(* 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 \<subseteq> B \<Longrightarrow> f i \<in> B"
by blast
lemma Int_Diff_disjoint: "A \<inter> B \<inter> (A - B) = {}"
by blast
lemma Int_Diff_Un: "A \<inter> B \<union> (A - B) = A"
by blast
lemma mono_Un: "mono A \<Longrightarrow> (\<Union>i\<le>n. A i) = A n"
unfolding mono_def by auto
lemma disjnt_equiv_class: "equiv A r \<Longrightarrow> disjnt (r``{a}) (r``{b}) \<longleftrightarrow> (a, b) \<notin> 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 \<Rightarrow> bool" where "disjoint \<equiv> pairwise disjnt"
lemma disjoint_def: "disjoint A \<longleftrightarrow> (\<forall>a\<in>A. \<forall>b\<in>A. a \<noteq> b \<longrightarrow> a \<inter> b = {})"
unfolding pairwise_def disjnt_def by auto
lemma disjointI:
"(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<inter> b = {}) \<Longrightarrow> disjoint A"
unfolding disjoint_def by auto
lemma disjointD:
"disjoint A \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<inter> b = {}"
unfolding disjoint_def by auto
lemma disjoint_image: "inj_on f (\<Union>A) \<Longrightarrow> disjoint A \<Longrightarrow> disjoint ((`) f ` A)"
unfolding inj_on_def disjoint_def by blast
lemma assumes "disjoint (A \<union> B)"
shows disjoint_unionD1: "disjoint A" and disjoint_unionD2: "disjoint B"
using assms by (simp_all add: disjoint_def)
lemma disjoint_INT:
assumes *: "\<And>i. i \<in> I \<Longrightarrow> disjoint (F i)"
shows "disjoint {\<Inter>i\<in>I. X i | X. \<forall>i\<in>I. X i \<in> F i}"
proof (safe intro!: disjointI del: equalityI)
fix A B :: "'a \<Rightarrow> 'b set" assume "(\<Inter>i\<in>I. A i) \<noteq> (\<Inter>i\<in>I. B i)"
then obtain i where "A i \<noteq> B i" "i \<in> I"
by auto
moreover assume "\<forall>i\<in>I. A i \<in> F i" "\<forall>i\<in>I. B i \<in> F i"
ultimately show "(\<Inter>i\<in>I. A i) \<inter> (\<Inter>i\<in>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
subsubsection "Family of Disjoint Sets"
definition disjoint_family_on :: "('i \<Rightarrow> 'a set) \<Rightarrow> 'i set \<Rightarrow> bool" where
"disjoint_family_on A S \<longleftrightarrow> (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})"
abbreviation "disjoint_family A \<equiv> 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 \<in> A" "disjnt C (B x)"
proof -
have "False" if *: "\<forall>x \<in> A. \<exists>y. y \<in> C \<and> y \<in> B x"
proof -
obtain g where g: "\<forall>x \<in> A. g x \<in> C \<and> g x \<in> 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 \<Longrightarrow> i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> i \<noteq> j \<Longrightarrow> A i \<inter> A j = {}"
by (auto simp: disjoint_family_on_def)
lemma disjoint_family_subset: "disjoint_family A \<Longrightarrow> (\<And>x. B x \<subseteq> A x) \<Longrightarrow> disjoint_family B"
by (force simp add: disjoint_family_on_def)
lemma disjoint_family_on_bisimulation:
assumes "disjoint_family_on f S"
and "\<And>n m. n \<in> S \<Longrightarrow> m \<in> S \<Longrightarrow> n \<noteq> m \<Longrightarrow> f n \<inter> f m = {} \<Longrightarrow> g n \<inter> g m = {}"
shows "disjoint_family_on g S"
using assms unfolding disjoint_family_on_def by auto
lemma disjoint_family_on_mono:
"A \<subseteq> B \<Longrightarrow> disjoint_family_on f B \<Longrightarrow> disjoint_family_on f A"
unfolding disjoint_family_on_def by auto
lemma disjoint_family_Suc:
"(\<And>n. A n \<subseteq> A (Suc n)) \<Longrightarrow> disjoint_family (\<lambda>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 \<Longrightarrow> disjoint (A ` I)"
unfolding disjoint_family_on_def disjoint_def by force
lemma disjoint_family_on_vimageI: "disjoint_family_on F I \<Longrightarrow> disjoint_family_on (\<lambda>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 \<in> I" "n \<in> I" and "n \<noteq> m"
with i[THEN inj_onD, of n m] show "A n \<inter> A m = {}"
by (intro disjointD[OF d]) auto
qed
lemma disjoint_UN:
assumes F: "\<And>i. i \<in> I \<Longrightarrow> disjoint (F i)" and *: "disjoint_family_on (\<lambda>i. \<Union>F i) I"
shows "disjoint (\<Union>i\<in>I. F i)"
proof (safe intro!: disjointI del: equalityI)
fix A B i j assume "A \<noteq> B" "A \<in> F i" "i \<in> I" "B \<in> F j" "j \<in> I"
show "A \<inter> 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 \<noteq> 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 \<inter> B = {}"
by auto
qed
qed
lemma distinct_list_bind:
assumes "distinct xs" "\<And>x. x \<in> set xs \<Longrightarrow> distinct (f x)"
"disjoint_family_on (set \<circ> 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: "\<And>i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)"
shows "bij_betw f (\<Union>i\<in>I. A i) (\<Union>i\<in>I. A' i)"
unfolding bij_betw_def
proof
from bij show eq: "f ` \<Union>(A ` I) = \<Union>(A' ` I)"
by (auto simp: bij_betw_def image_UN)
show "inj_on f (\<Union>(A ` I))"
proof (rule inj_onI, clarify)
fix i j x y assume A: "i \<in> I" "j \<in> I" "x \<in> A i" "y \<in> A j" and B: "f x = f y"
from A bij[of i] bij[of j] have "f x \<in> A' i" "f y \<in> A' j"
by (auto simp: bij_betw_def)
with B have "A' i \<inter> A' j \<noteq> {}" 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 \<Longrightarrow> disjoint B \<Longrightarrow> \<Union>C \<inter> \<Union>B = {} \<Longrightarrow> disjoint (C \<union> B)"
using disjoint_UN[of "{C, B}" "\<lambda>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 "\<not>finite A" "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> {}" "disjoint_family_on f A"
shows "\<not>finite (\<Union>(f ` A))"
proof -
define g where "g x = (SOME y. y \<in> f x)" for x
have g: "g x \<in> f x" if "x \<in> 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 \<in> A" "y \<in> A" "g x = g y" "x \<noteq> y"
with g[of x] g[of y] have "g x \<in> f x" "g x \<in> 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 \<subseteq> \<Union>(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 \<Rightarrow> 'a set) \<Rightarrow> nat \<Rightarrow> 'a set" where
"disjointed A n = A n - (\<Union>i\<in>{0..<n}. A i)"
lemma finite_UN_disjointed_eq: "(\<Union>i\<in>{0..<n}. disjointed A i) = (\<Union>i\<in>{0..<n}. A i)"
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: "(\<Union>i. disjointed A i) = (\<Union>i. A i)"
by (rule UN_finite2_eq [where k=0])
(simp add: finite_UN_disjointed_eq)
lemma less_disjoint_disjointed: "m < n \<Longrightarrow> disjointed A m \<inter> 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 \<subseteq> 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 \<Longrightarrow> 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 \<Rightarrow> 'a set set \<Rightarrow> bool"
where
"partition_on A P \<longleftrightarrow> \<Union>P = A \<and> disjoint P \<and> {} \<notin> P"
lemma partition_onI:
"\<Union>P = A \<Longrightarrow> (\<And>p q. p \<in> P \<Longrightarrow> q \<in> P \<Longrightarrow> p \<noteq> q \<Longrightarrow> disjnt p q) \<Longrightarrow> {} \<notin> P \<Longrightarrow> partition_on A P"
by (auto simp: partition_on_def pairwise_def)
lemma partition_onD1: "partition_on A P \<Longrightarrow> A = \<Union>P"
by (auto simp: partition_on_def)
lemma partition_onD2: "partition_on A P \<Longrightarrow> disjoint P"
by (auto simp: partition_on_def)
lemma partition_onD3: "partition_on A P \<Longrightarrow> {} \<notin> P"
by (auto simp: partition_on_def)
subsection \<open>Constructions of partitions\<close>
lemma partition_on_empty: "partition_on {} P \<longleftrightarrow> P = {}"
unfolding partition_on_def by fastforce
lemma partition_on_space: "A \<noteq> {} \<Longrightarrow> partition_on A {A}"
by (auto simp: partition_on_def disjoint_def)
lemma partition_on_singletons: "partition_on A ((\<lambda>x. {x}) ` A)"
by (auto simp: partition_on_def disjoint_def)
lemma partition_on_transform:
assumes P: "partition_on A P"
assumes F_UN: "\<Union>(F ` P) = F (\<Union>P)" and F_disjnt: "\<And>p q. p \<in> P \<Longrightarrow> q \<in> P \<Longrightarrow> disjnt p q \<Longrightarrow> disjnt (F p) (F q)"
shows "partition_on (F A) (F ` P - {{}})"
proof -
have "\<Union>(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 \<Longrightarrow> partition_on (B \<inter> A) ((\<inter>) B ` P - {{}})"
by (intro partition_on_transform) (auto simp: disjnt_def)
lemma partition_on_vimage: "partition_on A P \<Longrightarrow> 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 \<in> P \<Longrightarrow> q \<in> P \<Longrightarrow> disjnt p q \<Longrightarrow> 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} \<subseteq> Pow (Pow A)"
unfolding partition_on_def by auto
show "finite (Pow (Pow A))"
using assms by simp
qed
lemma finite_elements: "finite A \<Longrightarrow> partition_on A P \<Longrightarrow> 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 "\<Union>(A // r) = A" "{} \<notin> A // r"
by (auto simp: refl_on_def quotient_def)
fix p q assume "p \<in> A // r" "q \<in> A // r" "p \<noteq> q"
then obtain x y where "x \<in> A" "y \<in> 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). \<exists>p \<in> P. x \<in> p \<and> y \<in> p}"
proof (rule equivI)
have "A = \<Union>P" "disjoint P" "{} \<notin> P"
using P by (auto simp: partition_on_def)
then show "refl_on A {(x, y). \<exists>p\<in>P. x \<in> p \<and> y \<in> p}"
unfolding refl_on_def by auto
show "trans {(x, y). \<exists>p\<in>P. x \<in> p \<and> y \<in> 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). \<exists>p \<in> P. x \<in> p \<and> y \<in> p} = P"
unfolding quotient_def
proof safe
fix x assume "x \<in> A"
then obtain p where "p \<in> P" "x \<in> p" "\<And>q. q \<in> P \<Longrightarrow> x \<in> q \<Longrightarrow> p = q"
using P by (auto simp: partition_on_def disjoint_def)
then have "{y. \<exists>p\<in>P. x \<in> p \<and> y \<in> p} = p"
by (safe intro!: bexI[of _ p]) simp
then show "{(x, y). \<exists>p\<in>P. x \<in> p \<and> y \<in> p} `` {x} \<in> P"
by (simp add: \<open>p \<in> P\<close>)
next
fix p assume "p \<in> P"
then have "p \<noteq> {}"
using P by (auto simp: partition_on_def)
then obtain x where "x \<in> p"
by auto
then have "x \<in> A" "\<And>q. q \<in> P \<Longrightarrow> x \<in> q \<Longrightarrow> 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 \<in> (\<Union>x\<in>A. {{(x, y). \<exists>p\<in>P. x \<in> p \<and> y \<in> p} `` {x}})"
by (auto intro: \<open>x \<in> A\<close>)
qed
lemma partition_on_alt: "partition_on A P \<longleftrightarrow> (\<exists>r. equiv A r \<and> P = A // r)"
by (auto simp: partition_on_eq_quotient intro!: partition_on_quotient intro: equiv_partition_on)
end