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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 subst1_defs.
Require pr2_subst1.
Require pr3_defs.

   Section pr3_subst1_props. (***********************************************)

      Hints Resolve pr3_sing : ld.

      Lemma pr3_subst1: (c,e:?; v:?; i:?) (getl i c (CHead e (Bind Abbr) v)) ->
                        (t1,t2:?) (pr3 c t1 t2) ->
                        (w1:?) (subst1 i v t1 w1) ->
                        (EX w2 | (pr3 c w1 w2) & (subst1 i v t2 w2)).
      Proof.
      Intros until 2; XElim H0; Clear t1 t2; Intros.
(* case 1: pr3_refl *)
      XDEAuto 2.
(* case 2: pr3_single *)
      Pr2Subst1.
      LApply (H2 x); [ Clear H2; Intros H2 | XAuto ].
      XElim H2; XDEAuto 3.
      Qed.

   End pr3_subst1_props.

      Tactic Definition Pr3Subst1 := (
         Match Context With
            | [ H0: (getl ?1 ?2 (CHead ?3 (Bind Abbr) ?4));
                H1: (pr3 ?2 ?5 ?6); H3: (subst1 ?1 ?4 ?5 ?7) |- ? ] ->
               LApply (pr3_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
            | [ |- ? ] -> Pr2Subst1
         ).