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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 lift_gen.
Require subst0_gen.
Require subst0_confluence.
Require subst1_defs.
Require subst1_gen.

   Section subst1_confluence. (**********************************************)

      Hints Resolve sym_not_eq : ld.

      Theorem subst1_confluence_neq: (t0,t1,u1:?; i1:?) (subst1 i1 u1 t0 t1) ->
                                     (t2,u2:?; i2:?) (subst1 i2 u2 t0 t2) ->
                                     ~i1=i2 ->
                                     (EX t | (subst1 i2 u2 t1 t) &
                                             (subst1 i1 u1 t2 t)
                                     ).
      Proof.
      Intros until 1; XElim H; Clear t1; Intros.
(* case 1; subst1_refl *)
      XDEAuto 2.
(* case 2; subst1_single *)
      XElim H0; Intros; Try Subst0Confluence; XDEAuto 4.
      Qed.

      Theorem subst1_confluence_eq: (t0,t1,u:?; i:?) (subst1 i u t0 t1) ->
                                    (t2:?) (subst1 i u t0 t2) ->
                                    (EX t | (subst1 i u t1 t) &
                                            (subst1 i u t2 t)
                                    ).
      Proof.
      Intros until 1; XElim H; Intros.
(* case 1; subst1_refl *)
      XDEAuto 2.
(* case 2; subst1_single *)
      XElim H0; Intros;
      Try Subst0Confluence; Try Rewrite H0; XDEAuto 4.
      Qed.

      Hints Resolve sym_eq : ld.

      Theorem subst1_confluence_lift: (t0,t1,u:?; i:?) (subst1 i u t0 (lift (1) i t1)) ->
                                      (t2:?) (subst1 i u t0 (lift (1) i t2)) ->
                                      t1 = t2.
      Proof.
      Intros until 1; InsertEq H '(lift (1) i t1); XElim H; Clear y; Intros.
(* case 1: subst1_refl *)
      Rewrite H in H0; Clear H t0.
      Subst1Gen; SymEqual; LiftGen; XDEAuto 2.
(* case 2: subst1_single *)
      Rewrite H0 in H; Clear H0 t2.
      InsertEq H1 '(lift (1) i t3); XElim H0; Clear y; Intros.
(* case 2.1: subst1_refl *)
      Rewrite H0 in H; Clear H0 t0; Subst0Gen.
(* case 2.2: subst1_single *)
      Rewrite H1 in H0; Clear H1 t2; Subst0ConfluenceLift; XAuto.
      Qed.

   End subst1_confluence.

      Tactic Definition Subst1Confluence := (
         Match Context With
            | [ H0: (subst1 ?1 ?2 ?3 (lift (1) ?1 ?4));
                H1: (subst1 ?1 ?2 ?3 (lift (1) ?1 ?5)) |- ? ] ->
              LApply (subst1_confluence_lift ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
              LApply (H0 ?5); [ Clear H0; Intros | XAuto ]
            | [ H0: (subst1 ?1 ?2 ?3 ?4);
                H1: (subst1 ?1 ?2 ?3 ?5) |- ? ] ->
              LApply (subst1_confluence_eq ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
              LApply (H0 ?5); [ Clear H0; Intros H0 | XAuto ];
              XElim H0; Intros
            | [ H0: (subst0 ?1 ?2 ?3 ?4);
                H1: (subst1 ?1 ?2 ?3 ?5) |- ? ] ->
              LApply (subst1_confluence_eq ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
              LApply (H0 ?5); [ Clear H0; Intros H0 | XAuto ];
              XElim H0; Intros
            | [ H0: (subst1 ?1 ?2 ?3 ?4);
                H1: (subst1 ?5 ?6 ?3 ?7) |- ? ] ->
              LApply (subst1_confluence_neq ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
              LApply (H0 ?7 ?6 ?5); [ Clear H0 H1; Intros H0 | XAuto ];
              LApply H0; [ Clear H0; Intros H0 | XAuto ];
              XElim H0; Intros
            | [ H0: (subst0 ?1 ?2 ?3 ?4);
                H1: (subst1 ?5 ?6 ?3 ?7) |- ? ] ->
              LApply (subst1_confluence_neq ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
              LApply (H0 ?7 ?6 ?5); [ Clear H0 H1; Intros H0 | XAuto ];
              LApply H0; [ Clear H0; Intros H0 | XAuto ];
              XElim H0; Intros
         ).