src/HOL/Cardinals/Cardinal_Order_Relation.thy

tuned;

+
−(*  Title:      HOL/Cardinals/Cardinal_Order_Relation.thy
+
−    Author:     Andrei Popescu, TU Muenchen
+
−    Copyright   2012
+
−
+
−Cardinal-order relations.
+
−*)
+
−
+
−section \<open>Cardinal-Order Relations\<close>
+
−
+
−theory Cardinal_Order_Relation
+
−imports BNF_Cardinal_Order_Relation Wellorder_Constructions
+
−begin
+
−
+
−declare
+
−  card_order_on_well_order_on[simp]
+
−  card_of_card_order_on[simp]
+
−  card_of_well_order_on[simp]
+
−  Field_card_of[simp]
+
−  card_of_Card_order[simp]
+
−  card_of_Well_order[simp]
+
−  card_of_least[simp]
+
−  card_of_unique[simp]
+
−  card_of_mono1[simp]
+
−  card_of_mono2[simp]
+
−  card_of_cong[simp]
+
−  card_of_Field_ordLess[simp]
+
−  card_of_Field_ordIso[simp]
+
−  card_of_underS[simp]
+
−  ordLess_Field[simp]
+
−  card_of_empty[simp]
+
−  card_of_empty1[simp]
+
−  card_of_image[simp]
+
−  card_of_singl_ordLeq[simp]
+
−  Card_order_singl_ordLeq[simp]
+
−  card_of_Pow[simp]
+
−  Card_order_Pow[simp]
+
−  card_of_Plus1[simp]
+
−  Card_order_Plus1[simp]
+
−  card_of_Plus2[simp]
+
−  Card_order_Plus2[simp]
+
−  card_of_Plus_mono1[simp]
+
−  card_of_Plus_mono2[simp]
+
−  card_of_Plus_mono[simp]
+
−  card_of_Plus_cong2[simp]
+
−  card_of_Plus_cong[simp]
+
−  card_of_Un_Plus_ordLeq[simp]
+
−  card_of_Times1[simp]
+
−  card_of_Times2[simp]
+
−  card_of_Times3[simp]
+
−  card_of_Times_mono1[simp]
+
−  card_of_Times_mono2[simp]
+
−  card_of_ordIso_finite[simp]
+
−  card_of_Times_same_infinite[simp]
+
−  card_of_Times_infinite_simps[simp]
+
−  card_of_Plus_infinite1[simp]
+
−  card_of_Plus_infinite2[simp]
+
−  card_of_Plus_ordLess_infinite[simp]
+
−  card_of_Plus_ordLess_infinite_Field[simp]
+
−  infinite_cartesian_product[simp]
+
−  cardSuc_Card_order[simp]
+
−  cardSuc_greater[simp]
+
−  cardSuc_ordLeq[simp]
+
−  cardSuc_ordLeq_ordLess[simp]
+
−  cardSuc_mono_ordLeq[simp]
+
−  cardSuc_invar_ordIso[simp]
+
−  card_of_cardSuc_finite[simp]
+
−  cardSuc_finite[simp]
+
−  card_of_Plus_ordLeq_infinite_Field[simp]
+
−  curr_in[intro, simp]
+
−
+
−
+
−subsection \<open>Cardinal of a set\<close>
+
−
+
−lemma card_of_inj_rel: assumes INJ: "!! x y y'. \<lbrakk>(x,y) : R; (x,y') : R\<rbrakk> \<Longrightarrow> y = y'"
+
−shows "|{y. EX x. (x,y) : R}| <=o |{x. EX y. (x,y) : R}|"
+
−proof-
+
−  let ?Y = "{y. EX x. (x,y) : R}"  let ?X = "{x. EX y. (x,y) : R}"
+
−  let ?f = "% y. SOME x. (x,y) : R"
+
−  have "?f ` ?Y <= ?X" by (auto dest: someI)
+
−  moreover have "inj_on ?f ?Y"
+
−  unfolding inj_on_def proof(auto)
+
−    fix y1 x1 y2 x2
+
−    assume *: "(x1, y1) \<in> R" "(x2, y2) \<in> R" and **: "?f y1 = ?f y2"
+
−    hence "(?f y1,y1) : R" using someI[of "% x. (x,y1) : R"] by auto
+
−    moreover have "(?f y2,y2) : R" using * someI[of "% x. (x,y2) : R"] by auto
+
−    ultimately show "y1 = y2" using ** INJ by auto
+
−  qed
+
−  ultimately show "|?Y| <=o |?X|" using card_of_ordLeq by blast
+
−qed
+
−
+
−lemma card_of_unique2: "\<lbrakk>card_order_on B r; bij_betw f A B\<rbrakk> \<Longrightarrow> r =o |A|"
+
−using card_of_ordIso card_of_unique ordIso_equivalence by blast
+
−
+
−lemma internalize_card_of_ordLess:
+
−"( |A| <o r) = (\<exists>B < Field r. |A| =o |B| \<and> |B| <o r)"
+
−proof
+
−  assume "|A| <o r"
+
−  then obtain p where 1: "Field p < Field r \<and> |A| =o p \<and> p <o r"
+
−  using internalize_ordLess[of "|A|" r] by blast
+
−  hence "Card_order p" using card_of_Card_order Card_order_ordIso2 by blast
+
−  hence "|Field p| =o p" using card_of_Field_ordIso by blast
+
−  hence "|A| =o |Field p| \<and> |Field p| <o r"
+
−  using 1 ordIso_equivalence ordIso_ordLess_trans by blast
+
−  thus "\<exists>B < Field r. |A| =o |B| \<and> |B| <o r" using 1 by blast
+
−next
+
−  assume "\<exists>B < Field r. |A| =o |B| \<and> |B| <o r"
+
−  thus "|A| <o r" using ordIso_ordLess_trans by blast
+
−qed
+
−
+
−lemma internalize_card_of_ordLess2:
+
−"( |A| <o |C| ) = (\<exists>B < C. |A| =o |B| \<and> |B| <o |C| )"
+
−using internalize_card_of_ordLess[of "A" "|C|"] Field_card_of[of C] by auto
+
−
+
−lemma Card_order_omax:
+
−assumes "finite R" and "R \<noteq> {}" and "\<forall>r\<in>R. Card_order r"
+
−shows "Card_order (omax R)"
+
−proof-
+
−  have "\<forall>r\<in>R. Well_order r"
+
−  using assms unfolding card_order_on_def by simp
+
−  thus ?thesis using assms apply - apply(drule omax_in) by auto
+
−qed
+
−
+
−lemma Card_order_omax2:
+
−assumes "finite I" and "I \<noteq> {}"
+
−shows "Card_order (omax {|A i| | i. i \<in> I})"
+
−proof-
+
−  let ?R = "{|A i| | i. i \<in> I}"
+
−  have "finite ?R" and "?R \<noteq> {}" using assms by auto
+
−  moreover have "\<forall>r\<in>?R. Card_order r"
+
−  using card_of_Card_order by auto
+
−  ultimately show ?thesis by(rule Card_order_omax)
+
−qed
+
−
+
−
+
−subsection \<open>Cardinals versus set operations on arbitrary sets\<close>
+
−
+
−lemma card_of_set_type[simp]: "|UNIV::'a set| <o |UNIV::'a set set|"
+
−using card_of_Pow[of "UNIV::'a set"] by simp
+
−
+
−lemma card_of_Un1[simp]:
+
−shows "|A| \<le>o |A \<union> B| "
+
−using inj_on_id[of A] card_of_ordLeq[of A _] by fastforce
+
−
+
−lemma card_of_diff[simp]:
+
−shows "|A - B| \<le>o |A|"
+
−using inj_on_id[of "A - B"] card_of_ordLeq[of "A - B" _] by fastforce
+
−
+
−lemma subset_ordLeq_strict:
+
−assumes "A \<le> B" and "|A| <o |B|"
+
−shows "A < B"
+
−proof-
+
−  {assume "\<not>(A < B)"
+
−   hence "A = B" using assms(1) by blast
+
−   hence False using assms(2) not_ordLess_ordIso card_of_refl by blast
+
−  }
+
−  thus ?thesis by blast
+
−qed
+
−
+
−corollary subset_ordLeq_diff:
+
−assumes "A \<le> B" and "|A| <o |B|"
+
−shows "B - A \<noteq> {}"
+
−using assms subset_ordLeq_strict by blast
+
−
+
−lemma card_of_empty4:
+
−"|{}::'b set| <o |A::'a set| = (A \<noteq> {})"
+
−proof(intro iffI notI)
+
−  assume *: "|{}::'b set| <o |A|" and "A = {}"
+
−  hence "|A| =o |{}::'b set|"
+
−  using card_of_ordIso unfolding bij_betw_def inj_on_def by blast
+
−  hence "|{}::'b set| =o |A|" using ordIso_symmetric by blast
+
−  with * show False using not_ordLess_ordIso[of "|{}::'b set|" "|A|"] by blast
+
−next
+
−  assume "A \<noteq> {}"
+
−  hence "(\<not> (\<exists>f. inj_on f A \<and> f ` A \<subseteq> {}))"
+
−  unfolding inj_on_def by blast
+
−  thus "| {} | <o | A |"
+
−  using card_of_ordLess by blast
+
−qed
+
−
+
−lemma card_of_empty5:
+
−"|A| <o |B| \<Longrightarrow> B \<noteq> {}"
+
−using card_of_empty not_ordLess_ordLeq by blast
+
−
+
−lemma Well_order_card_of_empty:
+
−"Well_order r \<Longrightarrow> |{}| \<le>o r" by simp
+
−
+
−lemma card_of_UNIV[simp]:
+
−"|A :: 'a set| \<le>o |UNIV :: 'a set|"
+
−using card_of_mono1[of A] by simp
+
−
+
−lemma card_of_UNIV2[simp]:
+
−"Card_order r \<Longrightarrow> (r :: 'a rel) \<le>o |UNIV :: 'a set|"
+
−using card_of_UNIV[of "Field r"] card_of_Field_ordIso
+
−      ordIso_symmetric ordIso_ordLeq_trans by blast
+
−
+
−lemma card_of_Pow_mono[simp]:
+
−assumes "|A| \<le>o |B|"
+
−shows "|Pow A| \<le>o |Pow B|"
+
−proof-
+
−  obtain f where "inj_on f A \<and> f ` A \<le> B"
+
−  using assms card_of_ordLeq[of A B] by auto
+
−  hence "inj_on (image f) (Pow A) \<and> (image f) ` (Pow A) \<le> (Pow B)"
+
−  by (auto simp add: inj_on_image_Pow image_Pow_mono)
+
−  thus ?thesis using card_of_ordLeq[of "Pow A"] by metis
+
−qed
+
−
+
−lemma ordIso_Pow_mono[simp]:
+
−assumes "r \<le>o r'"
+
−shows "|Pow(Field r)| \<le>o |Pow(Field r')|"
+
−using assms card_of_mono2 card_of_Pow_mono by blast
+
−
+
−lemma card_of_Pow_cong[simp]:
+
−assumes "|A| =o |B|"
+
−shows "|Pow A| =o |Pow B|"
+
−proof-
+
−  obtain f where "bij_betw f A B"
+
−  using assms card_of_ordIso[of A B] by auto
+
−  hence "bij_betw (image f) (Pow A) (Pow B)"
+
−  by (auto simp add: bij_betw_image_Pow)
+
−  thus ?thesis using card_of_ordIso[of "Pow A"] by auto
+
−qed
+
−
+
−lemma ordIso_Pow_cong[simp]:
+
−assumes "r =o r'"
+
−shows "|Pow(Field r)| =o |Pow(Field r')|"
+
−using assms card_of_cong card_of_Pow_cong by blast
+
−
+
−corollary Card_order_Plus_empty1:
+
−"Card_order r \<Longrightarrow> r =o |(Field r) <+> {}|"
+
−using card_of_Plus_empty1 card_of_Field_ordIso ordIso_equivalence by blast
+
−
+
−corollary Card_order_Plus_empty2:
+
−"Card_order r \<Longrightarrow> r =o |{} <+> (Field r)|"
+
−using card_of_Plus_empty2 card_of_Field_ordIso ordIso_equivalence by blast
+
−
+
−lemma Card_order_Un1:
+
−shows "Card_order r \<Longrightarrow> |Field r| \<le>o |(Field r) \<union> B| "
+
−using card_of_Un1 card_of_Field_ordIso ordIso_symmetric ordIso_ordLeq_trans by auto
+
−
+
−lemma card_of_Un2[simp]:
+
−shows "|A| \<le>o |B \<union> A|"
+
−using inj_on_id[of A] card_of_ordLeq[of A _] by fastforce
+
−
+
−lemma Card_order_Un2:
+
−shows "Card_order r \<Longrightarrow> |Field r| \<le>o |A \<union> (Field r)| "
+
−using card_of_Un2 card_of_Field_ordIso ordIso_symmetric ordIso_ordLeq_trans by auto
+
−
+
−lemma Un_Plus_bij_betw:
+
−assumes "A Int B = {}"
+
−shows "\<exists>f. bij_betw f (A \<union> B) (A <+> B)"
+
−proof-
+
−  let ?f = "\<lambda> c. if c \<in> A then Inl c else Inr c"
+
−  have "bij_betw ?f (A \<union> B) (A <+> B)"
+
−  using assms by(unfold bij_betw_def inj_on_def, auto)
+
−  thus ?thesis by blast
+
−qed
+
−
+
−lemma card_of_Un_Plus_ordIso:
+
−assumes "A Int B = {}"
+
−shows "|A \<union> B| =o |A <+> B|"
+
−using assms card_of_ordIso[of "A \<union> B"] Un_Plus_bij_betw[of A B] by auto
+
−
+
−lemma card_of_Un_Plus_ordIso1:
+
−"|A \<union> B| =o |A <+> (B - A)|"
+
−using card_of_Un_Plus_ordIso[of A "B - A"] by auto
+
−
+
−lemma card_of_Un_Plus_ordIso2:
+
−"|A \<union> B| =o |(A - B) <+> B|"
+
−using card_of_Un_Plus_ordIso[of "A - B" B] by auto
+
−
+
−lemma card_of_Times_singl1: "|A| =o |A \<times> {b}|"
+
−proof-
+
−  have "bij_betw fst (A \<times> {b}) A" unfolding bij_betw_def inj_on_def by force
+
−  thus ?thesis using card_of_ordIso ordIso_symmetric by blast
+
−qed
+
−
+
−corollary Card_order_Times_singl1:
+
−"Card_order r \<Longrightarrow> r =o |(Field r) \<times> {b}|"
+
−using card_of_Times_singl1[of _ b] card_of_Field_ordIso ordIso_equivalence by blast
+
−
+
−lemma card_of_Times_singl2: "|A| =o |{b} \<times> A|"
+
−proof-
+
−  have "bij_betw snd ({b} \<times> A) A" unfolding bij_betw_def inj_on_def by force
+
−  thus ?thesis using card_of_ordIso ordIso_symmetric by blast
+
−qed
+
−
+
−corollary Card_order_Times_singl2:
+
−"Card_order r \<Longrightarrow> r =o |{a} \<times> (Field r)|"
+
−using card_of_Times_singl2[of _ a] card_of_Field_ordIso ordIso_equivalence by blast
+
−
+
−lemma card_of_Times_assoc: "|(A \<times> B) \<times> C| =o |A \<times> B \<times> C|"
+
−proof -
+
−  let ?f = "\<lambda>((a,b),c). (a,(b,c))"
+
−  have "A \<times> B \<times> C \<subseteq> ?f ` ((A \<times> B) \<times> C)"
+
−  proof
+
−    fix x assume "x \<in> A \<times> B \<times> C"
+
−    then obtain a b c where *: "a \<in> A" "b \<in> B" "c \<in> C" "x = (a, b, c)" by blast
+
−    let ?x = "((a, b), c)"
+
−    from * have "?x \<in> (A \<times> B) \<times> C" "x = ?f ?x" by auto
+
−    thus "x \<in> ?f ` ((A \<times> B) \<times> C)" by blast
+
−  qed
+
−  hence "bij_betw ?f ((A \<times> B) \<times> C) (A \<times> B \<times> C)"
+
−  unfolding bij_betw_def inj_on_def by auto
+
−  thus ?thesis using card_of_ordIso by blast
+
−qed
+
−
+
−corollary Card_order_Times3:
+
−"Card_order r \<Longrightarrow> |Field r| \<le>o |(Field r) \<times> (Field r)|"
+
−  by (rule card_of_Times3)
+
−
+
−lemma card_of_Times_cong1[simp]:
+
−assumes "|A| =o |B|"
+
−shows "|A \<times> C| =o |B \<times> C|"
+
−using assms by (simp add: ordIso_iff_ordLeq)
+
−
+
−lemma card_of_Times_cong2[simp]:
+
−assumes "|A| =o |B|"
+
−shows "|C \<times> A| =o |C \<times> B|"
+
−using assms by (simp add: ordIso_iff_ordLeq)
+
−
+
−lemma card_of_Times_mono[simp]:
+
−assumes "|A| \<le>o |B|" and "|C| \<le>o |D|"
+
−shows "|A \<times> C| \<le>o |B \<times> D|"
+
−using assms card_of_Times_mono1[of A B C] card_of_Times_mono2[of C D B]
+
−      ordLeq_transitive[of "|A \<times> C|"] by blast
+
−
+
−corollary ordLeq_Times_mono:
+
−assumes "r \<le>o r'" and "p \<le>o p'"
+
−shows "|(Field r) \<times> (Field p)| \<le>o |(Field r') \<times> (Field p')|"
+
−using assms card_of_mono2[of r r'] card_of_mono2[of p p'] card_of_Times_mono by blast
+
−
+
−corollary ordIso_Times_cong1[simp]:
+
−assumes "r =o r'"
+
−shows "|(Field r) \<times> C| =o |(Field r') \<times> C|"
+
−using assms card_of_cong card_of_Times_cong1 by blast
+
−
+
−corollary ordIso_Times_cong2:
+
−assumes "r =o r'"
+
−shows "|A \<times> (Field r)| =o |A \<times> (Field r')|"
+
−using assms card_of_cong card_of_Times_cong2 by blast
+
−
+
−lemma card_of_Times_cong[simp]:
+
−assumes "|A| =o |B|" and "|C| =o |D|"
+
−shows "|A \<times> C| =o |B \<times> D|"
+
−using assms
+
−by (auto simp add: ordIso_iff_ordLeq)
+
−
+
−corollary ordIso_Times_cong:
+
−assumes "r =o r'" and "p =o p'"
+
−shows "|(Field r) \<times> (Field p)| =o |(Field r') \<times> (Field p')|"
+
−using assms card_of_cong[of r r'] card_of_cong[of p p'] card_of_Times_cong by blast
+
−
+
−lemma card_of_Sigma_mono2:
+
−assumes "inj_on f (I::'i set)" and "f ` I \<le> (J::'j set)"
+
−shows "|SIGMA i : I. (A::'j \<Rightarrow> 'a set) (f i)| \<le>o |SIGMA j : J. A j|"
+
−proof-
+
−  let ?LEFT = "SIGMA i : I. A (f i)"
+
−  let ?RIGHT = "SIGMA j : J. A j"
+
−  obtain u where u_def: "u = (\<lambda>(i::'i,a::'a). (f i,a))" by blast
+
−  have "inj_on u ?LEFT \<and> u `?LEFT \<le> ?RIGHT"
+
−  using assms unfolding u_def inj_on_def by auto
+
−  thus ?thesis using card_of_ordLeq by blast
+
−qed
+
−
+
−lemma card_of_Sigma_mono:
+
−assumes INJ: "inj_on f I" and IM: "f ` I \<le> J" and
+
−        LEQ: "\<forall>j \<in> J. |A j| \<le>o |B j|"
+
−shows "|SIGMA i : I. A (f i)| \<le>o |SIGMA j : J. B j|"
+
−proof-
+
−  have "\<forall>i \<in> I. |A(f i)| \<le>o |B(f i)|"
+
−  using IM LEQ by blast
+
−  hence "|SIGMA i : I. A (f i)| \<le>o |SIGMA i : I. B (f i)|"
+
−  using card_of_Sigma_mono1[of I] by metis
+
−  moreover have "|SIGMA i : I. B (f i)| \<le>o |SIGMA j : J. B j|"
+
−  using INJ IM card_of_Sigma_mono2 by blast
+
−  ultimately show ?thesis using ordLeq_transitive by blast
+
−qed
+
−
+
−lemma ordLeq_Sigma_mono1:
+
−assumes "\<forall>i \<in> I. p i \<le>o r i"
+
−shows "|SIGMA i : I. Field(p i)| \<le>o |SIGMA i : I. Field(r i)|"
+
−using assms by (auto simp add: card_of_Sigma_mono1)
+
−
+
−lemma ordLeq_Sigma_mono:
+
−assumes "inj_on f I" and "f ` I \<le> J" and
+
−        "\<forall>j \<in> J. p j \<le>o r j"
+
−shows "|SIGMA i : I. Field(p(f i))| \<le>o |SIGMA j : J. Field(r j)|"
+
−using assms card_of_mono2 card_of_Sigma_mono
+
−      [of f I J "\<lambda> i. Field(p i)" "\<lambda> j. Field(r j)"] by metis
+
−
+
−lemma card_of_Sigma_cong1:
+
−assumes "\<forall>i \<in> I. |A i| =o |B i|"
+
−shows "|SIGMA i : I. A i| =o |SIGMA i : I. B i|"
+
−using assms by (auto simp add: card_of_Sigma_mono1 ordIso_iff_ordLeq)
+
−
+
−lemma card_of_Sigma_cong2:
+
−assumes "bij_betw f (I::'i set) (J::'j set)"
+
−shows "|SIGMA i : I. (A::'j \<Rightarrow> 'a set) (f i)| =o |SIGMA j : J. A j|"
+
−proof-
+
−  let ?LEFT = "SIGMA i : I. A (f i)"
+
−  let ?RIGHT = "SIGMA j : J. A j"
+
−  obtain u where u_def: "u = (\<lambda>(i::'i,a::'a). (f i,a))" by blast
+
−  have "bij_betw u ?LEFT ?RIGHT"
+
−  using assms unfolding u_def bij_betw_def inj_on_def by auto
+
−  thus ?thesis using card_of_ordIso by blast
+
−qed
+
−
+
−lemma card_of_Sigma_cong:
+
−assumes BIJ: "bij_betw f I J" and
+
−        ISO: "\<forall>j \<in> J. |A j| =o |B j|"
+
−shows "|SIGMA i : I. A (f i)| =o |SIGMA j : J. B j|"
+
−proof-
+
−  have "\<forall>i \<in> I. |A(f i)| =o |B(f i)|"
+
−  using ISO BIJ unfolding bij_betw_def by blast
+
−  hence "|SIGMA i : I. A (f i)| =o |SIGMA i : I. B (f i)|" by (rule card_of_Sigma_cong1)
+
−  moreover have "|SIGMA i : I. B (f i)| =o |SIGMA j : J. B j|"
+
−  using BIJ card_of_Sigma_cong2 by blast
+
−  ultimately show ?thesis using ordIso_transitive by blast
+
−qed
+
−
+
−lemma ordIso_Sigma_cong1:
+
−assumes "\<forall>i \<in> I. p i =o r i"
+
−shows "|SIGMA i : I. Field(p i)| =o |SIGMA i : I. Field(r i)|"
+
−using assms by (auto simp add: card_of_Sigma_cong1)
+
−
+
−lemma ordLeq_Sigma_cong:
+
−assumes "bij_betw f I J" and
+
−        "\<forall>j \<in> J. p j =o r j"
+
−shows "|SIGMA i : I. Field(p(f i))| =o |SIGMA j : J. Field(r j)|"
+
−using assms card_of_cong card_of_Sigma_cong
+
−      [of f I J "\<lambda> j. Field(p j)" "\<lambda> j. Field(r j)"] by blast
+
−
+
−lemma card_of_UNION_Sigma2:
+
−assumes
+
−"!! i j. \<lbrakk>{i,j} <= I; i ~= j\<rbrakk> \<Longrightarrow> A i Int A j = {}"
+
−shows
+
−"|\<Union>i\<in>I. A i| =o |Sigma I A|"
+
−proof-
+
−  let ?L = "\<Union>i\<in>I. A i"  let ?R = "Sigma I A"
+
−  have "|?L| <=o |?R|" using card_of_UNION_Sigma .
+
−  moreover have "|?R| <=o |?L|"
+
−  proof-
+
−    have "inj_on snd ?R"
+
−    unfolding inj_on_def using assms by auto
+
−    moreover have "snd ` ?R <= ?L" by auto
+
−    ultimately show ?thesis using card_of_ordLeq by blast
+
−  qed
+
−  ultimately show ?thesis by(simp add: ordIso_iff_ordLeq)
+
−qed
+
−
+
−corollary Plus_into_Times:
+
−assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
+
−        B2: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B"
+
−shows "\<exists>f. inj_on f (A <+> B) \<and> f ` (A <+> B) \<le> A \<times> B"
+
−using assms by (auto simp add: card_of_Plus_Times card_of_ordLeq)
+
−
+
−corollary Plus_into_Times_types:
+
−assumes A2: "(a1::'a) \<noteq> a2" and  B2: "(b1::'b) \<noteq> b2"
+
−shows "\<exists>(f::'a + 'b \<Rightarrow> 'a * 'b). inj f"
+
−using assms Plus_into_Times[of a1 a2 UNIV b1 b2 UNIV]
+
−by auto
+
−
+
−corollary Times_same_infinite_bij_betw:
+
−assumes "\<not>finite A"
+
−shows "\<exists>f. bij_betw f (A \<times> A) A"
+
−using assms by (auto simp add: card_of_ordIso)
+
−
+
−corollary Times_same_infinite_bij_betw_types:
+
−assumes INF: "\<not>finite(UNIV::'a set)"
+
−shows "\<exists>(f::('a * 'a) => 'a). bij f"
+
−using assms Times_same_infinite_bij_betw[of "UNIV::'a set"]
+
−by auto
+
−
+
−corollary Times_infinite_bij_betw:
+
−assumes INF: "\<not>finite A" and NE: "B \<noteq> {}" and INJ: "inj_on g B \<and> g ` B \<le> A"
+
−shows "(\<exists>f. bij_betw f (A \<times> B) A) \<and> (\<exists>h. bij_betw h (B \<times> A) A)"
+
−proof-
+
−  have "|B| \<le>o |A|" using INJ card_of_ordLeq by blast
+
−  thus ?thesis using INF NE
+
−  by (auto simp add: card_of_ordIso card_of_Times_infinite)
+
−qed
+
−
+
−corollary Times_infinite_bij_betw_types:
+
−assumes INF: "\<not>finite(UNIV::'a set)" and
+
−        BIJ: "inj(g::'b \<Rightarrow> 'a)"
+
−shows "(\<exists>(f::('b * 'a) => 'a). bij f) \<and> (\<exists>(h::('a * 'b) => 'a). bij h)"
+
−using assms Times_infinite_bij_betw[of "UNIV::'a set" "UNIV::'b set" g]
+
−by auto
+
−
+
−lemma card_of_Times_ordLeq_infinite:
+
−"\<lbrakk>\<not>finite C; |A| \<le>o |C|; |B| \<le>o |C|\<rbrakk>
+
− \<Longrightarrow> |A \<times> B| \<le>o |C|"
+
−by(simp add: card_of_Sigma_ordLeq_infinite)
+
−
+
−corollary Plus_infinite_bij_betw:
+
−assumes INF: "\<not>finite A" and INJ: "inj_on g B \<and> g ` B \<le> A"
+
−shows "(\<exists>f. bij_betw f (A <+> B) A) \<and> (\<exists>h. bij_betw h (B <+> A) A)"
+
−proof-
+
−  have "|B| \<le>o |A|" using INJ card_of_ordLeq by blast
+
−  thus ?thesis using INF
+
−  by (auto simp add: card_of_ordIso)
+
−qed
+
−
+
−corollary Plus_infinite_bij_betw_types:
+
−assumes INF: "\<not>finite(UNIV::'a set)" and
+
−        BIJ: "inj(g::'b \<Rightarrow> 'a)"
+
−shows "(\<exists>(f::('b + 'a) => 'a). bij f) \<and> (\<exists>(h::('a + 'b) => 'a). bij h)"
+
−using assms Plus_infinite_bij_betw[of "UNIV::'a set" g "UNIV::'b set"]
+
−by auto
+
−
+
−lemma card_of_Un_infinite:
+
−assumes INF: "\<not>finite A" and LEQ: "|B| \<le>o |A|"
+
−shows "|A \<union> B| =o |A| \<and> |B \<union> A| =o |A|"
+
−proof-
+
−  have "|A \<union> B| \<le>o |A <+> B|" by (rule card_of_Un_Plus_ordLeq)
+
−  moreover have "|A <+> B| =o |A|"
+
−  using assms by (metis card_of_Plus_infinite)
+
−  ultimately have "|A \<union> B| \<le>o |A|" using ordLeq_ordIso_trans by blast
+
−  hence "|A \<union> B| =o |A|" using card_of_Un1 ordIso_iff_ordLeq by blast
+
−  thus ?thesis using Un_commute[of B A] by auto
+
−qed
+
−
+
−lemma card_of_Un_infinite_simps[simp]:
+
−"\<lbrakk>\<not>finite A; |B| \<le>o |A| \<rbrakk> \<Longrightarrow> |A \<union> B| =o |A|"
+
−"\<lbrakk>\<not>finite A; |B| \<le>o |A| \<rbrakk> \<Longrightarrow> |B \<union> A| =o |A|"
+
−using card_of_Un_infinite by auto
+
−
+
−lemma card_of_Un_diff_infinite:
+
−assumes INF: "\<not>finite A" and LESS: "|B| <o |A|"
+
−shows "|A - B| =o |A|"
+
−proof-
+
−  obtain C where C_def: "C = A - B" by blast
+
−  have "|A \<union> B| =o |A|"
+
−  using assms ordLeq_iff_ordLess_or_ordIso card_of_Un_infinite by blast
+
−  moreover have "C \<union> B = A \<union> B" unfolding C_def by auto
+
−  ultimately have 1: "|C \<union> B| =o |A|" by auto
+
−  (*  *)
+
−  {assume *: "|C| \<le>o |B|"
+
−   moreover
+
−   {assume **: "finite B"
+
−    hence "finite C"
+
−    using card_of_ordLeq_finite * by blast
+
−    hence False using ** INF card_of_ordIso_finite 1 by blast
+
−   }
+
−   hence "\<not>finite B" by auto
+
−   ultimately have False
+
−   using card_of_Un_infinite 1 ordIso_equivalence(1,3) LESS not_ordLess_ordIso by metis
+
−  }
+
−  hence 2: "|B| \<le>o |C|" using card_of_Well_order ordLeq_total by blast
+
−  {assume *: "finite C"
+
−    hence "finite B" using card_of_ordLeq_finite 2 by blast
+
−    hence False using * INF card_of_ordIso_finite 1 by blast
+
−  }
+
−  hence "\<not>finite C" by auto
+
−  hence "|C| =o |A|"
+
−  using  card_of_Un_infinite 1 2 ordIso_equivalence(1,3) by metis
+
−  thus ?thesis unfolding C_def .
+
−qed
+
−
+
−corollary Card_order_Un_infinite:
+
−assumes INF: "\<not>finite(Field r)" and CARD: "Card_order r" and
+
−        LEQ: "p \<le>o r"
+
−shows "| (Field r) \<union> (Field p) | =o r \<and> | (Field p) \<union> (Field r) | =o r"
+
−proof-
+
−  have "| Field r \<union> Field p | =o | Field r | \<and>
+
−        | Field p \<union> Field r | =o | Field r |"
+
−  using assms by (auto simp add: card_of_Un_infinite)
+
−  thus ?thesis
+
−  using assms card_of_Field_ordIso[of r]
+
−        ordIso_transitive[of "|Field r \<union> Field p|"]
+
−        ordIso_transitive[of _ "|Field r|"] by blast
+
−qed
+
−
+
−corollary subset_ordLeq_diff_infinite:
+
−assumes INF: "\<not>finite B" and SUB: "A \<le> B" and LESS: "|A| <o |B|"
+
−shows "\<not>finite (B - A)"
+
−using assms card_of_Un_diff_infinite card_of_ordIso_finite by blast
+
−
+
−lemma card_of_Times_ordLess_infinite[simp]:
+
−assumes INF: "\<not>finite C" and
+
−        LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|"
+
−shows "|A \<times> B| <o |C|"
+
−proof(cases "A = {} \<or> B = {}")
+
−  assume Case1: "A = {} \<or> B = {}"
+
−  hence "A \<times> B = {}" by blast
+
−  moreover have "C \<noteq> {}" using
+
−  LESS1 card_of_empty5 by blast
+
−  ultimately show ?thesis by(auto simp add:  card_of_empty4)
+
−next
+
−  assume Case2: "\<not>(A = {} \<or> B = {})"
+
−  {assume *: "|C| \<le>o |A \<times> B|"
+
−   hence "\<not>finite (A \<times> B)" using INF card_of_ordLeq_finite by blast
+
−   hence 1: "\<not>finite A \<or> \<not>finite B" using finite_cartesian_product by blast
+
−   {assume Case21: "|A| \<le>o |B|"
+
−    hence "\<not>finite B" using 1 card_of_ordLeq_finite by blast
+
−    hence "|A \<times> B| =o |B|" using Case2 Case21
+
−    by (auto simp add: card_of_Times_infinite)
+
−    hence False using LESS2 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast
+
−   }
+
−   moreover
+
−   {assume Case22: "|B| \<le>o |A|"
+
−    hence "\<not>finite A" using 1 card_of_ordLeq_finite by blast
+
−    hence "|A \<times> B| =o |A|" using Case2 Case22
+
−    by (auto simp add: card_of_Times_infinite)
+
−    hence False using LESS1 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast
+
−   }
+
−   ultimately have False using ordLeq_total card_of_Well_order[of A]
+
−   card_of_Well_order[of B] by blast
+
−  }
+
−  thus ?thesis using ordLess_or_ordLeq[of "|A \<times> B|" "|C|"]
+
−  card_of_Well_order[of "A \<times> B"] card_of_Well_order[of "C"] by auto
+
−qed
+
−
+
−lemma card_of_Times_ordLess_infinite_Field[simp]:
+
−assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and
+
−        LESS1: "|A| <o r" and LESS2: "|B| <o r"
+
−shows "|A \<times> B| <o r"
+
−proof-
+
−  let ?C  = "Field r"
+
−  have 1: "r =o |?C| \<and> |?C| =o r" using r card_of_Field_ordIso
+
−  ordIso_symmetric by blast
+
−  hence "|A| <o |?C|"  "|B| <o |?C|"
+
−  using LESS1 LESS2 ordLess_ordIso_trans by blast+
+
−  hence  "|A \<times> B| <o |?C|" using INF
+
−  card_of_Times_ordLess_infinite by blast
+
−  thus ?thesis using 1 ordLess_ordIso_trans by blast
+
−qed
+
−
+
−lemma card_of_Un_ordLess_infinite[simp]:
+
−assumes INF: "\<not>finite C" and
+
−        LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|"
+
−shows "|A \<union> B| <o |C|"
+
−using assms card_of_Plus_ordLess_infinite card_of_Un_Plus_ordLeq
+
−      ordLeq_ordLess_trans by blast
+
−
+
−lemma card_of_Un_ordLess_infinite_Field[simp]:
+
−assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and
+
−        LESS1: "|A| <o r" and LESS2: "|B| <o r"
+
−shows "|A Un B| <o r"
+
−proof-
+
−  let ?C  = "Field r"
+
−  have 1: "r =o |?C| \<and> |?C| =o r" using r card_of_Field_ordIso
+
−  ordIso_symmetric by blast
+
−  hence "|A| <o |?C|"  "|B| <o |?C|"
+
−  using LESS1 LESS2 ordLess_ordIso_trans by blast+
+
−  hence  "|A Un B| <o |?C|" using INF
+
−  card_of_Un_ordLess_infinite by blast
+
−  thus ?thesis using 1 ordLess_ordIso_trans by blast
+
−qed
+
−
+
−lemma ordLeq_finite_Field:
+
−assumes "r \<le>o s" and "finite (Field s)"
+
−shows "finite (Field r)"
+
−by (metis assms card_of_mono2 card_of_ordLeq_infinite)
+
−
+
−lemma ordIso_finite_Field:
+
−assumes "r =o s"
+
−shows "finite (Field r) \<longleftrightarrow> finite (Field s)"
+
−by (metis assms ordIso_iff_ordLeq ordLeq_finite_Field)
+
−
+
−
+
−subsection \<open>Cardinals versus set operations involving infinite sets\<close>
+
−
+
−lemma finite_iff_cardOf_nat:
+
−"finite A = ( |A| <o |UNIV :: nat set| )"
+
−using infinite_iff_card_of_nat[of A]
+
−not_ordLeq_iff_ordLess[of "|A|" "|UNIV :: nat set|"]
+
−by fastforce
+
−
+
−lemma finite_ordLess_infinite2[simp]:
+
−assumes "finite A" and "\<not>finite B"
+
−shows "|A| <o |B|"
+
−using assms
+
−finite_ordLess_infinite[of "|A|" "|B|"]
+
−card_of_Well_order[of A] card_of_Well_order[of B]
+
−Field_card_of[of A] Field_card_of[of B] by auto
+
−
+
−lemma infinite_card_of_insert:
+
−assumes "\<not>finite A"
+
−shows "|insert a A| =o |A|"
+
−proof-
+
−  have iA: "insert a A = A \<union> {a}" by simp
+
−  show ?thesis
+
−  using infinite_imp_bij_betw2[OF assms] unfolding iA
+
−  by (metis bij_betw_inv card_of_ordIso)
+
−qed
+
−
+
−lemma card_of_Un_singl_ordLess_infinite1:
+
−assumes "\<not>finite B" and "|A| <o |B|"
+
−shows "|{a} Un A| <o |B|"
+
−proof-
+
−  have "|{a}| <o |B|" using assms by auto
+
−  thus ?thesis using assms card_of_Un_ordLess_infinite[of B] by blast
+
−qed
+
−
+
−lemma card_of_Un_singl_ordLess_infinite:
+
−assumes "\<not>finite B"
+
−shows "( |A| <o |B| ) = ( |{a} Un A| <o |B| )"
+
−using assms card_of_Un_singl_ordLess_infinite1[of B A]
+
−proof(auto)
+
−  assume "|insert a A| <o |B|"
+
−  moreover have "|A| <=o |insert a A|" using card_of_mono1[of A "insert a A"] by blast
+
−  ultimately show "|A| <o |B|" using ordLeq_ordLess_trans by blast
+
−qed
+
−
+
−
+
−subsection \<open>Cardinals versus lists\<close>
+
−
+
−text\<open>The next is an auxiliary operator, which shall be used for inductive
+
−proofs of facts concerning the cardinality of \<open>List\<close> :\<close>
+
−
+
−definition nlists :: "'a set \<Rightarrow> nat \<Rightarrow> 'a list set"
+
−where "nlists A n \<equiv> {l. set l \<le> A \<and> length l = n}"
+
−
+
−lemma lists_def2: "lists A = {l. set l \<le> A}"
+
−using in_listsI by blast
+
−
+
−lemma lists_UNION_nlists: "lists A = (\<Union>n. nlists A n)"
+
−unfolding lists_def2 nlists_def by blast
+
−
+
−lemma card_of_lists: "|A| \<le>o |lists A|"
+
−proof-
+
−  let ?h = "\<lambda> a. [a]"
+
−  have "inj_on ?h A \<and> ?h ` A \<le> lists A"
+
−  unfolding inj_on_def lists_def2 by auto
+
−  thus ?thesis by (metis card_of_ordLeq)
+
−qed
+
−
+
−lemma nlists_0: "nlists A 0 = {[]}"
+
−unfolding nlists_def by auto
+
−
+
−lemma nlists_not_empty:
+
−assumes "A \<noteq> {}"
+
−shows "nlists A n \<noteq> {}"
+
−proof(induct n, simp add: nlists_0)
+
−  fix n assume "nlists A n \<noteq> {}"
+
−  then obtain a and l where "a \<in> A \<and> l \<in> nlists A n" using assms by auto
+
−  hence "a # l \<in> nlists A (Suc n)" unfolding nlists_def by auto
+
−  thus "nlists A (Suc n) \<noteq> {}" by auto
+
−qed
+
−
+
−lemma Nil_in_lists: "[] \<in> lists A"
+
−unfolding lists_def2 by auto
+
−
+
−lemma lists_not_empty: "lists A \<noteq> {}"
+
−using Nil_in_lists by blast
+
−
+
−lemma card_of_nlists_Succ: "|nlists A (Suc n)| =o |A \<times> (nlists A n)|"
+
−proof-
+
−  let ?B = "A \<times> (nlists A n)"   let ?h = "\<lambda>(a,l). a # l"
+
−  have "inj_on ?h ?B \<and> ?h ` ?B \<le> nlists A (Suc n)"
+
−  unfolding inj_on_def nlists_def by auto
+
−  moreover have "nlists A (Suc n) \<le> ?h ` ?B"
+
−  proof(auto)
+
−    fix l assume "l \<in> nlists A (Suc n)"
+
−    hence 1: "length l = Suc n \<and> set l \<le> A" unfolding nlists_def by auto
+
−    then obtain a and l' where 2: "l = a # l'" by (auto simp: length_Suc_conv)
+
−    hence "a \<in> A \<and> set l' \<le> A \<and> length l' = n" using 1 by auto
+
−    thus "l \<in> ?h ` ?B"  using 2 unfolding nlists_def by auto
+
−  qed
+
−  ultimately have "bij_betw ?h ?B (nlists A (Suc n))"
+
−  unfolding bij_betw_def by auto
+
−  thus ?thesis using card_of_ordIso ordIso_symmetric by blast
+
−qed
+
−
+
−lemma card_of_nlists_infinite:
+
−assumes "\<not>finite A"
+
−shows "|nlists A n| \<le>o |A|"
+
−proof(induct n)
+
−  have "A \<noteq> {}" using assms by auto
+
−  thus "|nlists A 0| \<le>o |A|" by (simp add: nlists_0)
+
−next
+
−  fix n assume IH: "|nlists A n| \<le>o |A|"
+
−  have "|nlists A (Suc n)| =o |A \<times> (nlists A n)|"
+
−  using card_of_nlists_Succ by blast
+
−  moreover
+
−  {have "nlists A n \<noteq> {}" using assms nlists_not_empty[of A] by blast
+
−   hence "|A \<times> (nlists A n)| =o |A|"
+
−   using assms IH by (auto simp add: card_of_Times_infinite)
+
−  }
+
−  ultimately show "|nlists A (Suc n)| \<le>o |A|"
+
−  using ordIso_transitive ordIso_iff_ordLeq by blast
+
−qed
+
−
+
−lemma Card_order_lists: "Card_order r \<Longrightarrow> r \<le>o |lists(Field r) |"
+
−using card_of_lists card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
+
−
+
−lemma Union_set_lists: "\<Union>(set ` (lists A)) = A"
+
−  unfolding lists_def2
+
−proof(auto)
+
−  fix a assume "a \<in> A"
+
−  hence "set [a] \<le> A \<and> a \<in> set [a]" by auto
+
−  thus "\<exists>l. set l \<le> A \<and> a \<in> set l" by blast
+
−qed
+
−
+
−lemma inj_on_map_lists:
+
−assumes "inj_on f A"
+
−shows "inj_on (map f) (lists A)"
+
−using assms Union_set_lists[of A] inj_on_mapI[of f "lists A"] by auto
+
−
+
−lemma map_lists_mono:
+
−assumes "f ` A \<le> B"
+
−shows "(map f) ` (lists A) \<le> lists B"
+
−using assms unfolding lists_def2 by (auto, blast) (* lethal combination of methods :)  *)
+
−
+
−lemma map_lists_surjective:
+
−assumes "f ` A = B"
+
−shows "(map f) ` (lists A) = lists B"
+
−using assms unfolding lists_def2
+
−proof (auto, blast)
+
−  fix l' assume *: "set l' \<le> f ` A"
+
−  have "set l' \<le> f ` A \<longrightarrow> l' \<in> map f ` {l. set l \<le> A}"
+
−  proof(induct l', auto)
+
−    fix l a
+
−    assume "a \<in> A" and "set l \<le> A" and
+
−           IH: "f ` (set l) \<le> f ` A"
+
−    hence "set (a # l) \<le> A" by auto
+
−    hence "map f (a # l) \<in> map f ` {l. set l \<le> A}" by blast
+
−    thus "f a # map f l \<in> map f ` {l. set l \<le> A}" by auto
+
−  qed
+
−  thus "l' \<in> map f ` {l. set l \<le> A}" using * by auto
+
−qed
+
−
+
−lemma bij_betw_map_lists:
+
−assumes "bij_betw f A B"
+
−shows "bij_betw (map f) (lists A) (lists B)"
+
−using assms unfolding bij_betw_def
+
−by(auto simp add: inj_on_map_lists map_lists_surjective)
+
−
+
−lemma card_of_lists_mono[simp]:
+
−assumes "|A| \<le>o |B|"
+
−shows "|lists A| \<le>o |lists B|"
+
−proof-
+
−  obtain f where "inj_on f A \<and> f ` A \<le> B"
+
−  using assms card_of_ordLeq[of A B] by auto
+
−  hence "inj_on (map f) (lists A) \<and> (map f) ` (lists A) \<le> (lists B)"
+
−  by (auto simp add: inj_on_map_lists map_lists_mono)
+
−  thus ?thesis using card_of_ordLeq[of "lists A"] by metis
+
−qed
+
−
+
−lemma ordIso_lists_mono:
+
−assumes "r \<le>o r'"
+
−shows "|lists(Field r)| \<le>o |lists(Field r')|"
+
−using assms card_of_mono2 card_of_lists_mono by blast
+
−
+
−lemma card_of_lists_cong[simp]:
+
−assumes "|A| =o |B|"
+
−shows "|lists A| =o |lists B|"
+
−proof-
+
−  obtain f where "bij_betw f A B"
+
−  using assms card_of_ordIso[of A B] by auto
+
−  hence "bij_betw (map f) (lists A) (lists B)"
+
−  by (auto simp add: bij_betw_map_lists)
+
−  thus ?thesis using card_of_ordIso[of "lists A"] by auto
+
−qed
+
−
+
−lemma card_of_lists_infinite[simp]:
+
−assumes "\<not>finite A"
+
−shows "|lists A| =o |A|"
+
−proof-
+
−  have "|lists A| \<le>o |A|" unfolding lists_UNION_nlists
+
−  by (rule card_of_UNION_ordLeq_infinite[OF assms _ ballI[OF card_of_nlists_infinite[OF assms]]])
+
−    (metis infinite_iff_card_of_nat assms)
+
−  thus ?thesis using card_of_lists ordIso_iff_ordLeq by blast
+
−qed
+
−
+
−lemma Card_order_lists_infinite:
+
−assumes "Card_order r" and "\<not>finite(Field r)"
+
−shows "|lists(Field r)| =o r"
+
−using assms card_of_lists_infinite card_of_Field_ordIso ordIso_transitive by blast
+
−
+
−lemma ordIso_lists_cong:
+
−assumes "r =o r'"
+
−shows "|lists(Field r)| =o |lists(Field r')|"
+
−using assms card_of_cong card_of_lists_cong by blast
+
−
+
−corollary lists_infinite_bij_betw:
+
−assumes "\<not>finite A"
+
−shows "\<exists>f. bij_betw f (lists A) A"
+
−using assms card_of_lists_infinite card_of_ordIso by blast
+
−
+
−corollary lists_infinite_bij_betw_types:
+
−assumes "\<not>finite(UNIV :: 'a set)"
+
−shows "\<exists>(f::'a list \<Rightarrow> 'a). bij f"
+
−using assms assms lists_infinite_bij_betw[of "UNIV::'a set"]
+
−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
+
−
+
−lemma card_of_Fpow[simp]: "|A| \<le>o |Fpow A|"
+
−proof-
+
−  let ?h = "\<lambda> a. {a}"
+
−  have "inj_on ?h A \<and> ?h ` A \<le> Fpow A"
+
−  unfolding inj_on_def Fpow_def by auto
+
−  thus ?thesis using card_of_ordLeq by metis
+
−qed
+
−
+
−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"
+
−using assms proof(unfold Fpow_def, auto)
+
−  fix Y assume *: "Y \<le> f ` A" and **: "finite Y"
+
−  hence "\<forall>b \<in> Y. \<exists>a. a \<in> A \<and> f a = b" by auto
+
−  with bchoice[of Y "\<lambda>b a. a \<in> A \<and> f a = b"]
+
−  obtain g where 1: "\<forall>b \<in> Y. g b \<in> A \<and> f(g b) = b" by blast
+
−  obtain X where X_def: "X = g ` Y" by blast
+
−  have "f ` X = Y \<and> X \<le> A \<and> finite X"
+
−  by(unfold X_def, force simp add: ** 1)
+
−  thus "Y \<in> (image f) ` {X. X \<le> A \<and> finite X}" by auto
+
−qed
+
−
+
−lemma bij_betw_image_Fpow:
+
−assumes "bij_betw f A B"
+
−shows "bij_betw (image f) (Fpow A) (Fpow B)"
+
−using assms unfolding bij_betw_def
+
−by (auto simp add: inj_on_image_Fpow image_Fpow_surjective)
+
−
+
−lemma card_of_Fpow_mono[simp]:
+
−assumes "|A| \<le>o |B|"
+
−shows "|Fpow A| \<le>o |Fpow B|"
+
−proof-
+
−  obtain f where "inj_on f A \<and> f ` A \<le> B"
+
−  using assms card_of_ordLeq[of A B] by auto
+
−  hence "inj_on (image f) (Fpow A) \<and> (image f) ` (Fpow A) \<le> (Fpow B)"
+
−  by (auto simp add: inj_on_image_Fpow image_Fpow_mono)
+
−  thus ?thesis using card_of_ordLeq[of "Fpow A"] by auto
+
−qed
+
−
+
−lemma ordIso_Fpow_mono:
+
−assumes "r \<le>o r'"
+
−shows "|Fpow(Field r)| \<le>o |Fpow(Field r')|"
+
−using assms card_of_mono2 card_of_Fpow_mono by blast
+
−
+
−lemma card_of_Fpow_cong[simp]:
+
−assumes "|A| =o |B|"
+
−shows "|Fpow A| =o |Fpow B|"
+
−proof-
+
−  obtain f where "bij_betw f A B"
+
−  using assms card_of_ordIso[of A B] by auto
+
−  hence "bij_betw (image f) (Fpow A) (Fpow B)"
+
−  by (auto simp add: bij_betw_image_Fpow)
+
−  thus ?thesis using card_of_ordIso[of "Fpow A"] by auto
+
−qed
+
−
+
−lemma ordIso_Fpow_cong:
+
−assumes "r =o r'"
+
−shows "|Fpow(Field r)| =o |Fpow(Field r')|"
+
−using assms card_of_cong card_of_Fpow_cong by blast
+
−
+
−lemma card_of_Fpow_lists: "|Fpow A| \<le>o |lists A|"
+
−proof-
+
−  have "set ` (lists A) = Fpow A"
+
−  unfolding lists_def2 Fpow_def using finite_list finite_set by blast
+
−  thus ?thesis using card_of_ordLeq2[of "Fpow A"] Fpow_not_empty[of A] by blast
+
−qed
+
−
+
−lemma card_of_Fpow_infinite[simp]:
+
−assumes "\<not>finite A"
+
−shows "|Fpow A| =o |A|"
+
−using assms card_of_Fpow_lists card_of_lists_infinite card_of_Fpow
+
−      ordLeq_ordIso_trans ordIso_iff_ordLeq by blast
+
−
+
−corollary Fpow_infinite_bij_betw:
+
−assumes "\<not>finite A"
+
−shows "\<exists>f. bij_betw f (Fpow A) A"
+
−using assms card_of_Fpow_infinite card_of_ordIso by blast
+
−
+
−
+
−subsection \<open>The cardinal $\omega$ and the finite cardinals\<close>
+
−
+
−subsubsection \<open>First as well-orders\<close>
+
−
+
−lemma Field_natLess: "Field natLess = (UNIV::nat set)"
+
−by(unfold Field_def natLess_def, auto)
+
−
+
−lemma natLeq_well_order_on: "well_order_on UNIV natLeq"
+
−using natLeq_Well_order Field_natLeq by auto
+
−
+
−lemma natLeq_wo_rel: "wo_rel natLeq"
+
−unfolding wo_rel_def using natLeq_Well_order .
+
−
+
−lemma natLeq_ofilter_less: "ofilter natLeq {0 ..< n}"
+
−by(auto simp add: natLeq_wo_rel wo_rel.ofilter_def,
+
−   simp add: Field_natLeq, unfold under_def natLeq_def, auto)
+
−
+
−lemma natLeq_ofilter_leq: "ofilter natLeq {0 .. n}"
+
−by(auto simp add: natLeq_wo_rel wo_rel.ofilter_def,
+
−   simp add: Field_natLeq, unfold under_def natLeq_def, auto)
+
−
+
−lemma natLeq_UNIV_ofilter: "wo_rel.ofilter natLeq UNIV"
+
−using natLeq_wo_rel Field_natLeq wo_rel.Field_ofilter[of natLeq] by auto
+
−
+
−lemma closed_nat_set_iff:
+
−assumes "\<forall>(m::nat) n. n \<in> A \<and> m \<le> n \<longrightarrow> m \<in> A"
+
−shows "A = UNIV \<or> (\<exists>n. A = {0 ..< n})"
+
−proof-
+
−  {assume "A \<noteq> UNIV" hence "\<exists>n. n \<notin> A" by blast
+
−   moreover obtain n where n_def: "n = (LEAST n. n \<notin> A)" by blast
+
−   ultimately have 1: "n \<notin> A \<and> (\<forall>m. m < n \<longrightarrow> m \<in> A)"
+
−   using LeastI_ex[of "\<lambda> n. n \<notin> A"] n_def Least_le[of "\<lambda> n. n \<notin> A"] by fastforce
+
−   have "A = {0 ..< n}"
+
−   proof(auto simp add: 1)
+
−     fix m assume *: "m \<in> A"
+
−     {assume "n \<le> m" with assms * have "n \<in> A" by blast
+
−      hence False using 1 by auto
+
−     }
+
−     thus "m < n" by fastforce
+
−   qed
+
−   hence "\<exists>n. A = {0 ..< n}" by blast
+
−  }
+
−  thus ?thesis by blast
+
−qed
+
−
+
−lemma natLeq_ofilter_iff:
+
−"ofilter natLeq A = (A = UNIV \<or> (\<exists>n. A = {0 ..< n}))"
+
−proof(rule iffI)
+
−  assume "ofilter natLeq A"
+
−  hence "\<forall>m n. n \<in> A \<and> m \<le> n \<longrightarrow> m \<in> A" using natLeq_wo_rel
+
−  by(auto simp add: natLeq_def wo_rel.ofilter_def under_def)
+
−  thus "A = UNIV \<or> (\<exists>n. A = {0 ..< n})" using closed_nat_set_iff by blast
+
−next
+
−  assume "A = UNIV \<or> (\<exists>n. A = {0 ..< n})"
+
−  thus "ofilter natLeq A"
+
−  by(auto simp add: natLeq_ofilter_less natLeq_UNIV_ofilter)
+
−qed
+
−
+
−lemma natLeq_under_leq: "under natLeq n = {0 .. n}"
+
−unfolding under_def natLeq_def by auto
+
−
+
−lemma natLeq_on_ofilter_less_eq:
+
−"n \<le> m \<Longrightarrow> wo_rel.ofilter (natLeq_on m) {0 ..< n}"
+
−apply (auto simp add: natLeq_on_wo_rel wo_rel.ofilter_def)
+
−apply (simp add: Field_natLeq_on)
+
−by (auto simp add: under_def)
+
−
+
−lemma natLeq_on_ofilter_iff:
+
−"wo_rel.ofilter (natLeq_on m) A = (\<exists>n \<le> m. A = {0 ..< n})"
+
−proof(rule iffI)
+
−  assume *: "wo_rel.ofilter (natLeq_on m) A"
+
−  hence 1: "A \<le> {0..<m}"
+
−  by (auto simp add: natLeq_on_wo_rel wo_rel.ofilter_def under_def Field_natLeq_on)
+
−  hence "\<forall>n1 n2. n2 \<in> A \<and> n1 \<le> n2 \<longrightarrow> n1 \<in> A"
+
−  using * by(fastforce simp add: natLeq_on_wo_rel wo_rel.ofilter_def under_def)
+
−  hence "A = UNIV \<or> (\<exists>n. A = {0 ..< n})" using closed_nat_set_iff by blast
+
−  thus "\<exists>n \<le> m. A = {0 ..< n}" using 1 atLeastLessThan_less_eq by blast
+
−next
+
−  assume "(\<exists>n\<le>m. A = {0 ..< n})"
+
−  thus "wo_rel.ofilter (natLeq_on m) A" by (auto simp add: natLeq_on_ofilter_less_eq)
+
−qed
+
−
+
−corollary natLeq_on_ofilter:
+
−"ofilter(natLeq_on n) {0 ..< n}"
+
−by (auto simp add: natLeq_on_ofilter_less_eq)
+
−
+
−lemma natLeq_on_ofilter_less:
+
−"n < m \<Longrightarrow> ofilter (natLeq_on m) {0 .. n}"
+
−by(auto simp add: natLeq_on_wo_rel wo_rel.ofilter_def,
+
−   simp add: Field_natLeq_on, unfold under_def, auto)
+
−
+
−lemma natLeq_on_ordLess_natLeq: "natLeq_on n <o natLeq"
+
−using Field_natLeq Field_natLeq_on[of n]
+
−      finite_ordLess_infinite[of "natLeq_on n" natLeq]
+
−      natLeq_Well_order natLeq_on_Well_order[of n] by auto
+
−
+
−lemma natLeq_on_injective:
+
−"natLeq_on m = natLeq_on n \<Longrightarrow> m = n"
+
−using Field_natLeq_on[of m] Field_natLeq_on[of n]
+
−      atLeastLessThan_injective[of m n, unfolded atLeastLessThan_def] by blast
+
−
+
−lemma natLeq_on_injective_ordIso:
+
−"(natLeq_on m =o natLeq_on n) = (m = n)"
+
−proof(auto simp add: natLeq_on_Well_order ordIso_reflexive)
+
−  assume "natLeq_on m =o natLeq_on n"
+
−  then obtain f where "bij_betw f {x. x<m} {x. x<n}"
+
−  using Field_natLeq_on unfolding ordIso_def iso_def[abs_def] by auto
+
−  thus "m = n" using atLeastLessThan_injective2[of f m n]
+
−    unfolding atLeast_0 atLeastLessThan_def lessThan_def Int_UNIV_left by blast
+
−qed
+
−
+
−
+
−subsubsection \<open>Then as cardinals\<close>
+
−
+
−lemma ordIso_natLeq_infinite1:
+
−"|A| =o natLeq \<Longrightarrow> \<not>finite A"
+
−using ordIso_symmetric ordIso_imp_ordLeq infinite_iff_natLeq_ordLeq by blast
+
−
+
−lemma ordIso_natLeq_infinite2:
+
−"natLeq =o |A| \<Longrightarrow> \<not>finite A"
+
−using ordIso_imp_ordLeq infinite_iff_natLeq_ordLeq by blast
+
−
+
−lemma ordIso_natLeq_on_imp_finite:
+
−"|A| =o natLeq_on n \<Longrightarrow> finite A"
+
−unfolding ordIso_def iso_def[abs_def]
+
−by (auto simp: Field_natLeq_on bij_betw_finite)
+
−
+
−lemma natLeq_on_Card_order: "Card_order (natLeq_on n)"
+
−proof(unfold card_order_on_def,
+
−      auto simp add: natLeq_on_Well_order, simp add: Field_natLeq_on)
+
−  fix r assume "well_order_on {x. x < n} r"
+
−  thus "natLeq_on n \<le>o r"
+
−  using finite_atLeastLessThan natLeq_on_well_order_on
+
−        finite_well_order_on_ordIso ordIso_iff_ordLeq by blast
+
−qed
+
−
+
−corollary card_of_Field_natLeq_on:
+
−"|Field (natLeq_on n)| =o natLeq_on n"
+
−using Field_natLeq_on natLeq_on_Card_order
+
−      Card_order_iff_ordIso_card_of[of "natLeq_on n"]
+
−      ordIso_symmetric[of "natLeq_on n"] by blast
+
−
+
−corollary card_of_less:
+
−"|{0 ..< n}| =o natLeq_on n"
+
−using Field_natLeq_on card_of_Field_natLeq_on
+
−unfolding atLeast_0 atLeastLessThan_def lessThan_def Int_UNIV_left by auto
+
−
+
−lemma natLeq_on_ordLeq_less_eq:
+
−"((natLeq_on m) \<le>o (natLeq_on n)) = (m \<le> n)"
+
−proof
+
−  assume "natLeq_on m \<le>o natLeq_on n"
+
−  then obtain f where "inj_on f {x. x < m} \<and> f ` {x. x < m} \<le> {x. x < n}"
+
−  unfolding ordLeq_def using
+
−    embed_inj_on[OF natLeq_on_Well_order[of m], of "natLeq_on n", unfolded Field_natLeq_on]
+
−     embed_Field[OF natLeq_on_Well_order[of m], of "natLeq_on n", unfolded Field_natLeq_on] by blast
+
−  thus "m \<le> n" using atLeastLessThan_less_eq2
+
−    unfolding atLeast_0 atLeastLessThan_def lessThan_def Int_UNIV_left by blast
+
−next
+
−  assume "m \<le> n"
+
−  hence "inj_on id {0..<m} \<and> id ` {0..<m} \<le> {0..<n}" unfolding inj_on_def by auto
+
−  hence "|{0..<m}| \<le>o |{0..<n}|" using card_of_ordLeq by blast
+
−  thus "natLeq_on m \<le>o natLeq_on n"
+
−  using card_of_less ordIso_ordLeq_trans ordLeq_ordIso_trans ordIso_symmetric by blast
+
−qed
+
−
+
−lemma natLeq_on_ordLeq_less:
+
−"((natLeq_on m) <o (natLeq_on n)) = (m < n)"
+
−using not_ordLeq_iff_ordLess[OF natLeq_on_Well_order natLeq_on_Well_order, of n m]
+
−   natLeq_on_ordLeq_less_eq[of n m] by linarith
+
−
+
−lemma ordLeq_natLeq_on_imp_finite:
+
−assumes "|A| \<le>o natLeq_on n"
+
−shows "finite A"
+
−proof-
+
−  have "|A| \<le>o |{0 ..< n}|"
+
−  using assms card_of_less ordIso_symmetric ordLeq_ordIso_trans by blast
+
−  thus ?thesis by (auto simp add: card_of_ordLeq_finite)
+
−qed
+
−
+
−
+
−subsubsection \<open>"Backward compatibility" with the numeric cardinal operator for finite sets\<close>
+
−
+
−lemma finite_card_of_iff_card2:
+
−assumes FIN: "finite A" and FIN': "finite B"
+
−shows "( |A| \<le>o |B| ) = (card A \<le> card B)"
+
−using assms card_of_ordLeq[of A B] inj_on_iff_card_le[of A B] by blast
+
−
+
−lemma finite_imp_card_of_natLeq_on:
+
−assumes "finite A"
+
−shows "|A| =o natLeq_on (card A)"
+
−proof-
+
−  obtain h where "bij_betw h A {0 ..< card A}"
+
−  using assms ex_bij_betw_finite_nat by blast
+
−  thus ?thesis using card_of_ordIso card_of_less ordIso_equivalence by blast
+
−qed
+
−
+
−lemma finite_iff_card_of_natLeq_on:
+
−"finite A = (\<exists>n. |A| =o natLeq_on n)"
+
−using finite_imp_card_of_natLeq_on[of A]
+
−by(auto simp add: ordIso_natLeq_on_imp_finite)
+
−
+
−lemma finite_card_of_iff_card:
+
−assumes FIN: "finite A" and FIN': "finite B"
+
−shows "( |A| =o |B| ) = (card A = card B)"
+
−using assms card_of_ordIso[of A B] bij_betw_iff_card[of A B] by blast
+
−
+
−lemma finite_card_of_iff_card3:
+
−assumes FIN: "finite A" and FIN': "finite B"
+
−shows "( |A| <o |B| ) = (card A < card B)"
+
−proof-
+
−  have "( |A| <o |B| ) = (~ ( |B| \<le>o |A| ))" by simp
+
−  also have "... = (~ (card B \<le> card A))"
+
−  using assms by(simp add: finite_card_of_iff_card2)
+
−  also have "... = (card A < card B)" by auto
+
−  finally show ?thesis .
+
−qed
+
−
+
−lemma card_Field_natLeq_on:
+
−"card(Field(natLeq_on n)) = n"
+
−using Field_natLeq_on card_atLeastLessThan by auto
+
−
+
−
+
−subsection \<open>The successor of a cardinal\<close>
+
−
+
−lemma embed_implies_ordIso_Restr:
+
−assumes WELL: "Well_order r" and WELL': "Well_order r'" and EMB: "embed r' r f"
+
−shows "r' =o Restr r (f ` (Field r'))"
+
−using assms embed_implies_iso_Restr Well_order_Restr unfolding ordIso_def by blast
+
−
+
−lemma cardSuc_Well_order[simp]:
+
−"Card_order r \<Longrightarrow> Well_order(cardSuc r)"
+
−using cardSuc_Card_order unfolding card_order_on_def by blast
+
−
+
−lemma Field_cardSuc_not_empty:
+
−assumes "Card_order r"
+
−shows "Field (cardSuc r) \<noteq> {}"
+
−proof
+
−  assume "Field(cardSuc r) = {}"
+
−  hence "|Field(cardSuc r)| \<le>o r" using assms Card_order_empty[of r] by auto
+
−  hence "cardSuc r \<le>o r" using assms card_of_Field_ordIso
+
−  cardSuc_Card_order ordIso_symmetric ordIso_ordLeq_trans by blast
+
−  thus False using cardSuc_greater not_ordLess_ordLeq assms by blast
+
−qed
+
−
+
−lemma cardSuc_mono_ordLess[simp]:
+
−assumes CARD: "Card_order r" and CARD': "Card_order r'"
+
−shows "(cardSuc r <o cardSuc r') = (r <o r')"
+
−proof-
+
−  have 0: "Well_order r \<and> Well_order r' \<and> Well_order(cardSuc r) \<and> Well_order(cardSuc r')"
+
−  using assms by auto
+
−  thus ?thesis
+
−  using not_ordLeq_iff_ordLess not_ordLeq_iff_ordLess[of r r']
+
−  using cardSuc_mono_ordLeq[of r' r] assms by blast
+
−qed
+
−
+
−lemma cardSuc_natLeq_on_Suc:
+
−"cardSuc(natLeq_on n) =o natLeq_on(Suc n)"
+
−proof-
+
−  obtain r r' p where r_def: "r = natLeq_on n" and
+
−                      r'_def: "r' = cardSuc(natLeq_on n)"  and
+
−                      p_def: "p = natLeq_on(Suc n)" by blast
+
−  (* Preliminary facts:  *)
+
−  have CARD: "Card_order r \<and> Card_order r' \<and> Card_order p" unfolding r_def r'_def p_def
+
−  using cardSuc_ordLess_ordLeq natLeq_on_Card_order cardSuc_Card_order by blast
+
−  hence WELL: "Well_order r \<and> Well_order r' \<and>  Well_order p"
+
−  unfolding card_order_on_def by force
+
−  have FIELD: "Field r = {0..<n} \<and> Field p = {0..<(Suc n)}"
+
−  unfolding r_def p_def Field_natLeq_on atLeast_0 atLeastLessThan_def lessThan_def by simp
+
−  hence FIN: "finite (Field r)" by force
+
−  have "r <o r'" using CARD unfolding r_def r'_def using cardSuc_greater by blast
+
−  hence "|Field r| <o r'" using CARD card_of_Field_ordIso ordIso_ordLess_trans by blast
+
−  hence LESS: "|Field r| <o |Field r'|"
+
−  using CARD card_of_Field_ordIso ordLess_ordIso_trans ordIso_symmetric by blast
+
−  (* Main proof: *)
+
−  have "r' \<le>o p" using CARD unfolding r_def r'_def p_def
+
−  using natLeq_on_ordLeq_less cardSuc_ordLess_ordLeq by blast
+
−  moreover have "p \<le>o r'"
+
−  proof-
+
−    {assume "r' <o p"
+
−     then obtain f where 0: "embedS r' p f" unfolding ordLess_def by force
+
−     let ?q = "Restr p (f ` Field r')"
+
−     have 1: "embed r' p f" using 0 unfolding embedS_def by force
+
−     hence 2: "f ` Field r' < {0..<(Suc n)}"
+
−     using WELL FIELD 0 by (auto simp add: embedS_iff)
+
−     have "wo_rel.ofilter p (f ` Field r')" using embed_Field_ofilter 1 WELL by blast
+
−     then obtain m where "m \<le> Suc n" and 3: "f ` (Field r') = {0..<m}"
+
−     unfolding p_def by (auto simp add: natLeq_on_ofilter_iff)
+
−     hence 4: "m \<le> n" using 2 by force
+
−     (*  *)
+
−     have "bij_betw f (Field r') (f ` (Field r'))"
+
−     using WELL embed_inj_on[OF _ 1] unfolding bij_betw_def by blast
+
−     moreover have "finite(f ` (Field r'))" using 3 finite_atLeastLessThan[of 0 m] by force
+
−     ultimately have 5: "finite (Field r') \<and> card(Field r') = card (f ` (Field r'))"
+
−     using bij_betw_same_card bij_betw_finite by metis
+
−     hence "card(Field r') \<le> card(Field r)" using 3 4 FIELD by force
+
−     hence "|Field r'| \<le>o |Field r|" using FIN 5 finite_card_of_iff_card2 by blast
+
−     hence False using LESS not_ordLess_ordLeq by auto
+
−    }
+
−    thus ?thesis using WELL CARD by fastforce
+
−  qed
+
−  ultimately show ?thesis using ordIso_iff_ordLeq unfolding r'_def p_def by blast
+
−qed
+
−
+
−lemma card_of_Plus_ordLeq_infinite[simp]:
+
−assumes C: "\<not>finite C" and A: "|A| \<le>o |C|" and B: "|B| \<le>o |C|"
+
−shows "|A <+> B| \<le>o |C|"
+
−proof-
+
−  let ?r = "cardSuc |C|"
+
−  have "Card_order ?r \<and> \<not>finite (Field ?r)" using assms by simp
+
−  moreover have "|A| <o ?r" and "|B| <o ?r" using A B by auto
+
−  ultimately have "|A <+> B| <o ?r"
+
−  using card_of_Plus_ordLess_infinite_Field by blast
+
−  thus ?thesis using C by simp
+
−qed
+
−
+
−lemma card_of_Un_ordLeq_infinite[simp]:
+
−assumes C: "\<not>finite C" and A: "|A| \<le>o |C|" and B: "|B| \<le>o |C|"
+
−shows "|A Un B| \<le>o |C|"
+
−using assms card_of_Plus_ordLeq_infinite card_of_Un_Plus_ordLeq
+
−ordLeq_transitive by metis
+
−
+
−
+
−subsection \<open>Others\<close>
+
−
+
−lemma under_mono[simp]:
+
−assumes "Well_order r" and "(i,j) \<in> r"
+
−shows "under r i \<subseteq> under r j"
+
−using assms unfolding under_def order_on_defs
+
−trans_def by blast
+
−
+
−lemma underS_under:
+
−assumes "i \<in> Field r"
+
−shows "underS r i = under r i - {i}"
+
−using assms unfolding underS_def under_def by auto
+
−
+
−lemma relChain_under:
+
−assumes "Well_order r"
+
−shows "relChain r (\<lambda> i. under r i)"
+
−using assms unfolding relChain_def by auto
+
−
+
−lemma card_of_infinite_diff_finite:
+
−assumes "\<not>finite A" and "finite B"
+
−shows "|A - B| =o |A|"
+
−by (metis assms card_of_Un_diff_infinite finite_ordLess_infinite2)
+
−
+
−lemma infinite_card_of_diff_singl:
+
−assumes "\<not>finite A"
+
−shows "|A - {a}| =o |A|"
+
−by (metis assms card_of_infinite_diff_finite finite.emptyI finite_insert)
+
−
+
−lemma card_of_vimage:
+
−assumes "B \<subseteq> range f"
+
−shows "|B| \<le>o |f -` B|"
+
−apply(rule surj_imp_ordLeq[of _ f])
+
−using assms by (metis Int_absorb2 image_vimage_eq order_refl)
+
−
+
−lemma surj_card_of_vimage:
+
−assumes "surj f"
+
−shows "|B| \<le>o |f -` B|"
+
−by (metis assms card_of_vimage subset_UNIV)
+
−
+
−lemma infinite_Pow:
+
−assumes "\<not> finite A"
+
−shows "\<not> finite (Pow A)"
+
−proof-
+
−  have "|A| \<le>o |Pow A|" by (metis card_of_Pow ordLess_imp_ordLeq)
+
−  thus ?thesis by (metis assms finite_Pow_iff)
+
−qed
+
−
+
−(* bounded powerset *)
+
−definition Bpow where
+
−"Bpow r A \<equiv> {X . X \<subseteq> A \<and> |X| \<le>o r}"
+
−
+
−lemma Bpow_empty[simp]:
+
−assumes "Card_order r"
+
−shows "Bpow r {} = {{}}"
+
−using assms unfolding Bpow_def by auto
+
−
+
−lemma singl_in_Bpow:
+
−assumes rc: "Card_order r"
+
−and r: "Field r \<noteq> {}" and a: "a \<in> A"
+
−shows "{a} \<in> Bpow r A"
+
−proof-
+
−  have "|{a}| \<le>o r" using r rc by auto
+
−  thus ?thesis unfolding Bpow_def using a by auto
+
−qed
+
−
+
−lemma ordLeq_card_Bpow:
+
−assumes rc: "Card_order r" and r: "Field r \<noteq> {}"
+
−shows "|A| \<le>o |Bpow r A|"
+
−proof-
+
−  have "inj_on (\<lambda> a. {a}) A" unfolding inj_on_def by auto
+
−  moreover have "(\<lambda> a. {a}) ` A \<subseteq> Bpow r A"
+
−  using singl_in_Bpow[OF assms] by auto
+
−  ultimately show ?thesis unfolding card_of_ordLeq[symmetric] by blast
+
−qed
+
−
+
−lemma infinite_Bpow:
+
−assumes rc: "Card_order r" and r: "Field r \<noteq> {}"
+
−and A: "\<not>finite A"
+
−shows "\<not>finite (Bpow r A)"
+
−using ordLeq_card_Bpow[OF rc r]
+
−by (metis A card_of_ordLeq_infinite)
+
−
+
−definition Func_option where
+
− "Func_option A B \<equiv>
+
−  {f. (\<forall> a. f a \<noteq> None \<longleftrightarrow> a \<in> A) \<and> (\<forall> a \<in> A. case f a of Some b \<Rightarrow> b \<in> B |None \<Rightarrow> True)}"
+
−
+
−lemma card_of_Func_option_Func:
+
−"|Func_option A B| =o |Func A B|"
+
−proof (rule ordIso_symmetric, unfold card_of_ordIso[symmetric], intro exI)
+
−  let ?F = "\<lambda> f a. if a \<in> A then Some (f a) else None"
+
−  show "bij_betw ?F (Func A B) (Func_option A B)"
+
−  unfolding bij_betw_def unfolding inj_on_def proof(intro conjI ballI impI)
+
−    fix f g assume f: "f \<in> Func A B" and g: "g \<in> Func A B" and eq: "?F f = ?F g"
+
−    show "f = g"
+
−    proof(rule ext)
+
−      fix a
+
−      show "f a = g a"
+
−      proof(cases "a \<in> A")
+
−        case True
+
−        have "Some (f a) = ?F f a" using True by auto
+
−        also have "... = ?F g a" using eq unfolding fun_eq_iff by(rule allE)
+
−        also have "... = Some (g a)" using True by auto
+
−        finally have "Some (f a) = Some (g a)" .
+
−        thus ?thesis by simp
+
−      qed(insert f g, unfold Func_def, auto)
+
−    qed
+
−  next
+
−    show "?F ` Func A B = Func_option A B"
+
−    proof safe
+
−      fix f assume f: "f \<in> Func_option A B"
+
−      define g where [abs_def]: "g a = (case f a of Some b \<Rightarrow> b | None \<Rightarrow> undefined)" for a
+
−      have "g \<in> Func A B"
+
−      using f unfolding g_def Func_def Func_option_def by force+
+
−      moreover have "f = ?F g"
+
−      proof(rule ext)
+
−        fix a show "f a = ?F g a"
+
−        using f unfolding Func_option_def g_def by (cases "a \<in> A") force+
+
−      qed
+
−      ultimately show "f \<in> ?F ` (Func A B)" by blast
+
−    qed(unfold Func_def Func_option_def, auto)
+
−  qed
+
−qed
+
−
+
−(* partial-function space: *)
+
−definition Pfunc where
+
−"Pfunc A B \<equiv>
+
− {f. (\<forall>a. f a \<noteq> None \<longrightarrow> a \<in> A) \<and>
+
−     (\<forall>a. case f a of None \<Rightarrow> True | Some b \<Rightarrow> b \<in> B)}"
+
−
+
−lemma Func_Pfunc:
+
−"Func_option A B \<subseteq> Pfunc A B"
+
−unfolding Func_option_def Pfunc_def by auto
+
−
+
−lemma Pfunc_Func_option:
+
−"Pfunc A B = (\<Union>A' \<in> Pow A. Func_option A' B)"
+
−proof safe
+
−  fix f assume f: "f \<in> Pfunc A B"
+
−  show "f \<in> (\<Union>A'\<in>Pow A. Func_option A' B)"
+
−  proof (intro UN_I)
+
−    let ?A' = "{a. f a \<noteq> None}"
+
−    show "?A' \<in> Pow A" using f unfolding Pow_def Pfunc_def by auto
+
−    show "f \<in> Func_option ?A' B" using f unfolding Func_option_def Pfunc_def by auto
+
−  qed
+
−next
+
−  fix f A' assume "f \<in> Func_option A' B" and "A' \<subseteq> A"
+
−  thus "f \<in> Pfunc A B" unfolding Func_option_def Pfunc_def by auto
+
−qed
+
−
+
−lemma card_of_Func_mono:
+
−fixes A1 A2 :: "'a set" and B :: "'b set"
+
−assumes A12: "A1 \<subseteq> A2" and B: "B \<noteq> {}"
+
−shows "|Func A1 B| \<le>o |Func A2 B|"
+
−proof-
+
−  obtain bb where bb: "bb \<in> B" using B by auto
+
−  define F where [abs_def]: "F f1 a =
+
−    (if a \<in> A2 then (if a \<in> A1 then f1 a else bb) else undefined)" for f1 :: "'a \<Rightarrow> 'b" and a
+
−  show ?thesis unfolding card_of_ordLeq[symmetric] proof(intro exI[of _ F] conjI)
+
−    show "inj_on F (Func A1 B)" unfolding inj_on_def proof safe
+
−      fix f g assume f: "f \<in> Func A1 B" and g: "g \<in> Func A1 B" and eq: "F f = F g"
+
−      show "f = g"
+
−      proof(rule ext)
+
−        fix a show "f a = g a"
+
−        proof(cases "a \<in> A1")
+
−          case True
+
−          thus ?thesis using eq A12 unfolding F_def fun_eq_iff
+
−          by (elim allE[of _ a]) auto
+
−        qed(insert f g, unfold Func_def, fastforce)
+
−      qed
+
−    qed
+
−  qed(insert bb, unfold Func_def F_def, force)
+
−qed
+
−
+
−lemma card_of_Func_option_mono:
+
−fixes A1 A2 :: "'a set" and B :: "'b set"
+
−assumes A12: "A1 \<subseteq> A2" and B: "B \<noteq> {}"
+
−shows "|Func_option A1 B| \<le>o |Func_option A2 B|"
+
−by (metis card_of_Func_mono[OF A12 B] card_of_Func_option_Func
+
−  ordIso_ordLeq_trans ordLeq_ordIso_trans ordIso_symmetric)
+
−
+
−lemma card_of_Pfunc_Pow_Func_option:
+
−assumes "B \<noteq> {}"
+
−shows "|Pfunc A B| \<le>o |Pow A \<times> Func_option A B|"
+
−proof-
+
−  have "|Pfunc A B| =o |\<Union>A' \<in> Pow A. Func_option A' B|" (is "_ =o ?K")
+
−    unfolding Pfunc_Func_option by(rule card_of_refl)
+
−  also have "?K \<le>o |Sigma (Pow A) (\<lambda> A'. Func_option A' B)|" using card_of_UNION_Sigma .
+
−  also have "|Sigma (Pow A) (\<lambda> A'. Func_option A' B)| \<le>o |Pow A \<times> Func_option A B|"
+
−    apply(rule card_of_Sigma_mono1) using card_of_Func_option_mono[OF _ assms] by auto
+
−  finally show ?thesis .
+
−qed
+
−
+
−lemma Bpow_ordLeq_Func_Field:
+
−assumes rc: "Card_order r" and r: "Field r \<noteq> {}" and A: "\<not>finite A"
+
−shows "|Bpow r A| \<le>o |Func (Field r) A|"
+
−proof-
+
−  let ?F = "\<lambda> f. {x | x a. f a = x \<and> a \<in> Field r}"
+
−  {fix X assume "X \<in> Bpow r A - {{}}"
+
−   hence XA: "X \<subseteq> A" and "|X| \<le>o r"
+
−   and X: "X \<noteq> {}" unfolding Bpow_def by auto
+
−   hence "|X| \<le>o |Field r|" by (metis Field_card_of card_of_mono2)
+
−   then obtain F where 1: "X = F ` (Field r)"
+
−   using card_of_ordLeq2[OF X] by metis
+
−   define f where [abs_def]: "f i = (if i \<in> Field r then F i else undefined)" for i
+
−   have "\<exists> f \<in> Func (Field r) A. X = ?F f"
+
−   apply (intro bexI[of _ f]) using 1 XA unfolding Func_def f_def by auto
+
−  }
+
−  hence "Bpow r A - {{}} \<subseteq> ?F ` (Func (Field r) A)" by auto
+
−  hence "|Bpow r A - {{}}| \<le>o |Func (Field r) A|"
+
−  by (rule surj_imp_ordLeq)
+
−  moreover
+
−  {have 2: "\<not>finite (Bpow r A)" using infinite_Bpow[OF rc r A] .
+
−   have "|Bpow r A| =o |Bpow r A - {{}}|"
+
−     by (metis 2 infinite_card_of_diff_singl ordIso_symmetric)
+
−  }
+
−  ultimately show ?thesis by (metis ordIso_ordLeq_trans)
+
−qed
+
−
+
−lemma empty_in_Func[simp]:
+
−"B \<noteq> {} \<Longrightarrow> (\<lambda>x. undefined) \<in> Func {} B"
+
−unfolding Func_def by auto
+
−
+
−lemma Func_mono[simp]:
+
−assumes "B1 \<subseteq> B2"
+
−shows "Func A B1 \<subseteq> Func A B2"
+
−using assms unfolding Func_def by force
+
−
+
−lemma Pfunc_mono[simp]:
+
−assumes "A1 \<subseteq> A2" and "B1 \<subseteq> B2"
+
−shows "Pfunc A B1 \<subseteq> Pfunc A B2"
+
−using assms unfolding Pfunc_def
+
−apply safe
+
−apply (case_tac "x a", auto)
+
−apply (metis in_mono option.simps(5))
+
−done
+
−
+
−lemma card_of_Func_UNIV_UNIV:
+
−"|Func (UNIV::'a set) (UNIV::'b set)| =o |UNIV::('a \<Rightarrow> 'b) set|"
+
−using card_of_Func_UNIV[of "UNIV::'b set"] by auto
+
−
+
−lemma ordLeq_Func:
+
−assumes "{b1,b2} \<subseteq> B" "b1 \<noteq> b2"
+
−shows "|A| \<le>o |Func A B|"
+
−unfolding card_of_ordLeq[symmetric] proof(intro exI conjI)
+
−  let ?F = "\<lambda> aa a. if a \<in> A then (if a = aa then b1 else b2) else undefined"
+
−  show "inj_on ?F A" using assms unfolding inj_on_def fun_eq_iff by auto
+
−  show "?F ` A \<subseteq> Func A B" using assms unfolding Func_def by auto
+
−qed
+
−
+
−lemma infinite_Func:
+
−assumes A: "\<not>finite A" and B: "{b1,b2} \<subseteq> B" "b1 \<noteq> b2"
+
−shows "\<not>finite (Func A B)"
+
−using ordLeq_Func[OF B] by (metis A card_of_ordLeq_finite)
+
−
+
−
+
−subsection \<open>Infinite cardinals are limit ordinals\<close>
+
−
+
−lemma card_order_infinite_isLimOrd:
+
−assumes c: "Card_order r" and i: "\<not>finite (Field r)"
+
−shows "isLimOrd r"
+
−proof-
+
−  have 0: "wo_rel r" and 00: "Well_order r"
+
−  using c unfolding card_order_on_def wo_rel_def by auto
+
−  hence rr: "Refl r" by (metis wo_rel.REFL)
+
−  show ?thesis unfolding wo_rel.isLimOrd_def[OF 0] wo_rel.isSuccOrd_def[OF 0] proof safe
+
−    fix j assume j: "j \<in> Field r" and jm: "\<forall>i\<in>Field r. (i, j) \<in> r"
+
−    define p where "p = Restr r (Field r - {j})"
+
−    have fp: "Field p = Field r - {j}"
+
−    unfolding p_def apply(rule Refl_Field_Restr2[OF rr]) by auto
+
−    have of: "ofilter r (Field p)" unfolding wo_rel.ofilter_def[OF 0] proof safe
+
−      fix a x assume a: "a \<in> Field p" and "x \<in> under r a"
+
−      hence x: "(x,a) \<in> r" "x \<in> Field r" unfolding under_def Field_def by auto
+
−      moreover have a: "a \<noteq> j" "a \<in> Field r" "(a,j) \<in> r" using a jm  unfolding fp by auto
+
−      ultimately have "x \<noteq> j" using j jm  by (metis 0 wo_rel.max2_def wo_rel.max2_equals1)
+
−      thus "x \<in> Field p" using x unfolding fp by auto
+
−    qed(unfold p_def Field_def, auto)
+
−    have "p <o r" using j ofilter_ordLess[OF 00 of] unfolding fp p_def[symmetric] by auto
+
−    hence 2: "|Field p| <o r" using c by (metis BNF_Cardinal_Order_Relation.ordLess_Field)
+
−    have "|Field p| =o |Field r|" unfolding fp using i by (metis infinite_card_of_diff_singl)
+
−    also have "|Field r| =o r"
+
−    using c by (metis card_of_unique ordIso_symmetric)
+
−    finally have "|Field p| =o r" .
+
−    with 2 show False by (metis not_ordLess_ordIso)
+
−  qed
+
−qed
+
−
+
−lemma insert_Chain:
+
−assumes "Refl r" "C \<in> Chains r" and "i \<in> Field r" and "\<And>j. j \<in> C \<Longrightarrow> (j,i) \<in> r \<or> (i,j) \<in> r"
+
−shows "insert i C \<in> Chains r"
+
−using assms unfolding Chains_def by (auto dest: refl_onD)
+
−
+
−lemma Collect_insert: "{R j |j. j \<in> insert j1 J} = insert (R j1) {R j |j. j \<in> J}"
+
−by auto
+
−
+
−lemma Field_init_seg_of[simp]:
+
−"Field init_seg_of = UNIV"
+
−unfolding Field_def init_seg_of_def by auto
+
−
+
−lemma refl_init_seg_of[intro, simp]: "refl init_seg_of"
+
−unfolding refl_on_def Field_def by auto
+
−
+
−lemma regularCard_all_ex:
+
−assumes r: "Card_order r"   "regularCard r"
+
−and As: "\<And> i j b. b \<in> B \<Longrightarrow> (i,j) \<in> r \<Longrightarrow> P i b \<Longrightarrow> P j b"
+
−and Bsub: "\<forall> b \<in> B. \<exists> i \<in> Field r. P i b"
+
−and cardB: "|B| <o r"
+
−shows "\<exists> i \<in> Field r. \<forall> b \<in> B. P i b"
+
−proof-
+
−  let ?As = "\<lambda>i. {b \<in> B. P i b}"
+
−  have "EX i : Field r. B \<le> ?As i"
+
−  apply(rule regularCard_UNION) using assms unfolding relChain_def by auto
+
−  thus ?thesis by auto
+
−qed
+
−
+
−lemma relChain_stabilize:
+
−assumes rc: "relChain r As" and AsB: "(\<Union>i \<in> Field r. As i) \<subseteq> B" and Br: "|B| <o r"
+
−and ir: "\<not>finite (Field r)" and cr: "Card_order r"
+
−shows "\<exists> i \<in> Field r. As (succ r i) = As i"
+
−proof(rule ccontr, auto)
+
−  have 0: "wo_rel r" and 00: "Well_order r"
+
−  unfolding wo_rel_def by (metis card_order_on_well_order_on cr)+
+
−  have L: "isLimOrd r" using ir cr by (metis card_order_infinite_isLimOrd)
+
−  have AsBs: "(\<Union>i \<in> Field r. As (succ r i)) \<subseteq> B"
+
−  using AsB L apply safe by (metis "0" UN_I set_mp wo_rel.isLimOrd_succ)
+
−  assume As_s: "\<forall>i\<in>Field r. As (succ r i) \<noteq> As i"
+
−  have 1: "\<forall>i j. (i,j) \<in> r \<and> i \<noteq> j \<longrightarrow> As i \<subset> As j"
+
−  proof safe
+
−    fix i j assume 1: "(i, j) \<in> r" "i \<noteq> j" and Asij: "As i = As j"
+
−    hence rij: "(succ r i, j) \<in> r" by (metis "0" wo_rel.succ_smallest)
+
−    hence "As (succ r i) \<subseteq> As j" using rc unfolding relChain_def by auto
+
−    moreover
+
−    {have "(i,succ r i) \<in> r" apply(rule wo_rel.succ_in[OF 0])
+
−     using 1 unfolding aboveS_def by auto
+
−     hence "As i \<subset> As (succ r i)" using As_s rc rij unfolding relChain_def Field_def by auto
+
−    }
+
−    ultimately show False unfolding Asij by auto
+
−  qed (insert rc, unfold relChain_def, auto)
+
−  hence "\<forall> i \<in> Field r. \<exists> a. a \<in> As (succ r i) - As i"
+
−  using wo_rel.succ_in[OF 0] AsB
+
−  by(metis 0 card_order_infinite_isLimOrd cr ir psubset_imp_ex_mem
+
−            wo_rel.isLimOrd_aboveS wo_rel.succ_diff)
+
−  then obtain f where f: "\<forall> i \<in> Field r. f i \<in> As (succ r i) - As i" by metis
+
−  have "inj_on f (Field r)" unfolding inj_on_def proof safe
+
−    fix i j assume ij: "i \<in> Field r" "j \<in> Field r" and fij: "f i = f j"
+
−    show "i = j"
+
−    proof(cases rule: wo_rel.cases_Total3[OF 0], safe)
+
−      assume "(i, j) \<in> r" and ijd: "i \<noteq> j"
+
−      hence rij: "(succ r i, j) \<in> r" by (metis "0" wo_rel.succ_smallest)
+
−      hence "As (succ r i) \<subseteq> As j" using rc unfolding relChain_def by auto
+
−      thus "i = j" using ij ijd fij f by auto
+
−    next
+
−      assume "(j, i) \<in> r" and ijd: "i \<noteq> j"
+
−      hence rij: "(succ r j, i) \<in> r" by (metis "0" wo_rel.succ_smallest)
+
−      hence "As (succ r j) \<subseteq> As i" using rc unfolding relChain_def by auto
+
−      thus "j = i" using ij ijd fij f by fastforce
+
−    qed(insert ij, auto)
+
−  qed
+
−  moreover have "f ` (Field r) \<subseteq> B" using f AsBs by auto
+
−  moreover have "|B| <o |Field r|" using Br cr by (metis card_of_unique ordLess_ordIso_trans)
+
−  ultimately show False unfolding card_of_ordLess[symmetric] by auto
+
−qed
+
−
+
−
+
−subsection \<open>Regular vs. stable cardinals\<close>
+
−
+
−definition stable :: "'a rel \<Rightarrow> bool"
+
−where
+
−"stable r \<equiv> \<forall>(A::'a set) (F :: 'a \<Rightarrow> 'a set).
+
−               |A| <o r \<and> (\<forall>a \<in> A. |F a| <o r)
+
−               \<longrightarrow> |SIGMA a : A. F a| <o r"
+
−
+
−lemma regularCard_stable:
+
−assumes cr: "Card_order r" and ir: "\<not>finite (Field r)" and reg: "regularCard r"
+
−shows "stable r"
+
−unfolding stable_def proof safe
+
−  fix A :: "'a set" and F :: "'a \<Rightarrow> 'a set" assume A: "|A| <o r" and F: "\<forall>a\<in>A. |F a| <o r"
+
−  {assume "r \<le>o |Sigma A F|"
+
−   hence "|Field r| \<le>o |Sigma A F|" using card_of_Field_ordIso[OF cr]
+
−   by (metis Field_card_of card_of_cong ordLeq_iff_ordLess_or_ordIso ordLeq_ordLess_trans)
+
−   moreover have Fi: "Field r \<noteq> {}" using ir by auto
+
−   ultimately obtain f where f: "f ` Sigma A F = Field r" using card_of_ordLeq2 by metis
+
−   have r: "wo_rel r" using cr unfolding card_order_on_def wo_rel_def by auto
+
−   {fix a assume a: "a \<in> A"
+
−    define L where "L = {(a,u) | u. u \<in> F a}"
+
−    have fL: "f ` L \<subseteq> Field r" using f a unfolding L_def by auto
+
−    have "|L| =o |F a|" unfolding L_def card_of_ordIso[symmetric]
+
−    apply(rule exI[of _ snd]) unfolding bij_betw_def inj_on_def by (auto simp: image_def)
+
−    hence "|L| <o r" using F a ordIso_ordLess_trans[of "|L|" "|F a|"] unfolding L_def by auto
+
−    hence "|f ` L| <o r" using ordLeq_ordLess_trans[OF card_of_image, of "L"] unfolding L_def by auto
+
−    hence "\<not> cofinal (f ` L) r" using reg fL unfolding regularCard_def by (metis not_ordLess_ordIso)
+
−    then obtain k where k: "k \<in> Field r" and "\<forall> l \<in> L. \<not> (f l \<noteq> k \<and> (k, f l) \<in> r)"
+
−    unfolding cofinal_def image_def by auto
+
−    hence "\<exists> k \<in> Field r. \<forall> l \<in> L. (f l, k) \<in> r" using r by (metis fL image_subset_iff wo_rel.in_notinI)
+
−    hence "\<exists> k \<in> Field r. \<forall> u \<in> F a. (f (a,u), k) \<in> r" unfolding L_def by auto
+
−   }
+
−   then obtain gg where gg: "\<forall> a \<in> A. \<forall> u \<in> F a. (f (a,u), gg a) \<in> r" by metis
+
−   obtain j0 where j0: "j0 \<in> Field r" using Fi by auto
+
−   define g where [abs_def]: "g a = (if F a \<noteq> {} then gg a else j0)" for a
+
−   have g: "\<forall> a \<in> A. \<forall> u \<in> F a. (f (a,u),g a) \<in> r" using gg unfolding g_def by auto
+
−   hence 1: "Field r \<subseteq> (\<Union>a \<in> A. under r (g a))"
+
−   using f[symmetric] unfolding under_def image_def by auto
+
−   have gA: "g ` A \<subseteq> Field r" using gg j0 unfolding Field_def g_def by auto
+
−   moreover have "cofinal (g ` A) r" unfolding cofinal_def proof safe
+
−     fix i assume "i \<in> Field r"
+
−     then obtain j where ij: "(i,j) \<in> r" "i \<noteq> j" using cr ir by (metis infinite_Card_order_limit)
+
−     hence "j \<in> Field r" by (metis card_order_on_def cr well_order_on_domain)
+
−     then obtain a where a: "a \<in> A" and j: "(j, g a) \<in> r" using 1 unfolding under_def by auto
+
−     hence "(i, g a) \<in> r" using ij wo_rel.TRANS[OF r] unfolding trans_def by blast
+
−     moreover have "i \<noteq> g a"
+
−     using ij j r unfolding wo_rel_def unfolding well_order_on_def linear_order_on_def
+
−     partial_order_on_def antisym_def by auto
+
−     ultimately show "\<exists>j\<in>g ` A. i \<noteq> j \<and> (i, j) \<in> r" using a by auto
+
−   qed
+
−   ultimately have "|g ` A| =o r" using reg unfolding regularCard_def by auto
+
−   moreover have "|g ` A| \<le>o |A|" by (metis card_of_image)
+
−   ultimately have False using A by (metis not_ordLess_ordIso ordLeq_ordLess_trans)
+
−  }
+
−  thus "|Sigma A F| <o r"
+
−  using cr not_ordLess_iff_ordLeq by (metis card_of_Well_order card_order_on_well_order_on)
+
−qed
+
−
+
−lemma stable_regularCard:
+
−assumes cr: "Card_order r" and ir: "\<not>finite (Field r)" and st: "stable r"
+
−shows "regularCard r"
+
−unfolding regularCard_def proof safe
+
−  fix K assume K: "K \<subseteq> Field r" and cof: "cofinal K r"
+
−  have "|K| \<le>o r" using K by (metis card_of_Field_ordIso card_of_mono1 cr ordLeq_ordIso_trans)
+
−  moreover
+
−  {assume Kr: "|K| <o r"
+
−   then obtain f where "\<forall> a \<in> Field r. f a \<in> K \<and> a \<noteq> f a \<and> (a, f a) \<in> r"
+
−   using cof unfolding cofinal_def by metis
+
−   hence "Field r \<subseteq> (\<Union>a \<in> K. underS r a)" unfolding underS_def by auto
+
−   hence "r \<le>o |\<Union>a \<in> K. underS r a|" using cr
+
−   by (metis Card_order_iff_ordLeq_card_of card_of_mono1 ordLeq_transitive)
+
−   also have "|\<Union>a \<in> K. underS r a| \<le>o |SIGMA a: K. underS r a|" by (rule card_of_UNION_Sigma)
+
−   also
+
−   {have "\<forall> a \<in> K. |underS r a| <o r" using K by (metis card_of_underS cr subsetD)
+
−    hence "|SIGMA a: K. underS r a| <o r" using st Kr unfolding stable_def by auto
+
−   }
+
−   finally have "r <o r" .
+
−   hence False by (metis ordLess_irreflexive)
+
−  }
+
−  ultimately show "|K| =o r" by (metis ordLeq_iff_ordLess_or_ordIso)
+
−qed
+
−
+
−(* Note that below the types of A and F are now unconstrained: *)
+
−lemma stable_elim:
+
−assumes ST: "stable r" and A_LESS: "|A| <o r" and
+
−        F_LESS: "\<And> a. a \<in> A \<Longrightarrow> |F a| <o r"
+
−shows "|SIGMA a : A. F a| <o r"
+
−proof-
+
−  obtain A' where 1: "A' < Field r \<and> |A'| <o r" and 2: " |A| =o |A'|"
+
−  using internalize_card_of_ordLess[of A r] A_LESS by blast
+
−  then obtain G where 3: "bij_betw G A' A"
+
−  using card_of_ordIso  ordIso_symmetric by blast
+
−  (*  *)
+
−  {fix a assume "a \<in> A"
+
−   hence "\<exists>B'. B' \<le> Field r \<and> |F a| =o |B'| \<and> |B'| <o r"
+
−   using internalize_card_of_ordLess[of "F a" r] F_LESS by blast
+
−  }
+
−  then obtain F' where
+
−  temp: "\<forall>a \<in> A. F' a \<le> Field r \<and> |F a| =o |F' a| \<and> |F' a| <o r"
+
−  using bchoice[of A "\<lambda> a B'. B' \<le> Field r \<and> |F a| =o |B'| \<and> |B'| <o r"] by blast
+
−  hence 4: "\<forall>a \<in> A. F' a \<le> Field r \<and> |F' a| <o r" by auto
+
−  have 5: "\<forall>a \<in> A. |F' a| =o |F a|" using temp ordIso_symmetric by auto
+
−  (*  *)
+
−  have "\<forall>a' \<in> A'. F'(G a') \<le> Field r \<and> |F'(G a')| <o r"
+
−  using 3 4 bij_betw_ball[of G A' A] by auto
+
−  hence "|SIGMA a' : A'. F'(G a')| <o r"
+
−  using ST 1 unfolding stable_def by auto
+
−  moreover have "|SIGMA a' : A'. F'(G a')| =o |SIGMA a : A. F a|"
+
−  using card_of_Sigma_cong[of G A' A F' F] 5 3 by blast
+
−  ultimately show ?thesis using ordIso_symmetric ordIso_ordLess_trans by blast
+
−qed
+
−
+
−lemma stable_natLeq: "stable natLeq"
+
−proof(unfold stable_def, safe)
+
−  fix A :: "'a set" and F :: "'a \<Rightarrow> 'a set"
+
−  assume "|A| <o natLeq" and "\<forall>a\<in>A. |F a| <o natLeq"
+
−  hence "finite A \<and> (\<forall>a \<in> A. finite(F a))"
+
−  by (auto simp add: finite_iff_ordLess_natLeq)
+
−  hence "finite(Sigma A F)" by (simp only: finite_SigmaI[of A F])
+
−  thus "|Sigma A F | <o natLeq"
+
−  by (auto simp add: finite_iff_ordLess_natLeq)
+
−qed
+
−
+
−corollary regularCard_natLeq: "regularCard natLeq"
+
−using stable_regularCard[OF natLeq_Card_order _ stable_natLeq] Field_natLeq by simp
+
−
+
−lemma stable_cardSuc:
+
−assumes CARD: "Card_order r" and INF: "\<not>finite (Field r)"
+
−shows "stable(cardSuc r)"
+
−using infinite_cardSuc_regularCard regularCard_stable
+
−by (metis CARD INF cardSuc_Card_order cardSuc_finite)
+
−
+
−lemma stable_UNION:
+
−assumes ST: "stable r" and A_LESS: "|A| <o r" and
+
−        F_LESS: "\<And> a. a \<in> A \<Longrightarrow> |F a| <o r"
+
−shows "|\<Union>a \<in> A. F a| <o r"
+
−proof-
+
−  have "|\<Union>a \<in> A. F a| \<le>o |SIGMA a : A. F a|"
+
−  using card_of_UNION_Sigma by blast
+
−  thus ?thesis using assms stable_elim ordLeq_ordLess_trans by blast
+
−qed
+
−
+
−lemma stable_ordIso1:
+
−assumes ST: "stable r" and ISO: "r' =o r"
+
−shows "stable r'"
+
−proof(unfold stable_def, auto)
+
−  fix A::"'b set" and F::"'b \<Rightarrow> 'b set"
+
−  assume "|A| <o r'" and "\<forall>a \<in> A. |F a| <o r'"
+
−  hence "( |A| <o r) \<and> (\<forall>a \<in> A. |F a| <o r)"
+
−  using ISO ordLess_ordIso_trans by blast
+
−  hence "|SIGMA a : A. F a| <o r" using assms stable_elim by blast
+
−  thus "|SIGMA a : A. F a| <o r'"
+
−  using ISO ordIso_symmetric ordLess_ordIso_trans by blast
+
−qed
+
−
+
−lemma stable_ordIso2:
+
−assumes ST: "stable r" and ISO: "r =o r'"
+
−shows "stable r'"
+
−using assms stable_ordIso1 ordIso_symmetric by blast
+
−
+
−lemma stable_ordIso:
+
−assumes "r =o r'"
+
−shows "stable r = stable r'"
+
−using assms stable_ordIso1 stable_ordIso2 by blast
+
−
+
−lemma stable_nat: "stable |UNIV::nat set|"
+
−using stable_natLeq card_of_nat stable_ordIso by auto
+
−
+
−text\<open>Below, the type of "A" is not important -- we just had to choose an appropriate
+
−   type to make "A" possible. What is important is that arbitrarily large
+
−   infinite sets of stable cardinality exist.\<close>
+
−
+
−lemma infinite_stable_exists:
+
−assumes CARD: "\<forall>r \<in> R. Card_order (r::'a rel)"
+
−shows "\<exists>(A :: (nat + 'a set)set).
+
−          \<not>finite A \<and> stable |A| \<and> (\<forall>r \<in> R. r <o |A| )"
+
−proof-
+
−  have "\<exists>(A :: (nat + 'a set)set).
+
−          \<not>finite A \<and> stable |A| \<and> |UNIV::'a set| <o |A|"
+
−  proof(cases "finite (UNIV::'a set)")
+
−    assume Case1: "finite (UNIV::'a set)"
+
−    let ?B = "UNIV::nat set"
+
−    have "\<not>finite(?B <+> {})" using finite_Plus_iff by blast
+
−    moreover
+
−    have "stable |?B|" using stable_natLeq card_of_nat stable_ordIso1 by blast
+
−    hence "stable |?B <+> {}|" using stable_ordIso card_of_Plus_empty1 by blast
+
−    moreover
+
−    have "\<not>finite(Field |?B| ) \<and> finite(Field |UNIV::'a set| )" using Case1 by simp
+
−    hence "|UNIV::'a set| <o |?B|" by (simp add: finite_ordLess_infinite)
+
−    hence "|UNIV::'a set| <o |?B <+> {}|" using card_of_Plus_empty1 ordLess_ordIso_trans by blast
+
−    ultimately show ?thesis by blast
+
−  next
+
−    assume Case1: "\<not>finite (UNIV::'a set)"
+
−    hence *: "\<not>finite(Field |UNIV::'a set| )" by simp
+
−    let ?B = "Field(cardSuc |UNIV::'a set| )"
+
−    have 0: "|?B| =o |{} <+> ?B|" using card_of_Plus_empty2 by blast
+
−    have 1: "\<not>finite ?B" using Case1 card_of_cardSuc_finite by blast
+
−    hence 2: "\<not>finite({} <+> ?B)" using 0 card_of_ordIso_finite by blast
+
−    have "|?B| =o cardSuc |UNIV::'a set|"
+
−    using card_of_Card_order cardSuc_Card_order card_of_Field_ordIso by blast
+
−    moreover have "stable(cardSuc |UNIV::'a set| )"
+
−    using stable_cardSuc * card_of_Card_order by blast
+
−    ultimately have "stable |?B|" using stable_ordIso by blast
+
−    hence 3: "stable |{} <+> ?B|" using stable_ordIso 0 by blast
+
−    have "|UNIV::'a set| <o cardSuc |UNIV::'a set|"
+
−    using card_of_Card_order cardSuc_greater by blast
+
−    moreover have "|?B| =o  cardSuc |UNIV::'a set|"
+
−    using card_of_Card_order cardSuc_Card_order  card_of_Field_ordIso by blast
+
−    ultimately have "|UNIV::'a set| <o |?B|"
+
−    using ordIso_symmetric ordLess_ordIso_trans by blast
+
−    hence "|UNIV::'a set| <o |{} <+> ?B|" using 0 ordLess_ordIso_trans by blast
+
−    thus ?thesis using 2 3 by blast
+
−  qed
+
−  thus ?thesis using CARD card_of_UNIV2 ordLeq_ordLess_trans by blast
+
−qed
+
−
+
−corollary infinite_regularCard_exists:
+
−assumes CARD: "\<forall>r \<in> R. Card_order (r::'a rel)"
+
−shows "\<exists>(A :: (nat + 'a set)set).
+
−          \<not>finite A \<and> regularCard |A| \<and> (\<forall>r \<in> R. r <o |A| )"
+
−using infinite_stable_exists[OF CARD] stable_regularCard by (metis Field_card_of card_of_card_order_on)
+
−
+
−end