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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_defs.
Require subst0_gen.
Require subst0_lift.

   Section subst0_subst0. (**************************************************)

      Tactic Definition IH := (
         Match Context With
            | [ H1: (u1,u:?; i:?) (subst0 i u u1 ?1) -> ?;
                H2: (subst0 ?2 ?3 ?4 ?1) |- ? ] ->
               LApply (H1 ?4 ?3 ?2); [ Clear H1; Intros H1 | XAuto ];
               XElim H1; Intros
            | [ H1: (u1,u:?; i:?) (subst0 i u ?1 u1) -> ?;
                H2: (subst0 ?2 ?3 ?1 ?4) |- ? ] ->
               LApply (H1 ?4 ?3 ?2); [ Clear H1; Intros H1 | XAuto ];
               XElim H1; Intros
         ).

      Theorem subst0_subst0: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
                             (u1,u:?; i:?) (subst0 i u u1 u2) ->
                             (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t t2)).
      Proof.
      Intros until 1; XElim H; Intros.
(* case 1 : subst0_lref *)
      Arith5 i0 i; XDEAuto 3.
(* case 2 : subst0_fst *)
      IH; XDEAuto 4.
(* case 3 : subst0_snd *)
      IH; SRwBackIn H2; XDEAuto 4.
(* case 4 : subst0_both *)
      Repeat IH; SRwBackIn H4; XDEAuto 4.
      Qed.

      Theorem subst0_subst0_back: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
                                  (u1,u:?; i:?) (subst0 i u u2 u1) ->
                                  (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t2 t)).
      Proof.
      Intros until 1; XElim H; Intros.
(* case 1 : subst0_lref *)
      Arith5 i0 i; XDEAuto 3.
(* case 2 : subst0_fst *)
      IH; XDEAuto 4.
(* case 3 : subst0_snd *)
      IH; SRwBackIn H2; XDEAuto 4.
(* case 4 : subst0_both *)
      Repeat IH; SRwBackIn H4; XDEAuto 4.
      Qed.

      Theorem subst0_trans: (t2,t1,v:?; i:?) (subst0 i v t1 t2) ->
                            (t3:?) (subst0 i v t2 t3) ->
                            (subst0 i v t1 t3).
      Proof.
      Intros until 1; XElim H; Intros;
      Subst0Gen; Try Rewrite H1; Try Rewrite H3; XAuto.
      Qed.

   End subst0_subst0.

      Tactic Definition Subst0Subst0 := (
         Match Context With
            | [ H1: (subst0 ?0 ?1 ?2 ?3); H2: (subst0 ?4 ?5 ?6 ?1) |- ? ] ->
               LApply (subst0_subst0 ?2 ?3 ?1 ?0); [ Intros H_x | XAuto ];
               LApply (H_x ?6 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
               XElim H_x; Intros
            | [ H1: (subst0 ?0 ?1 ?2 ?3); H2: (subst0 ?4 ?5 ?1 ?6) |- ? ] ->
               LApply (subst0_subst0_back ?2 ?3 ?1 ?0); [ Intros H_x | XAuto ];
               LApply (H_x ?6 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
               XElim H_x; Intros
         ).