
               LApply (H_x ?1 ?3 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
               XElim H_x; Intros
            | [ _: (drop ?1 (0) ?2 (CTail ?3 ?4 ?5));
                _: (cpr0 ?6 ?2) |- ? ] ->
               LApply (cpr0_drop_back ?2 ?6); [ Intros H_x | XAuto ];
               LApply (H_x ?1 ?3 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
               XElim H_x; Intros
            | [ _: (drop ?1 (0) (CTail ?2 ?7 ?8) (CTail ?3 ?4 ?5));
                _: (cpr0 ?2 ?6) |- ? ] ->
               LApply (cpr0_drop (CTail ?2 ?7 ?8) (CTail ?6 ?7 ?8)); [ Intros H_x | XAuto ];
               LApply (H_x ?1 ?3 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
               XElim H_x; Intros
            | [ _: (drop ?1 (0) (CTail ?2 ?7 ?8) (CTail ?3 ?4 ?5));
                _: (cpr0 ?6 ?2) |- ? ] ->
               LApply (cpr0_drop_back (CTail ?2 ?7 ?8) (CTail ?6 ?7 ?8)); [ Intros H_x | XAuto ];
               LApply (H_x ?1 ?3 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
               XElim H_x; Intros.