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products/sources/formale Sprachen/Isabelle/HOL/Library/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 33 kB

Quelle Equipollence.thy Sprache: Isabelle

section \<open>Equipollence and Other Relations Connected with Cardinality\<close>

theory "Equipollence"
  imports FuncSet Countable_Set
begin

subsection\<open>Eqpoll\<close>

definition eqpoll :: "'a set \ 'b set \ bool" (infixl \\\ 50)
  where "eqpoll A B \ \f. bij_betw f A B"

definition lepoll :: "'a set \ 'b set \ bool" (infixl \\\ 50)
  where "lepoll A B \ \f. inj_on f A \ f ` A \ B"

definition lesspoll :: "'a set \ 'b set \ bool" (infixl \\\ 50)
  where "A \ B == A \ B \ ~(A \ B)"

lemma lepoll_def': "lepoll A B \ \f. inj_on f A \ f \ A \ B"
  by (simp add: Pi_iff image_subset_iff lepoll_def)

lemma eqpoll_empty_iff_empty [simp]: "A \ {} \ A={}"
  by (simp add: bij_betw_iff_bijections eqpoll_def)

lemma lepoll_empty_iff_empty [simp]: "A \ {} \ A = {}"
  by (auto simp: lepoll_def)

lemma not_lesspoll_empty: "\ A \ {}"
  by (simp add: lesspoll_def)

(*The HOL Light CARD_LE_RELATIONAL_FULL*)
lemma lepoll_relational_full:
  assumes "\y. y \ B \ \x. x \ A \ R x y"
    and "\x y y'. \x \ A; y \ B; y' \ B; R x y; R x y'\ \ y = y'"
  shows "B \ A"
proof -
  obtain f where f: "\y. y \ B \ f y \ A \ R (f y) y"
    using assms by metis
  with assms have "inj_on f B"
    by (metis inj_onI)
  with f show ?thesis
    unfolding lepoll_def by blast
qed

lemma eqpoll_iff_card_of_ordIso: "A \ B \ ordIso2 (card_of A) (card_of B)"
  by (simp add: card_of_ordIso eqpoll_def)

lemma eqpoll_refl [iff]: "A \ A"
  by (simp add: card_of_refl eqpoll_iff_card_of_ordIso)

lemma eqpoll_finite_iff: "A \ B \ finite A \ finite B"
  by (meson bij_betw_finite eqpoll_def)

lemma eqpoll_iff_card:
  assumes "finite A" "finite B"
  shows  "A \ B \ card A = card B"
  using assms by (auto simp: bij_betw_iff_card eqpoll_def)

lemma eqpoll_singleton_iff: "A \ {x} \ (\u. A = {u})"
  by (metis card.infinite card_1_singleton_iff eqpoll_finite_iff eqpoll_iff_card not_less_eq_eq)

lemma eqpoll_doubleton_iff: "A \ {x,y} \ (\u v. A = {u,v} \ (u=v \ x=y))"
proof (cases "x=y")
  case True
  then show ?thesis
    by (simp add: eqpoll_singleton_iff)
next
  case False
  then show ?thesis
    by (smt (verit, ccfv_threshold) card_1_singleton_iff card_Suc_eq_finite eqpoll_finite_iff
        eqpoll_iff_card finite.insertI singleton_iff)
qed

lemma lepoll_antisym:
  assumes "A \ B" "B \ A" shows "A \ B"
  using assms unfolding eqpoll_def lepoll_def by (metis Schroeder_Bernstein)

lemma lepoll_trans [trans]:
  assumes "A \ B" " B \ C" shows "A \ C"
proof -
  obtain f g where fg: "inj_on f A" "inj_on g B" and "f : A \ B" "g \ B \ C"
    by (metis assms lepoll_def')
  then have "g \ f \ A \ C"
    by auto
  with fg show ?thesis
    unfolding lepoll_def
    by (metis \<open>f \<in> A \<rightarrow> B\<close> comp_inj_on image_subset_iff_funcset inj_on_subset)
qed

lemma lepoll_trans1 [trans]: "\A \ B; B \ C\ \ A \ C"
  by (meson card_of_ordLeq eqpoll_iff_card_of_ordIso lepoll_def lepoll_trans ordIso_iff_ordLeq)

lemma lepoll_trans2 [trans]: "\A \ B; B \ C\ \ A \ C"
  by (metis bij_betw_def eqpoll_def lepoll_def lepoll_trans order_refl)

lemma eqpoll_sym: "A \ B \ B \ A"
  unfolding eqpoll_def
  using bij_betw_the_inv_into by auto

lemma eqpoll_trans [trans]: "\A \ B; B \ C\ \ A \ C"
  unfolding eqpoll_def using bij_betw_trans by blast

lemma eqpoll_imp_lepoll: "A \ B \ A \ B"
  unfolding eqpoll_def lepoll_def by (metis bij_betw_def order_refl)

lemma subset_imp_lepoll: "A \ B \ A \ B"
  by (force simp: lepoll_def)

lemma lepoll_refl [iff]: "A \ A"
  by (simp add: subset_imp_lepoll)

lemma lepoll_iff: "A \ B \ (\g. A \ g ` B)"
  unfolding lepoll_def
proof safe
  fix g assume "A \ g ` B"
  then show "\f. inj_on f A \ f ` A \ 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 empty_lepoll [iff]: "{} \ A"
  by (simp add: lepoll_iff)

lemma subset_image_lepoll: "B \ f ` A \ B \ A"
  by (auto simp: lepoll_iff)

lemma image_lepoll: "f ` A \ A"
  by (auto simp: lepoll_iff)

lemma infinite_le_lepoll: "infinite A \ (UNIV::nat set) \ A"
  by (simp add: infinite_iff_countable_subset lepoll_def)

lemma lepoll_Pow_self: "A \ Pow A"
  unfolding lepoll_def inj_def
  proof (intro exI conjI)
    show "inj_on (\x. {x}) A"
      by (auto simp: inj_on_def)
qed auto

lemma eqpoll_iff_bijections:
   "A \ B \ (\f g. (\x \ A. f x \ B \ g(f x) = x) \ (\y \ B. g y \ A \ f(g y) = y))"
    by (auto simp: eqpoll_def bij_betw_iff_bijections)

lemma lepoll_restricted_funspace:
   "{f. f ` A \ B \ {x. f x \ k x} \ A \ finite {x. f x \ k x}} \ Fpow (A \ B)"
proof -
  have *: "\U \ Fpow (A \ B). f = (\x. if \y. (x, y) \ U then SOME y. (x,y) \ U else k x)"
    if "f ` A \ B" "{x. f x \ k x} \ A" "finite {x. f x \ k x}" for f
    apply (rule_tac x="(\x. (x, f x)) ` {x. f x \ k x}" in bexI)
    using that by (auto simp: image_def Fpow_def)
  show ?thesis
    apply (rule subset_image_lepoll [where f = "\U x. if \y. (x,y) \ U then @y. (x,y) \ U else k x"])
    using * by (auto simp: image_def)
qed

lemma singleton_lepoll: "{x} \ insert y A"
  by (force simp: lepoll_def)

lemma singleton_eqpoll: "{x} \ {y}"
  by (blast intro: lepoll_antisym singleton_lepoll)

lemma subset_singleton_iff_lepoll: "(\x. S \ {x}) \ S \ {()}"
  using lepoll_iff by fastforce

lemma infinite_insert_lepoll:
  assumes "infinite A" shows "insert a A \ A"
proof -
  obtain f :: "nat \ 'a" where "inj f" and f: "range f \ A"
    using assms infinite_countable_subset by blast
  let ?g = "(\z. if z=a then f 0 else if z \ 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 \ A"
      using f by auto
  qed
qed

lemma infinite_insert_eqpoll: "infinite A \ insert a A \ 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 \ A"
proof -
  have "B \ (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

lemma countable_lepoll: "\countable A; B \ A\ \ countable B"
  by (meson countable_image countable_subset lepoll_iff)

lemma countable_eqpoll: "\countable A; B \ A\ \ countable B"
  using countable_lepoll eqpoll_imp_lepoll by blast

subsection\<open>The strict relation\<close>

lemma lesspoll_not_refl [iff]: "~ (i \ i)"
  by (simp add: lepoll_antisym lesspoll_def)

lemma lesspoll_imp_lepoll: "A \ B ==> A \ B"
by (unfold lesspoll_def, blast)

lemma lepoll_iff_leqpoll: "A \ B \ A \ B | A \ B"
  using eqpoll_imp_lepoll lesspoll_def by blast

lemma lesspoll_trans [trans]: "\X \ Y; Y \ Z\ \ X \ Z"
  by (meson eqpoll_sym lepoll_antisym lepoll_trans lepoll_trans1 lesspoll_def)

lemma lesspoll_trans1 [trans]: "\X \ Y; Y \ Z\ \ X \ Z"
  by (meson eqpoll_sym lepoll_antisym lepoll_trans lepoll_trans1 lesspoll_def)

lemma lesspoll_trans2 [trans]: "\X \ Y; Y \ Z\ \ X \ Z"
  by (meson eqpoll_imp_lepoll eqpoll_sym lepoll_antisym lepoll_trans lesspoll_def)

lemma eq_lesspoll_trans [trans]: "\X \ Y; Y \ Z\ \ X \ Z"
  using eqpoll_imp_lepoll lesspoll_trans1 by blast

lemma lesspoll_eq_trans [trans]: "\X \ Y; Y \ Z\ \ X \ Z"
  using eqpoll_imp_lepoll lesspoll_trans2 by blast

lemma lesspoll_Pow_self: "A \ Pow A"
  unfolding lesspoll_def bij_betw_def eqpoll_def
  by (meson lepoll_Pow_self Cantors_theorem)

lemma finite_lesspoll_infinite:
  assumes "infinite A" "finite B" shows "B \ A"
  by (meson assms eqpoll_finite_iff finite_lepoll_infinite lesspoll_def)

lemma countable_lesspoll: "\countable A; B \ A\ \ countable B"
  using countable_lepoll lesspoll_def by blast

lemma lepoll_iff_card_le: "\finite A; finite B\ \ A \ B \ card A \ card B"
  by (simp add: inj_on_iff_card_le lepoll_def)

lemma lepoll_iff_finite_card: "A \ {.. finite A \ card A \ n"
  by (metis card_lessThan finite_lessThan finite_surj lepoll_iff lepoll_iff_card_le)

lemma eqpoll_iff_finite_card: "A \ {.. finite A \ card A = n"
  by (metis card_lessThan eqpoll_finite_iff eqpoll_iff_card finite_lessThan)

lemma lesspoll_iff_finite_card: "A \ {.. finite A \ card A < n"
  by (metis eqpoll_iff_finite_card lepoll_iff_finite_card lesspoll_def order_less_le)

subsection\<open>Mapping by an injection\<close>

lemma inj_on_image_eqpoll_self: "inj_on f A \ f ` A \ 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 \ B \ A \ 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 \ f ` B \ A \ 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 \ B \ A \ 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 \ f ` B \ A \ 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 \ B \ A \ 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 \ f ` B \ A \ 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 \ insert v B" "u \ A" "v \ B" shows "A \ B"
proof -
  obtain f where inj: "inj_on f (insert u A)" and fim: "f ` (insert u A) \ insert v B"
    by (meson assms lepoll_def)
  show ?thesis
    unfolding lepoll_def
  proof (intro exI conjI)
    let ?g = "\x\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 \ 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: "\insert u A \ insert v B; u \ A; v \ B\ \ A \ B"
  by (meson insert_lepoll_insertD eqpoll_imp_lepoll eqpoll_sym lepoll_antisym)

lemma insert_lepoll_cong:
  assumes "A \ B" "b \ B" shows "insert a A \ insert b B"
proof -
  obtain f where f: "inj_on f A" "f ` A \ B"
    by (meson assms lepoll_def)
  let ?f = "\u \ 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 \ insert b B"
      using f \<open>b \<notin> B\<close> by auto
  qed
qed

lemma insert_eqpoll_cong:
     "\A \ B; a \ A; b \ B\ \ insert a A \ insert b B"
  by (meson eqpoll_imp_lepoll eqpoll_sym insert_lepoll_cong lepoll_antisym)

lemma insert_eqpoll_insert_iff:
     "\a \ A; b \ B\ \ insert a A \ insert b B \ A \ B"
  by (meson insert_eqpoll_insertD insert_eqpoll_cong)

lemma insert_lepoll_insert_iff:
     " \a \ A; b \ B\ \ (insert a A \ insert b B) \ (A \ B)"
  by (meson insert_lepoll_insertD insert_lepoll_cong)

lemma less_imp_insert_lepoll:
  assumes "A \ B" shows "insert a A \ B"
proof -
  obtain f where "inj_on f A" "f ` A \ B"
    using assms by (metis bij_betw_def eqpoll_def lepoll_def lesspoll_def psubset_eq)
  then obtain b where b: "b \ B" "b \ 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 \ B"
      using \<open>f ` A \<subset> B\<close>  by (auto simp: b)
  qed
qed

lemma finite_insert_lepoll: "finite A \ (insert a A \ A) \ (a \ A)"
proof (induction A rule: finite_induct)
  case (insert x A)
  then show ?case
    by (metis insertI2 insert_lepoll_insert_iff insert_subsetI lepoll_trans subsetI subset_imp_lepoll)
qed auto

subsection\<open>Binary sums and unions\<close>

lemma Un_lepoll_mono:
  assumes "A \ C" "B \ D" "disjnt C D" shows "A \ B \ C \ D"
proof -
  obtain f g where inj: "inj_on f A" "inj_on g B" and fg: "f ` A \ C" "g ` B \ D"
    by (meson assms lepoll_def)
  have "inj_on (\x. if x \ A then f x else g x) (A \ 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="\x. if x \ A then f x else g x" in exI) auto
qed

lemma Un_eqpoll_cong: "\A \ C; B \ D; disjnt A B; disjnt C D\ \ A \ B \ C \ D"
  by (meson Un_lepoll_mono eqpoll_imp_lepoll eqpoll_sym lepoll_antisym)

lemma sum_lepoll_mono:
  assumes "A \ C" "B \ D" shows "A <+> B \ C <+> D"
proof -
  obtain f g where "inj_on f A" "f ` A \ C" "inj_on g B" "g ` B \ D"
    by (meson assms lepoll_def)
  then show ?thesis
    unfolding lepoll_def
    by (rule_tac x="case_sum (Inl \ f) (Inr \ g)" in exI) (force simp: inj_on_def)
qed

lemma sum_eqpoll_cong: "\A \ C; B \ D\ \ A <+> B \ 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 \ A \ A"
  unfolding lepoll_def inj_def
proof (intro exI conjI)
  show "inj_on (\x. (x,x)) A"
    by (auto simp: inj_on_def)
qed auto

lemma times_commute_eqpoll: "A \ B \ B \ A"
  unfolding eqpoll_def
  by (force intro: bij_betw_byWitness [where f = "\(x,y). (y,x)" and f' = "\(x,y). (y,x)"])

lemma times_assoc_eqpoll: "(A \ B) \ C \ A \ (B \ C)"
  unfolding eqpoll_def
  by (force intro: bij_betw_byWitness [where f = "\((x,y),z). (x,(y,z))" and f' = "\(x,(y,z)). ((x,y),z)"])

lemma times_singleton_eqpoll: "{a} \ A \ A"
proof -
  have "{a} \ A = (\x. (a,x)) ` A"
    by auto
  also have "\ \ 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 \ C" "B \ D" shows "A \ B \ C \ D"
proof -
  obtain f g where "inj_on f A" "f ` A \ C" "inj_on g B" "g ` B \ D"
    by (meson assms lepoll_def)
  then show ?thesis
    unfolding lepoll_def
    by (rule_tac x="\(x,y). (f x, g y)" in exI) (auto simp: inj_on_def)
qed

lemma times_eqpoll_cong: "\A \ C; B \ D\ \ A \ B \ C \ D"
  by (metis eqpoll_imp_lepoll eqpoll_sym lepoll_antisym times_lepoll_mono)

lemma
  assumes "B \ {}" shows lepoll_times1: "A \ A \ B" and lepoll_times2: "A \ B \ A"
  using assms lepoll_iff by fastforce+

lemma times_0_eqpoll: "{} \ A \ {}"
  by (simp add: eqpoll_iff_bijections)

lemma Sigma_inj_lepoll_mono:
  assumes h: "inj_on h A" "h ` A \ C" and "\x. x \ A \ B x \ D (h x)"
  shows "Sigma A B \ Sigma C D"
proof -
  have "\x. x \ A \ \f. inj_on f (B x) \ f ` (B x) \ D (h x)"
    by (meson assms lepoll_def)
  then obtain f where  "\x. x \ A \ inj_on (f x) (B x) \ f x ` B x \ D (h x)"
    by metis
  with h show ?thesis
    unfolding lepoll_def inj_on_def
    by (rule_tac x="\(x,y). (h x, f x y)" in exI) force
qed

lemma Sigma_lepoll_mono:
  assumes "A \ C" "\x. x \ A \ B x \ D x" shows "Sigma A B \ Sigma C D"
  using Sigma_inj_lepoll_mono [of id] assms by auto

lemma sum_times_distrib_eqpoll: "(A <+> B) \ C \ (A \ C) <+> (B \ C)"
  unfolding eqpoll_def
proof
  show "bij_betw (\(x,z). case_sum(\y. Inl(y,z)) (\y. Inr(y,z)) x) ((A <+> B) \ C) (A \ C <+> B \ C)"
    by (rule bij_betw_byWitness [where f' = "case_sum (\(x,z). (Inl x, z)) (\(y,z). (Inr y, z))"]) auto
qed

lemma Sigma_eqpoll_cong:
  assumes h: "bij_betw h A C"  and BD: "\x. x \ A \ B x \ D (h x)"
  shows "Sigma A B \ Sigma C D"
proof (intro lepoll_antisym)
  show "Sigma A B \ Sigma C D"
    by (metis Sigma_inj_lepoll_mono bij_betw_def eqpoll_imp_lepoll subset_refl assms)
  have "inj_on (inv_into A h) C \ inv_into A h ` C \ A"
    by (metis bij_betw_def bij_betw_inv_into h set_eq_subset)
  then show "Sigma C D \ Sigma A B"
    by (smt (verit, best) BD Sigma_inj_lepoll_mono bij_betw_inv_into_right eqpoll_sym h image_subset_iff lepoll_refl lepoll_trans2)
qed

lemma prod_insert_eqpoll:
  assumes "a \ A" shows "insert a A \ B \ B <+> A \ B"
  unfolding eqpoll_def
  proof
  show "bij_betw (\(x,y). if x=a then Inl y else Inr (x,y)) (insert a A \ B) (B <+> A \ B)"
    by (rule bij_betw_byWitness [where f' = "case_sum (\y. (a,y)) id"]) (auto simp: assms)
qed

subsection\<open>General Unions\<close>

lemma Union_eqpoll_Times:
  assumes B: "\x. x \ A \ F x \ B" and disj: "pairwise (\x y. disjnt (F x) (F y)) A"
  shows "(\x\A. F x) \ A \ B"
proof (rule lepoll_antisym)
  obtain b where b: "\x. x \ A \ bij_betw (b x) (F x) B"
    using B unfolding eqpoll_def by metis
  show "\(F ` A) \ A \ 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
      by (smt (verit, best) disj disjnt_iff pairwiseD that(1,2) theI_unique)
    let ?f = "\z. (\ z, b (\ z) z)"
    show "inj_on ?f (\(F ` A))"
      unfolding inj_on_def
      by clarify (metis \<chi> b bij_betw_inv_into_left)
    show "?f ` \(F ` A) \ A \ B"
      using \<chi> b bij_betwE by blast
  qed
  show "A \ B \ \(F ` A)"
    unfolding lepoll_def
  proof (intro exI conjI)
    let ?f = "\(x,y). inv_into (F x) (b x) y"
    have *: "inv_into (F x) (b x) y \ F x" if "x \ A" "y \ B" for x y
      by (metis b bij_betw_imp_surj_on inv_into_into that)
    then show "inj_on ?f (A \ B)"
      unfolding inj_on_def
      by clarsimp (metis (mono_tags, lifting) b bij_betw_inv_into_right disj disjnt_iff pairwiseD)
    show "?f ` (A \ B) \ \ (F ` A)"
      by clarsimp (metis b bij_betw_imp_surj_on inv_into_into)
  qed
qed

lemma UN_lepoll_UN:
  assumes A: "\x. x \ A \ B x \ C x"
    and disj: "pairwise (\x y. disjnt (C x) (C y)) A"
  shows "\ (B`A) \ \ (C`A)"
proof -
  obtain f where f: "\x. x \ A \ inj_on (f x) (B x) \ f x ` (B x) \ (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 = "\z. (f (\ z) z)"
    show "inj_on ?f (\(B ` A))"
      using disj f unfolding inj_on_def disjnt_iff pairwise_def image_subset_iff
      by (metis UN_iff \<chi>)
    show "?f ` \ (B ` A) \ \ (C ` A)"
      using \<chi> f unfolding image_subset_iff by blast
  qed
qed

lemma UN_eqpoll_UN:
  assumes A: "\x. x \ A \ B x \ C x"
    and B: "pairwise (\x y. disjnt (B x) (B y)) A"
    and C: "pairwise (\x y. disjnt (C x) (C y)) A"
  shows "(\x\A. B x) \ (\x\A. C x)"
proof (rule lepoll_antisym)
  show "\ (B ` A) \ \ (C ` A)"
    by (meson A C UN_lepoll_UN eqpoll_imp_lepoll)
  show "\ (C ` A) \ \ (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} \\<^sub>E B) \ B"
proof -
  have 1: "x = y"
    if "x \ {a} \\<^sub>E B" "y \ {a} \\<^sub>E B" "x a = y a" for x y
    by (metis IntD2 PiE_def extensionalityI singletonD that)
  have 2: "x \ (\h. h a) ` ({a} \\<^sub>E B)" if "x \ B" for x
    using that by (rule_tac x="\z\{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:
  assumes "B \ B'" shows "A \\<^sub>E B \ A \\<^sub>E B'"
proof -
  obtain f where f: "inj_on f B" "f ` B \ B'"
    by (meson assms lepoll_def)
  let ?G = "\g. \z \ A. f(g z)"
  have "inj_on ?G (A \\<^sub>E B)"
    using f by (smt (verit, best) PiE_ext PiE_mem inj_on_def restrict_apply')
  moreover have "?G ` (A \\<^sub>E B) \ (A \\<^sub>E B')"
    using f by fastforce
  ultimately show ?thesis
    by (meson lepoll_def)
qed

lemma lepoll_funcset_left:
  assumes "B \ {}" "A \ A'"
  shows "A \\<^sub>E B \ A' \\<^sub>E B"
proof -
  obtain b where "b \ B"
    using assms by blast
  obtain f where "inj_on f A" and fim: "f ` A \ A'"
    using assms by (auto simp: lepoll_def)
  then obtain h where h: "\x. x \ A \ h (f x) = x"
    using the_inv_into_f_f by fastforce
  let ?F = "\g. \u \ A'. if h u \ 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 \ A \\<^sub>E B" and l: "l \ A \\<^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"
      by (smt (verit, best) PiE_arb fim h image_subset_iff k l restrict_apply')
  next
    show "?F ` (A \\<^sub>E B) \ A' \\<^sub>E B"
      using \<open>b \<in> B\<close> by force
  qed
qed

lemma lepoll_funcset:
   "\B \ {}; A \ A'; B \ B'\ \ A \\<^sub>E B \ A' \\<^sub>E B'"
  by (rule lepoll_trans [OF lepoll_funcset_right lepoll_funcset_left]) auto

lemma lepoll_PiE:
  assumes "\i. i \ A \ B i \ C i"
  shows "PiE A B \ PiE A C"
proof -
  obtain f where f: "\i. i \ A \ inj_on (f i) (B i) \ (f i) ` B i \ C i"
    using assms unfolding lepoll_def by metis
  let ?G = "\g. \i \ A. f i (g i)"
  have "inj_on ?G (PiE A B)"
    by (smt (verit, ccfv_SIG) PiE_ext PiE_iff f inj_on_def restrict_apply')
  moreover have "?G ` (PiE A B) \ (PiE A C)"
    using f by fastforce
  ultimately show ?thesis
    by (meson lepoll_def)
qed

lemma card_le_PiE_subindex:
  assumes "A \ A'" "Pi\<^sub>E A' B \ {}"
  shows "PiE A B \ PiE A' B"
proof -
  have "\x. x \ A' \ \y. y \ B x"
    using assms by blast
  then obtain g where g: "\x. x \ A' \ g x \ B x"
    by metis
  let ?F = "\f x. if x \ A then f x else if x \ A' then g x else undefined"
  have "Pi\<^sub>E A B \ (\f. restrict f A) ` Pi\<^sub>E A' B"
  proof
    show "f \ Pi\<^sub>E A B \ f \ (\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 \ (\f. \i \ A. f i) ` Pi\<^sub>E A' B"
    by (simp add: subset_imp_lepoll)
  also have "\ \ 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 \ B \ {x. f x \ k x} \ A}" (is "finite ?F")
proof (rule finite_subset)
  show "finite ((\U x. if \y. (x,y) \ U then @y. (x,y) \ U else k x) ` Pow(A \ B))" (is "finite ?G")
    using assms by auto
  show "?F \ ?G"
  proof
    fix f
    assume "f \ ?F"
    then show "f \ ?G"
      by (rule_tac x="(\x. (x,f x)) ` {x. f x \ k x}" in image_eqI) (auto simp: fun_eq_iff image_def)
  qed
qed

proposition finite_PiE_iff:
   "finite(PiE I S) \ PiE I S = {} \ finite {i \ I. ~(\a. S i \ {a})} \ (\i \ I. finite(S i))"
(is "?lhs = ?rhs")
proof (cases "PiE I S = {}")
  case False
  define J where "J \ {i \ I. \a. S i \ {a}}"
  show ?thesis
  proof
    assume L: ?lhs
    have "infinite (Pi\<^sub>E I S)" if "infinite J"
    proof -
      have "(UNIV::nat set) \ (UNIV::(nat\bool) set)"
      proof -
        have "\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 "\ = (UNIV::nat set) \\<^sub>E (UNIV::bool set)"
        by auto
      also have "\ \ J \\<^sub>E (UNIV::bool set)"
        by (metis empty_not_UNIV infinite_le_lepoll lepoll_funcset_left that)
      also have "\ \ Pi\<^sub>E J S"
      proof -
        have *: "(UNIV::bool set) \ S i" if "i \ I" and \

: "\a. \ S i \ {a}" for i
        proof -
          obtain a b where "{a,b} \ S i" "a \ b"
            by (metis \<section> empty_subsetI insert_subset subsetI)
          then show ?thesis
            by (metis Set.set_insert UNIV_bool insert_iff insert_lepoll_cong insert_subset singleton_lepoll)
        qed
        show ?thesis
          by (auto simp: * J_def intro: lepoll_PiE)
      qed
      also have "\ \ Pi\<^sub>E I S"
        using False by (auto simp: J_def intro: card_le_PiE_subindex)
      finally have "(UNIV::nat set) \ Pi\<^sub>E I S" .
      then show ?thesis
        by (simp add: infinite_le_lepoll)
    qed
    moreover have "finite (S i)" if "i \ I" for i
    proof (rule finite_subset)
      obtain f where f: "f \ PiE I S"
        using False by blast
      show "S i \ (\f. f i) ` Pi\<^sub>E I S"
      proof
        show "s \ (\f. f i) ` Pi\<^sub>E I S" if "s \ S i" for s
          using that f \<open>i \<in> I\<close>
          by (rule_tac x="\j. if j = i then s else f j" in image_eqI) auto
      qed
    next
      show "finite ((\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 "\i \ I - J. \a. S i = {a}"
      using False J_def by blast
    then obtain a where a: "\i \ I - J. S i = {a i}"
      by metis
    let ?F = "{f. f ` J \ (\i \ J. S i) \ {i. f i \ (if i \ I then a i else undefined)} \ J}"
    have *: "finite (Pi\<^sub>E I S)"
      if "finite J" and "\i\I. finite (S i)"
    proof (rule finite_subset)
      have "\f j. \f \ Pi\<^sub>E I S; j \ J\ \ f j \ \ (S ` J)"
        using J_def by blast
      moreover
      have "\f j. \f \ Pi\<^sub>E I S; f j \ (if j \ I then a j else undefined)\ \ j \ J"
        by (metis DiffI PiE_E a singletonD)
      ultimately show "Pi\<^sub>E I S \ ?F" by force
      show "finite ?F"
      proof (rule finite_restricted_funspace [OF \<open>finite J\<close>])
        show "finite (\ (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 \\<^sub>E S) \ (\a. S \ {a}) \ I = {} \ finite I \ finite S"
  by (fastforce simp: finite_PiE_iff PiE_eq_empty_iff dest: subset_singletonD)

subsection \<open>Misc other resultd\<close>

lemma lists_lepoll_mono:
  assumes "A \ B" shows "lists A \ lists B"
proof -
  obtain f where f: "inj_on f A" "f ` A \ B"
    by (meson assms lepoll_def)
  moreover have "inj_on (map f) (lists A)"
    using f unfolding inj_on_def
    by clarsimp (metis list.inj_map_strong)
  ultimately show ?thesis
    unfolding lepoll_def by force
qed

lemma lepoll_lists: "A \ lists A"
  unfolding lepoll_def inj_on_def by(rule_tac x="\x. [x]" in exI) auto

text \<open>Dedekind's definition of infinite set\<close>

lemma infinite_iff_psubset: "infinite A \ (\B. B \ A \ A\B)"
proof
  assume "infinite A"
  then obtain f :: "nat \ 'a" where "inj f" and f: "range f \ A"
    by (meson infinite_countable_subset)
  define C where "C \ A - range f"
  have C: "A = range f \ C" "range f \ C = {}"
    using f by (auto simp: C_def)
  have *: "range (f \ Suc) \ range f"
    using inj_eq [OF \<open>inj f\<close>] by (fastforce simp: set_eq_iff)
  have "range f \ C \ range (f \ Suc) \ C"
  proof (intro Un_eqpoll_cong)
    show "range f \ range (f \ Suc)"
      by (meson \<open>inj f\<close> eqpoll_refl inj_Suc inj_compose inj_on_image_eqpoll_2)
    show "disjnt (range f) C"
      by (simp add: C disjnt_def)
    then show "disjnt (range (f \ Suc)) C"
      using "*" disjnt_subset1 by blast
  qed auto
  moreover have "range (f \ Suc) \ C \ A"
    using "*" f C_def by blast
  ultimately show "\B\A. A \ B"
    by (metis C(1))
next
  assume "\B\A. A \ B" then show "infinite A"
    by (metis card_subset_eq eqpoll_finite_iff eqpoll_iff_card psubsetE)
qed

lemma infinite_iff_psubset_le: "infinite A \ (\B. B \ A \ A \ B)"
  by (meson eqpoll_imp_lepoll infinite_iff_psubset lepoll_antisym psubsetE subset_imp_lepoll)

end

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