(*#* #stop file *)

Require subst0_gen.
Require subst1_defs.

   Section subst1_gen_lift. (************************************************)

      Theorem subst1_gen_lift_lt : (u,t1,x:?; i,h,d:?) (subst1 i (lift h d u) (lift h (S (plus i d)) t1) x) ->
                                   (EX t2 | x = (lift h (S (plus i d)) t2) & (subst1 i u t1 t2)).
      Intros; XElim H; Clear x; Intros;
      Try Subst0Gen; XEAuto.
      Qed.

      Theorem subst1_gen_lift_eq : (t,u,x:?; h,d,i:?)
                                   (le d i) -> (lt i (plus d h)) ->
                                   (subst1 i u (lift h d t) x) ->
                                   x = (lift h d t).
      Intros; XElim H1; Clear x; Intros;
      Try Subst0Gen; XAuto.
      Qed.

      Theorem subst1_gen_lift_ge : (u,t1,x:?; i,h,d:?) (subst1 i u (lift h d t1) x) ->
                                   (le (plus d h) i) ->
                                   (EX t2 | x = (lift h d t2) & (subst1 (minus i h) u t1 t2)).
      Intros; XElim H; Clear x; Intros;
      Try Subst0Gen; XEAuto.
      Qed.

   End subst1_gen_lift.

      Tactic Definition Subst1Gen :=
         Match Context With
            | [ H: (subst1 ?0 (lift ?1 ?2 ?3) (lift ?1 (S (plus ?0 ?2)) ?4) ?5) |- ? ] ->
               LApply (subst1_gen_lift_lt ?3 ?4 ?5 ?0 ?1 ?2); [ Clear H; Intros H | XAuto ];
               XElim H; Intros
            | [ H: (subst1 ?0 ?1 (lift (1) ?0 ?2) ?3) |- ? ] ->
               LApply (subst1_gen_lift_eq ?2 ?1 ?3 (1) ?0 ?0); [ Intros H_x | XAuto ];
               LApply H_x; [ Clear H_x; Intros H_x | Rewrite plus_sym; XAuto ];
               LApply H_x; [ Clear H H_x; Intros | XAuto ]
            | [ H0: (subst1 ?0 ?1 (lift (1) ?4 ?2) ?3); H1: (lt ?4 ?0) |- ? ] ->
               LApply (subst1_gen_lift_ge ?1 ?2 ?3 ?0 (1) ?4); [ Clear H0; Intros H0 | XAuto ];
               LApply H0; [ Clear H0; Intros H0 | Rewrite plus_sym; XAuto ];
               XElim H0; Intros
            | _ -> Subst1GenBase.