--- a/src/HOL/Cardinals/Cardinal_Order_Relation.thy Thu Jan 24 10:04:32 2019 +0100
+++ b/src/HOL/Cardinals/Cardinal_Order_Relation.thy Thu Jan 24 14:44:52 2019 +0000
@@ -886,22 +886,7 @@
using lists_UNIV by auto
-subsection \<open>Cardinals versus the set-of-finite-sets operator\<close>
-
-definition Fpow :: "'a set \<Rightarrow> 'a set set"
-where "Fpow A \<equiv> {X. X \<le> A \<and> finite X}"
-
-lemma Fpow_mono: "A \<le> B \<Longrightarrow> Fpow A \<le> Fpow B"
-unfolding Fpow_def by auto
-
-lemma empty_in_Fpow: "{} \<in> Fpow A"
-unfolding Fpow_def by auto
-
-lemma Fpow_not_empty: "Fpow A \<noteq> {}"
-using empty_in_Fpow by blast
-
-lemma Fpow_subset_Pow: "Fpow A \<le> Pow A"
-unfolding Fpow_def by auto
+subsection \<open>Cardinals versus the finite powerset operator\<close>
lemma card_of_Fpow[simp]: "|A| \<le>o |Fpow A|"
proof-
@@ -914,20 +899,6 @@
lemma Card_order_Fpow: "Card_order r \<Longrightarrow> r \<le>o |Fpow(Field r) |"
using card_of_Fpow card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
-lemma Fpow_Pow_finite: "Fpow A = Pow A Int {A. finite A}"
-unfolding Fpow_def Pow_def by blast
-
-lemma inj_on_image_Fpow:
-assumes "inj_on f A"
-shows "inj_on (image f) (Fpow A)"
-using assms Fpow_subset_Pow[of A] subset_inj_on[of "image f" "Pow A"]
- inj_on_image_Pow by blast
-
-lemma image_Fpow_mono:
-assumes "f ` A \<le> B"
-shows "(image f) ` (Fpow A) \<le> Fpow B"
-using assms by(unfold Fpow_def, auto)
-
lemma image_Fpow_surjective:
assumes "f ` A = B"
shows "(image f) ` (Fpow A) = Fpow B"
--- a/src/HOL/Cardinals/Fun_More.thy Thu Jan 24 10:04:32 2019 +0100
+++ b/src/HOL/Cardinals/Fun_More.thy Thu Jan 24 14:44:52 2019 +0000
@@ -58,32 +58,6 @@
subsection \<open>Properties involving finite and infinite sets\<close>
-(*3*)lemma inj_on_image_Pow:
-assumes "inj_on f A"
-shows "inj_on (image f) (Pow A)"
-unfolding Pow_def inj_on_def proof(clarsimp)
- fix X Y assume *: "X \<le> A" and **: "Y \<le> A" and
- ***: "f ` X = f ` Y"
- show "X = Y"
- proof(auto)
- fix x assume ****: "x \<in> X"
- with *** obtain y where "y \<in> Y \<and> f x = f y" by blast
- with **** * ** assms show "x \<in> Y"
- unfolding inj_on_def by auto
- next
- fix y assume ****: "y \<in> Y"
- with *** obtain x where "x \<in> X \<and> f x = f y" by atomize_elim force
- with **** * ** assms show "y \<in> X"
- unfolding inj_on_def by auto
- qed
-qed
-
-(*2*)lemma bij_betw_image_Pow:
-assumes "bij_betw f A B"
-shows "bij_betw (image f) (Pow A) (Pow B)"
-using assms unfolding bij_betw_def
-by (auto simp add: inj_on_image_Pow image_Pow_surj)
-
(* unused *)
(*1*)lemma bij_betw_inv_into_RIGHT:
assumes BIJ: "bij_betw f A A'" and SUB: "B' \<le> A'"
--- a/src/HOL/Finite_Set.thy Thu Jan 24 10:04:32 2019 +0100
+++ b/src/HOL/Finite_Set.thy Thu Jan 24 14:44:52 2019 +0000
@@ -2182,4 +2182,35 @@
for S :: "'a::linordered_ring set"
by (rule ccontr) (auto simp: abs_eq_iff vimage_def dest: inf_img_fin_dom)
+subsection \<open>The finite powerset operator\<close>
+
+definition Fpow :: "'a set \<Rightarrow> 'a set set"
+where "Fpow A \<equiv> {X. X \<subseteq> A \<and> finite X}"
+
+lemma Fpow_mono: "A \<subseteq> B \<Longrightarrow> Fpow A \<subseteq> Fpow B"
+unfolding Fpow_def by auto
+
+lemma empty_in_Fpow: "{} \<in> Fpow A"
+unfolding Fpow_def by auto
+
+lemma Fpow_not_empty: "Fpow A \<noteq> {}"
+using empty_in_Fpow by blast
+
+lemma Fpow_subset_Pow: "Fpow A \<subseteq> Pow A"
+unfolding Fpow_def by auto
+
+lemma Fpow_Pow_finite: "Fpow A = Pow A Int {A. finite A}"
+unfolding Fpow_def Pow_def by blast
+
+lemma inj_on_image_Fpow:
+ assumes "inj_on f A"
+ shows "inj_on (image f) (Fpow A)"
+ using assms Fpow_subset_Pow[of A] subset_inj_on[of "image f" "Pow A"]
+ inj_on_image_Pow by blast
+
+lemma image_Fpow_mono:
+ assumes "f ` A \<subseteq> B"
+ shows "(image f) ` (Fpow A) \<subseteq> Fpow B"
+ using assms by(unfold Fpow_def, auto)
+
end
--- a/src/HOL/Fun.thy Thu Jan 24 10:04:32 2019 +0100
+++ b/src/HOL/Fun.thy Thu Jan 24 14:44:52 2019 +0000
@@ -292,6 +292,13 @@
by (rule linorder_cases) (auto dest: assms simp: that)
qed
+
+lemma inj_on_image_Pow: "inj_on f A \<Longrightarrow>inj_on (image f) (Pow A)"
+ unfolding Pow_def inj_on_def by blast
+
+lemma bij_betw_image_Pow: "bij_betw f A B \<Longrightarrow> bij_betw (image f) (Pow A) (Pow B)"
+ by (auto simp add: bij_betw_def inj_on_image_Pow image_Pow_surj)
+
lemma surj_def: "surj f \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)"
by auto
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Equipollence.thy Thu Jan 24 14:44:52 2019 +0000
@@ -0,0 +1,345 @@
+section \<open>Equipollence and Other Relations Connected with Cardinality\<close>
+
+theory "Equipollence"
+ imports FuncSet
+begin
+
+subsection\<open>Eqpoll\<close>
+
+definition eqpoll :: "'a set \<Rightarrow> 'b set \<Rightarrow> bool" (infixl "\<approx>" 50)
+ where "eqpoll A B \<equiv> \<exists>f. bij_betw f A B"
+
+definition lepoll :: "'a set \<Rightarrow> 'b set \<Rightarrow> bool" (infixl "\<lesssim>" 50)
+ where "lepoll A B \<equiv> \<exists>f. inj_on f A \<and> f ` A \<subseteq> B"
+
+definition lesspoll :: "'a set \<Rightarrow> 'b set \<Rightarrow> bool" (infixl \<open>\<prec>\<close> 50)
+ where "A \<prec> B == A \<lesssim> B \<and> ~(A \<approx> B)"
+
+lemma lepoll_empty_iff_empty [simp]: "A \<lesssim> {} \<longleftrightarrow> A = {}"
+ by (auto simp: lepoll_def)
+
+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_finite_iff: "A \<approx> B \<Longrightarrow> finite A \<longleftrightarrow> finite B"
+ by (meson bij_betw_finite eqpoll_def)
+
+lemma eqpoll_iff_card:
+ assumes "finite A" "finite B"
+ shows "A \<approx> B \<longleftrightarrow> card A = card B"
+ using assms by (auto simp: bij_betw_iff_card eqpoll_def)
+
+lemma lepoll_antisym:
+ assumes "A \<lesssim> B" "B \<lesssim> A" shows "A \<approx> B"
+ using assms unfolding eqpoll_def lepoll_def by (metis Schroeder_Bernstein)
+
+lemma lepoll_trans [trans]: "\<lbrakk>A \<lesssim> B; B \<lesssim> C\<rbrakk> \<Longrightarrow> A \<lesssim> C"
+ apply (clarsimp simp: lepoll_def)
+ apply (rename_tac f g)
+ apply (rule_tac x="g \<circ> f" in exI)
+ apply (auto simp: image_subset_iff inj_on_def)
+ done
+
+lemma lepoll_trans1 [trans]: "\<lbrakk>A \<approx> B; B \<lesssim> C\<rbrakk> \<Longrightarrow> A \<lesssim> C"
+ by (meson card_of_ordLeq eqpoll_iff_card_of_ordIso lepoll_def lepoll_trans ordIso_iff_ordLeq)
+
+lemma lepoll_trans2 [trans]: "\<lbrakk>A \<lesssim> B; B \<approx> C\<rbrakk> \<Longrightarrow> A \<lesssim> C"
+ apply (clarsimp simp: eqpoll_def lepoll_def bij_betw_def)
+ apply (rename_tac f g)
+ apply (rule_tac x="g \<circ> f" in exI)
+ apply (auto simp: image_subset_iff inj_on_def)
+ done
+
+lemma eqpoll_sym: "A \<approx> B \<Longrightarrow> B \<approx> A"
+ unfolding eqpoll_def
+ using bij_betw_the_inv_into by auto
+
+lemma eqpoll_trans [trans]: "\<lbrakk>A \<approx> B; B \<approx> C\<rbrakk> \<Longrightarrow> A \<approx> C"
+ unfolding eqpoll_def using bij_betw_trans by blast
+
+lemma eqpoll_imp_lepoll: "A \<approx> B \<Longrightarrow> A \<lesssim> B"
+ unfolding eqpoll_def lepoll_def by (metis bij_betw_def order_refl)
+
+lemma subset_imp_lepoll: "A \<subseteq> B \<Longrightarrow> A \<lesssim> B"
+ by (force simp: lepoll_def)
+
+lemma lepoll_iff: "A \<lesssim> B \<longleftrightarrow> (\<exists>g. A \<subseteq> g ` B)"
+ unfolding lepoll_def
+proof safe
+ fix g assume "A \<subseteq> g ` B"
+ then show "\<exists>f. inj_on f A \<and> f ` A \<subseteq> B"
+ 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 subset_image_lepoll: "B \<subseteq> f ` A \<Longrightarrow> B \<lesssim> A"
+ by (auto simp: lepoll_iff)
+
+lemma image_lepoll: "f ` A \<lesssim> A"
+ by (auto simp: lepoll_iff)
+
+lemma infinite_le_lepoll: "infinite A \<longleftrightarrow> (UNIV::nat set) \<lesssim> A"
+apply (auto simp: lepoll_def)
+ apply (simp add: infinite_countable_subset)
+ using infinite_iff_countable_subset by blast
+
+
+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")
+proof
+ assume L: ?lhs
+ then show ?rhs
+ apply (rule_tac x="the_inv_into A f" in exI)
+ apply (auto simp: bij_betw_def f_the_inv_into_f the_inv_into_f_f the_inv_into_into)
+ done
+next
+ assume ?rhs
+ then show ?lhs
+ by (auto simp: bij_betw_def inj_on_def image_def; metis)
+qed
+
+lemma eqpoll_iff_bijections:
+ "A \<approx> B \<longleftrightarrow> (\<exists>f 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))"
+ by (auto simp: eqpoll_def bij_betw_iff_bijections)
+
+lemma lepoll_restricted_funspace:
+ "{f. f ` A \<subseteq> B \<and> {x. f x \<noteq> k x} \<subseteq> A \<and> finite {x. f x \<noteq> k x}} \<lesssim> Fpow (A \<times> B)"
+proof -
+ have *: "\<exists>U \<in> Fpow (A \<times> B). f = (\<lambda>x. if \<exists>y. (x, y) \<in> U then SOME y. (x,y) \<in> U else k x)"
+ if "f ` A \<subseteq> B" "{x. f x \<noteq> k x} \<subseteq> A" "finite {x. f x \<noteq> k x}" for f
+ apply (rule_tac x="(\<lambda>x. (x, f x)) ` {x. f x \<noteq> k x}" in bexI)
+ using that by (auto simp: image_def Fpow_def)
+ show ?thesis
+ apply (rule subset_image_lepoll [where f = "\<lambda>U x. if \<exists>y. (x,y) \<in> U then @y. (x,y) \<in> U else k x"])
+ using * by (auto simp: image_def)
+qed
+
+subsection\<open>The strict relation\<close>
+
+
+lemma lesspoll_not_refl [iff]: "~ (i \<prec> i)"
+ by (simp add: lepoll_antisym lesspoll_def)
+
+lemma lesspoll_imp_lepoll: "A \<prec> B ==> A \<lesssim> B"
+by (unfold lesspoll_def, blast)
+
+lemma lepoll_iff_leqpoll: "A \<lesssim> B \<longleftrightarrow> A \<prec> B | A \<approx> B"
+ using eqpoll_imp_lepoll lesspoll_def by blast
+
+lemma lesspoll_trans [trans]: "\<lbrakk>X \<prec> Y; Y \<prec> Z\<rbrakk> \<Longrightarrow> X \<prec> Z"
+ by (meson eqpoll_sym lepoll_antisym lepoll_trans lepoll_trans1 lesspoll_def)
+
+lemma lesspoll_trans1 [trans]: "\<lbrakk>X \<lesssim> Y; Y \<prec> Z\<rbrakk> \<Longrightarrow> X \<prec> Z"
+ by (meson eqpoll_sym lepoll_antisym lepoll_trans lepoll_trans1 lesspoll_def)
+
+lemma lesspoll_trans2 [trans]: "\<lbrakk>X \<prec> Y; Y \<lesssim> Z\<rbrakk> \<Longrightarrow> X \<prec> Z"
+ by (meson eqpoll_imp_lepoll eqpoll_sym lepoll_antisym lepoll_trans lesspoll_def)
+
+lemma eq_lesspoll_trans [trans]: "\<lbrakk>X \<approx> Y; Y \<prec> Z\<rbrakk> \<Longrightarrow> X \<prec> Z"
+ using eqpoll_imp_lepoll lesspoll_trans1 by blast
+
+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 PiE_sing_eqpoll_self: "({a} \<rightarrow>\<^sub>E B) \<approx> B"
+proof -
+ have 1: "x = y"
+ if "x \<in> {a} \<rightarrow>\<^sub>E B" "y \<in> {a} \<rightarrow>\<^sub>E B" "x a = y a" for x y
+ by (metis IntD2 PiE_def extensionalityI singletonD that)
+ have 2: "x \<in> (\<lambda>h. h a) ` ({a} \<rightarrow>\<^sub>E B)" if "x \<in> B" for x
+ using that by (rule_tac x="\<lambda>z\<in>{a}. x" in image_eqI) auto
+ show ?thesis
+ unfolding eqpoll_def bij_betw_def inj_on_def
+ by (force intro: 1 2)
+qed
+
+lemma lepoll_funcset_right:
+ "B \<lesssim> B' \<Longrightarrow> A \<rightarrow>\<^sub>E B \<lesssim> A \<rightarrow>\<^sub>E B'"
+ apply (auto simp: lepoll_def inj_on_def)
+ apply (rule_tac x = "\<lambda>g. \<lambda>z \<in> A. f(g z)" in exI)
+ apply (auto simp: fun_eq_iff)
+ apply (metis PiE_E)
+ by blast
+
+lemma lepoll_funcset_left:
+ assumes "B \<noteq> {}" "A \<lesssim> A'"
+ shows "A \<rightarrow>\<^sub>E B \<lesssim> A' \<rightarrow>\<^sub>E B"
+proof -
+ obtain b where "b \<in> B"
+ using assms by blast
+ obtain f where "inj_on f A" and fim: "f ` A \<subseteq> A'"
+ using assms by (auto simp: lepoll_def)
+ then obtain h where h: "\<And>x. x \<in> A \<Longrightarrow> h (f x) = x"
+ using the_inv_into_f_f by fastforce
+ let ?F = "\<lambda>g. \<lambda>u \<in> A'. if h u \<in> A then g(h u) else b"
+ show ?thesis
+ unfolding lepoll_def inj_on_def
+ proof (intro exI conjI ballI impI ext)
+ fix k l x
+ assume k: "k \<in> A \<rightarrow>\<^sub>E B" and l: "l \<in> A \<rightarrow>\<^sub>E B" and "?F k = ?F l"
+ then have "?F k (f x) = ?F l (f x)"
+ by simp
+ then show "k x = l x"
+ apply (auto simp: h split: if_split_asm)
+ apply (metis PiE_arb h k l)
+ apply (metis (full_types) PiE_E h k l)
+ using fim k l by fastforce
+ next
+ show "?F ` (A \<rightarrow>\<^sub>E B) \<subseteq> A' \<rightarrow>\<^sub>E B"
+ using \<open>b \<in> B\<close> by force
+ qed
+qed
+
+lemma lepoll_funcset:
+ "\<lbrakk>B \<noteq> {}; A \<lesssim> A'; B \<lesssim> B'\<rbrakk> \<Longrightarrow> A \<rightarrow>\<^sub>E B \<lesssim> A' \<rightarrow>\<^sub>E B'"
+ by (rule lepoll_trans [OF lepoll_funcset_right lepoll_funcset_left]) auto
+
+lemma lepoll_PiE:
+ assumes "\<And>i. i \<in> A \<Longrightarrow> B i \<lesssim> C i"
+ shows "PiE A B \<lesssim> PiE A C"
+proof -
+ obtain f where f: "\<And>i. i \<in> A \<Longrightarrow> inj_on (f i) (B i) \<and> (f i) ` B i \<subseteq> C i"
+ using assms unfolding lepoll_def by metis
+ then show ?thesis
+ unfolding lepoll_def
+ apply (rule_tac x = "\<lambda>g. \<lambda>i \<in> A. f i (g i)" in exI)
+ apply (auto simp: inj_on_def)
+ apply (rule PiE_ext, auto)
+ apply (metis (full_types) PiE_mem restrict_apply')
+ by blast
+qed
+
+
+lemma card_le_PiE_subindex:
+ assumes "A \<subseteq> A'" "Pi\<^sub>E A' B \<noteq> {}"
+ shows "PiE A B \<lesssim> PiE A' B"
+proof -
+ have "\<And>x. x \<in> A' \<Longrightarrow> \<exists>y. y \<in> B x"
+ using assms by blast
+ then obtain g where g: "\<And>x. x \<in> A' \<Longrightarrow> g x \<in> B x"
+ by metis
+ let ?F = "\<lambda>f x. if x \<in> A then f x else if x \<in> A' then g x else undefined"
+ have "Pi\<^sub>E A B \<subseteq> (\<lambda>f. restrict f A) ` Pi\<^sub>E A' B"
+ proof
+ show "f \<in> Pi\<^sub>E A B \<Longrightarrow> f \<in> (\<lambda>f. restrict f A) ` Pi\<^sub>E A' B" for f
+ using \<open>A \<subseteq> A'\<close>
+ by (rule_tac x="?F f" in image_eqI) (auto simp: g fun_eq_iff)
+ qed
+ then have "Pi\<^sub>E A B \<lesssim> (\<lambda>f. \<lambda>i \<in> A. f i) ` Pi\<^sub>E A' B"
+ by (simp add: subset_imp_lepoll)
+ also have "\<dots> \<lesssim> PiE A' B"
+ by (rule image_lepoll)
+ finally show ?thesis .
+qed
+
+
+lemma finite_restricted_funspace:
+ assumes "finite A" "finite B"
+ shows "finite {f. f ` A \<subseteq> B \<and> {x. f x \<noteq> k x} \<subseteq> A}" (is "finite ?F")
+proof (rule finite_subset)
+ show "finite ((\<lambda>U x. if \<exists>y. (x,y) \<in> U then @y. (x,y) \<in> U else k x) ` Pow(A \<times> B))" (is "finite ?G")
+ using assms by auto
+ show "?F \<subseteq> ?G"
+ proof
+ fix f
+ assume "f \<in> ?F"
+ then show "f \<in> ?G"
+ by (rule_tac x="(\<lambda>x. (x,f x)) ` {x. f x \<noteq> k x}" in image_eqI) (auto simp: fun_eq_iff image_def)
+ qed
+qed
+
+
+proposition finite_PiE_iff:
+ "finite(PiE I S) \<longleftrightarrow> PiE I S = {} \<or> finite {i \<in> I. ~(\<exists>a. S i \<subseteq> {a})} \<and> (\<forall>i \<in> I. finite(S i))"
+ (is "?lhs = ?rhs")
+proof (cases "PiE I S = {}")
+ case False
+ define J where "J \<equiv> {i \<in> I. \<nexists>a. S i \<subseteq> {a}}"
+ show ?thesis
+ proof
+ assume L: ?lhs
+ have "infinite (Pi\<^sub>E I S)" if "infinite J"
+ proof -
+ have "(UNIV::nat set) \<lesssim> (UNIV::(nat\<Rightarrow>bool) set)"
+ proof -
+ have "\<forall>N::nat set. inj_on (=) N"
+ by (simp add: inj_on_def)
+ then show ?thesis
+ by (meson infinite_iff_countable_subset infinite_le_lepoll top.extremum)
+ qed
+ also have "\<dots> = (UNIV::nat set) \<rightarrow>\<^sub>E (UNIV::bool set)"
+ by auto
+ also have "\<dots> \<lesssim> J \<rightarrow>\<^sub>E (UNIV::bool set)"
+ apply (rule lepoll_funcset_left)
+ using infinite_le_lepoll that by auto
+ also have "\<dots> \<lesssim> Pi\<^sub>E J S"
+ proof -
+ have *: "(UNIV::bool set) \<lesssim> S i" if "i \<in> I" and "\<forall>a. \<not> S i \<subseteq> {a}" for i
+ proof -
+ obtain a b where "{a,b} \<subseteq> S i" "a \<noteq> b"
+ by (metis \<open>\<forall>a. \<not> S i \<subseteq> {a}\<close> all_not_in_conv empty_subsetI insertCI insert_subset set_eq_subset subsetI)
+ then show ?thesis
+ apply (clarsimp simp: lepoll_def inj_on_def)
+ apply (rule_tac x="\<lambda>x. if x then a else b" in exI, auto)
+ done
+ qed
+ show ?thesis
+ by (auto simp: * J_def intro: lepoll_PiE)
+ qed
+ also have "\<dots> \<lesssim> Pi\<^sub>E I S"
+ using False by (auto simp: J_def intro: card_le_PiE_subindex)
+ finally have "(UNIV::nat set) \<lesssim> Pi\<^sub>E I S" .
+ then show ?thesis
+ by (simp add: infinite_le_lepoll)
+ qed
+ moreover have "finite (S i)" if "i \<in> I" for i
+ proof (rule finite_subset)
+ obtain f where f: "f \<in> PiE I S"
+ using False by blast
+ show "S i \<subseteq> (\<lambda>f. f i) ` Pi\<^sub>E I S"
+ proof
+ show "s \<in> (\<lambda>f. f i) ` Pi\<^sub>E I S" if "s \<in> S i" for s
+ using that f \<open>i \<in> I\<close>
+ by (rule_tac x="\<lambda>j. if j = i then s else f j" in image_eqI) auto
+ qed
+ next
+ show "finite ((\<lambda>x. x i) ` Pi\<^sub>E I S)"
+ using L by blast
+ qed
+ ultimately show ?rhs
+ using L
+ by (auto simp: J_def False)
+ next
+ assume R: ?rhs
+ have "\<forall>i \<in> I - J. \<exists>a. S i = {a}"
+ using False J_def by blast
+ then obtain a where a: "\<forall>i \<in> I - J. S i = {a i}"
+ by metis
+ let ?F = "{f. f ` J \<subseteq> (\<Union>i \<in> J. S i) \<and> {i. f i \<noteq> (if i \<in> I then a i else undefined)} \<subseteq> J}"
+ have *: "finite (Pi\<^sub>E I S)"
+ if "finite J" and "\<forall>i\<in>I. finite (S i)"
+ proof (rule finite_subset)
+ show "Pi\<^sub>E I S \<subseteq> ?F"
+ apply safe
+ using J_def apply blast
+ by (metis DiffI PiE_E a singletonD)
+ show "finite ?F"
+ proof (rule finite_restricted_funspace [OF \<open>finite J\<close>])
+ show "finite (\<Union> (S ` J))"
+ using that J_def by blast
+ qed
+ qed
+ show ?lhs
+ using R by (auto simp: * J_def)
+ qed
+qed auto
+
+
+corollary finite_funcset_iff:
+ "finite(I \<rightarrow>\<^sub>E S) \<longleftrightarrow> (\<exists>a. S \<subseteq> {a}) \<or> I = {} \<or> finite I \<and> finite S"
+ apply (auto simp: finite_PiE_iff PiE_eq_empty_iff dest: not_finite_existsD)
+ using finite.simps by auto
+
+end
--- a/src/HOL/Library/Library.thy Thu Jan 24 10:04:32 2019 +0100
+++ b/src/HOL/Library/Library.thy Thu Jan 24 14:44:52 2019 +0000
@@ -23,6 +23,7 @@
Discrete
Disjoint_Sets
Dlist
+ Equipollence
Extended
Extended_Nat
Extended_Nonnegative_Real