Require subst0_lift.
Require drop_props.
Require pr0_lift.
Require pr2_defs.

(*#* #caption "main properties of predicate \\texttt{pr2}" *)
(*#* #clauses pr2_props *)

   Section pr2_lift. (*******************************************************)

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

      Theorem pr2_lift : (c,e:?; h,d:?) (drop h d c e) ->
                         (t1,t2:?) (pr2 e t1 t2) ->
                         (pr2 c (lift h d t1) (lift h d t2)).

(*#* #stop file *)

      Intros until 2; XElim H0; Intros.
(* case 1 : pr2_pr0 *)
      XAuto.
(* case 2 : pr2_delta *)
      Apply (lt_le_e i d); Intros; DropDis.
(* case 2.1 : i < d *)
      Rewrite minus_x_Sy in H0; [ Idtac | XAuto ].
      DropGenBase; Rewrite H0 in H; Simpl in H; XEAuto.
(* case 2.2 : i >= d *)
      XEAuto.
      Qed.

   End pr2_lift.

      Hints Resolve pr2_lift : ltlc.