(*#* #stop file *)

Require subst1_defs.
Require subst1_confluence.
Require drop_props.
Require pr0_subst1.
Require pr2_defs.

   Section pr2_subst1_props. (***********************************************)

      Theorem pr2_delta1: (c,d:?; u:?; i:?) (drop i (0) c (CTail d (Bind Abbr) u)) ->
                          (t1,t2:?) (pr0 t1 t2) -> (t:?) (subst1 i u t2 t) -> 
			  (pr2 c t1 t).
      Intros; XElim H1; Clear t; XEAuto.
      Qed.

      Hints Resolve pr2_delta1 : ltlc.

      Theorem pr2_subst1: (c,e:?; v:?; i:?) (drop i (0) c (CTail e (Bind Abbr) v)) ->
                          (t1,t2:?) (pr2 c t1 t2) ->
                          (w1:?) (subst1 i v t1 w1) ->
                          (EX w2 | (pr2 c w1 w2) & (subst1 i v t2 w2)).
     Intros until 2; XElim H0; Intros; 
     Pr0Subst1.
(* case 1: pr2_free *)
     XEAuto.
(* case 2: pr2_delta *)
     Apply (neq_eq_e i i0); Intros.
(* case 2.1: i <> i0 *)
     Subst1Confluence; XEAuto.
(* case 2.2: i = i0 *)
     Rewrite <- H4 in H0; Rewrite <- H4 in H2; Clear H4 i0.
     DropDis; Inversion H0; Rewrite H6 in H3; Clear H0 H5 H6 e v.
     Subst1Confluence; XEAuto.
     Qed.

   End pr2_subst1_props.

      Hints Resolve pr2_delta1 : ltlc.

      Tactic Definition Pr2Subst1 :=
         Match Context With
            | [ H0: (drop ?1 (0) ?2 (CTail ?3 (Bind Abbr) ?4));
                H1: (pr2 ?2 ?5 ?6); H3: (subst1 ?1 ?4 ?5 ?7) |- ? ] ->
               LApply (pr2_subst1 ?2 ?3 ?4 ?1); [ Intros H_x | XAuto ];
               LApply (H_x ?5 ?6); [ Clear H_x H1; Intros H1 | XAuto ];
               LApply (H1 ?7); [ Clear H1; Intros H1 | XAuto ];
               XElim H1; Intros.