Require subst0_subst0.
Require pr0_subst0.
Require pr2_lift.
Require pr3_defs.

(*#* #caption "main properties of predicate \\texttt{pr3}" *)
(*#* #clauses *)

(*#* #stop file *)

   Section pr3_context. (****************************************************)

      Theorem pr3_pr0_pr2_t : (u1,u2:?) (pr0 u1 u2) ->
                              (c:?; t1,t2:?; k:?) (pr2 (CTail c k u2) t1 t2) ->
                              (pr3 (CTail c k u1) t1 t2).
      Intros.
      Inversion H0; Clear H0; XAuto.
      NewInduction i.
(* case 1 : pr2_delta i = 0 *)
      DropGenBase; Inversion H0; Clear H0 H3 H4 c k.
      Rewrite H5 in H; Clear H5 u2.
      Pr0Subst0; XEAuto.
(* case 2 : pr2_delta i > 0 *)
      NewInduction k; DropGenBase; XEAuto.
      Qed.

      Theorem pr3_pr2_pr2_t : (c:?; u1,u2:?) (pr2 c u1 u2) ->
                              (t1,t2:?; k:?) (pr2 (CTail c k u2) t1 t2) ->
                              (pr3 (CTail c k u1) t1 t2).
      Intros until 1; Inversion H; Clear H; Intros.
(* case 1 : pr2_pr0 *)
      EApply pr3_pr0_pr2_t; [ Apply H0 | XAuto ].
(* case 2 : pr2_delta *)
      Inversion H; [ XAuto | NewInduction i0 ].
(* case 2.1 : i0 = 0 *)
      DropGenBase; Inversion H2; Clear H2.
      Rewrite <- H5; Rewrite H6 in H; Rewrite <- H7 in H3; Clear H5 H6 H7 d0 k u0.
      Subst0Subst0; Arith9'In H4 i; XDEAuto 7.
(* case 2.2 : i0 > 0 *)
      Clear IHi0; NewInduction k; DropGenBase; XEAuto.
      Qed.

      Theorem pr3_pr2_pr3_t : (c:?; u2,t1,t2:?; k:?)
                              (pr3 (CTail c k u2) t1 t2) ->
                              (u1:?) (pr2 c u1 u2) ->
                              (pr3 (CTail c k u1) t1 t2).
      Intros until 1; XElim H; Intros.
(* case 1 : pr3_r *)
      XAuto.
(* case 2 : pr3_u *)
      EApply pr3_t.
      EApply pr3_pr2_pr2_t; [ Apply H2 | Apply H ].
      XAuto.
      Qed.

(*#* #start file *)

(*#* #caption "reduction inside context items" *)
(*#* #cap #cap t1, t2 #alpha c in E, u1 in V1, u2 in V2, k in z *)

      Theorem pr3_pr3_pr3_t : (c:?; u1,u2:?) (pr3 c u1 u2) ->
                              (t1,t2:?; k:?) (pr3 (CTail c k u2) t1 t2) ->
                              (pr3 (CTail c k u1) t1 t2).

(*#* #stop file *)

      Intros until 1; XElim H; Intros.
(* case 1 : pr3_r *)
      XAuto.
(* case 2 : pr3_u *)
      EApply pr3_pr2_pr3_t; [ Apply H1; XAuto | XAuto ].
      Qed.

   End pr3_context.

      Tactic Definition Pr3Context :=
         Match Context With
            | [ H1: (pr0 ?2 ?3); H2: (pr2 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
               LApply (pr3_pr0_pr2_t ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
               LApply (H1 ?1 ?5 ?6 ?4); [ Clear H1 H2; Intros | XAuto ]
            | [ H1: (pr0 ?2 ?3); H2: (pr3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
               LApply (pr3_pr2_pr3_t ?1 ?3 ?5 ?6 ?4); [ Clear H2; Intros H2 | XAuto ];
               LApply (H2 ?2); [ Clear H1 H2; Intros | XAuto ]
            | [ H1: (pr2 ?1 ?2 ?3); H2: (pr2 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
               LApply (pr3_pr2_pr2_t ?1 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
               LApply (H1 ?5 ?6 ?4); [ Clear H1 H2; Intros | XAuto ]
            | [ H1: (pr2 ?1 ?2 ?3); H2: (pr3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
               LApply (pr3_pr2_pr3_t ?1 ?3 ?5 ?6 ?4); [ Clear H2; Intros H2 | XAuto ];
               LApply (H2 ?2); [ Clear H1 H2; Intros | XAuto ]
            | [ H1: (pr3 ?1 ?2 ?3); H2: (pr3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
               LApply (pr3_pr3_pr3_t ?1 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
               LApply (H1 ?5 ?6 ?4); [ Clear H1 H2; Intros | XAuto ].

   Section pr3_lift. (*******************************************************)

(*#* #start file *)

(*#* #caption "conguence with lift" *)
(*#* #cap #cap c, t1, t2 #alpha e in D, d in i *)

      Theorem pr3_lift : (c,e:?; h,d:?) (drop h d c e) ->
                         (t1,t2:?) (pr3 e t1 t2) ->
                         (pr3 c (lift h d t1) (lift h d t2)).

(*#* #stop file *)

      Intros until 2; XElim H0; Intros; XEAuto.
      Qed.

   End pr3_lift.

      Hints Resolve pr3_lift : ltlc.