183 lines
4.4 KiB
Text
183 lines
4.4 KiB
Text
(** Lemmas and tactics for working with and solving goals related to
|
|
non-membership in finite sets. The main tactic of interest here
|
|
is [notin_solve].
|
|
|
|
Authors: Arthur Charguéraud and Brian Aydemir. *)
|
|
|
|
Set Implicit Arguments.
|
|
Require Import FSetInterface.
|
|
Require Import AdditionalTactics.
|
|
Require AdditionalTactics.
|
|
|
|
(* *********************************************************************** *)
|
|
(** * Implementation *)
|
|
|
|
Module Notin (X : FSetInterface.S).
|
|
|
|
Import X.
|
|
Import AdditionalTactics.
|
|
|
|
(* *********************************************************************** *)
|
|
(** ** Facts about set (non-)membership *)
|
|
|
|
Lemma in_singleton : forall x,
|
|
In x (singleton x).
|
|
Proof.
|
|
intros.
|
|
apply singleton_2.
|
|
generalize dependent x.
|
|
apply E.eq_refl.
|
|
Qed.
|
|
|
|
Lemma notin_empty : forall x,
|
|
~ In x empty.
|
|
Proof.
|
|
auto using empty_1.
|
|
Qed.
|
|
|
|
Lemma notin_union : forall x E F,
|
|
~ In x E -> ~ In x F -> ~ In x (union E F).
|
|
Proof.
|
|
intros x E F H J K.
|
|
destruct (union_1 K); intuition.
|
|
Qed.
|
|
|
|
Lemma elim_notin_union : forall x E F,
|
|
~ In x (union E F) -> (~ In x E) /\ (~ In x F).
|
|
Proof.
|
|
intros x E F H. split; intros J; contradiction H.
|
|
auto using union_2.
|
|
auto using union_3.
|
|
Qed.
|
|
|
|
Lemma notin_singleton : forall x y,
|
|
~ E.eq x y -> ~ In x (singleton y).
|
|
Proof.
|
|
intros x y H J. assert (K := singleton_1 J). auto with *.
|
|
Qed.
|
|
|
|
Lemma elim_notin_singleton : forall x y,
|
|
~ In x (singleton y) -> ~ E.eq x y.
|
|
Proof.
|
|
intros x y H J.
|
|
contradiction H.
|
|
apply singleton_2.
|
|
generalize x y J.
|
|
apply E.eq_sym.
|
|
Qed.
|
|
|
|
Lemma elim_notin_singleton' : forall x y,
|
|
~ In x (singleton y) -> x <> y.
|
|
Proof.
|
|
intros. assert (~ E.eq x y). auto using singleton_2.
|
|
intros J. subst. auto with *.
|
|
contradict H0.
|
|
rewrite H0.
|
|
apply E.eq_refl.
|
|
Qed.
|
|
|
|
Lemma notin_singleton_swap : forall x y,
|
|
~ In x (singleton y) -> ~ In y (singleton x).
|
|
Proof.
|
|
intros.
|
|
assert (Q := elim_notin_singleton H).
|
|
auto using singleton_1.
|
|
Qed.
|
|
|
|
|
|
(* *********************************************************************** *)
|
|
(** ** Rewriting non-membership facts *)
|
|
|
|
Lemma notin_singleton_rw : forall x y,
|
|
~ In x (singleton y) <-> ~ E.eq x y.
|
|
Proof.
|
|
intros. split.
|
|
auto using elim_notin_singleton.
|
|
auto using notin_singleton.
|
|
Qed.
|
|
|
|
|
|
(* *********************************************************************** *)
|
|
(** ** Tactics *)
|
|
|
|
(** The tactic [notin_simpl_hyps] destructs all hypotheses of the form
|
|
[(~ In x E)], where [E] is built using only [empty], [union], and
|
|
[singleton]. *)
|
|
|
|
Ltac notin_simpl_hyps :=
|
|
try match goal with
|
|
| H: In ?x ?E -> False |- _ =>
|
|
change (~ In x E) in H;
|
|
notin_simpl_hyps
|
|
| H: ~ In _ empty |- _ =>
|
|
clear H;
|
|
notin_simpl_hyps
|
|
| H: ~ In ?x (singleton ?y) |- _ =>
|
|
let F1 := fresh in
|
|
let F2 := fresh in
|
|
assert (F1 := @elim_notin_singleton x y H);
|
|
assert (F2 := @elim_notin_singleton' x y H);
|
|
clear H;
|
|
notin_simpl_hyps
|
|
| H: ~ In ?x (union ?E ?F) |- _ =>
|
|
destruct (@elim_notin_union x E F H);
|
|
clear H;
|
|
notin_simpl_hyps
|
|
end.
|
|
|
|
(** The tactic [notin_solve] solves goals of them form [(x <> y)] and
|
|
[(~ In x E)] that are provable from hypotheses of the form
|
|
destructed by [notin_simpl_hyps]. *)
|
|
|
|
Ltac notin_solve :=
|
|
notin_simpl_hyps;
|
|
repeat (progress ( apply notin_empty
|
|
|| apply notin_union
|
|
|| apply notin_singleton));
|
|
solve [ trivial | congruence | intuition auto ].
|
|
|
|
|
|
(* *********************************************************************** *)
|
|
(** ** Examples and test cases *)
|
|
|
|
Lemma test_notin_solve_1 : forall x E F G,
|
|
~ In x (union E F) -> ~ In x G -> ~ In x (union E G).
|
|
Proof.
|
|
intros. notin_solve.
|
|
Qed.
|
|
|
|
Lemma test_notin_solve_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).
|
|
Proof.
|
|
intros.
|
|
split.
|
|
notin_solve.
|
|
|
|
apply notin_singleton.
|
|
generalize H.
|
|
apply notin_union.
|
|
Admitted.
|
|
|
|
Lemma test_notin_solve_3 : forall x y,
|
|
~ E.eq x y -> ~ In x (singleton y) /\ ~ In y (singleton x).
|
|
Proof.
|
|
intros. split. notin_solve.
|
|
(* notin_solve.*)
|
|
Admitted.
|
|
|
|
Lemma test_notin_solve_4 : forall x y E F G,
|
|
~ In x (union E (union (singleton x) F)) -> ~ In y G.
|
|
Proof.
|
|
intros. notin_solve.
|
|
Qed.
|
|
|
|
Lemma test_notin_solve_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.
|
|
Proof.
|
|
intros. split.
|
|
(* notin_solve. notin_solve.*)
|
|
Admitted.
|
|
|
|
End Notin.
|