# HG changeset patch
# User paulson <lp15@cam.ac.uk>
# Date 1573561985 0
# Node ID ec7cc76e88e51773b81ee19e6dca85f0265c3b87
# Parent  038727567817a16541e03c343372e2f398be03c7
New library material from the AFP entry ZFC_in_HOL

diff -r 038727567817 -r ec7cc76e88e5 src/HOL/Conditionally_Complete_Lattices.thy
--- a/src/HOL/Conditionally_Complete_Lattices.thy	Mon Nov 11 07:16:17 2019 +0000
+++ b/src/HOL/Conditionally_Complete_Lattices.thy	Tue Nov 12 12:33:05 2019 +0000
@@ -337,7 +337,8 @@
 lemma cINF_superset_mono: "A \<noteq> {} \<Longrightarrow> bdd_below (g ` B) \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> g x \<le> f x) \<Longrightarrow> \<Sqinter>(g ` B) \<le> \<Sqinter>(f ` A)"
   by (rule cINF_mono) auto
 
-lemma cSUP_subset_mono: "A \<noteq> {} \<Longrightarrow> bdd_above (g ` B) \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<le> g x) \<Longrightarrow> \<Squnion>(f ` A) \<le> \<Squnion>(g ` B)"
+lemma cSUP_subset_mono: 
+  "\<lbrakk>A \<noteq> {}; bdd_above (g ` B); A \<subseteq> B; \<And>x. x \<in> A \<Longrightarrow> f x \<le> g x\<rbrakk> \<Longrightarrow> \<Squnion> (f ` A) \<le> \<Squnion> (g ` B)"
   by (rule cSUP_mono) auto
 
 lemma less_eq_cInf_inter: "bdd_below A \<Longrightarrow> bdd_below B \<Longrightarrow> A \<inter> B \<noteq> {} \<Longrightarrow> inf (Inf A) (Inf B) \<le> Inf (A \<inter> B)"
diff -r 038727567817 -r ec7cc76e88e5 src/HOL/Library/Equipollence.thy
--- a/src/HOL/Library/Equipollence.thy	Mon Nov 11 07:16:17 2019 +0000
+++ b/src/HOL/Library/Equipollence.thy	Tue Nov 12 12:33:05 2019 +0000
@@ -1,7 +1,7 @@
 section \<open>Equipollence and Other Relations Connected with Cardinality\<close>
 
 theory "Equipollence"
-  imports FuncSet
+  imports FuncSet 
 begin
 
 subsection\<open>Eqpoll\<close>
@@ -21,6 +21,9 @@
 lemma eqpoll_iff_card_of_ordIso: "A \<approx> B \<longleftrightarrow> ordIso2 (card_of A) (card_of B)"
   by (simp add: card_of_ordIso eqpoll_def)
 
+lemma eqpoll_refl [iff]: "A \<approx> A"
+  by (simp add: card_of_refl eqpoll_iff_card_of_ordIso)
+
 lemma eqpoll_finite_iff: "A \<approx> B \<Longrightarrow> finite A \<longleftrightarrow> finite B"
   by (meson bij_betw_finite eqpoll_def)
 
@@ -63,6 +66,9 @@
 lemma subset_imp_lepoll: "A \<subseteq> B \<Longrightarrow> A \<lesssim> B"
   by (force simp: lepoll_def)
 
+lemma lepoll_refl [iff]: "A \<lesssim> A"
+  by (simp add: subset_imp_lepoll)
+
 lemma lepoll_iff: "A \<lesssim> B \<longleftrightarrow> (\<exists>g. A \<subseteq> g ` B)"
   unfolding lepoll_def
 proof safe
@@ -71,6 +77,9 @@
     by (rule_tac x="inv_into B g" in exI) (auto simp: inv_into_into inj_on_inv_into)
 qed (metis image_mono the_inv_into_onto)
 
+lemma empty_lepoll [iff]: "{} \<lesssim> A"
+  by (simp add: lepoll_iff)
+
 lemma subset_image_lepoll: "B \<subseteq> f ` A \<Longrightarrow> B \<lesssim> A"
   by (auto simp: lepoll_iff)
 
@@ -82,6 +91,13 @@
   apply (simp add: infinite_countable_subset)
   using infinite_iff_countable_subset by blast
 
+lemma lepoll_Pow_self: "A \<lesssim> Pow A"
+  unfolding lepoll_def inj_def
+  proof (intro exI conjI)
+    show "inj_on (\<lambda>x. {x}) A"
+      by (auto simp: inj_on_def)
+qed auto
+
 lemma bij_betw_iff_bijections:
   "bij_betw f A B \<longleftrightarrow> (\<exists>g. (\<forall>x \<in> A. f x \<in> B \<and> g(f x) = x) \<and> (\<forall>y \<in> B. g y \<in> A \<and> f(g y) = y))"
   (is "?lhs = ?rhs")
@@ -127,6 +143,35 @@
   by (metis (no_types, hide_lams) image_empty image_insert lepoll_iff that)
 qed
 
+lemma infinite_insert_lepoll:
+  assumes "infinite A" shows "insert a A \<lesssim> A"
+proof -
+  obtain f :: "nat \<Rightarrow> 'a" where "inj f" and f: "range f \<subseteq> A"
+    using assms infinite_countable_subset by blast
+  let ?g = "(\<lambda>z. if z=a then f 0 else if z \<in> range f then f (Suc (inv f z)) else z)"
+  show ?thesis
+    unfolding lepoll_def
+  proof (intro exI conjI)
+    show "inj_on ?g (insert a A)"
+      using inj_on_eq_iff [OF \<open>inj f\<close>]
+      by (auto simp: inj_on_def)
+    show "?g ` insert a A \<subseteq> A"
+      using f by auto
+  qed
+qed
+
+lemma infinite_insert_eqpoll: "infinite A \<Longrightarrow> insert a A \<approx> A"
+  by (simp add: lepoll_antisym infinite_insert_lepoll subset_imp_lepoll subset_insertI)
+
+lemma finite_lepoll_infinite:
+  assumes "infinite A" "finite B" shows "B \<lesssim> A"
+proof -
+  have "B \<lesssim> (UNIV::nat set)"
+    unfolding lepoll_def
+    using finite_imp_inj_to_nat_seg [OF \<open>finite B\<close>] by blast
+  then show ?thesis
+    using \<open>infinite A\<close> infinite_le_lepoll lepoll_trans by auto
+qed
 
 subsection\<open>The strict relation\<close>
 
@@ -154,7 +199,300 @@
 lemma lesspoll_eq_trans [trans]: "\<lbrakk>X \<prec> Y; Y \<approx> Z\<rbrakk> \<Longrightarrow> X \<prec> Z"
   using eqpoll_imp_lepoll lesspoll_trans2 by blast
 
-subsection\<open>Cartesian products\<close>
+lemma lesspoll_Pow_self: "A \<prec> Pow A"
+  unfolding lesspoll_def bij_betw_def eqpoll_def
+  by (meson lepoll_Pow_self Cantors_paradox)
+
+lemma finite_lesspoll_infinite:
+  assumes "infinite A" "finite B" shows "B \<prec> A"
+  by (meson assms eqpoll_finite_iff finite_lepoll_infinite lesspoll_def)
+
+subsection\<open>Mapping by an injection\<close>
+
+lemma inj_on_image_eqpoll_self: "inj_on f A \<Longrightarrow> f ` A \<approx> A"
+  by (meson bij_betw_def eqpoll_def eqpoll_sym)
+
+lemma inj_on_image_lepoll_1 [simp]:
+  assumes "inj_on f A" shows "f ` A \<lesssim> B \<longleftrightarrow> A \<lesssim> B"
+  by (meson assms image_lepoll lepoll_def lepoll_trans order_refl)
+
+lemma inj_on_image_lepoll_2 [simp]:
+  assumes "inj_on f B" shows "A \<lesssim> f ` B \<longleftrightarrow> A \<lesssim> B"
+  by (meson assms eq_iff image_lepoll lepoll_def lepoll_trans)
+
+lemma inj_on_image_lesspoll_1 [simp]:
+  assumes "inj_on f A" shows "f ` A \<prec> B \<longleftrightarrow> A \<prec> B"
+  by (meson assms image_lepoll le_less lepoll_def lesspoll_trans1)
+
+lemma inj_on_image_lesspoll_2 [simp]:
+  assumes "inj_on f B" shows "A \<prec> f ` B \<longleftrightarrow> A \<prec> B"
+  by (meson assms eqpoll_sym inj_on_image_eqpoll_self lesspoll_eq_trans)
+
+lemma inj_on_image_eqpoll_1 [simp]:
+  assumes "inj_on f A" shows "f ` A \<approx> B \<longleftrightarrow> A \<approx> B"
+  by (metis assms eqpoll_trans inj_on_image_eqpoll_self eqpoll_sym)
+
+lemma inj_on_image_eqpoll_2 [simp]:
+  assumes "inj_on f B" shows "A \<approx> f ` B \<longleftrightarrow> A \<approx> B"
+  by (metis assms inj_on_image_eqpoll_1 eqpoll_sym)
+
+subsection \<open>Inserting elements into sets\<close>
+
+lemma insert_lepoll_insertD:
+  assumes "insert u A \<lesssim> insert v B" "u \<notin> A" "v \<notin> B" shows "A \<lesssim> B"
+proof -
+  obtain f where inj: "inj_on f (insert u A)" and fim: "f ` (insert u A) \<subseteq> insert v B"
+    by (meson assms lepoll_def)
+  show ?thesis
+    unfolding lepoll_def
+  proof (intro exI conjI)
+    let ?g = "\<lambda>x\<in>A. if f x = v then f u else f x"
+    show "inj_on ?g A"
+      using inj \<open>u \<notin> A\<close> by (auto simp: inj_on_def)
+    show "?g ` A \<subseteq> B"
+      using fim \<open>u \<notin> A\<close> image_subset_iff inj inj_on_image_mem_iff by fastforce
+  qed
+qed
+
+lemma insert_eqpoll_insertD: "\<lbrakk>insert u A \<approx> insert v B; u \<notin> A; v \<notin> B\<rbrakk> \<Longrightarrow> A \<approx> B"
+  by (meson insert_lepoll_insertD eqpoll_imp_lepoll eqpoll_sym lepoll_antisym)
+
+lemma insert_lepoll_cong:
+  assumes "A \<lesssim> B" "b \<notin> B" shows "insert a A \<lesssim> insert b B"
+proof -
+  obtain f where f: "inj_on f A" "f ` A \<subseteq> B"
+    by (meson assms lepoll_def)
+  let ?f = "\<lambda>u \<in> insert a A. if u=a then b else f u"
+  show ?thesis
+    unfolding lepoll_def
+  proof (intro exI conjI)
+    show "inj_on ?f (insert a A)"
+      using f \<open>b \<notin> B\<close> by (auto simp: inj_on_def)
+    show "?f ` insert a A \<subseteq> insert b B"
+      using f \<open>b \<notin> B\<close> by auto
+  qed
+qed
+
+lemma insert_eqpoll_cong:
+     "\<lbrakk>A \<approx> B; a \<notin> A; b \<notin> B\<rbrakk> \<Longrightarrow> insert a A \<approx> insert b B"
+  apply (rule lepoll_antisym)
+  apply (simp add: eqpoll_imp_lepoll insert_lepoll_cong)+
+  by (meson eqpoll_imp_lepoll eqpoll_sym insert_lepoll_cong)
+
+lemma insert_eqpoll_insert_iff:
+     "\<lbrakk>a \<notin> A; b \<notin> B\<rbrakk> \<Longrightarrow> insert a A \<approx> insert b B  \<longleftrightarrow>  A \<approx> B"
+  by (meson insert_eqpoll_insertD insert_eqpoll_cong)
+
+lemma insert_lepoll_insert_iff:
+     " \<lbrakk>a \<notin> A; b \<notin> B\<rbrakk> \<Longrightarrow> (insert a A \<lesssim> insert b B) \<longleftrightarrow> (A \<lesssim> B)"
+  by (meson insert_lepoll_insertD insert_lepoll_cong)
+
+lemma less_imp_insert_lepoll:
+  assumes "A \<prec> B" shows "insert a A \<lesssim> B"
+proof -
+  obtain f where "inj_on f A" "f ` A \<subset> B"
+    using assms by (metis bij_betw_def eqpoll_def lepoll_def lesspoll_def psubset_eq)
+  then obtain b where b: "b \<in> B" "b \<notin> f ` A"
+    by auto
+  show ?thesis
+    unfolding lepoll_def
+  proof (intro exI conjI)
+    show "inj_on (f(a:=b)) (insert a A)"
+      using b \<open>inj_on f A\<close> by (auto simp: inj_on_def)
+    show "(f(a:=b)) ` insert a A \<subseteq> B"
+      using \<open>f ` A \<subset> B\<close>  by (auto simp: b)
+  qed
+qed
+
+lemma finite_insert_lepoll: "finite A \<Longrightarrow> (insert a A \<lesssim> A) \<longleftrightarrow> (a \<in> A)"
+proof (induction A rule: finite_induct)
+  case (insert x A)
+  then show ?case
+    apply (auto simp: insert_absorb)
+    by (metis insert_commute insert_iff insert_lepoll_insertD)
+qed auto
+
+
+subsection\<open>Binary sums and unions\<close>
+
+lemma Un_lepoll_mono:
+  assumes "A \<lesssim> C" "B \<lesssim> D" "disjnt C D" shows "A \<union> B \<lesssim> C \<union> D"
+proof -
+  obtain f g where inj: "inj_on f A" "inj_on g B" and fg: "f ` A \<subseteq> C" "g ` B \<subseteq> D"
+    by (meson assms lepoll_def)
+  have "inj_on (\<lambda>x. if x \<in> A then f x else g x) (A \<union> B)"
+    using inj \<open>disjnt C D\<close> fg unfolding disjnt_iff
+    by (fastforce intro: inj_onI dest: inj_on_contraD split: if_split_asm)
+  with fg show ?thesis
+    unfolding lepoll_def
+    by (rule_tac x="\<lambda>x. if x \<in> A then f x else g x" in exI) auto
+qed
+
+lemma Un_eqpoll_cong: "\<lbrakk>A \<approx> C; B \<approx> D; disjnt A B; disjnt C D\<rbrakk> \<Longrightarrow> A \<union> B \<approx> C \<union> D"
+  by (meson Un_lepoll_mono eqpoll_imp_lepoll eqpoll_sym lepoll_antisym)
+
+lemma sum_lepoll_mono:
+  assumes "A \<lesssim> C" "B \<lesssim> D" shows "A <+> B \<lesssim> C <+> D"
+proof -
+  obtain f g where "inj_on f A" "f ` A \<subseteq> C" "inj_on g B" "g ` B \<subseteq> D"
+    by (meson assms lepoll_def)
+  then show ?thesis
+    unfolding lepoll_def
+    by (rule_tac x="case_sum (Inl \<circ> f) (Inr \<circ> g)" in exI) (force simp: inj_on_def)
+qed
+
+lemma sum_eqpoll_cong: "\<lbrakk>A \<approx> C; B \<approx> D\<rbrakk> \<Longrightarrow> A <+> B \<approx> C <+> D"
+  by (meson eqpoll_imp_lepoll eqpoll_sym lepoll_antisym sum_lepoll_mono)
+
+subsection\<open>Binary Cartesian products\<close>
+
+lemma times_square_lepoll: "A \<lesssim> A \<times> A"
+  unfolding lepoll_def inj_def
+proof (intro exI conjI)
+  show "inj_on (\<lambda>x. (x,x)) A"
+    by (auto simp: inj_on_def)
+qed auto
+
+lemma times_commute_eqpoll: "A \<times> B \<approx> B \<times> A"
+  unfolding eqpoll_def
+  by (force intro: bij_betw_byWitness [where f = "\<lambda>(x,y). (y,x)" and f' = "\<lambda>(x,y). (y,x)"])
+
+lemma times_assoc_eqpoll: "(A \<times> B) \<times> C \<approx> A \<times> (B \<times> C)"
+  unfolding eqpoll_def
+  by (force intro: bij_betw_byWitness [where f = "\<lambda>((x,y),z). (x,(y,z))" and f' = "\<lambda>(x,(y,z)). ((x,y),z)"])
+
+lemma times_singleton_eqpoll: "{a} \<times> A \<approx> A"
+proof -
+  have "{a} \<times> A = (\<lambda>x. (a,x)) ` A"
+    by auto
+  also have "\<dots>  \<approx> A"
+    proof (rule inj_on_image_eqpoll_self)
+      show "inj_on (Pair a) A"
+        by (auto simp: inj_on_def)
+    qed
+    finally show ?thesis .
+qed
+
+lemma times_lepoll_mono:
+  assumes "A \<lesssim> C" "B \<lesssim> D" shows "A \<times> B \<lesssim> C \<times> D"
+proof -
+  obtain f g where "inj_on f A" "f ` A \<subseteq> C" "inj_on g B" "g ` B \<subseteq> D"
+    by (meson assms lepoll_def)
+  then show ?thesis
+    unfolding lepoll_def
+    by (rule_tac x="\<lambda>(x,y). (f x, g y)" in exI) (auto simp: inj_on_def)
+qed
+
+lemma times_eqpoll_cong: "\<lbrakk>A \<approx> C; B \<approx> D\<rbrakk> \<Longrightarrow> A \<times> B \<approx> C \<times> D"
+  by (metis eqpoll_imp_lepoll eqpoll_sym lepoll_antisym times_lepoll_mono)
+
+lemma
+  assumes "B \<noteq> {}" shows lepoll_times1: "A \<lesssim> A \<times> B" and lepoll_times2:  "A \<lesssim> B \<times> A"
+  using assms lepoll_iff by fastforce+
+
+lemma times_0_eqpoll: "{} \<times> A \<approx> {}"
+  by (simp add: eqpoll_iff_bijections)
+
+lemma Sigma_lepoll_mono:
+  assumes "A \<subseteq> C" "\<And>x. x \<in> A \<Longrightarrow> B x \<lesssim> D x" shows "Sigma A B \<lesssim> Sigma C D"
+proof -
+  have "\<And>x. x \<in> A \<Longrightarrow> \<exists>f. inj_on f (B x) \<and> f ` (B x) \<subseteq> D x"
+    by (meson assms lepoll_def)
+  then obtain f where  "\<And>x. x \<in> A \<Longrightarrow> inj_on (f x) (B x) \<and> f x ` B x \<subseteq> D x"
+    by metis
+  with \<open>A \<subseteq> C\<close> show ?thesis
+    unfolding lepoll_def inj_on_def
+    by (rule_tac x="\<lambda>(x,y). (x, f x y)" in exI) force
+qed
+
+lemma sum_times_distrib_eqpoll: "(A <+> B) \<times> C \<approx> (A \<times> C) <+> (B \<times> C)"
+  unfolding eqpoll_def
+proof
+  show "bij_betw (\<lambda>(x,z). case_sum(\<lambda>y. Inl(y,z)) (\<lambda>y. Inr(y,z)) x) ((A <+> B) \<times> C) (A \<times> C <+> B \<times> C)"
+    by (rule bij_betw_byWitness [where f' = "case_sum (\<lambda>(x,z). (Inl x, z)) (\<lambda>(y,z). (Inr y, z))"]) auto
+qed
+
+lemma prod_insert_eqpoll:
+  assumes "a \<notin> A" shows "insert a A \<times> B \<approx> B <+> A \<times> B"
+  unfolding eqpoll_def
+  proof
+  show "bij_betw (\<lambda>(x,y). if x=a then Inl y else Inr (x,y)) (insert a A \<times> B) (B <+> A \<times> B)"
+    by (rule bij_betw_byWitness [where f' = "case_sum (\<lambda>y. (a,y)) id"]) (auto simp: assms)
+qed
+
+subsection\<open>General Unions\<close>
+
+lemma Union_eqpoll_Times:
+  assumes B: "\<And>x. x \<in> A \<Longrightarrow> F x \<approx> B" and disj: "pairwise (\<lambda>x y. disjnt (F x) (F y)) A"
+  shows "(\<Union>x\<in>A. F x) \<approx> A \<times> B"
+proof (rule lepoll_antisym)
+  obtain b where b: "\<And>x. x \<in> A \<Longrightarrow> bij_betw (b x) (F x) B"
+    using B unfolding eqpoll_def by metis
+  show "\<Union>(F ` A) \<lesssim> A \<times> B"
+    unfolding lepoll_def
+  proof (intro exI conjI)
+    define \<chi> where "\<chi> \<equiv> \<lambda>z. THE x. x \<in> A \<and> z \<in> F x"
+    have \<chi>: "\<chi> z = x" if "x \<in> A" "z \<in> F x" for x z
+      unfolding \<chi>_def
+      apply (rule the_equality)
+      apply (simp add: that)
+      by (metis disj disjnt_iff pairwiseD that)
+    let ?f = "\<lambda>z. (\<chi> z, b (\<chi> z) z)"
+    show "inj_on ?f (\<Union>(F ` A))"
+      unfolding inj_on_def
+      by clarify (metis \<chi> b bij_betw_inv_into_left)
+    show "?f ` \<Union>(F ` A) \<subseteq> A \<times> B"
+      using \<chi> b bij_betwE by blast
+  qed
+  show "A \<times> B \<lesssim> \<Union>(F ` A)"
+    unfolding lepoll_def
+  proof (intro exI conjI)
+    let ?f = "\<lambda>(x,y). inv_into (F x) (b x) y"
+    have *: "inv_into (F x) (b x) y \<in> F x" if "x \<in> A" "y \<in> B" for x y
+      by (metis b bij_betw_imp_surj_on inv_into_into that)
+    then show "inj_on ?f (A \<times> B)"
+      unfolding inj_on_def
+      by clarsimp (metis (mono_tags, lifting) b bij_betw_inv_into_right disj disjnt_iff pairwiseD)
+    show "?f ` (A \<times> B) \<subseteq> \<Union> (F ` A)"
+      by clarsimp (metis b bij_betw_imp_surj_on inv_into_into)
+  qed
+qed
+
+lemma UN_lepoll_UN:
+  assumes A: "\<And>x. x \<in> A \<Longrightarrow> B x \<lesssim> C x"
+    and disj: "pairwise (\<lambda>x y. disjnt (C x) (C y)) A"
+  shows "\<Union> (B`A) \<lesssim> \<Union> (C`A)"
+proof -
+  obtain f where f: "\<And>x. x \<in> A \<Longrightarrow> inj_on (f x) (B x) \<and> f x ` (B x) \<subseteq> (C x)"
+    using A unfolding lepoll_def by metis
+  show ?thesis
+    unfolding lepoll_def
+  proof (intro exI conjI)
+    define \<chi> where "\<chi> \<equiv> \<lambda>z. @x. x \<in> A \<and> z \<in> B x"
+    have \<chi>: "\<chi> z \<in> A \<and> z \<in> B (\<chi> z)" if "x \<in> A" "z \<in> B x" for x z
+      unfolding \<chi>_def by (metis (mono_tags, lifting) someI_ex that)
+    let ?f = "\<lambda>z. (f (\<chi> z) z)"
+    show "inj_on ?f (\<Union>(B ` A))"
+      using disj f unfolding inj_on_def disjnt_iff pairwise_def image_subset_iff
+      by (metis UN_iff \<chi>)
+    show "?f ` \<Union> (B ` A) \<subseteq> \<Union> (C ` A)"
+      using \<chi> f unfolding image_subset_iff by blast
+  qed
+qed
+
+lemma UN_eqpoll_UN:
+  assumes A: "\<And>x. x \<in> A \<Longrightarrow> B x \<approx> C x"
+    and B: "pairwise (\<lambda>x y. disjnt (B x) (B y)) A"
+    and C: "pairwise (\<lambda>x y. disjnt (C x) (C y)) A"
+  shows "(\<Union>x\<in>A. B x) \<approx> (\<Union>x\<in>A. C x)"
+proof (rule lepoll_antisym)
+  show "\<Union> (B ` A) \<lesssim> \<Union> (C ` A)"
+    by (meson A C UN_lepoll_UN eqpoll_imp_lepoll)
+  show "\<Union> (C ` A) \<lesssim> \<Union> (B ` A)"
+    by (simp add: A B UN_lepoll_UN eqpoll_imp_lepoll eqpoll_sym)
+qed
+
+subsection\<open>General Cartesian products (Pi)\<close>
 
 lemma PiE_sing_eqpoll_self: "({a} \<rightarrow>\<^sub>E B) \<approx> B"
 proof -