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(*  Title:      HOL/Library/Disjoint_Sets.thy
    Author:     Johannes Hölzl, TU München
*)

section \<open>Partitions and Disjoint Sets\<close>

theory Disjoint_Sets
  imports Main
begin

lemma range_subsetD: "range f \ B \ f i \ B"
  by blast

lemma Int_Diff_disjoint: "A \ B \ (A - B) = {}"
  by blast

lemma Int_Diff_Un: "A \ B \ (A - B) = A"
  by blast

lemma mono_Un: "mono A \ (\i\n. A i) = A n"
  unfolding mono_def by auto

lemma disjnt_equiv_class: "equiv A r \ disjnt (r``{a}) (r``{b}) \ (a, b) \ r"
  by (auto dest: equiv_class_self simp: equiv_class_eq_iff disjnt_def)

subsection \<open>Set of Disjoint Sets\<close>

abbreviation disjoint :: "'a set set \ bool" where "disjoint \ pairwise disjnt"

lemma disjoint_def: "disjoint A \ (\a\A. \b\A. a \ b \ a \ b = {})"
  unfolding pairwise_def disjnt_def by auto

lemma disjointI:
  "(\a b. a \ A \ b \ A \ a \ b \ a \ b = {}) \ disjoint A"
  unfolding disjoint_def by auto

lemma disjointD:
  "disjoint A \ a \ A \ b \ A \ a \ b \ a \ b = {}"
  unfolding disjoint_def by auto

lemma disjoint_image: "inj_on f (\A) \ disjoint A \ disjoint ((`) f ` A)"
  unfolding inj_on_def disjoint_def by blast

lemma assumes "disjoint (A \ B)"
      shows   disjoint_unionD1: "disjoint A" and disjoint_unionD2: "disjoint B"
  using assms by (simp_all add: disjoint_def)

lemma disjoint_INT:
  assumes *: "\i. i \ I \ disjoint (F i)"
  shows "disjoint {\i\I. X i | X. \i\I. X i \ F i}"
proof (safe intro!: disjointI del: equalityI)
  fix A B :: "'a \ 'b set" assume "(\i\I. A i) \ (\i\I. B i)"
  then obtain i where "A i \ B i" "i \ I"
    by auto
  moreover assume "\i\I. A i \ F i" "\i\I. B i \ F i"
  ultimately show "(\i\I. A i) \ (\i\I. B i) = {}"
    using *[OF \<open>i\<in>I\<close>, THEN disjointD, of "A i" "B i"]
    by (auto simp flip: INT_Int_distrib)
qed

lemma diff_Union_pairwise_disjoint:
  assumes "pairwise disjnt \" "\ \ \"
  shows "\\ - \\ = \(\ - \)"
proof -
  have "False"
    if x: "x \ A" "x \ B" and AB: "A \ \" "A \ \" "B \ \" for x A B
  proof -
    have "A \ B = {}"
      using assms disjointD AB by blast
  with x show ?thesis
    by blast
  qed
  then show ?thesis by auto
qed

lemma Int_Union_pairwise_disjoint:
  assumes "pairwise disjnt (\ \ \)"
  shows "\\ \ \\ = \(\ \ \)"
proof -
  have "False"
    if x: "x \ A" "x \ B" and AB: "A \ \" "A \ \" "B \ \" for x A B
  proof -
    have "A \ B = {}"
      using assms disjointD AB by blast
  with x show ?thesis
    by blast
  qed
  then show ?thesis by auto
qed

lemma psubset_Union_pairwise_disjoint:
  assumes \<B>: "pairwise disjnt \<B>" and "\<A> \<subset> \<B> - {{}}"
  shows "\\ \ \\"
  unfolding psubset_eq
proof
  show "\\ \ \\"
    using assms by blast
  have "\ \ \" "\(\ - \ \ (\ - {{}})) \ {}"
    using assms by blast+
  then show "\\ \ \\"
    using diff_Union_pairwise_disjoint [OF \<B>] by blast
qed

subsubsection "Family of Disjoint Sets"

definition disjoint_family_on :: "('i \ 'a set) \ 'i set \ bool" where
  "disjoint_family_on A S \ (\m\S. \n\S. m \ n \ A m \ A n = {})"

abbreviation "disjoint_family A \ disjoint_family_on A UNIV"

lemma disjoint_family_elem_disjnt:
  assumes "infinite A" "finite C"
      and df: "disjoint_family_on B A"
  obtains x where "x \ A" "disjnt C (B x)"
proof -
  have "False" if *: "\x \ A. \y. y \ C \ y \ B x"
  proof -
    obtain g where g: "\x \ A. g x \ C \ g x \ B x"
      using * by metis
    with df have "inj_on g A"
      by (fastforce simp add: inj_on_def disjoint_family_on_def)
    then have "infinite (g ` A)"
      using \<open>infinite A\<close> finite_image_iff by blast
    then show False
      by (meson \<open>finite C\<close> finite_subset g image_subset_iff)
  qed
  then show ?thesis
    by (force simp: disjnt_iff intro: that)
qed

lemma disjoint_family_onD:
  "disjoint_family_on A I \ i \ I \ j \ I \ i \ j \ A i \ A j = {}"
  by (auto simp: disjoint_family_on_def)

lemma disjoint_family_subset: "disjoint_family A \ (\x. B x \ A x) \ disjoint_family B"
  by (force simp add: disjoint_family_on_def)

lemma disjoint_family_on_bisimulation:
  assumes "disjoint_family_on f S"
  and "\n m. n \ S \ m \ S \ n \ m \ f n \ f m = {} \ g n \ g m = {}"
  shows "disjoint_family_on g S"
  using assms unfolding disjoint_family_on_def by auto

lemma disjoint_family_on_mono:
  "A \ B \ disjoint_family_on f B \ disjoint_family_on f A"
  unfolding disjoint_family_on_def by auto

lemma disjoint_family_Suc:
  "(\n. A n \ A (Suc n)) \ disjoint_family (\i. A (Suc i) - A i)"
  using lift_Suc_mono_le[of A]
  by (auto simp add: disjoint_family_on_def)
     (metis insert_absorb insert_subset le_SucE le_antisym not_le_imp_less less_imp_le)

lemma disjoint_family_on_disjoint_image:
  "disjoint_family_on A I \ disjoint (A ` I)"
  unfolding disjoint_family_on_def disjoint_def by force

lemma disjoint_family_on_vimageI: "disjoint_family_on F I \ disjoint_family_on (\i. f -` F i) I"
  by (auto simp: disjoint_family_on_def)

lemma disjoint_image_disjoint_family_on:
  assumes d: "disjoint (A ` I)" and i: "inj_on A I"
  shows "disjoint_family_on A I"
  unfolding disjoint_family_on_def
proof (intro ballI impI)
  fix n m assume nm: "m \ I" "n \ I" and "n \ m"
  with i[THEN inj_onD, of n m] show "A n \ A m = {}"
    by (intro disjointD[OF d]) auto
qed

lemma disjoint_UN:
  assumes F: "\i. i \ I \ disjoint (F i)" and *: "disjoint_family_on (\i. \(F i)) I"
  shows "disjoint (\i\I. F i)"
proof (safe intro!: disjointI del: equalityI)
  fix A B i j assume "A \ B" "A \ F i" "i \ I" "B \ F j" "j \ I"
  show "A \ B = {}"
  proof cases
    assume "i = j" with F[of i] \<open>i \<in> I\<close> \<open>A \<in> F i\<close> \<open>B \<in> F j\<close> \<open>A \<noteq> B\<close> show "A \<inter> B = {}"
      by (auto dest: disjointD)
  next
    assume "i \ j"
    with * \<open>i\<in>I\<close> \<open>j\<in>I\<close> have "(\<Union>(F i)) \<inter> (\<Union>(F j)) = {}"
      by (rule disjoint_family_onD)
    with \<open>A\<in>F i\<close> \<open>i\<in>I\<close> \<open>B\<in>F j\<close> \<open>j\<in>I\<close>
    show "A \ B = {}"
      by auto
  qed
qed

lemma distinct_list_bind:
  assumes "distinct xs" "\x. x \ set xs \ distinct (f x)"
          "disjoint_family_on (set \ f) (set xs)"
  shows   "distinct (List.bind xs f)"
  using assms
  by (induction xs)
     (auto simp: disjoint_family_on_def distinct_map inj_on_def set_list_bind)

lemma bij_betw_UNION_disjoint:
  assumes disj: "disjoint_family_on A' I"
  assumes bij: "\i. i \ I \ bij_betw f (A i) (A' i)"
  shows   "bij_betw f (\i\I. A i) (\i\I. A' i)"
unfolding bij_betw_def
proof
  from bij show eq: "f ` \(A ` I) = \(A' ` I)"
    by (auto simp: bij_betw_def image_UN)
  show "inj_on f (\(A ` I))"
  proof (rule inj_onI, clarify)
    fix i j x y assume A: "i \ I" "j \ I" "x \ A i" "y \ A j" and B: "f x = f y"
    from A bij[of i] bij[of j] have "f x \ A' i" "f y \ A' j"
      by (auto simp: bij_betw_def)
    with B have "A' i \ A' j \ {}" by auto
    with disj A have "i = j" unfolding disjoint_family_on_def by blast
    with A B bij[of i] show "x = y" by (auto simp: bij_betw_def dest: inj_onD)
  qed
qed

lemma disjoint_union: "disjoint C \ disjoint B \ \C \ \B = {} \ disjoint (C \ B)"
  using disjoint_UN[of "{C, B}" "\x. x"] by (auto simp add: disjoint_family_on_def)

text \<open>
  The union of an infinite disjoint family of non-empty sets is infinite.
\<close>
lemma infinite_disjoint_family_imp_infinite_UNION:
  assumes "\finite A" "\x. x \ A \ f x \ {}" "disjoint_family_on f A"
  shows   "\finite (\(f ` A))"
proof -
  define g where "g x = (SOME y. y \ f x)" for x
  have g: "g x \ f x" if "x \ A" for x
    unfolding g_def by (rule someI_ex, insert assms(2) that) blast
  have inj_on_g: "inj_on g A"
  proof (rule inj_onI, rule ccontr)
    fix x y assume A: "x \ A" "y \ A" "g x = g y" "x \ y"
    with g[of x] g[of y] have "g x \ f x" "g x \ f y" by auto
    with A \<open>x \<noteq> y\<close> assms show False
      by (auto simp: disjoint_family_on_def inj_on_def)
  qed
  from g have "g ` A \ \(f ` A)" by blast
  moreover from inj_on_g \<open>\<not>finite A\<close> have "\<not>finite (g ` A)"
    using finite_imageD by blast
  ultimately show ?thesis using finite_subset by blast
qed


subsection \<open>Construct Disjoint Sequences\<close>

definition disjointed :: "(nat \ 'a set) \ nat \ 'a set" where
  "disjointed A n = A n - (\i\{0..

lemma finite_UN_disjointed_eq: "(\i\{0..i\{0..
proof (induct n)
  case 0 show ?case by simp
next
  case (Suc n)
  thus ?case by (simp add: atLeastLessThanSuc disjointed_def)
qed

lemma UN_disjointed_eq: "(\i. disjointed A i) = (\i. A i)"
  by (rule UN_finite2_eq [where k=0])
     (simp add: finite_UN_disjointed_eq)

lemma less_disjoint_disjointed: "m < n \ disjointed A m \ disjointed A n = {}"
  by (auto simp add: disjointed_def)

lemma disjoint_family_disjointed: "disjoint_family (disjointed A)"
  by (simp add: disjoint_family_on_def)
     (metis neq_iff Int_commute less_disjoint_disjointed)

lemma disjointed_subset: "disjointed A n \ A n"
  by (auto simp add: disjointed_def)

lemma disjointed_0[simp]: "disjointed A 0 = A 0"
  by (simp add: disjointed_def)

lemma disjointed_mono: "mono A \ disjointed A (Suc n) = A (Suc n) - A n"
  using mono_Un[of A] by (simp add: disjointed_def atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)

subsection \<open>Partitions\<close>

text \<open>
  Partitions \<^term>\<open>P\<close> of a set \<^term>\<open>A\<close>. We explicitly disallow empty sets.
\<close>

definition partition_on :: "'a set \ 'a set set \ bool"
where
  "partition_on A P \ \P = A \ disjoint P \ {} \ P"

lemma partition_onI:
  "\P = A \ (\p q. p \ P \ q \ P \ p \ q \ disjnt p q) \ {} \ P \ partition_on A P"
  by (auto simp: partition_on_def pairwise_def)

lemma partition_onD1: "partition_on A P \ A = \P"
  by (auto simp: partition_on_def)

lemma partition_onD2: "partition_on A P \ disjoint P"
  by (auto simp: partition_on_def)

lemma partition_onD3: "partition_on A P \ {} \ P"
  by (auto simp: partition_on_def)

subsection \<open>Constructions of partitions\<close>

lemma partition_on_empty: "partition_on {} P \ P = {}"
  unfolding partition_on_def by fastforce

lemma partition_on_space: "A \ {} \ partition_on A {A}"
  by (auto simp: partition_on_def disjoint_def)

lemma partition_on_singletons: "partition_on A ((\x. {x}) ` A)"
  by (auto simp: partition_on_def disjoint_def)

lemma partition_on_transform:
  assumes P: "partition_on A P"
  assumes F_UN: "\(F ` P) = F (\P)" and F_disjnt: "\p q. p \ P \ q \ P \ disjnt p q \ disjnt (F p) (F q)"
  shows "partition_on (F A) (F ` P - {{}})"
proof -
  have "\(F ` P - {{}}) = F A"
    unfolding P[THEN partition_onD1] F_UN[symmetric] by auto
  with P show ?thesis
    by (auto simp add: partition_on_def pairwise_def intro!: F_disjnt)
qed

lemma partition_on_restrict: "partition_on A P \ partition_on (B \ A) ((\) B ` P - {{}})"
  by (intro partition_on_transform) (auto simp: disjnt_def)

lemma partition_on_vimage: "partition_on A P \ partition_on (f -` A) ((-`) f ` P - {{}})"
  by (intro partition_on_transform) (auto simp: disjnt_def)

lemma partition_on_inj_image:
  assumes P: "partition_on A P" and f: "inj_on f A"
  shows "partition_on (f ` A) ((`) f ` P - {{}})"
proof (rule partition_on_transform[OF P])
  show "p \ P \ q \ P \ disjnt p q \ disjnt (f ` p) (f ` q)" for p q
    using f[THEN inj_onD] P[THEN partition_onD1] by (auto simp: disjnt_def)
qed auto

subsection \<open>Finiteness of partitions\<close>

lemma finitely_many_partition_on:
  assumes "finite A"
  shows "finite {P. partition_on A P}"
proof (rule finite_subset)
  show "{P. partition_on A P} \ Pow (Pow A)"
    unfolding partition_on_def by auto
  show "finite (Pow (Pow A))"
    using assms by simp
qed

lemma finite_elements: "finite A \ partition_on A P \ finite P"
  using partition_onD1[of A P] by (simp add: finite_UnionD)

subsection \<open>Equivalence of partitions and equivalence classes\<close>

lemma partition_on_quotient:
  assumes r: "equiv A r"
  shows "partition_on A (A // r)"
proof (rule partition_onI)
  from r have "refl_on A r"
    by (auto elim: equivE)
  then show "\(A // r) = A" "{} \ A // r"
    by (auto simp: refl_on_def quotient_def)

  fix p q assume "p \ A // r" "q \ A // r" "p \ q"
  then obtain x y where "x \ A" "y \ A" "p = r `` {x}" "q = r `` {y}"
    by (auto simp: quotient_def)
  with r equiv_class_eq_iff[OF r, of x y] \<open>p \<noteq> q\<close> show "disjnt p q"
    by (auto simp: disjnt_equiv_class)
qed

lemma equiv_partition_on:
  assumes P: "partition_on A P"
  shows "equiv A {(x, y). \p \ P. x \ p \ y \ p}"
proof (rule equivI)
  have "A = \P" "disjoint P" "{} \ P"
    using P by (auto simp: partition_on_def)
  then show "refl_on A {(x, y). \p\P. x \ p \ y \ p}"
    unfolding refl_on_def by auto
  show "trans {(x, y). \p\P. x \ p \ y \ p}"
    using \<open>disjoint P\<close> by (auto simp: trans_def disjoint_def)
qed (auto simp: sym_def)

lemma partition_on_eq_quotient:
  assumes P: "partition_on A P"
  shows "A // {(x, y). \p \ P. x \ p \ y \ p} = P"
  unfolding quotient_def
proof safe
  fix x assume "x \ A"
  then obtain p where "p \ P" "x \ p" "\q. q \ P \ x \ q \ p = q"
    using P by (auto simp: partition_on_def disjoint_def)
  then have "{y. \p\P. x \ p \ y \ p} = p"
    by (safe intro!: bexI[of _ p]) simp
  then show "{(x, y). \p\P. x \ p \ y \ p} `` {x} \ P"
    by (simp add: \<open>p \<in> P\<close>)
next
  fix p assume "p \ P"
  then have "p \ {}"
    using P by (auto simp: partition_on_def)
  then obtain x where "x \ p"
    by auto
  then have "x \ A" "\q. q \ P \ x \ q \ p = q"
    using P \<open>p \<in> P\<close> by (auto simp: partition_on_def disjoint_def)
  with \<open>p\<in>P\<close> \<open>x \<in> p\<close> have "{y. \<exists>p\<in>P. x \<in> p \<and> y \<in> p} = p"
    by (safe intro!: bexI[of _ p]) simp
  then show "p \ (\x\A. {{(x, y). \p\P. x \ p \ y \ p} `` {x}})"
    by (auto intro: \<open>x \<in> A\<close>)
qed

lemma partition_on_alt: "partition_on A P \ (\r. equiv A r \ P = A // r)"
  by (auto simp: partition_on_eq_quotient intro!: partition_on_quotient intro: equiv_partition_on)

end

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Datei: Sprache: Isabelle

Original von: Isabelle©

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^© Kompilation durch diese Firma

Original von: Isabelle^©