184 lines
4.4 KiB
Coq
184 lines
4.4 KiB
Coq
(** Lemmas and tactics for working with and solving goals related to
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non-membership in finite sets. The main tactic of interest here
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is [notin_solve].
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Authors: Arthur Charguéraud and Brian Aydemir. *)
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Set Implicit Arguments.
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Require Import FSetInterface.
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(* *********************************************************************** *)
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(** * Implementation *)
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Module Notin (X : FSetInterface.S).
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Import X.
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(* *********************************************************************** *)
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(** ** Facts about set (non-)membership *)
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Lemma in_singleton : forall x,
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In x (singleton x).
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Proof.
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intros.
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apply singleton_2.
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generalize dependent x.
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apply E.eq_refl.
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Qed.
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Lemma notin_empty : forall x,
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~ In x empty.
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Proof.
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auto using empty_1.
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Qed.
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Lemma notin_union : forall x E F,
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~ In x E -> ~ In x F -> ~ In x (union E F).
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Proof.
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intros x E F H J K.
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destruct (union_1 K); intuition.
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Qed.
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Lemma elim_notin_union : forall x E F,
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~ In x (union E F) -> (~ In x E) /\ (~ In x F).
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Proof.
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intros x E F H. split; intros J; contradiction H.
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auto using union_2.
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auto using union_3.
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Qed.
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Lemma notin_singleton : forall x y,
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~ E.eq x y -> ~ In x (singleton y).
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Proof.
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intros x y H J. assert (K := singleton_1 J). auto with *.
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Qed.
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Lemma elim_notin_singleton : forall x y,
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~ In x (singleton y) -> ~ E.eq x y.
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Proof.
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intros x y H J.
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contradiction H.
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apply singleton_2.
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generalize x y J.
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apply E.eq_sym.
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Qed.
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Lemma elim_notin_singleton' : forall x y,
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~ In x (singleton y) -> x <> y.
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Proof.
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intros. assert (~ E.eq x y). auto using singleton_2.
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intros J. subst. auto with *.
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contradict H0.
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rewrite H0.
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apply E.eq_refl.
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Qed.
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Lemma notin_singleton_swap : forall x y,
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~ In x (singleton y) -> ~ In y (singleton x).
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Proof.
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intros.
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assert (Q := elim_notin_singleton H).
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auto using singleton_1.
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Qed.
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(* *********************************************************************** *)
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(** ** Rewriting non-membership facts *)
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Lemma notin_singleton_rw : forall x y,
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~ In x (singleton y) <-> ~ E.eq x y.
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Proof.
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intros. split.
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auto using elim_notin_singleton.
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auto using notin_singleton.
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Qed.
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(* *********************************************************************** *)
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(** ** Tactics *)
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(** The tactic [notin_simpl_hyps] destructs all hypotheses of the form
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[(~ In x E)], where [E] is built using only [empty], [union], and
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[singleton]. *)
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Ltac notin_simpl_hyps :=
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try match goal with
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| H: In ?x ?E -> False |- _ =>
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change (~ In x E) in H;
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notin_simpl_hyps
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| H: ~ In _ empty |- _ =>
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clear H;
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notin_simpl_hyps
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| H: ~ In ?x (singleton ?y) |- _ =>
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let F1 := fresh in
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let F2 := fresh in
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assert (F1 := @elim_notin_singleton x y H);
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assert (F2 := @elim_notin_singleton' x y H);
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clear H;
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notin_simpl_hyps
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| H: ~ In ?x (union ?E ?F) |- _ =>
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destruct (@elim_notin_union x E F H);
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clear H;
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notin_simpl_hyps
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end.
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(** The tactic [notin_solve] solves goals of them form [(x <> y)] and
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[(~ In x E)] that are provable from hypotheses of the form
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destructed by [notin_simpl_hyps]. *)
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Ltac notin_solve :=
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notin_simpl_hyps;
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repeat (progress ( apply notin_empty
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|| apply notin_union
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|| apply notin_singleton));
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solve [ trivial | congruence | intuition auto ].
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(* *********************************************************************** *)
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(** ** Examples and test cases *)
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Lemma test_notin_solve_1 : forall x E F G,
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~ In x (union E F) -> ~ In x G -> ~ In x (union E G).
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Proof.
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intros. notin_solve.
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Qed.
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Lemma test_notin_solve_2 : forall x y E F G,
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~ In x (union E (union (singleton y) F)) -> ~ In x G ->
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~ In x (singleton y) /\ ~ In y (singleton x).
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Proof.
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intros.
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split.
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notin_solve.
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(*
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apply notin_singleton.
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generalize H.
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apply notin_union.
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*)
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Admitted.
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Lemma test_notin_solve_3 : forall x y,
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~ E.eq x y -> ~ In x (singleton y) /\ ~ In y (singleton x).
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Proof.
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intros. split. notin_solve.
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(* notin_solve.*)
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Admitted.
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Lemma test_notin_solve_4 : forall x y E F G,
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~ In x (union E (union (singleton x) F)) -> ~ In y G.
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Proof.
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intros. notin_solve.
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Qed.
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Lemma test_notin_solve_5 : forall x y E F,
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~ In x (union E (union (singleton y) F)) -> ~ In y E ->
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~ E.eq y x /\ ~ E.eq x y.
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Proof.
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intros. split.
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(* notin_solve. notin_solve.*)
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Admitted.
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End Notin.
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