Require subst0_subst0.
Require pr0_subst0.
Require cpr0_defs.
Require pr2_lift.
Require pr2_gen.
Require pr3_defs.

(*#* #caption "main properties of the relation $\\PrT{}{}{}$" *)
(*#* #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 H4 H5 H6 c k t.
      Rewrite H7 in H; Clear H7 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_free *)
      EApply pr3_pr0_pr2_t; [ Apply H0 | XAuto ].
(* case 2 : pr2_delta *)
      Inversion H; [ XAuto | NewInduction i0 ]. 
(* case 2.1 : i0 = 0 *)
      DropGenBase; Inversion H4; Clear H3 H4 H7 t t4.
      Rewrite <- H9; Rewrite H10 in H; Rewrite <- H11 in H6; Clear H9 H10 H11 d0 k u0.
      Subst0Subst0; Arith9'In H4 i; Clear H2 H H6 u2.
      Pr0Subst0; Apply pr3_sing with t2:=x0; XEAuto.
(* 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_refl *)
      XAuto.
(* case 2 : pr3_sing *)
      EApply pr3_t.
      EApply pr3_pr2_pr2_t; [ Apply H2 | Apply H ].
      XAuto.
      Qed.

(*#* #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).
      Intros until 1; XElim H; Intros.
(* case 1 : pr3_refl *)
      XAuto.
(* case 2 : pr3_sing *)
      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); [ Intros H_x | XAuto ];
               LApply (H_x ?1 ?5 ?6 ?4); [ Clear H_x 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 H2; Intros | XAuto ]
            | [ H1: (pr2 ?1 ?2 ?3); H2: (pr2 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
               LApply (pr3_pr2_pr2_t ?1 ?2 ?3); [ Intros H_x | XAuto ];
               LApply (H_x ?5 ?6 ?4); [ Clear H_x 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 H2; Intros | XAuto ]
            | [ H1: (pr3 ?1 ?2 ?3); H2: (pr3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
               LApply (pr3_pr3_pr3_t ?1 ?2 ?3); [ Intros H_x | XAuto ];
               LApply (H_x ?5 ?6 ?4); [ Clear H_x H2; Intros | XAuto ].

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

(*#* #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)).
      Intros until 2; XElim H0; Intros; XEAuto.
      Qed.

   End pr3_lift.

      Hints Resolve pr3_lift : ltlc.

   Section pr3_cpr0. (*******************************************************)

      Theorem pr3_cpr0_t: (c1,c2:?) (cpr0 c2 c1) -> (t1,t2:?) (pr3 c1 t1 t2) ->
                          (pr3 c2 t1 t2).
      Intros until 1; XElim H; Intros.
(* case 1 : cpr0_refl *)
      XAuto.
(* case 2 : cpr0_comp *)
      Pr3Context; Clear H1.
      XElim H2; Intros.
(* case 2.1 : pr3_refl *)
      XAuto.
(* case 2.2 : pr3_sing *)
      EApply pr3_t; [ Idtac | XEAuto ]. Clear H2 H3 c1 c2 t1 t2 t4 u2.
      Inversion_clear H1.
(* case 2.2.1 : pr2_free *)
      XAuto.
(* case 2.2.1 : pr2_delta *)
      Cpr0Drop; Pr0Subst0; Apply pr3_sing with t2:=x; XEAuto.
      Qed.

   End pr3_cpr0.