(* *********************************************************************** *)
(** * Tactics for non-membership *)

(** [destruct_notin] decomposes all hypotheses of the form [~ In x s]. *)

Ltac destruct_notin :=
  match goal with
    | H : In ?x ?s -> False |- _ =>
      change (~ In x s) in H;
      destruct_notin
    | |- In ?x ?s -> False =>
      change (~ In x s);
      destruct_notin
    | H : ~ In _ empty |- _ =>
      clear H;
      destruct_notin
    | H : ~ In ?y (add ?x ?s) |- _ =>
      let J1 := fresh "NotInTac" in
      let J2 := fresh "NotInTac" in
      pose proof H as J1;
      pose proof H as J2;
      apply notin_add_1 in H;
      apply notin_add_1' in J1;
      apply notin_add_2 in J2;
      destruct_notin
    | H : ~ In ?y (singleton ?x) |- _ =>
      let J := fresh "NotInTac" in
      pose proof H as J;
      apply notin_singleton_1 in H;
      apply notin_singleton_1' in J;
      destruct_notin
    | H : ~ In ?y (remove ?x ?s) |- _ =>
      let J := fresh "NotInTac" in
      apply notin_remove_1 in H;
      destruct H as [J | J];
      destruct_notin
    | H : ~ In ?x (union ?s ?s') |- _ =>
      let J := fresh "NotInTac" in
      pose proof H as J;
      apply notin_union_1 in H;
      apply notin_union_2 in J;
      destruct_notin
    | H : ~ In ?x (inter ?s ?s') |- _ =>
      let J := fresh "NotInTac" in
      apply notin_inter_1 in H;
      destruct H as [J | J];
      destruct_notin
    | H : ~ In ?x (diff ?s ?s') |- _ =>
      let J := fresh "NotInTac" in
      apply notin_diff_1 in H;
      destruct H as [J | J];
      destruct_notin
    | _ =>
      idtac
  end.Ltac solve_notin :=
  intros;
  destruct_notin;
  repeat first [ apply notin_union_3
               | apply notin_add_3
               | apply notin_singleton_2
               | apply notin_empty_1
               ];
  auto with Auto_notin;
  try tauto;
  fail "Not solvable by [solve_notin]; try [destruct_notin]".

(* Specifications: *)
Lemma test_solve_notin_1 : forall x E F G, ~ In x (union E F) -> ~ In x G -> ~ In x (union E G).
Lemma test_solve_notin_2 : forall x y E F G, ~ In x (union E (union (singleton y) F)) -> ~ In x G -> ~ In x (singleton y) /\ ~ In y (singleton x).
Lemma test_solve_notin_3 : forall x y, ~ E.eq x y -> ~ In x (singleton y) /\ ~ In y (singleton x).
Lemma test_solve_notin_4 : forall x y E F G, ~ In x (union E (union (singleton x) F)) -> ~ In y G.
Lemma test_solve_notin_5 : forall x y E F, ~ In x (union E (union (singleton y) F)) -> ~ In y E -> ~ E.eq y x /\ ~ E.eq x y.
Lemma test_solve_notin_6 : forall x y E, ~ In x (add y E) -> ~ E.eq x y /\ ~ In x E.
Lemma test_solve_notin_7 : forall x, ~ In x (singleton x) -> False.