(*#* #stop file *)

Require subst0_subst0.
Require pr0_subst0.
Require cpr0_defs.
Require pr3_defs.
Require pr3_props.
Require pr3_confluence.
Require pc3_defs.

   Section pc3_trans. (******************************************************)

      Theorem pc3_t: (t2,c,t1:?) (pc3 c t1 t2) ->
                     (t3:?) (pc3 c t2 t3) -> (pc3 c t1 t3).
      Intros; Repeat Pc3Unfold; Pr3Confluence; XEAuto.
      Qed.

      Theorem pc3_pr2_u2: (c:?; t0,t1:?) (pr2 c t0 t1) ->
                          (t2:?) (pc3 c t0 t2) -> (pc3 c t1 t2).
      Intros; Apply (pc3_t t0); XAuto.
      Qed.

      Theorem pc3_tail_12: (c:?; u1,u2:?) (pc3 c u1 u2) ->
                           (k:?; t1,t2:?) (pc3 (CTail c k u2) t1 t2) ->
                           (pc3 c (TTail k u1 t1) (TTail k u2 t2)).
      Intros.
      EApply pc3_t; [ Apply pc3_tail_1 | Apply pc3_tail_2 ]; XAuto.
      Qed.

      Theorem pc3_tail_21: (c:?; u1,u2:?) (pc3 c u1 u2) ->
                           (k:?; t1,t2:?) (pc3 (CTail c k u1) t1 t2) ->
                           (pc3 c (TTail k u1 t1) (TTail k u2 t2)).
      Intros.
      EApply pc3_t; [ Apply pc3_tail_2 | Apply pc3_tail_1 ]; XAuto.
      Qed.

   End pc3_trans.

      Hints Resolve pc3_t pc3_tail_12 pc3_tail_21 : ltlc.

      Tactic Definition Pc3T :=
         Match Context With
            | [ _: (pr3 ?1 ?2 (TTail ?3 ?4 ?5)); _: (pc3 ?1 ?6 ?4) |- ? ] ->
               LApply (pc3_t (TTail ?3 ?4 ?5) ?1 ?2); [ Intros H_x | XAuto ];
               LApply (H_x (TTail ?3 ?6 ?5)); [ Clear H_x; Intros | Apply pc3_s; XAuto ]
            | [ _: (pc3 ?1 ?2 ?3); _: (pr3 ?1 ?3 ?4) |- ? ] ->
               LApply (pc3_t ?3 ?1 ?2); [ Intros H_x | XAuto ];
               LApply (H_x ?4); [ Clear H_x; Intros | XAuto ]
            | [ _: (pc3 ?1 ?2 ?3); _: (pc3 ?1 ?4 ?3) |- ? ] ->
               LApply (pc3_t ?3 ?1 ?2); [ Intros H_x | XAuto ];
               LApply (H_x ?4); [ Clear H_x; Intros | XAuto ].

   Section pc3_context. (****************************************************)

      Theorem pc3_pr0_pr2_t: (u1,u2:?) (pr0 u2 u1) ->
                             (c:?; t1,t2:?; k:?) (pr2 (CTail c k u2) t1 t2) ->
                             (pc3 (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; Apply pc3_pr3_t with t0:=x; XEAuto.
(* case 2: pr2_delta i > 0 *)
      NewInduction k; DropGenBase; XEAuto.
      Qed.

      Theorem pc3_pr2_pr2_t: (c:?; u1,u2:?) (pr2 c u2 u1) ->
                             (t1,t2:?; k:?) (pr2 (CTail c k u2) t1 t2) ->
                             (pc3 (CTail c k u1) t1 t2).
      Intros until 1; Inversion H; Clear H; Intros.
(* case 1: pr2_free *)
      EApply pc3_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.
      Pr0Subst0; Subst0Subst0; Arith9'In H6 i.
      EApply pc3_pr2_u.
      EApply pr2_delta; XEAuto.
      Apply pc3_pr2_x; EApply pr2_delta; [ Idtac | XEAuto | XEAuto ]; XEAuto.
(* case 2.2: i0 > 0 *)
      Clear IHi0; NewInduction k; DropGenBase; XEAuto.
      Qed.

      Theorem pc3_pr2_pr3_t: (c:?; u2,t1,t2:?; k:?)
                             (pr3 (CTail c k u2) t1 t2) ->
                             (u1:?) (pr2 c u2 u1) ->
                             (pc3 (CTail c k u1) t1 t2).
      Intros until 1; XElim H; Intros.
(* case 1: pr3_refl *)
      XAuto.
(* case 2: pr3_sing *)
      EApply pc3_t.
      EApply pc3_pr2_pr2_t; [ Apply H2 | Apply H ].
      XAuto.
      Qed.

      Theorem pc3_pr3_pc3_t: (c:?; u1,u2:?) (pr3 c u2 u1) ->
                             (t1,t2:?; k:?) (pc3 (CTail c k u2) t1 t2) ->
                             (pc3 (CTail c k u1) t1 t2).
      Intros until 1; XElim H; Intros.
(* case 1: pr3_refl *)
      XAuto.
(* case 2: pr3_sing *)
      Apply H1; Pc3Unfold.
      EApply pc3_t; [ Idtac | Apply pc3_s ]; EApply pc3_pr2_pr3_t; XEAuto.
      Qed.

   End pc3_context.

      Tactic Definition Pc3Context :=
         Match Context With
            | [ H1: (pr0 ?3 ?2); H2: (pr2 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
               LApply (pc3_pr0_pr2_t ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
               LApply (H1 ?1 ?5 ?6 ?4); [ Clear H1 H2; Intros | XAuto ]
            | [ H1: (pr0 ?3 ?2); H2: (pr3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
               LApply (pc3_pr2_pr3_t ?1 ?3 ?5 ?6 ?4); [ Clear H2; Intros H2 | XAuto ];
               LApply (H2 ?2); [ Clear H1 H2; Intros | XAuto ]
            | [ H1: (pr2 ?1 ?3 ?2); H2: (pr2 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
               LApply (pc3_pr2_pr2_t ?1 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
               LApply (H1 ?5 ?6 ?4); [ Clear H1 H2; Intros | XAuto ]
            | [ H1: (pr2 ?1 ?3 ?2); H2: (pr3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
               LApply (pc3_pr2_pr3_t ?1 ?3 ?5 ?6 ?4); [ Clear H2; Intros H2 | XAuto ];
               LApply (H2 ?2); [ Clear H1 H2; Intros | XAuto ]
            | [ H1: (pr3 ?1 ?3 ?2); H2: (pc3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
               LApply (pc3_pr3_pc3_t ?1 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
               LApply (H1 ?5 ?6 ?4); [ Clear H1 H2; Intros | XAuto ]
            | _ -> Pr3Context.

   Section pc3_lift. (*******************************************************)

      Theorem pc3_lift: (c,e:?; h,d:?) (drop h d c e) ->
                        (t1,t2:?) (pc3 e t1 t2) ->
                        (pc3 c (lift h d t1) (lift h d t2)).

      Intros.
      Pc3Unfold.
      EApply pc3_pr3_t; (EApply pr3_lift; [ XEAuto | Apply H1 Orelse Apply H2 ]).
      Qed.

   End pc3_lift.

      Hints Resolve pc3_lift : ltlc.

   Section pc3_cpr0. (*******************************************************)

      Remark pc3_cpr0_t_aux: (c1,c2:?) (cpr0 c1 c2) ->
                             (k:?; u,t1,t2:?) (pr3 (CTail c1 k u) t1 t2) ->
                             (pc3 (CTail c2 k u) t1 t2).
      Intros; XElim H0; Intros.
(* case 1.1: pr3_refl *)
      XAuto.
(* case 1.2: pr3_sing *)
      EApply pc3_t; [ Idtac | XEAuto ]. Clear H2 t1 t2.
      Inversion_clear H0.
(* case 1.2.1: pr2_free *)
      XAuto.
(* case 1.2.2: pr2_delta *)
      Cpr0Drop; Pr0Subst0.
      EApply pc3_pr2_u; [ EApply pr2_delta; XEAuto | XAuto ].
      Qed.

      Theorem pc3_cpr0_t: (c1,c2:?) (cpr0 c1 c2) ->
                          (t1,t2:?) (pr3 c1 t1 t2) ->
                          (pc3 c2 t1 t2).
      Intros until 1; XElim H; Intros.
(* case 1: cpr0_refl *)
      XAuto.
(* case 2: cpr0_comp *)
      Pc3Context; Pc3Unfold.
      EApply pc3_t; [ Idtac | Apply pc3_s ]; EApply pc3_cpr0_t_aux; XEAuto.
      Qed.

      Theorem pc3_cpr0: (c1,c2:?) (cpr0 c1 c2) -> (t1,t2:?) (pc3 c1 t1 t2) ->
                        (pc3 c2 t1 t2).
      Intros; Pc3Unfold.
      EApply pc3_t; [ Idtac | Apply pc3_s ]; EApply pc3_cpr0_t; XEAuto.
      Qed.

   End pc3_cpr0.

      Hints Resolve pc3_cpr0 : ltlc.

   Section pc3_ind_left. (***************************************************)
   
      Inductive pc3_left [c:C] : T -> T -> Prop :=
         | pc3_left_r : (t:?) (pc3_left c t t)
	 | pc3_left_ur: (t1,t2:?) (pr2 c t1 t2) -> (t3:?) (pc3_left c t2 t3) ->
	                (pc3_left c t1 t3)
	 | pc3_left_ux: (t1,t2:?) (pr2 c t1 t2) -> (t3:?) (pc3_left c t1 t3) ->
	                (pc3_left c t2 t3).
			
      Hint pc3_left: ltlc := Constructors pc3_left.

      Remark pc3_left_pr3: (c:?; t1,t2:?) (pr3 c t1 t2) -> (pc3_left c t1 t2).
      Intros; XElim H; XEAuto.
      Qed.

      Remark pc3_left_trans: (c:?; t1,t2:?) (pc3_left c t1 t2) -> 
                             (t3:?) (pc3_left c t2 t3) -> (pc3_left c t1 t3).
      Intros until 1; XElim H; XEAuto.
      Qed.

      Hints Resolve pc3_left_trans : ltlc.

      Remark pc3_left_sym: (c:?; t1,t2:?) (pc3_left c t1 t2) -> 
                           (pc3_left c t2 t1).
      Intros; XElim H; XEAuto.
      Qed.

      Hints Resolve pc3_left_sym pc3_left_pr3 : ltlc.

      Remark pc3_left_pc3: (c:?; t1,t2:?) (pc3 c t1 t2) -> (pc3_left c t1 t2).
      Intros; Pc3Unfold; XEAuto.
      Qed.

      Remark pc3_pc3_left: (c:?; t1,t2:?) (pc3_left c t1 t2) -> (pc3 c t1 t2).
      Intros; XElim H; XEAuto.
      Qed.
      
      Hints Resolve pc3_left_pc3 pc3_pc3_left : ltlc.

      Theorem pc3_ind_left: (c:C; P:(T->T->Prop))
                            ((t:T) (P t t)) ->
			    ((t1,t2:T) (pr2 c t1 t2) -> (t3:T) (pc3 c t2 t3) -> (P t2 t3) -> (P t1 t3)) ->
			    ((t1,t2:T) (pr2 c t1 t2) -> (t3:T) (pc3 c t1 t3) -> (P t1 t3) -> (P t2 t3)) ->
	                    (t,t0:T) (pc3 c t t0) -> (P t t0).
      Intros; ElimType (pc3_left c t t0); XEAuto.
      Qed.

   End pc3_ind_left.