(* *********************************************************************** *)
(** * Tactic support for [disjoint] and [uniq] *)

(** [destruct_uniq] decomposes all [uniq] and [disjoint] hypotheses. *)

Ltac destruct_uniq :=
  match goal with
    | H : uniq nil |- _ =>
      clear H;
      destruct_uniq
    | H : uniq (?x ~ ?a) |- _ =>
      clear H;
      destruct_uniq
    | H : uniq ((?x, ?a) :: ?E) |- _ =>
      let J := fresh "UniqTac" in
      pose proof H as J;
      apply uniq_cons_1 in H;
      apply uniq_cons_2 in J;
      autorewrite with rewr_dom in J;
      destruct_uniq
    | H : uniq (?E ++ ?F) |- _ =>
      let J1 := fresh "UniqTac" in
      let J2 := fresh "UniqTac" in
      pose proof H as J1;
      pose proof H as J2;
      apply uniq_app_1 in H;
      apply uniq_app_2 in J1;
      apply uniq_app_3 in J2;
      destruct_uniq
    | H : uniq (map ?f ?E) |- _ =>
      apply uniq_map_1 in H;
      destruct_uniq
    | H : disjoint nil ?E |- _ =>
      clear H;
      destruct_uniq
    | H : disjoint (?x ~ ?a) ?F |- _ =>
      apply disjoint_one_1 in H;
      autorewrite with rewr_dom in H;
      destruct_uniq
    | H : disjoint ((?x, ?a) :: ?E) ?F |- _ =>
      let J := fresh "UniqTac" in
      pose proof H as J;
      apply disjoint_cons_1 in H;
      apply disjoint_cons_2 in J;
      autorewrite with rewr_dom in H;
      destruct_uniq
    | H : disjoint (?E ++ ?F) ?G |- _ =>
      let J := fresh "UniqTac" in
      pose proof H as J;
      apply disjoint_app_1 in H;
      apply disjoint_app_2 in J;
      destruct_uniq
    | H : disjoint (map ?f ?E) ?F |- _ =>
      apply disjoint_map_1 in H;
      destruct_uniq
    | H : disjoint ?E nil |- _ =>
      clear H;
      destruct_uniq
    | H : disjoint ?F (?x ~ ?a) |- _ =>
      apply disjoint_sym_1 in H;
      destruct_uniq
    | H : disjoint ?F ((?x, ?a) :: ?E) |- _ =>
      apply disjoint_sym_1 in H;
      destruct_uniq
    | H : disjoint ?G (?E ++ ?F) |- _ =>
      apply disjoint_sym_1 in H;
      destruct_uniq
    | H : disjoint ?F (map ?f ?E) |- _ =>
      apply disjoint_sym_1 in H;
      destruct_uniq
    | _ =>
      idtac
  end.(** [solve_uniq] attempts to solve goals by first decomposing
    hypotheses about [disjoint] and [uniq] and then trying some
    simple, if perhaps slow, heuristics. *)

Ltac solve_uniq :=
  intros;
  destruct_uniq;
  repeat first [ apply uniq_push
               | apply uniq_cons_3
               | apply uniq_app_4
               | apply uniq_one_1
               | apply uniq_nil ];
  auto;
  try tauto;
  unfold disjoint in *;
  try fsetdec;
  fail "Not solvable by [solve_uniq]; try [destruct_uniq]".

Lemma fresh_mid_tail : uniq (F ++ (x ~ a) ++ E) -> ~ In x (dom E).