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(*                                                                          *)
(*                The formal system \lambda\delta                           *)
(*                                                                          *)
(*                Author: Ferruccio Guidi                                   *)
(*                                                                          *)
(*                This file is distributed under the terms of the           *)
(*                GNU General Public License Version 2                      *)
(*                                                                          *)
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Require subst0_gen.
Require subst1_defs.

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

      Lemma 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)).
      Proof.
      Intros; XElim H; Clear x; Intros;
      Try Subst0Gen; XDEAuto 3.
      Qed.

      Lemma 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).
      Proof.
      Intros; XElim H1; Clear x; Intros;
      Try Subst0Gen; XAuto.
      Qed.

      Lemma 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)).
      Proof.
      Intros; XElim H; Clear x; Intros;
      Try Subst0Gen; XDEAuto 3.
      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
         ).