(* Title: HOL/Library/Set_Idioms.thy
Author: Lawrence Paulson (but borrowed from HOL Light)
*)
section \<open>Set Idioms\<close>
theory Set_Idioms
imports Countable_Set
begin
subsection\<open>Idioms for being a suitable union/intersection of something\<close>
definition union_of :: "('a set set \<Rightarrow> bool) \<Rightarrow> ('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
(infixr "union'_of" 60)
where "P union_of Q \<equiv> \<lambda>S. \<exists>\<U>. P \<U> \<and> \<U> \<subseteq> Collect Q \<and> \<Union>\<U> = S"
definition intersection_of :: "('a set set \<Rightarrow> bool) \<Rightarrow> ('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
(infixr "intersection'_of" 60)
where "P intersection_of Q \<equiv> \<lambda>S. \<exists>\<U>. P \<U> \<and> \<U> \<subseteq> Collect Q \<and> \<Inter>\<U> = S"
definition arbitrary:: "'a set set \<Rightarrow> bool" where "arbitrary \<U> \<equiv> True"
lemma union_of_inc: "\<lbrakk>P {S}; Q S\<rbrakk> \<Longrightarrow> (P union_of Q) S"
by (auto simp: union_of_def)
lemma intersection_of_inc:
"\<lbrakk>P {S}; Q S\<rbrakk> \<Longrightarrow> (P intersection_of Q) S"
by (auto simp: intersection_of_def)
lemma union_of_mono:
"\<lbrakk>(P union_of Q) S; \<And>x. Q x \<Longrightarrow> Q' x\<rbrakk> \<Longrightarrow> (P union_of Q') S"
by (auto simp: union_of_def)
lemma intersection_of_mono:
"\<lbrakk>(P intersection_of Q) S; \<And>x. Q x \<Longrightarrow> Q' x\<rbrakk> \<Longrightarrow> (P intersection_of Q') S"
by (auto simp: intersection_of_def)
lemma all_union_of:
"(\<forall>S. (P union_of Q) S \<longrightarrow> R S) \<longleftrightarrow> (\<forall>T. P T \<and> T \<subseteq> Collect Q \<longrightarrow> R(\<Union>T))"
by (auto simp: union_of_def)
lemma all_intersection_of:
"(\<forall>S. (P intersection_of Q) S \<longrightarrow> R S) \<longleftrightarrow> (\<forall>T. P T \<and> T \<subseteq> Collect Q \<longrightarrow> R(\<Inter>T))"
by (auto simp: intersection_of_def)
lemma intersection_ofE:
"\<lbrakk>(P intersection_of Q) S; \<And>T. \<lbrakk>P T; T \<subseteq> Collect Q\<rbrakk> \<Longrightarrow> R(\<Inter>T)\<rbrakk> \<Longrightarrow> R S"
by (auto simp: intersection_of_def)
lemma union_of_empty:
"P {} \<Longrightarrow> (P union_of Q) {}"
by (auto simp: union_of_def)
lemma intersection_of_empty:
"P {} \<Longrightarrow> (P intersection_of Q) UNIV"
by (auto simp: intersection_of_def)
text\<open> The arbitrary and finite cases\<close>
lemma arbitrary_union_of_alt:
"(arbitrary union_of Q) S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>U. Q U \<and> x \<in> U \<and> U \<subseteq> S)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
by (force simp: union_of_def arbitrary_def)
next
assume ?rhs
then have "{U. Q U \<and> U \<subseteq> S} \<subseteq> Collect Q" "\<Union>{U. Q U \<and> U \<subseteq> S} = S"
by auto
then show ?lhs
unfolding union_of_def arbitrary_def by blast
qed
lemma arbitrary_union_of_empty [simp]: "(arbitrary union_of P) {}"
by (force simp: union_of_def arbitrary_def)
lemma arbitrary_intersection_of_empty [simp]:
"(arbitrary intersection_of P) UNIV"
by (force simp: intersection_of_def arbitrary_def)
lemma arbitrary_union_of_inc:
"P S \<Longrightarrow> (arbitrary union_of P) S"
by (force simp: union_of_inc arbitrary_def)
lemma arbitrary_intersection_of_inc:
"P S \<Longrightarrow> (arbitrary intersection_of P) S"
by (force simp: intersection_of_inc arbitrary_def)
lemma arbitrary_union_of_complement:
"(arbitrary union_of P) S \<longleftrightarrow> (arbitrary intersection_of (\<lambda>S. P(- S))) (- S)" (is "?lhs = ?rhs")
proof
assume ?lhs
then obtain \<U> where "\<U> \<subseteq> Collect P" "S = \<Union>\<U>"
by (auto simp: union_of_def arbitrary_def)
then show ?rhs
unfolding intersection_of_def arbitrary_def
by (rule_tac x="uminus ` \<U>" in exI) auto
next
assume ?rhs
then obtain \<U> where "\<U> \<subseteq> {S. P (- S)}" "\<Inter>\<U> = - S"
by (auto simp: union_of_def intersection_of_def arbitrary_def)
then show ?lhs
unfolding union_of_def arbitrary_def
by (rule_tac x="uminus ` \<U>" in exI) auto
qed
lemma arbitrary_intersection_of_complement:
"(arbitrary intersection_of P) S \<longleftrightarrow> (arbitrary union_of (\<lambda>S. P(- S))) (- S)"
by (simp add: arbitrary_union_of_complement)
lemma arbitrary_union_of_idempot [simp]:
"arbitrary union_of arbitrary union_of P = arbitrary union_of P"
proof -
have 1: "\<exists>\<U>'\<subseteq>Collect P. \<Union>\<U>' = \<Union>\<U>" if "\<U> \<subseteq> {S. \<exists>\<V>\<subseteq>Collect P. \<Union>\<V> = S}" for \<U>
proof -
let ?\<W> = "{V. \<exists>\<V>. \<V>\<subseteq>Collect P \<and> V \<in> \<V> \<and> (\<exists>S \<in> \<U>. \<Union>\<V> = S)}"
have *: "\<And>x U. \<lbrakk>x \<in> U; U \<in> \<U>\<rbrakk> \<Longrightarrow> x \<in> \<Union>?\<W>"
using that
apply simp
apply (drule subsetD, assumption, auto)
done
show ?thesis
apply (rule_tac x="{V. \<exists>\<V>. \<V>\<subseteq>Collect P \<and> V \<in> \<V> \<and> (\<exists>S \<in> \<U>. \<Union>\<V> = S)}" in exI)
using that by (blast intro: *)
qed
have 2: "\<exists>\<U>'\<subseteq>{S. \<exists>\<U>\<subseteq>Collect P. \<Union>\<U> = S}. \<Union>\<U>' = \<Union>\<U>" if "\<U> \<subseteq> Collect P" for \<U>
by (metis (mono_tags, lifting) union_of_def arbitrary_union_of_inc that)
show ?thesis
unfolding union_of_def arbitrary_def by (force simp: 1 2)
qed
lemma arbitrary_intersection_of_idempot:
"arbitrary intersection_of arbitrary intersection_of P = arbitrary intersection_of P" (is "?lhs = ?rhs")
proof -
have "- ?lhs = - ?rhs"
unfolding arbitrary_intersection_of_complement by simp
then show ?thesis
by simp
qed
lemma arbitrary_union_of_Union:
"(\<And>S. S \<in> \<U> \<Longrightarrow> (arbitrary union_of P) S) \<Longrightarrow> (arbitrary union_of P) (\<Union>\<U>)"
by (metis union_of_def arbitrary_def arbitrary_union_of_idempot mem_Collect_eq subsetI)
lemma arbitrary_union_of_Un:
"\<lbrakk>(arbitrary union_of P) S; (arbitrary union_of P) T\<rbrakk>
\<Longrightarrow> (arbitrary union_of P) (S \<union> T)"
using arbitrary_union_of_Union [of "{S,T}"] by auto
lemma arbitrary_intersection_of_Inter:
"(\<And>S. S \<in> \<U> \<Longrightarrow> (arbitrary intersection_of P) S) \<Longrightarrow> (arbitrary intersection_of P) (\<Inter>\<U>)"
by (metis intersection_of_def arbitrary_def arbitrary_intersection_of_idempot mem_Collect_eq subsetI)
lemma arbitrary_intersection_of_Int:
"\<lbrakk>(arbitrary intersection_of P) S; (arbitrary intersection_of P) T\<rbrakk>
\<Longrightarrow> (arbitrary intersection_of P) (S \<inter> T)"
using arbitrary_intersection_of_Inter [of "{S,T}"] by auto
lemma arbitrary_union_of_Int_eq:
"(\<forall>S T. (arbitrary union_of P) S \<and> (arbitrary union_of P) T
\<longrightarrow> (arbitrary union_of P) (S \<inter> T))
\<longleftrightarrow> (\<forall>S T. P S \<and> P T \<longrightarrow> (arbitrary union_of P) (S \<inter> T))" (is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
by (simp add: arbitrary_union_of_inc)
next
assume R: ?rhs
show ?lhs
proof clarify
fix S :: "'a set" and T :: "'a set"
assume "(arbitrary union_of P) S" and "(arbitrary union_of P) T"
then obtain \<U> \<V> where *: "\<U> \<subseteq> Collect P" "\<Union>\<U> = S" "\<V> \<subseteq> Collect P" "\<Union>\<V> = T"
by (auto simp: union_of_def)
then have "(arbitrary union_of P) (\<Union>C\<in>\<U>. \<Union>D\<in>\<V>. C \<inter> D)"
using R by (blast intro: arbitrary_union_of_Union)
then show "(arbitrary union_of P) (S \<inter> T)"
by (simp add: Int_UN_distrib2 *)
qed
qed
lemma arbitrary_intersection_of_Un_eq:
"(\<forall>S T. (arbitrary intersection_of P) S \<and> (arbitrary intersection_of P) T
\<longrightarrow> (arbitrary intersection_of P) (S \<union> T)) \<longleftrightarrow>
(\<forall>S T. P S \<and> P T \<longrightarrow> (arbitrary intersection_of P) (S \<union> T))"
apply (simp add: arbitrary_intersection_of_complement)
using arbitrary_union_of_Int_eq [of "\<lambda>S. P (- S)"]
by (metis (no_types, lifting) arbitrary_def double_compl union_of_inc)
lemma finite_union_of_empty [simp]: "(finite union_of P) {}"
by (simp add: union_of_empty)
lemma finite_intersection_of_empty [simp]: "(finite intersection_of P) UNIV"
by (simp add: intersection_of_empty)
lemma finite_union_of_inc:
"P S \<Longrightarrow> (finite union_of P) S"
by (simp add: union_of_inc)
lemma finite_intersection_of_inc:
"P S \<Longrightarrow> (finite intersection_of P) S"
by (simp add: intersection_of_inc)
lemma finite_union_of_complement:
"(finite union_of P) S \<longleftrightarrow> (finite intersection_of (\<lambda>S. P(- S))) (- S)"
unfolding union_of_def intersection_of_def
apply safe
apply (rule_tac x="uminus ` \<U>" in exI, fastforce)+
done
lemma finite_intersection_of_complement:
"(finite intersection_of P) S \<longleftrightarrow> (finite union_of (\<lambda>S. P(- S))) (- S)"
by (simp add: finite_union_of_complement)
lemma finite_union_of_idempot [simp]:
"finite union_of finite union_of P = finite union_of P"
proof -
have "(finite union_of P) S" if S: "(finite union_of finite union_of P) S" for S
proof -
obtain \<U> where "finite \<U>" "S = \<Union>\<U>" and \<U>: "\<forall>U\<in>\<U>. \<exists>\<U>. finite \<U> \<and> (\<U> \<subseteq> Collect P) \<and> \<Union>\<U> = U"
using S unfolding union_of_def by (auto simp: subset_eq)
then obtain f where "\<forall>U\<in>\<U>. finite (f U) \<and> (f U \<subseteq> Collect P) \<and> \<Union>(f U) = U"
by metis
then show ?thesis
unfolding union_of_def \<open>S = \<Union>\<U>\<close>
by (rule_tac x = "snd ` Sigma \<U> f" in exI) (fastforce simp: \<open>finite \<U>\<close>)
qed
moreover
have "(finite union_of finite union_of P) S" if "(finite union_of P) S" for S
by (simp add: finite_union_of_inc that)
ultimately show ?thesis
by force
qed
lemma finite_intersection_of_idempot [simp]:
"finite intersection_of finite intersection_of P = finite intersection_of P"
by (force simp: finite_intersection_of_complement)
lemma finite_union_of_Union:
"\<lbrakk>finite \<U>; \<And>S. S \<in> \<U> \<Longrightarrow> (finite union_of P) S\<rbrakk> \<Longrightarrow> (finite union_of P) (\<Union>\<U>)"
using finite_union_of_idempot [of P]
by (metis mem_Collect_eq subsetI union_of_def)
lemma finite_union_of_Un:
"\<lbrakk>(finite union_of P) S; (finite union_of P) T\<rbrakk> \<Longrightarrow> (finite union_of P) (S \<union> T)"
by (auto simp: union_of_def)
lemma finite_intersection_of_Inter:
"\<lbrakk>finite \<U>; \<And>S. S \<in> \<U> \<Longrightarrow> (finite intersection_of P) S\<rbrakk> \<Longrightarrow> (finite intersection_of P) (\<Inter>\<U>)"
using finite_intersection_of_idempot [of P]
by (metis intersection_of_def mem_Collect_eq subsetI)
lemma finite_intersection_of_Int:
"\<lbrakk>(finite intersection_of P) S; (finite intersection_of P) T\<rbrakk>
\<Longrightarrow> (finite intersection_of P) (S \<inter> T)"
by (auto simp: intersection_of_def)
lemma finite_union_of_Int_eq:
"(\<forall>S T. (finite union_of P) S \<and> (finite union_of P) T \<longrightarrow> (finite union_of P) (S \<inter> T))
\<longleftrightarrow> (\<forall>S T. P S \<and> P T \<longrightarrow> (finite union_of P) (S \<inter> T))"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
by (simp add: finite_union_of_inc)
next
assume R: ?rhs
show ?lhs
proof clarify
fix S :: "'a set" and T :: "'a set"
assume "(finite union_of P) S" and "(finite union_of P) T"
then obtain \<U> \<V> where *: "\<U> \<subseteq> Collect P" "\<Union>\<U> = S" "finite \<U>" "\<V> \<subseteq> Collect P" "\<Union>\<V> = T" "finite \<V>"
by (auto simp: union_of_def)
then have "(finite union_of P) (\<Union>C\<in>\<U>. \<Union>D\<in>\<V>. C \<inter> D)"
using R
by (blast intro: finite_union_of_Union)
then show "(finite union_of P) (S \<inter> T)"
by (simp add: Int_UN_distrib2 *)
qed
qed
lemma finite_intersection_of_Un_eq:
"(\<forall>S T. (finite intersection_of P) S \<and>
(finite intersection_of P) T
\<longrightarrow> (finite intersection_of P) (S \<union> T)) \<longleftrightarrow>
(\<forall>S T. P S \<and> P T \<longrightarrow> (finite intersection_of P) (S \<union> T))"
apply (simp add: finite_intersection_of_complement)
using finite_union_of_Int_eq [of "\<lambda>S. P (- S)"]
by (metis (no_types, lifting) double_compl)
abbreviation finite' :: "'a set \<Rightarrow> bool"
where "finite' A \<equiv> finite A \<and> A \<noteq> {}"
lemma finite'_intersection_of_Int:
"\<lbrakk>(finite' intersection_of P) S; (finite' intersection_of P) T\<rbrakk>
\<Longrightarrow> (finite' intersection_of P) (S \<inter> T)"
by (auto simp: intersection_of_def)
lemma finite'_intersection_of_inc:
"P S \<Longrightarrow> (finite' intersection_of P) S"
by (simp add: intersection_of_inc)
subsection \<open>The ``Relative to'' operator\<close>
text\<open>A somewhat cheap but handy way of getting localized forms of various topological concepts
(open, closed, borel, fsigma, gdelta etc.)\<close>
definition relative_to :: "['a set \<Rightarrow> bool, 'a set, 'a set] \<Rightarrow> bool" (infixl "relative'_to" 55)
where "P relative_to S \<equiv> \<lambda>T. \<exists>U. P U \<and> S \<inter> U = T"
lemma relative_to_UNIV [simp]: "(P relative_to UNIV) S \<longleftrightarrow> P S"
by (simp add: relative_to_def)
lemma relative_to_imp_subset:
"(P relative_to S) T \<Longrightarrow> T \<subseteq> S"
by (auto simp: relative_to_def)
lemma all_relative_to: "(\<forall>S. (P relative_to U) S \<longrightarrow> Q S) \<longleftrightarrow> (\<forall>S. P S \<longrightarrow> Q(U \<inter> S))"
by (auto simp: relative_to_def)
lemma relative_toE: "\<lbrakk>(P relative_to U) S; \<And>S. P S \<Longrightarrow> Q(U \<inter> S)\<rbrakk> \<Longrightarrow> Q S"
by (auto simp: relative_to_def)
lemma relative_to_inc:
"P S \<Longrightarrow> (P relative_to U) (U \<inter> S)"
by (auto simp: relative_to_def)
lemma relative_to_relative_to [simp]:
"P relative_to S relative_to T = P relative_to (S \<inter> T)"
unfolding relative_to_def
by auto
lemma relative_to_compl:
"S \<subseteq> U \<Longrightarrow> ((P relative_to U) (U - S) \<longleftrightarrow> ((\<lambda>c. P(- c)) relative_to U) S)"
unfolding relative_to_def
by (metis Diff_Diff_Int Diff_eq double_compl inf.absorb_iff2)
lemma relative_to_subset_trans:
"\<lbrakk>(P relative_to U) S; S \<subseteq> T; T \<subseteq> U\<rbrakk> \<Longrightarrow> (P relative_to T) S"
unfolding relative_to_def by auto
lemma relative_to_mono:
"\<lbrakk>(P relative_to U) S; \<And>S. P S \<Longrightarrow> Q S\<rbrakk> \<Longrightarrow> (Q relative_to U) S"
unfolding relative_to_def by auto
lemma relative_to_subset_inc: "\<lbrakk>S \<subseteq> U; P S\<rbrakk> \<Longrightarrow> (P relative_to U) S"
unfolding relative_to_def by auto
lemma relative_to_Int:
"\<lbrakk>(P relative_to S) C; (P relative_to S) D; \<And>X Y. \<lbrakk>P X; P Y\<rbrakk> \<Longrightarrow> P(X \<inter> Y)\<rbrakk>
\<Longrightarrow> (P relative_to S) (C \<inter> D)"
unfolding relative_to_def by auto
lemma relative_to_Un:
"\<lbrakk>(P relative_to S) C; (P relative_to S) D; \<And>X Y. \<lbrakk>P X; P Y\<rbrakk> \<Longrightarrow> P(X \<union> Y)\<rbrakk>
\<Longrightarrow> (P relative_to S) (C \<union> D)"
unfolding relative_to_def by auto
lemma arbitrary_union_of_relative_to:
"((arbitrary union_of P) relative_to U) = (arbitrary union_of (P relative_to U))" (is "?lhs = ?rhs")
proof -
have "?rhs S" if L: "?lhs S" for S
proof -
obtain \<U> where "S = U \<inter> \<Union>\<U>" "\<U> \<subseteq> Collect P"
using L unfolding relative_to_def union_of_def by auto
then show ?thesis
unfolding relative_to_def union_of_def arbitrary_def
by (rule_tac x="(\<lambda>X. U \<inter> X) ` \<U>" in exI) auto
qed
moreover have "?lhs S" if R: "?rhs S" for S
proof -
obtain \<U> where "S = \<Union>\<U>" "\<forall>T\<in>\<U>. \<exists>V. P V \<and> U \<inter> V = T"
using R unfolding relative_to_def union_of_def by auto
then obtain f where f: "\<And>T. T \<in> \<U> \<Longrightarrow> P (f T)" "\<And>T. T \<in> \<U> \<Longrightarrow> U \<inter> (f T) = T"
by metis
then have "\<exists>\<U>'\<subseteq>Collect P. \<Union>\<U>' = \<Union> (f ` \<U>)"
by (metis image_subset_iff mem_Collect_eq)
moreover have eq: "U \<inter> \<Union> (f ` \<U>) = \<Union>\<U>"
using f by auto
ultimately show ?thesis
unfolding relative_to_def union_of_def arbitrary_def \<open>S = \<Union>\<U>\<close>
by metis
qed
ultimately show ?thesis
by blast
qed
lemma finite_union_of_relative_to:
"((finite union_of P) relative_to U) = (finite union_of (P relative_to U))" (is "?lhs = ?rhs")
proof -
have "?rhs S" if L: "?lhs S" for S
proof -
obtain \<U> where "S = U \<inter> \<Union>\<U>" "\<U> \<subseteq> Collect P" "finite \<U>"
using L unfolding relative_to_def union_of_def by auto
then show ?thesis
unfolding relative_to_def union_of_def
by (rule_tac x="(\<lambda>X. U \<inter> X) ` \<U>" in exI) auto
qed
moreover have "?lhs S" if R: "?rhs S" for S
proof -
obtain \<U> where "S = \<Union>\<U>" "\<forall>T\<in>\<U>. \<exists>V. P V \<and> U \<inter> V = T" "finite \<U>"
using R unfolding relative_to_def union_of_def by auto
then obtain f where f: "\<And>T. T \<in> \<U> \<Longrightarrow> P (f T)" "\<And>T. T \<in> \<U> \<Longrightarrow> U \<inter> (f T) = T"
by metis
then have "\<exists>\<U>'\<subseteq>Collect P. \<Union>\<U>' = \<Union> (f ` \<U>)"
by (metis image_subset_iff mem_Collect_eq)
moreover have eq: "U \<inter> \<Union> (f ` \<U>) = \<Union>\<U>"
using f by auto
ultimately show ?thesis
using \<open>finite \<U>\<close> f
unfolding relative_to_def union_of_def \<open>S = \<Union>\<U>\<close>
by (rule_tac x="\<Union> (f ` \<U>)" in exI) (metis finite_imageI image_subsetI mem_Collect_eq)
qed
ultimately show ?thesis
by blast
qed
lemma countable_union_of_relative_to:
"((countable union_of P) relative_to U) = (countable union_of (P relative_to U))" (is "?lhs = ?rhs")
proof -
have "?rhs S" if L: "?lhs S" for S
proof -
obtain \<U> where "S = U \<inter> \<Union>\<U>" "\<U> \<subseteq> Collect P" "countable \<U>"
using L unfolding relative_to_def union_of_def by auto
then show ?thesis
unfolding relative_to_def union_of_def
by (rule_tac x="(\<lambda>X. U \<inter> X) ` \<U>" in exI) auto
qed
moreover have "?lhs S" if R: "?rhs S" for S
proof -
obtain \<U> where "S = \<Union>\<U>" "\<forall>T\<in>\<U>. \<exists>V. P V \<and> U \<inter> V = T" "countable \<U>"
using R unfolding relative_to_def union_of_def by auto
then obtain f where f: "\<And>T. T \<in> \<U> \<Longrightarrow> P (f T)" "\<And>T. T \<in> \<U> \<Longrightarrow> U \<inter> (f T) = T"
by metis
then have "\<exists>\<U>'\<subseteq>Collect P. \<Union>\<U>' = \<Union> (f ` \<U>)"
by (metis image_subset_iff mem_Collect_eq)
moreover have eq: "U \<inter> \<Union> (f ` \<U>) = \<Union>\<U>"
using f by auto
ultimately show ?thesis
using \<open>countable \<U>\<close> f
unfolding relative_to_def union_of_def \<open>S = \<Union>\<U>\<close>
by (rule_tac x="\<Union> (f ` \<U>)" in exI) (metis countable_image image_subsetI mem_Collect_eq)
qed
ultimately show ?thesis
by blast
qed
lemma arbitrary_intersection_of_relative_to:
"((arbitrary intersection_of P) relative_to U) = ((arbitrary intersection_of (P relative_to U)) relative_to U)" (is "?lhs = ?rhs")
proof -
have "?rhs S" if L: "?lhs S" for S
proof -
obtain \<U> where \<U>: "S = U \<inter> \<Inter>\<U>" "\<U> \<subseteq> Collect P"
using L unfolding relative_to_def intersection_of_def by auto
show ?thesis
unfolding relative_to_def intersection_of_def arbitrary_def
proof (intro exI conjI)
show "U \<inter> (\<Inter>X\<in>\<U>. U \<inter> X) = S" "(\<inter>) U ` \<U> \<subseteq> {T. \<exists>Ua. P Ua \<and> U \<inter> Ua = T}"
using \<U> by blast+
qed auto
qed
moreover have "?lhs S" if R: "?rhs S" for S
proof -
obtain \<U> where "S = U \<inter> \<Inter>\<U>" "\<forall>T\<in>\<U>. \<exists>V. P V \<and> U \<inter> V = T"
using R unfolding relative_to_def intersection_of_def by auto
then obtain f where f: "\<And>T. T \<in> \<U> \<Longrightarrow> P (f T)" "\<And>T. T \<in> \<U> \<Longrightarrow> U \<inter> (f T) = T"
by metis
then have "f ` \<U> \<subseteq> Collect P"
by auto
moreover have eq: "U \<inter> \<Inter>(f ` \<U>) = U \<inter> \<Inter>\<U>"
using f by auto
ultimately show ?thesis
unfolding relative_to_def intersection_of_def arbitrary_def \<open>S = U \<inter> \<Inter>\<U>\<close>
by auto
qed
ultimately show ?thesis
by blast
qed
lemma finite_intersection_of_relative_to:
"((finite intersection_of P) relative_to U) = ((finite intersection_of (P relative_to U)) relative_to U)" (is "?lhs = ?rhs")
proof -
have "?rhs S" if L: "?lhs S" for S
proof -
obtain \<U> where \<U>: "S = U \<inter> \<Inter>\<U>" "\<U> \<subseteq> Collect P" "finite \<U>"
using L unfolding relative_to_def intersection_of_def by auto
show ?thesis
unfolding relative_to_def intersection_of_def
proof (intro exI conjI)
show "U \<inter> (\<Inter>X\<in>\<U>. U \<inter> X) = S" "(\<inter>) U ` \<U> \<subseteq> {T. \<exists>Ua. P Ua \<and> U \<inter> Ua = T}"
using \<U> by blast+
show "finite ((\<inter>) U ` \<U>)"
by (simp add: \<open>finite \<U>\<close>)
qed auto
qed
moreover have "?lhs S" if R: "?rhs S" for S
proof -
obtain \<U> where "S = U \<inter> \<Inter>\<U>" "\<forall>T\<in>\<U>. \<exists>V. P V \<and> U \<inter> V = T" "finite \<U>"
using R unfolding relative_to_def intersection_of_def by auto
then obtain f where f: "\<And>T. T \<in> \<U> \<Longrightarrow> P (f T)" "\<And>T. T \<in> \<U> \<Longrightarrow> U \<inter> (f T) = T"
by metis
then have "f ` \<U> \<subseteq> Collect P"
by auto
moreover have eq: "U \<inter> \<Inter> (f ` \<U>) = U \<inter> \<Inter> \<U>"
using f by auto
ultimately show ?thesis
unfolding relative_to_def intersection_of_def \<open>S = U \<inter> \<Inter>\<U>\<close>
using \<open>finite \<U>\<close>
by auto
qed
ultimately show ?thesis
by blast
qed
lemma countable_intersection_of_relative_to:
"((countable intersection_of P) relative_to U) = ((countable intersection_of (P relative_to U)) relative_to U)" (is "?lhs = ?rhs")
proof -
have "?rhs S" if L: "?lhs S" for S
proof -
obtain \<U> where \<U>: "S = U \<inter> \<Inter>\<U>" "\<U> \<subseteq> Collect P" "countable \<U>"
using L unfolding relative_to_def intersection_of_def by auto
show ?thesis
unfolding relative_to_def intersection_of_def
proof (intro exI conjI)
show "U \<inter> (\<Inter>X\<in>\<U>. U \<inter> X) = S" "(\<inter>) U ` \<U> \<subseteq> {T. \<exists>Ua. P Ua \<and> U \<inter> Ua = T}"
using \<U> by blast+
show "countable ((\<inter>) U ` \<U>)"
by (simp add: \<open>countable \<U>\<close>)
qed auto
qed
moreover have "?lhs S" if R: "?rhs S" for S
proof -
obtain \<U> where "S = U \<inter> \<Inter>\<U>" "\<forall>T\<in>\<U>. \<exists>V. P V \<and> U \<inter> V = T" "countable \<U>"
using R unfolding relative_to_def intersection_of_def by auto
then obtain f where f: "\<And>T. T \<in> \<U> \<Longrightarrow> P (f T)" "\<And>T. T \<in> \<U> \<Longrightarrow> U \<inter> (f T) = T"
by metis
then have "f ` \<U> \<subseteq> Collect P"
by auto
moreover have eq: "U \<inter> \<Inter> (f ` \<U>) = U \<inter> \<Inter> \<U>"
using f by auto
ultimately show ?thesis
unfolding relative_to_def intersection_of_def \<open>S = U \<inter> \<Inter>\<U>\<close>
using \<open>countable \<U>\<close> countable_image
by auto
qed
ultimately show ?thesis
by blast
qed
lemma countable_union_of_empty [simp]: "(countable union_of P) {}"
by (simp add: union_of_empty)
lemma countable_intersection_of_empty [simp]: "(countable intersection_of P) UNIV"
by (simp add: intersection_of_empty)
lemma countable_union_of_inc: "P S \<Longrightarrow> (countable union_of P) S"
by (simp add: union_of_inc)
lemma countable_intersection_of_inc: "P S \<Longrightarrow> (countable intersection_of P) S"
by (simp add: intersection_of_inc)
lemma countable_union_of_complement:
"(countable union_of P) S \<longleftrightarrow> (countable intersection_of (\<lambda>S. P(-S))) (-S)"
(is "?lhs=?rhs")
proof
assume ?lhs
then obtain \<U> where "countable \<U>" and \<U>: "\<U> \<subseteq> Collect P" "\<Union>\<U> = S"
by (metis union_of_def)
define \<U>' where "\<U>' \<equiv> (\<lambda>C. -C) ` \<U>"
have "\<U>' \<subseteq> {S. P (- S)}" "\<Inter>\<U>' = -S"
using \<U>'_def \<U> by auto
then show ?rhs
unfolding intersection_of_def by (metis \<U>'_def \<open>countable \<U>\<close> countable_image)
next
assume ?rhs
then obtain \<U> where "countable \<U>" and \<U>: "\<U> \<subseteq> {S. P (- S)}" "\<Inter>\<U> = -S"
by (metis intersection_of_def)
define \<U>' where "\<U>' \<equiv> (\<lambda>C. -C) ` \<U>"
have "\<U>' \<subseteq> Collect P" "\<Union> \<U>' = S"
using \<U>'_def \<U> by auto
then show ?lhs
unfolding union_of_def
by (metis \<U>'_def \<open>countable \<U>\<close> countable_image)
qed
lemma countable_intersection_of_complement:
"(countable intersection_of P) S \<longleftrightarrow> (countable union_of (\<lambda>S. P(- S))) (- S)"
by (simp add: countable_union_of_complement)
lemma countable_union_of_explicit:
assumes "P {}"
shows "(countable union_of P) S \<longleftrightarrow>
(\<exists>T. (\<forall>n::nat. P(T n)) \<and> \<Union>(range T) = S)" (is "?lhs=?rhs")
proof
assume ?lhs
then obtain \<U> where "countable \<U>" and \<U>: "\<U> \<subseteq> Collect P" "\<Union>\<U> = S"
by (metis union_of_def)
then show ?rhs
by (metis SUP_bot Sup_empty assms from_nat_into mem_Collect_eq range_from_nat_into subsetD)
next
assume ?rhs
then show ?lhs
by (metis countableI_type countable_image image_subset_iff mem_Collect_eq union_of_def)
qed
lemma countable_union_of_ascending:
assumes empty: "P {}" and Un: "\<And>T U. \<lbrakk>P T; P U\<rbrakk> \<Longrightarrow> P(T \<union> U)"
shows "(countable union_of P) S \<longleftrightarrow>
(\<exists>T. (\<forall>n. P(T n)) \<and> (\<forall>n. T n \<subseteq> T(Suc n)) \<and> \<Union>(range T) = S)" (is "?lhs=?rhs")
proof
assume ?lhs
then obtain T where T: "\<And>n::nat. P(T n)" "\<Union>(range T) = S"
by (meson empty countable_union_of_explicit)
have "P (\<Union> (T ` {..n}))" for n
by (induction n) (auto simp: atMost_Suc Un T)
with T show ?rhs
by (rule_tac x="\<lambda>n. \<Union>k\<le>n. T k" in exI) force
next
assume ?rhs
then show ?lhs
using empty countable_union_of_explicit by auto
qed
lemma countable_union_of_idem [simp]:
"countable union_of countable union_of P = countable union_of P" (is "?lhs=?rhs")
proof
fix S
show "(countable union_of countable union_of P) S = (countable union_of P) S"
proof
assume L: "?lhs S"
then obtain \<U> where "countable \<U>" and \<U>: "\<U> \<subseteq> Collect (countable union_of P)" "\<Union>\<U> = S"
by (metis union_of_def)
then have "\<forall>U \<in> \<U>. \<exists>\<V>. countable \<V> \<and> \<V> \<subseteq> Collect P \<and> U = \<Union>\<V>"
by (metis Ball_Collect union_of_def)
then obtain \<F> where \<F>: "\<forall>U \<in> \<U>. countable (\<F> U) \<and> \<F> U \<subseteq> Collect P \<and> U = \<Union>(\<F> U)"
by metis
have "countable (\<Union> (\<F> ` \<U>))"
using \<F> \<open>countable \<U>\<close> by blast
moreover have "\<Union> (\<F> ` \<U>) \<subseteq> Collect P"
by (simp add: Sup_le_iff \<F>)
moreover have "\<Union> (\<Union> (\<F> ` \<U>)) = S"
by auto (metis Union_iff \<F> \<U>(2))+
ultimately show "?rhs S"
by (meson union_of_def)
qed (simp add: countable_union_of_inc)
qed
lemma countable_intersection_of_idem [simp]:
"countable intersection_of countable intersection_of P =
countable intersection_of P"
by (force simp: countable_intersection_of_complement)
lemma countable_union_of_Union:
"\<lbrakk>countable \<U>; \<And>S. S \<in> \<U> \<Longrightarrow> (countable union_of P) S\<rbrakk>
\<Longrightarrow> (countable union_of P) (\<Union> \<U>)"
by (metis Ball_Collect countable_union_of_idem union_of_def)
lemma countable_union_of_UN:
"\<lbrakk>countable I; \<And>i. i \<in> I \<Longrightarrow> (countable union_of P) (U i)\<rbrakk>
\<Longrightarrow> (countable union_of P) (\<Union>i\<in>I. U i)"
by (metis (mono_tags, lifting) countable_image countable_union_of_Union imageE)
lemma countable_union_of_Un:
"\<lbrakk>(countable union_of P) S; (countable union_of P) T\<rbrakk>
\<Longrightarrow> (countable union_of P) (S \<union> T)"
by (smt (verit) Union_Un_distrib countable_Un le_sup_iff union_of_def)
lemma countable_intersection_of_Inter:
"\<lbrakk>countable \<U>; \<And>S. S \<in> \<U> \<Longrightarrow> (countable intersection_of P) S\<rbrakk>
\<Longrightarrow> (countable intersection_of P) (\<Inter> \<U>)"
by (metis countable_intersection_of_idem intersection_of_def mem_Collect_eq subsetI)
lemma countable_intersection_of_INT:
"\<lbrakk>countable I; \<And>i. i \<in> I \<Longrightarrow> (countable intersection_of P) (U i)\<rbrakk>
\<Longrightarrow> (countable intersection_of P) (\<Inter>i\<in>I. U i)"
by (metis (mono_tags, lifting) countable_image countable_intersection_of_Inter imageE)
lemma countable_intersection_of_inter:
"\<lbrakk>(countable intersection_of P) S; (countable intersection_of P) T\<rbrakk>
\<Longrightarrow> (countable intersection_of P) (S \<inter> T)"
by (simp add: countable_intersection_of_complement countable_union_of_Un)
lemma countable_union_of_Int:
assumes S: "(countable union_of P) S" and T: "(countable union_of P) T"
and Int: "\<And>S T. P S \<and> P T \<Longrightarrow> P(S \<inter> T)"
shows "(countable union_of P) (S \<inter> T)"
proof -
obtain \<U> where "countable \<U>" and \<U>: "\<U> \<subseteq> Collect P" "\<Union>\<U> = S"
using S by (metis union_of_def)
obtain \<V> where "countable \<V>" and \<V>: "\<V> \<subseteq> Collect P" "\<Union>\<V> = T"
using T by (metis union_of_def)
have "\<And>U V. \<lbrakk>U \<in> \<U>; V \<in> \<V>\<rbrakk> \<Longrightarrow> (countable union_of P) (U \<inter> V)"
using \<U> \<V> by (metis Ball_Collect countable_union_of_inc local.Int)
then have "(countable union_of P) (\<Union>U\<in>\<U>. \<Union>V\<in>\<V>. U \<inter> V)"
by (meson \<open>countable \<U>\<close> \<open>countable \<V>\<close> countable_union_of_UN)
moreover have "S \<inter> T = (\<Union>U\<in>\<U>. \<Union>V\<in>\<V>. U \<inter> V)"
by (simp add: \<U> \<V>)
ultimately show ?thesis
by presburger
qed
lemma countable_intersection_of_union:
assumes S: "(countable intersection_of P) S" and T: "(countable intersection_of P) T"
and Un: "\<And>S T. P S \<and> P T \<Longrightarrow> P(S \<union> T)"
shows "(countable intersection_of P) (S \<union> T)"
by (metis (mono_tags, lifting) Compl_Int S T Un compl_sup countable_intersection_of_complement countable_union_of_Int)
end