--- a/src/HOL/BNF_Cardinal_Order_Relation.thy Fri Jan 13 22:47:40 2023 +0000
+++ b/src/HOL/BNF_Cardinal_Order_Relation.thy Sat Jan 14 16:53:54 2023 +0000
@@ -9,7 +9,7 @@
section \<open>Cardinal-Order Relations as Needed by Bounded Natural Functors\<close>
theory BNF_Cardinal_Order_Relation
-imports Zorn BNF_Wellorder_Constructions
+ imports Zorn BNF_Wellorder_Constructions
begin
text\<open>In this section, we define cardinal-order relations to be minim well-orders
@@ -42,20 +42,20 @@
strict order-embedding relation, \<open><o\<close>), among all the well-orders on its field.\<close>
definition card_order_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
-where
-"card_order_on A r \<equiv> well_order_on A r \<and> (\<forall>r'. well_order_on A r' \<longrightarrow> r \<le>o r')"
+ where
+ "card_order_on A r \<equiv> well_order_on A r \<and> (\<forall>r'. well_order_on A r' \<longrightarrow> r \<le>o r')"
abbreviation "Card_order r \<equiv> card_order_on (Field r) r"
abbreviation "card_order r \<equiv> card_order_on UNIV r"
lemma card_order_on_well_order_on:
-assumes "card_order_on A r"
-shows "well_order_on A r"
-using assms unfolding card_order_on_def by simp
+ assumes "card_order_on A r"
+ shows "well_order_on A r"
+ using assms unfolding card_order_on_def by simp
lemma card_order_on_Card_order:
-"card_order_on A r \<Longrightarrow> A = Field r \<and> Card_order r"
-unfolding card_order_on_def using well_order_on_Field by blast
+ "card_order_on A r \<Longrightarrow> A = Field r \<and> Card_order r"
+ unfolding card_order_on_def using well_order_on_Field by blast
text\<open>The existence of a cardinal relation on any given set (which will mean
that any set has a cardinal) follows from two facts:
@@ -68,48 +68,47 @@
\<close>
theorem card_order_on: "\<exists>r. card_order_on A r"
-proof-
- obtain R where R_def: "R = {r. well_order_on A r}" by blast
- have 1: "R \<noteq> {} \<and> (\<forall>r \<in> R. Well_order r)"
- using well_order_on[of A] R_def well_order_on_Well_order by blast
- hence "\<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
- using exists_minim_Well_order[of R] by auto
- thus ?thesis using R_def unfolding card_order_on_def by auto
+proof -
+ define R where "R \<equiv> {r. well_order_on A r}"
+ have "R \<noteq> {} \<and> (\<forall>r \<in> R. Well_order r)"
+ using well_order_on[of A] R_def well_order_on_Well_order by blast
+ with exists_minim_Well_order[of R] show ?thesis
+ by (auto simp: R_def card_order_on_def)
qed
lemma card_order_on_ordIso:
-assumes CO: "card_order_on A r" and CO': "card_order_on A r'"
-shows "r =o r'"
-using assms unfolding card_order_on_def
-using ordIso_iff_ordLeq by blast
+ assumes CO: "card_order_on A r" and CO': "card_order_on A r'"
+ shows "r =o r'"
+ using assms unfolding card_order_on_def
+ using ordIso_iff_ordLeq by blast
lemma Card_order_ordIso:
-assumes CO: "Card_order r" and ISO: "r' =o r"
-shows "Card_order r'"
-using ISO unfolding ordIso_def
+ assumes CO: "Card_order r" and ISO: "r' =o r"
+ shows "Card_order r'"
+ using ISO unfolding ordIso_def
proof(unfold card_order_on_def, auto)
fix p' assume "well_order_on (Field r') p'"
hence 0: "Well_order p' \<and> Field p' = Field r'"
- using well_order_on_Well_order by blast
+ using well_order_on_Well_order by blast
obtain f where 1: "iso r' r f" and 2: "Well_order r \<and> Well_order r'"
- using ISO unfolding ordIso_def by auto
+ using ISO unfolding ordIso_def by auto
hence 3: "inj_on f (Field r') \<and> f ` (Field r') = Field r"
- by (auto simp add: iso_iff embed_inj_on)
+ by (auto simp add: iso_iff embed_inj_on)
let ?p = "dir_image p' f"
have 4: "p' =o ?p \<and> Well_order ?p"
- using 0 2 3 by (auto simp add: dir_image_ordIso Well_order_dir_image)
- moreover have "Field ?p = Field r"
- using 0 3 by (auto simp add: dir_image_Field)
+ using 0 2 3 by (auto simp add: dir_image_ordIso Well_order_dir_image)
+ moreover have "Field ?p = Field r"
+ using 0 3 by (auto simp add: dir_image_Field)
ultimately have "well_order_on (Field r) ?p" by auto
hence "r \<le>o ?p" using CO unfolding card_order_on_def by auto
thus "r' \<le>o p'"
- using ISO 4 ordLeq_ordIso_trans ordIso_ordLeq_trans ordIso_symmetric by blast
+ using ISO 4 ordLeq_ordIso_trans ordIso_ordLeq_trans ordIso_symmetric by blast
qed
lemma Card_order_ordIso2:
-assumes CO: "Card_order r" and ISO: "r =o r'"
-shows "Card_order r'"
-using assms Card_order_ordIso ordIso_symmetric by blast
+ assumes CO: "Card_order r" and ISO: "r =o r'"
+ shows "Card_order r'"
+ using assms Card_order_ordIso ordIso_symmetric by blast
subsection \<open>Cardinal of a set\<close>
@@ -119,220 +118,213 @@
that order isomorphism shall be the true identity of cardinals.\<close>
definition card_of :: "'a set \<Rightarrow> 'a rel" ("|_|" )
-where "card_of A = (SOME r. card_order_on A r)"
+ where "card_of A = (SOME r. card_order_on A r)"
lemma card_of_card_order_on: "card_order_on A |A|"
-unfolding card_of_def by (auto simp add: card_order_on someI_ex)
+ unfolding card_of_def by (auto simp add: card_order_on someI_ex)
lemma card_of_well_order_on: "well_order_on A |A|"
-using card_of_card_order_on card_order_on_def by blast
+ using card_of_card_order_on card_order_on_def by blast
lemma Field_card_of: "Field |A| = A"
-using card_of_card_order_on[of A] unfolding card_order_on_def
-using well_order_on_Field by blast
+ using card_of_card_order_on[of A] unfolding card_order_on_def
+ using well_order_on_Field by blast
lemma card_of_Card_order: "Card_order |A|"
-by (simp only: card_of_card_order_on Field_card_of)
+ by (simp only: card_of_card_order_on Field_card_of)
corollary ordIso_card_of_imp_Card_order:
-"r =o |A| \<Longrightarrow> Card_order r"
-using card_of_Card_order Card_order_ordIso by blast
+ "r =o |A| \<Longrightarrow> Card_order r"
+ using card_of_Card_order Card_order_ordIso by blast
lemma card_of_Well_order: "Well_order |A|"
-using card_of_Card_order unfolding card_order_on_def by auto
+ using card_of_Card_order unfolding card_order_on_def by auto
lemma card_of_refl: "|A| =o |A|"
-using card_of_Well_order ordIso_reflexive by blast
+ using card_of_Well_order ordIso_reflexive by blast
lemma card_of_least: "well_order_on A r \<Longrightarrow> |A| \<le>o r"
-using card_of_card_order_on unfolding card_order_on_def by blast
+ using card_of_card_order_on unfolding card_order_on_def by blast
lemma card_of_ordIso:
-"(\<exists>f. bij_betw f A B) = ( |A| =o |B| )"
+ "(\<exists>f. bij_betw f A B) = ( |A| =o |B| )"
proof(auto)
fix f assume *: "bij_betw f A B"
then obtain r where "well_order_on B r \<and> |A| =o r"
- using Well_order_iso_copy card_of_well_order_on by blast
+ using Well_order_iso_copy card_of_well_order_on by blast
hence "|B| \<le>o |A|" using card_of_least
- ordLeq_ordIso_trans ordIso_symmetric by blast
+ ordLeq_ordIso_trans ordIso_symmetric by blast
moreover
{let ?g = "inv_into A f"
- have "bij_betw ?g B A" using * bij_betw_inv_into by blast
- then obtain r where "well_order_on A r \<and> |B| =o r"
- using Well_order_iso_copy card_of_well_order_on by blast
- hence "|A| \<le>o |B|" using card_of_least
- ordLeq_ordIso_trans ordIso_symmetric by blast
+ have "bij_betw ?g B A" using * bij_betw_inv_into by blast
+ then obtain r where "well_order_on A r \<and> |B| =o r"
+ using Well_order_iso_copy card_of_well_order_on by blast
+ hence "|A| \<le>o |B|"
+ using card_of_least ordLeq_ordIso_trans ordIso_symmetric by blast
}
ultimately show "|A| =o |B|" using ordIso_iff_ordLeq by blast
next
assume "|A| =o |B|"
then obtain f where "iso ( |A| ) ( |B| ) f"
- unfolding ordIso_def by auto
+ unfolding ordIso_def by auto
hence "bij_betw f A B" unfolding iso_def Field_card_of by simp
thus "\<exists>f. bij_betw f A B" by auto
qed
lemma card_of_ordLeq:
-"(\<exists>f. inj_on f A \<and> f ` A \<le> B) = ( |A| \<le>o |B| )"
+ "(\<exists>f. inj_on f A \<and> f ` A \<le> B) = ( |A| \<le>o |B| )"
proof(auto)
fix f assume *: "inj_on f A" and **: "f ` A \<le> B"
{assume "|B| <o |A|"
- hence "|B| \<le>o |A|" using ordLeq_iff_ordLess_or_ordIso by blast
- then obtain g where "embed ( |B| ) ( |A| ) g"
- unfolding ordLeq_def by auto
- hence 1: "inj_on g B \<and> g ` B \<le> A" using embed_inj_on[of "|B|" "|A|" "g"]
- card_of_Well_order[of "B"] Field_card_of[of "B"] Field_card_of[of "A"]
- embed_Field[of "|B|" "|A|" g] by auto
- obtain h where "bij_betw h A B"
- using * ** 1 Schroeder_Bernstein[of f] by fastforce
- hence "|A| =o |B|" using card_of_ordIso by blast
- hence "|A| \<le>o |B|" using ordIso_iff_ordLeq by auto
+ hence "|B| \<le>o |A|" using ordLeq_iff_ordLess_or_ordIso by blast
+ then obtain g where "embed ( |B| ) ( |A| ) g"
+ unfolding ordLeq_def by auto
+ hence 1: "inj_on g B \<and> g ` B \<le> A" using embed_inj_on[of "|B|" "|A|" "g"]
+ card_of_Well_order[of "B"] Field_card_of[of "B"] Field_card_of[of "A"]
+ embed_Field[of "|B|" "|A|" g] by auto
+ obtain h where "bij_betw h A B"
+ using * ** 1 Schroeder_Bernstein[of f] by fastforce
+ hence "|A| \<le>o |B|" using card_of_ordIso ordIso_iff_ordLeq by auto
}
thus "|A| \<le>o |B|" using ordLess_or_ordLeq[of "|B|" "|A|"]
- by (auto simp: card_of_Well_order)
+ by (auto simp: card_of_Well_order)
next
assume *: "|A| \<le>o |B|"
- obtain f where "embed ( |A| ) ( |B| ) f"
- using * unfolding ordLeq_def by auto
- hence "inj_on f A \<and> f ` A \<le> B" using embed_inj_on[of "|A|" "|B|" f]
- card_of_Well_order[of "A"] Field_card_of[of "A"] Field_card_of[of "B"]
- embed_Field[of "|A|" "|B|" f] by auto
+ obtain f where "embed |A| |B| f"
+ using * unfolding ordLeq_def by auto
+ hence "inj_on f A \<and> f ` A \<le> B"
+ using embed_inj_on[of "|A|" "|B|"] card_of_Well_order embed_Field[of "|A|" "|B|"]
+ by (auto simp: Field_card_of)
thus "\<exists>f. inj_on f A \<and> f ` A \<le> B" by auto
qed
lemma card_of_ordLeq2:
-"A \<noteq> {} \<Longrightarrow> (\<exists>g. g ` B = A) = ( |A| \<le>o |B| )"
-using card_of_ordLeq[of A B] inj_on_iff_surj[of A B] by auto
+ "A \<noteq> {} \<Longrightarrow> (\<exists>g. g ` B = A) = ( |A| \<le>o |B| )"
+ using card_of_ordLeq[of A B] inj_on_iff_surj[of A B] by auto
lemma card_of_ordLess:
-"(\<not>(\<exists>f. inj_on f A \<and> f ` A \<le> B)) = ( |B| <o |A| )"
-proof-
+ "(\<not>(\<exists>f. inj_on f A \<and> f ` A \<le> B)) = ( |B| <o |A| )"
+proof -
have "(\<not>(\<exists>f. inj_on f A \<and> f ` A \<le> B)) = (\<not> |A| \<le>o |B| )"
- using card_of_ordLeq by blast
+ using card_of_ordLeq by blast
also have "\<dots> = ( |B| <o |A| )"
- using card_of_Well_order[of A] card_of_Well_order[of B]
- not_ordLeq_iff_ordLess by blast
+ using not_ordLeq_iff_ordLess by (auto intro: card_of_Well_order)
finally show ?thesis .
qed
lemma card_of_ordLess2:
-"B \<noteq> {} \<Longrightarrow> (\<not>(\<exists>f. f ` A = B)) = ( |A| <o |B| )"
-using card_of_ordLess[of B A] inj_on_iff_surj[of B A] by auto
+ "B \<noteq> {} \<Longrightarrow> (\<not>(\<exists>f. f ` A = B)) = ( |A| <o |B| )"
+ using card_of_ordLess[of B A] inj_on_iff_surj[of B A] by auto
lemma card_of_ordIsoI:
-assumes "bij_betw f A B"
-shows "|A| =o |B|"
-using assms unfolding card_of_ordIso[symmetric] by auto
+ assumes "bij_betw f A B"
+ shows "|A| =o |B|"
+ using assms unfolding card_of_ordIso[symmetric] by auto
lemma card_of_ordLeqI:
-assumes "inj_on f A" and "\<And> a. a \<in> A \<Longrightarrow> f a \<in> B"
-shows "|A| \<le>o |B|"
-using assms unfolding card_of_ordLeq[symmetric] by auto
+ assumes "inj_on f A" and "\<And> a. a \<in> A \<Longrightarrow> f a \<in> B"
+ shows "|A| \<le>o |B|"
+ using assms unfolding card_of_ordLeq[symmetric] by auto
lemma card_of_unique:
-"card_order_on A r \<Longrightarrow> r =o |A|"
-by (simp only: card_order_on_ordIso card_of_card_order_on)
+ "card_order_on A r \<Longrightarrow> r =o |A|"
+ by (simp only: card_order_on_ordIso card_of_card_order_on)
lemma card_of_mono1:
-"A \<le> B \<Longrightarrow> |A| \<le>o |B|"
-using inj_on_id[of A] card_of_ordLeq[of A B] by fastforce
+ "A \<le> B \<Longrightarrow> |A| \<le>o |B|"
+ using inj_on_id[of A] card_of_ordLeq[of A B] by fastforce
lemma card_of_mono2:
-assumes "r \<le>o r'"
-shows "|Field r| \<le>o |Field r'|"
-proof-
+ assumes "r \<le>o r'"
+ shows "|Field r| \<le>o |Field r'|"
+proof -
obtain f where
- 1: "well_order_on (Field r) r \<and> well_order_on (Field r) r \<and> embed r r' f"
- using assms unfolding ordLeq_def
- by (auto simp add: well_order_on_Well_order)
+ 1: "well_order_on (Field r) r \<and> well_order_on (Field r) r \<and> embed r r' f"
+ using assms unfolding ordLeq_def
+ by (auto simp add: well_order_on_Well_order)
hence "inj_on f (Field r) \<and> f ` (Field r) \<le> Field r'"
- by (auto simp add: embed_inj_on embed_Field)
+ by (auto simp add: embed_inj_on embed_Field)
thus "|Field r| \<le>o |Field r'|" using card_of_ordLeq by blast
qed
lemma card_of_cong: "r =o r' \<Longrightarrow> |Field r| =o |Field r'|"
-by (simp add: ordIso_iff_ordLeq card_of_mono2)
-
-lemma card_of_Field_ordLess: "Well_order r \<Longrightarrow> |Field r| \<le>o r"
-using card_of_least card_of_well_order_on well_order_on_Well_order by blast
+ by (simp add: ordIso_iff_ordLeq card_of_mono2)
lemma card_of_Field_ordIso:
-assumes "Card_order r"
-shows "|Field r| =o r"
-proof-
+ assumes "Card_order r"
+ shows "|Field r| =o r"
+proof -
have "card_order_on (Field r) r"
- using assms card_order_on_Card_order by blast
+ using assms card_order_on_Card_order by blast
moreover have "card_order_on (Field r) |Field r|"
- using card_of_card_order_on by blast
+ using card_of_card_order_on by blast
ultimately show ?thesis using card_order_on_ordIso by blast
qed
lemma Card_order_iff_ordIso_card_of:
-"Card_order r = (r =o |Field r| )"
-using ordIso_card_of_imp_Card_order card_of_Field_ordIso ordIso_symmetric by blast
+ "Card_order r = (r =o |Field r| )"
+ using ordIso_card_of_imp_Card_order card_of_Field_ordIso ordIso_symmetric by blast
lemma Card_order_iff_ordLeq_card_of:
-"Card_order r = (r \<le>o |Field r| )"
-proof-
+ "Card_order r = (r \<le>o |Field r| )"
+proof -
have "Card_order r = (r =o |Field r| )"
- unfolding Card_order_iff_ordIso_card_of by simp
- also have "... = (r \<le>o |Field r| \<and> |Field r| \<le>o r)"
- unfolding ordIso_iff_ordLeq by simp
- also have "... = (r \<le>o |Field r| )"
- using card_of_Field_ordLess
- by (auto simp: card_of_Field_ordLess ordLeq_Well_order_simp)
+ unfolding Card_order_iff_ordIso_card_of by simp
+ also have "\<dots> = (r \<le>o |Field r| \<and> |Field r| \<le>o r)"
+ unfolding ordIso_iff_ordLeq by simp
+ also have "\<dots> = (r \<le>o |Field r| )"
+ using card_of_least
+ by (auto simp: card_of_least ordLeq_Well_order_simp)
finally show ?thesis .
qed
lemma Card_order_iff_Restr_underS:
-assumes "Well_order r"
-shows "Card_order r = (\<forall>a \<in> Field r. Restr r (underS r a) <o |Field r| )"
-using assms unfolding Card_order_iff_ordLeq_card_of
-using ordLeq_iff_ordLess_Restr card_of_Well_order by blast
+ assumes "Well_order r"
+ shows "Card_order r = (\<forall>a \<in> Field r. Restr r (underS r a) <o |Field r| )"
+ using assms ordLeq_iff_ordLess_Restr card_of_Well_order
+ unfolding Card_order_iff_ordLeq_card_of by blast
lemma card_of_underS:
-assumes r: "Card_order r" and a: "a \<in> Field r"
-shows "|underS r a| <o r"
-proof-
+ assumes r: "Card_order r" and a: "a \<in> Field r"
+ shows "|underS r a| <o r"
+proof -
let ?A = "underS r a" let ?r' = "Restr r ?A"
have 1: "Well_order r"
- using r unfolding card_order_on_def by simp
+ using r unfolding card_order_on_def by simp
have "Well_order ?r'" using 1 Well_order_Restr by auto
- moreover have "card_order_on (Field ?r') |Field ?r'|"
- using card_of_card_order_on .
- ultimately have "|Field ?r'| \<le>o ?r'"
- unfolding card_order_on_def by simp
+ with card_of_card_order_on have "|Field ?r'| \<le>o ?r'"
+ unfolding card_order_on_def by auto
moreover have "Field ?r' = ?A"
- using 1 wo_rel.underS_ofilter Field_Restr_ofilter
- unfolding wo_rel_def by fastforce
+ using 1 wo_rel.underS_ofilter Field_Restr_ofilter
+ unfolding wo_rel_def by fastforce
ultimately have "|?A| \<le>o ?r'" by simp
also have "?r' <o |Field r|"
- using 1 a r Card_order_iff_Restr_underS by blast
+ using 1 a r Card_order_iff_Restr_underS by blast
also have "|Field r| =o r"
- using r ordIso_symmetric unfolding Card_order_iff_ordIso_card_of by auto
+ using r ordIso_symmetric unfolding Card_order_iff_ordIso_card_of by auto
finally show ?thesis .
qed
lemma ordLess_Field:
-assumes "r <o r'"
-shows "|Field r| <o r'"
-proof-
+ assumes "r <o r'"
+ shows "|Field r| <o r'"
+proof -
have "well_order_on (Field r) r" using assms unfolding ordLess_def
- by (auto simp add: well_order_on_Well_order)
+ by (auto simp add: well_order_on_Well_order)
hence "|Field r| \<le>o r" using card_of_least by blast
thus ?thesis using assms ordLeq_ordLess_trans by blast
qed
lemma internalize_card_of_ordLeq:
-"( |A| \<le>o r) = (\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r)"
+ "( |A| \<le>o r) = (\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r)"
proof
assume "|A| \<le>o r"
then obtain p where 1: "Field p \<le> Field r \<and> |A| =o p \<and> p \<le>o r"
- using internalize_ordLeq[of "|A|" r] by blast
+ using internalize_ordLeq[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| \<le>o r"
- using 1 ordIso_equivalence ordIso_ordLeq_trans by blast
+ using 1 ordIso_equivalence ordIso_ordLeq_trans by blast
thus "\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r" using 1 by blast
next
assume "\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r"
@@ -340,8 +332,8 @@
qed
lemma internalize_card_of_ordLeq2:
-"( |A| \<le>o |C| ) = (\<exists>B \<le> C. |A| =o |B| \<and> |B| \<le>o |C| )"
-using internalize_card_of_ordLeq[of "A" "|C|"] Field_card_of[of C] by auto
+ "( |A| \<le>o |C| ) = (\<exists>B \<le> C. |A| =o |B| \<and> |B| \<le>o |C| )"
+ using internalize_card_of_ordLeq[of "A" "|C|"] Field_card_of[of C] by auto
subsection \<open>Cardinals versus set operations on arbitrary sets\<close>
@@ -353,123 +345,113 @@
\<close>
lemma card_of_empty: "|{}| \<le>o |A|"
-using card_of_ordLeq inj_on_id by blast
+ using card_of_ordLeq inj_on_id by blast
lemma card_of_empty1:
-assumes "Well_order r \<or> Card_order r"
-shows "|{}| \<le>o r"
-proof-
+ assumes "Well_order r \<or> Card_order r"
+ shows "|{}| \<le>o r"
+proof -
have "Well_order r" using assms unfolding card_order_on_def by auto
- hence "|Field r| <=o r"
- using assms card_of_Field_ordLess by blast
+ hence "|Field r| \<le>o r"
+ using assms card_of_least by blast
moreover have "|{}| \<le>o |Field r|" by (simp add: card_of_empty)
ultimately show ?thesis using ordLeq_transitive by blast
qed
corollary Card_order_empty:
-"Card_order r \<Longrightarrow> |{}| \<le>o r" by (simp add: card_of_empty1)
+ "Card_order r \<Longrightarrow> |{}| \<le>o r" by (simp add: card_of_empty1)
lemma card_of_empty2:
-assumes LEQ: "|A| =o |{}|"
-shows "A = {}"
-using assms card_of_ordIso[of A] bij_betw_empty2 by blast
+ assumes "|A| =o |{}|"
+ shows "A = {}"
+ using assms card_of_ordIso[of A] bij_betw_empty2 by blast
lemma card_of_empty3:
-assumes LEQ: "|A| \<le>o |{}|"
-shows "A = {}"
-using assms
-by (simp add: ordIso_iff_ordLeq card_of_empty1 card_of_empty2
- ordLeq_Well_order_simp)
+ assumes "|A| \<le>o |{}|"
+ shows "A = {}"
+ using assms
+ by (simp add: ordIso_iff_ordLeq card_of_empty1 card_of_empty2
+ ordLeq_Well_order_simp)
lemma card_of_empty_ordIso:
-"|{}::'a set| =o |{}::'b set|"
-using card_of_ordIso unfolding bij_betw_def inj_on_def by blast
+ "|{}::'a set| =o |{}::'b set|"
+ using card_of_ordIso unfolding bij_betw_def inj_on_def by blast
lemma card_of_image:
-"|f ` A| <=o |A|"
-proof(cases "A = {}", simp add: card_of_empty)
- assume "A \<noteq> {}"
+ "|f ` A| \<le>o |A|"
+proof(cases "A = {}")
+ case False
hence "f ` A \<noteq> {}" by auto
- thus "|f ` A| \<le>o |A|"
- using card_of_ordLeq2[of "f ` A" A] by auto
-qed
+ thus ?thesis
+ using card_of_ordLeq2[of "f ` A" A] by auto
+qed (simp add: card_of_empty)
lemma surj_imp_ordLeq:
-assumes "B \<subseteq> f ` A"
-shows "|B| \<le>o |A|"
-proof-
- have "|B| <=o |f ` A|" using assms card_of_mono1 by auto
+ assumes "B \<subseteq> f ` A"
+ shows "|B| \<le>o |A|"
+proof -
+ have "|B| \<le>o |f ` A|" using assms card_of_mono1 by auto
thus ?thesis using card_of_image ordLeq_transitive by blast
qed
lemma card_of_singl_ordLeq:
-assumes "A \<noteq> {}"
-shows "|{b}| \<le>o |A|"
-proof-
+ assumes "A \<noteq> {}"
+ shows "|{b}| \<le>o |A|"
+proof -
obtain a where *: "a \<in> A" using assms by auto
let ?h = "\<lambda> b'::'b. if b' = b then a else undefined"
have "inj_on ?h {b} \<and> ?h ` {b} \<le> A"
- using * unfolding inj_on_def by auto
+ using * unfolding inj_on_def by auto
thus ?thesis unfolding card_of_ordLeq[symmetric] by (intro exI)
qed
corollary Card_order_singl_ordLeq:
-"\<lbrakk>Card_order r; Field r \<noteq> {}\<rbrakk> \<Longrightarrow> |{b}| \<le>o r"
-using card_of_singl_ordLeq[of "Field r" b]
- card_of_Field_ordIso[of r] ordLeq_ordIso_trans by blast
+ "\<lbrakk>Card_order r; Field r \<noteq> {}\<rbrakk> \<Longrightarrow> |{b}| \<le>o r"
+ using card_of_singl_ordLeq[of "Field r" b]
+ card_of_Field_ordIso[of r] ordLeq_ordIso_trans by blast
lemma card_of_Pow: "|A| <o |Pow A|"
-using card_of_ordLess2[of "Pow A" A] Cantors_paradox[of A]
- Pow_not_empty[of A] by auto
+ using card_of_ordLess2[of "Pow A" A] Cantors_paradox[of A]
+ Pow_not_empty[of A] by auto
corollary Card_order_Pow:
-"Card_order r \<Longrightarrow> r <o |Pow(Field r)|"
-using card_of_Pow card_of_Field_ordIso ordIso_ordLess_trans ordIso_symmetric by blast
+ "Card_order r \<Longrightarrow> r <o |Pow(Field r)|"
+ using card_of_Pow card_of_Field_ordIso ordIso_ordLess_trans ordIso_symmetric by blast
-lemma card_of_Plus1: "|A| \<le>o |A <+> B|"
-proof-
- have "Inl ` A \<le> A <+> B" by auto
- thus ?thesis using inj_Inl[of A] card_of_ordLeq by blast
-qed
+lemma card_of_Plus1: "|A| \<le>o |A <+> B|" and card_of_Plus2: "|B| \<le>o |A <+> B|"
+ using card_of_ordLeq by force+
corollary Card_order_Plus1:
-"Card_order r \<Longrightarrow> r \<le>o |(Field r) <+> B|"
-using card_of_Plus1 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
-
-lemma card_of_Plus2: "|B| \<le>o |A <+> B|"
-proof-
- have "Inr ` B \<le> A <+> B" by auto
- thus ?thesis using inj_Inr[of B] card_of_ordLeq by blast
-qed
+ "Card_order r \<Longrightarrow> r \<le>o |(Field r) <+> B|"
+ using card_of_Plus1 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
corollary Card_order_Plus2:
-"Card_order r \<Longrightarrow> r \<le>o |A <+> (Field r)|"
-using card_of_Plus2 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
+ "Card_order r \<Longrightarrow> r \<le>o |A <+> (Field r)|"
+ using card_of_Plus2 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
lemma card_of_Plus_empty1: "|A| =o |A <+> {}|"
-proof-
+proof -
have "bij_betw Inl A (A <+> {})" unfolding bij_betw_def inj_on_def by auto
thus ?thesis using card_of_ordIso by auto
qed
lemma card_of_Plus_empty2: "|A| =o |{} <+> A|"
-proof-
+proof -
have "bij_betw Inr A ({} <+> A)" unfolding bij_betw_def inj_on_def by auto
thus ?thesis using card_of_ordIso by auto
qed
lemma card_of_Plus_commute: "|A <+> B| =o |B <+> A|"
-proof-
- let ?f = "\<lambda>(c::'a + 'b). case c of Inl a \<Rightarrow> Inr a
- | Inr b \<Rightarrow> Inl b"
+proof -
+ let ?f = "\<lambda>c. case c of Inl a \<Rightarrow> Inr a | Inr b \<Rightarrow> Inl b"
have "bij_betw ?f (A <+> B) (B <+> A)"
- unfolding bij_betw_def inj_on_def by force
+ unfolding bij_betw_def inj_on_def by force
thus ?thesis using card_of_ordIso by blast
qed
lemma card_of_Plus_assoc:
-fixes A :: "'a set" and B :: "'b set" and C :: "'c set"
-shows "|(A <+> B) <+> C| =o |A <+> B <+> C|"
+ fixes A :: "'a set" and B :: "'b set" and C :: "'c set"
+ shows "|(A <+> B) <+> C| =o |A <+> B <+> C|"
proof -
define f :: "('a + 'b) + 'c \<Rightarrow> 'a + 'b + 'c"
where [abs_def]: "f k =
@@ -487,21 +469,11 @@
proof(cases x)
case (Inl a)
hence "a \<in> A" "x = f (Inl (Inl a))"
- using x unfolding f_def by auto
+ using x unfolding f_def by auto
thus ?thesis by auto
next
- case (Inr bc) note 1 = Inr show ?thesis
- proof(cases bc)
- case (Inl b)
- hence "b \<in> B" "x = f (Inl (Inr b))"
- using x 1 unfolding f_def by auto
- thus ?thesis by auto
- next
- case (Inr c)
- hence "c \<in> C" "x = f (Inr c)"
- using x 1 unfolding f_def by auto
- thus ?thesis by auto
- qed
+ case (Inr bc) with x show ?thesis
+ by (cases bc) (force simp: f_def)+
qed
qed
hence "bij_betw f ((A <+> B) <+> C) (A <+> B <+> C)"
@@ -510,159 +482,134 @@
qed
lemma card_of_Plus_mono1:
-assumes "|A| \<le>o |B|"
-shows "|A <+> C| \<le>o |B <+> C|"
-proof-
- obtain f where 1: "inj_on f A \<and> f ` A \<le> B"
- using assms card_of_ordLeq[of A] by fastforce
- obtain g where g_def:
- "g = (\<lambda>d. case d of Inl a \<Rightarrow> Inl(f a) | Inr (c::'c) \<Rightarrow> Inr c)" by blast
+ assumes "|A| \<le>o |B|"
+ shows "|A <+> C| \<le>o |B <+> C|"
+proof -
+ obtain f where f: "inj_on f A \<and> f ` A \<le> B"
+ using assms card_of_ordLeq[of A] by fastforce
+ define g where "g \<equiv> \<lambda>d. case d of Inl a \<Rightarrow> Inl(f a) | Inr (c::'c) \<Rightarrow> Inr c"
have "inj_on g (A <+> C) \<and> g ` (A <+> C) \<le> (B <+> C)"
- proof-
- {fix d1 and d2 assume "d1 \<in> A <+> C \<and> d2 \<in> A <+> C" and
- "g d1 = g d2"
- hence "d1 = d2" using 1 unfolding inj_on_def g_def by force
- }
- moreover
- {fix d assume "d \<in> A <+> C"
- hence "g d \<in> B <+> C" using 1
- by(cases d) (auto simp add: g_def)
- }
- ultimately show ?thesis unfolding inj_on_def by auto
- qed
+ using f unfolding inj_on_def g_def by force
thus ?thesis using card_of_ordLeq by blast
qed
corollary ordLeq_Plus_mono1:
-assumes "r \<le>o r'"
-shows "|(Field r) <+> C| \<le>o |(Field r') <+> C|"
-using assms card_of_mono2 card_of_Plus_mono1 by blast
+ assumes "r \<le>o r'"
+ shows "|(Field r) <+> C| \<le>o |(Field r') <+> C|"
+ using assms card_of_mono2 card_of_Plus_mono1 by blast
lemma card_of_Plus_mono2:
-assumes "|A| \<le>o |B|"
-shows "|C <+> A| \<le>o |C <+> B|"
-using assms card_of_Plus_mono1[of A B C]
- card_of_Plus_commute[of C A] card_of_Plus_commute[of B C]
- ordIso_ordLeq_trans[of "|C <+> A|"] ordLeq_ordIso_trans[of "|C <+> A|"]
-by blast
+ assumes "|A| \<le>o |B|"
+ shows "|C <+> A| \<le>o |C <+> B|"
+ using card_of_Plus_mono1[OF assms]
+ by (blast intro: card_of_Plus_commute ordIso_ordLeq_trans ordLeq_ordIso_trans)
corollary ordLeq_Plus_mono2:
-assumes "r \<le>o r'"
-shows "|A <+> (Field r)| \<le>o |A <+> (Field r')|"
-using assms card_of_mono2 card_of_Plus_mono2 by blast
+ assumes "r \<le>o r'"
+ shows "|A <+> (Field r)| \<le>o |A <+> (Field r')|"
+ using assms card_of_mono2 card_of_Plus_mono2 by blast
lemma card_of_Plus_mono:
-assumes "|A| \<le>o |B|" and "|C| \<le>o |D|"
-shows "|A <+> C| \<le>o |B <+> D|"
-using assms card_of_Plus_mono1[of A B C] card_of_Plus_mono2[of C D B]
- ordLeq_transitive[of "|A <+> C|"] by blast
+ assumes "|A| \<le>o |B|" and "|C| \<le>o |D|"
+ shows "|A <+> C| \<le>o |B <+> D|"
+ using assms card_of_Plus_mono1[of A B C] card_of_Plus_mono2[of C D B]
+ ordLeq_transitive by blast
corollary ordLeq_Plus_mono:
-assumes "r \<le>o r'" and "p \<le>o p'"
-shows "|(Field r) <+> (Field p)| \<le>o |(Field r') <+> (Field p')|"
-using assms card_of_mono2[of r r'] card_of_mono2[of p p'] card_of_Plus_mono by blast
+ assumes "r \<le>o r'" and "p \<le>o p'"
+ shows "|(Field r) <+> (Field p)| \<le>o |(Field r') <+> (Field p')|"
+ using assms card_of_mono2[of r r'] card_of_mono2[of p p'] card_of_Plus_mono by blast
lemma card_of_Plus_cong1:
-assumes "|A| =o |B|"
-shows "|A <+> C| =o |B <+> C|"
-using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono1)
+ assumes "|A| =o |B|"
+ shows "|A <+> C| =o |B <+> C|"
+ using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono1)
corollary ordIso_Plus_cong1:
-assumes "r =o r'"
-shows "|(Field r) <+> C| =o |(Field r') <+> C|"
-using assms card_of_cong card_of_Plus_cong1 by blast
+ assumes "r =o r'"
+ shows "|(Field r) <+> C| =o |(Field r') <+> C|"
+ using assms card_of_cong card_of_Plus_cong1 by blast
lemma card_of_Plus_cong2:
-assumes "|A| =o |B|"
-shows "|C <+> A| =o |C <+> B|"
-using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono2)
+ assumes "|A| =o |B|"
+ shows "|C <+> A| =o |C <+> B|"
+ using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono2)
corollary ordIso_Plus_cong2:
-assumes "r =o r'"
-shows "|A <+> (Field r)| =o |A <+> (Field r')|"
-using assms card_of_cong card_of_Plus_cong2 by blast
+ assumes "r =o r'"
+ shows "|A <+> (Field r)| =o |A <+> (Field r')|"
+ using assms card_of_cong card_of_Plus_cong2 by blast
lemma card_of_Plus_cong:
-assumes "|A| =o |B|" and "|C| =o |D|"
-shows "|A <+> C| =o |B <+> D|"
-using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono)
+ assumes "|A| =o |B|" and "|C| =o |D|"
+ shows "|A <+> C| =o |B <+> D|"
+ using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono)
corollary ordIso_Plus_cong:
-assumes "r =o r'" and "p =o p'"
-shows "|(Field r) <+> (Field p)| =o |(Field r') <+> (Field p')|"
-using assms card_of_cong[of r r'] card_of_cong[of p p'] card_of_Plus_cong by blast
+ assumes "r =o r'" and "p =o p'"
+ shows "|(Field r) <+> (Field p)| =o |(Field r') <+> (Field p')|"
+ using assms card_of_cong[of r r'] card_of_cong[of p p'] card_of_Plus_cong by blast
lemma card_of_Un_Plus_ordLeq:
-"|A \<union> B| \<le>o |A <+> B|"
-proof-
- let ?f = "\<lambda> c. if c \<in> A then Inl c else Inr c"
- have "inj_on ?f (A \<union> B) \<and> ?f ` (A \<union> B) \<le> A <+> B"
- unfolding inj_on_def by auto
- thus ?thesis using card_of_ordLeq by blast
+ "|A \<union> B| \<le>o |A <+> B|"
+proof -
+ let ?f = "\<lambda> c. if c \<in> A then Inl c else Inr c"
+ have "inj_on ?f (A \<union> B) \<and> ?f ` (A \<union> B) \<le> A <+> B"
+ unfolding inj_on_def by auto
+ thus ?thesis using card_of_ordLeq by blast
qed
lemma card_of_Times1:
-assumes "A \<noteq> {}"
-shows "|B| \<le>o |B \<times> A|"
-proof(cases "B = {}", simp add: card_of_empty)
- assume *: "B \<noteq> {}"
+ assumes "A \<noteq> {}"
+ shows "|B| \<le>o |B \<times> A|"
+proof(cases "B = {}")
+ case False
have "fst `(B \<times> A) = B" using assms by auto
thus ?thesis using inj_on_iff_surj[of B "B \<times> A"]
- card_of_ordLeq[of B "B \<times> A"] * by blast
-qed
+ card_of_ordLeq False by blast
+qed (simp add: card_of_empty)
lemma card_of_Times_commute: "|A \<times> B| =o |B \<times> A|"
-proof-
- let ?f = "\<lambda>(a::'a,b::'b). (b,a)"
- have "bij_betw ?f (A \<times> B) (B \<times> A)"
- unfolding bij_betw_def inj_on_def by auto
+proof -
+ have "bij_betw (\<lambda>(a,b). (b,a)) (A \<times> B) (B \<times> A)"
+ unfolding bij_betw_def inj_on_def by auto
thus ?thesis using card_of_ordIso by blast
qed
lemma card_of_Times2:
-assumes "A \<noteq> {}" shows "|B| \<le>o |A \<times> B|"
-using assms card_of_Times1[of A B] card_of_Times_commute[of B A]
- ordLeq_ordIso_trans by blast
+ assumes "A \<noteq> {}" shows "|B| \<le>o |A \<times> B|"
+ using assms card_of_Times1[of A B] card_of_Times_commute[of B A]
+ ordLeq_ordIso_trans by blast
corollary Card_order_Times1:
-"\<lbrakk>Card_order r; B \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |(Field r) \<times> B|"
-using card_of_Times1[of B] card_of_Field_ordIso
- ordIso_ordLeq_trans ordIso_symmetric by blast
+ "\<lbrakk>Card_order r; B \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |(Field r) \<times> B|"
+ using card_of_Times1[of B] card_of_Field_ordIso
+ ordIso_ordLeq_trans ordIso_symmetric by blast
corollary Card_order_Times2:
-"\<lbrakk>Card_order r; A \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |A \<times> (Field r)|"
-using card_of_Times2[of A] card_of_Field_ordIso
- ordIso_ordLeq_trans ordIso_symmetric by blast
+ "\<lbrakk>Card_order r; A \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |A \<times> (Field r)|"
+ using card_of_Times2[of A] card_of_Field_ordIso
+ ordIso_ordLeq_trans ordIso_symmetric by blast
lemma card_of_Times3: "|A| \<le>o |A \<times> A|"
-using card_of_Times1[of A]
-by(cases "A = {}", simp add: card_of_empty, blast)
+ using card_of_Times1[of A]
+ by(cases "A = {}", simp add: card_of_empty)
lemma card_of_Plus_Times_bool: "|A <+> A| =o |A \<times> (UNIV::bool set)|"
-proof-
+proof -
let ?f = "\<lambda>c::'a + 'a. case c of Inl a \<Rightarrow> (a,True)
|Inr a \<Rightarrow> (a,False)"
have "bij_betw ?f (A <+> A) (A \<times> (UNIV::bool set))"
- proof-
- {fix c1 and c2 assume "?f c1 = ?f c2"
- hence "c1 = c2"
- by(cases c1; cases c2) auto
- }
+ proof -
+ have "\<And>c1 c2. ?f c1 = ?f c2 \<Longrightarrow> c1 = c2"
+ by (force split: sum.split_asm)
moreover
- {fix c assume "c \<in> A <+> A"
- hence "?f c \<in> A \<times> (UNIV::bool set)"
- by(cases c) auto
- }
+ have "\<And>c. c \<in> A <+> A \<Longrightarrow> ?f c \<in> A \<times> (UNIV::bool set)"
+ by (force split: sum.split_asm)
moreover
- {fix a bl assume *: "(a,bl) \<in> A \<times> (UNIV::bool set)"
- have "(a,bl) \<in> ?f ` ( A <+> A)"
- proof(cases bl)
- assume bl hence "?f(Inl a) = (a,bl)" by auto
- thus ?thesis using * by force
- next
- assume "\<not> bl" hence "?f(Inr a) = (a,bl)" by auto
- thus ?thesis using * by force
- qed
+ {fix a bl assume "(a,bl) \<in> A \<times> (UNIV::bool set)"
+ hence "(a,bl) \<in> ?f ` ( A <+> A)"
+ by (cases bl) (force split: sum.split_asm)+
}
ultimately show ?thesis unfolding bij_betw_def inj_on_def by auto
qed
@@ -670,127 +617,107 @@
qed
lemma card_of_Times_mono1:
-assumes "|A| \<le>o |B|"
-shows "|A \<times> C| \<le>o |B \<times> C|"
-proof-
- obtain f where 1: "inj_on f A \<and> f ` A \<le> B"
- using assms card_of_ordLeq[of A] by fastforce
- obtain g where g_def:
- "g = (\<lambda>(a,c::'c). (f a,c))" by blast
+ assumes "|A| \<le>o |B|"
+ shows "|A \<times> C| \<le>o |B \<times> C|"
+proof -
+ obtain f where f: "inj_on f A \<and> f ` A \<le> B"
+ using assms card_of_ordLeq[of A] by fastforce
+ define g where "g \<equiv> (\<lambda>(a,c::'c). (f a,c))"
have "inj_on g (A \<times> C) \<and> g ` (A \<times> C) \<le> (B \<times> C)"
- using 1 unfolding inj_on_def using g_def by auto
+ using f unfolding inj_on_def using g_def by auto
thus ?thesis using card_of_ordLeq by blast
qed
corollary ordLeq_Times_mono1:
-assumes "r \<le>o r'"
-shows "|(Field r) \<times> C| \<le>o |(Field r') \<times> C|"
-using assms card_of_mono2 card_of_Times_mono1 by blast
+ assumes "r \<le>o r'"
+ shows "|(Field r) \<times> C| \<le>o |(Field r') \<times> C|"
+ using assms card_of_mono2 card_of_Times_mono1 by blast
lemma card_of_Times_mono2:
-assumes "|A| \<le>o |B|"
-shows "|C \<times> A| \<le>o |C \<times> B|"
-using assms card_of_Times_mono1[of A B C]
- card_of_Times_commute[of C A] card_of_Times_commute[of B C]
- ordIso_ordLeq_trans[of "|C \<times> A|"] ordLeq_ordIso_trans[of "|C \<times> A|"]
-by blast
+ assumes "|A| \<le>o |B|"
+ shows "|C \<times> A| \<le>o |C \<times> B|"
+ using assms card_of_Times_mono1[of A B C]
+ by (blast intro: card_of_Times_commute ordIso_ordLeq_trans ordLeq_ordIso_trans)
corollary ordLeq_Times_mono2:
-assumes "r \<le>o r'"
-shows "|A \<times> (Field r)| \<le>o |A \<times> (Field r')|"
-using assms card_of_mono2 card_of_Times_mono2 by blast
+ assumes "r \<le>o r'"
+ shows "|A \<times> (Field r)| \<le>o |A \<times> (Field r')|"
+ using assms card_of_mono2 card_of_Times_mono2 by blast
lemma card_of_Sigma_mono1:
-assumes "\<forall>i \<in> I. |A i| \<le>o |B i|"
-shows "|SIGMA i : I. A i| \<le>o |SIGMA i : I. B i|"
-proof-
+ assumes "\<forall>i \<in> I. |A i| \<le>o |B i|"
+ shows "|SIGMA i : I. A i| \<le>o |SIGMA i : I. B i|"
+proof -
have "\<forall>i. i \<in> I \<longrightarrow> (\<exists>f. inj_on f (A i) \<and> f ` (A i) \<le> B i)"
- using assms by (auto simp add: card_of_ordLeq)
+ using assms by (auto simp add: card_of_ordLeq)
with choice[of "\<lambda> i f. i \<in> I \<longrightarrow> inj_on f (A i) \<and> f ` (A i) \<le> B i"]
- obtain F where 1: "\<forall>i \<in> I. inj_on (F i) (A i) \<and> (F i) ` (A i) \<le> B i"
+ obtain F where F: "\<forall>i \<in> I. inj_on (F i) (A i) \<and> (F i) ` (A i) \<le> B i"
by atomize_elim (auto intro: bchoice)
- obtain g where g_def: "g = (\<lambda>(i,a::'b). (i,F i a))" by blast
+ define g where "g \<equiv> (\<lambda>(i,a::'b). (i,F i a))"
have "inj_on g (Sigma I A) \<and> g ` (Sigma I A) \<le> (Sigma I B)"
- using 1 unfolding inj_on_def using g_def by force
+ using F unfolding inj_on_def using g_def by force
thus ?thesis using card_of_ordLeq by blast
qed
lemma card_of_UNION_Sigma:
-"|\<Union>i \<in> I. A i| \<le>o |SIGMA i : I. A i|"
-using Ex_inj_on_UNION_Sigma [of A I] card_of_ordLeq by blast
+ "|\<Union>i \<in> I. A i| \<le>o |SIGMA i : I. A i|"
+ using Ex_inj_on_UNION_Sigma [of A I] card_of_ordLeq by blast
lemma card_of_bool:
-assumes "a1 \<noteq> a2"
-shows "|UNIV::bool set| =o |{a1,a2}|"
-proof-
- let ?f = "\<lambda> bl. case bl of True \<Rightarrow> a1 | False \<Rightarrow> a2"
+ assumes "a1 \<noteq> a2"
+ shows "|UNIV::bool set| =o |{a1,a2}|"
+proof -
+ let ?f = "\<lambda> bl. if bl then a1 else a2"
have "bij_betw ?f UNIV {a1,a2}"
- proof-
- {fix bl1 and bl2 assume "?f bl1 = ?f bl2"
- hence "bl1 = bl2" using assms by (cases bl1, cases bl2) auto
- }
+ proof -
+ have "\<And>bl1 bl2. ?f bl1 = ?f bl2 \<Longrightarrow> bl1 = bl2"
+ using assms by (force split: if_split_asm)
moreover
- {fix bl have "?f bl \<in> {a1,a2}" by (cases bl) auto
- }
- moreover
- {fix a assume *: "a \<in> {a1,a2}"
- have "a \<in> ?f ` UNIV"
- proof(cases "a = a1")
- assume "a = a1"
- hence "?f True = a" by auto thus ?thesis by blast
- next
- assume "a \<noteq> a1" hence "a = a2" using * by auto
- hence "?f False = a" by auto thus ?thesis by blast
- qed
- }
- ultimately show ?thesis unfolding bij_betw_def inj_on_def by blast
+ have "\<And>bl. ?f bl \<in> {a1,a2}"
+ using assms by (force split: if_split_asm)
+ ultimately show ?thesis unfolding bij_betw_def inj_on_def by force
qed
thus ?thesis using card_of_ordIso by blast
qed
lemma card_of_Plus_Times_aux:
-assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
- LEQ: "|A| \<le>o |B|"
-shows "|A <+> B| \<le>o |A \<times> B|"
-proof-
+ assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
+ LEQ: "|A| \<le>o |B|"
+ shows "|A <+> B| \<le>o |A \<times> B|"
+proof -
have 1: "|UNIV::bool set| \<le>o |A|"
- using A2 card_of_mono1[of "{a1,a2}"] card_of_bool[of a1 a2]
- ordIso_ordLeq_trans[of "|UNIV::bool set|"] by blast
- (* *)
+ using A2 card_of_mono1[of "{a1,a2}"] card_of_bool[of a1 a2]
+ by (blast intro: ordIso_ordLeq_trans)
have "|A <+> B| \<le>o |B <+> B|"
- using LEQ card_of_Plus_mono1 by blast
+ using LEQ card_of_Plus_mono1 by blast
moreover have "|B <+> B| =o |B \<times> (UNIV::bool set)|"
- using card_of_Plus_Times_bool by blast
+ using card_of_Plus_Times_bool by blast
moreover have "|B \<times> (UNIV::bool set)| \<le>o |B \<times> A|"
- using 1 by (simp add: card_of_Times_mono2)
+ using 1 by (simp add: card_of_Times_mono2)
moreover have " |B \<times> A| =o |A \<times> B|"
- using card_of_Times_commute by blast
+ using card_of_Times_commute by blast
ultimately show "|A <+> B| \<le>o |A \<times> B|"
- using ordLeq_ordIso_trans[of "|A <+> B|" "|B <+> B|" "|B \<times> (UNIV::bool set)|"]
- ordLeq_transitive[of "|A <+> B|" "|B \<times> (UNIV::bool set)|" "|B \<times> A|"]
- ordLeq_ordIso_trans[of "|A <+> B|" "|B \<times> A|" "|A \<times> B|"]
- by blast
+ by (blast intro: ordLeq_transitive dest: ordLeq_ordIso_trans)
qed
lemma card_of_Plus_Times:
-assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
- B2: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B"
-shows "|A <+> B| \<le>o |A \<times> B|"
-proof-
+ assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and B2: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B"
+ shows "|A <+> B| \<le>o |A \<times> B|"
+proof -
{assume "|A| \<le>o |B|"
- hence ?thesis using assms by (auto simp add: card_of_Plus_Times_aux)
+ hence ?thesis using assms by (auto simp add: card_of_Plus_Times_aux)
}
moreover
{assume "|B| \<le>o |A|"
- hence "|B <+> A| \<le>o |B \<times> A|"
- using assms by (auto simp add: card_of_Plus_Times_aux)
- hence ?thesis
- using card_of_Plus_commute card_of_Times_commute
- ordIso_ordLeq_trans ordLeq_ordIso_trans by blast
+ hence "|B <+> A| \<le>o |B \<times> A|"
+ using assms by (auto simp add: card_of_Plus_Times_aux)
+ hence ?thesis
+ using card_of_Plus_commute card_of_Times_commute
+ ordIso_ordLeq_trans ordLeq_ordIso_trans by blast
}
ultimately show ?thesis
- using card_of_Well_order[of A] card_of_Well_order[of B]
- ordLeq_total[of "|A|"] by blast
+ using card_of_Well_order[of A] card_of_Well_order[of B]
+ ordLeq_total[of "|A|"] by blast
qed
lemma card_of_Times_Plus_distrib:
@@ -802,27 +729,27 @@
qed
lemma card_of_ordLeq_finite:
-assumes "|A| \<le>o |B|" and "finite B"
-shows "finite A"
-using assms unfolding ordLeq_def
-using embed_inj_on[of "|A|" "|B|"] embed_Field[of "|A|" "|B|"]
- Field_card_of[of "A"] Field_card_of[of "B"] inj_on_finite[of _ "A" "B"] by fastforce
+ assumes "|A| \<le>o |B|" and "finite B"
+ shows "finite A"
+ using assms unfolding ordLeq_def
+ using embed_inj_on[of "|A|" "|B|"] embed_Field[of "|A|" "|B|"]
+ Field_card_of[of "A"] Field_card_of[of "B"] inj_on_finite[of _ "A" "B"] by fastforce
lemma card_of_ordLeq_infinite:
-assumes "|A| \<le>o |B|" and "\<not> finite A"
-shows "\<not> finite B"
-using assms card_of_ordLeq_finite by auto
+ assumes "|A| \<le>o |B|" and "\<not> finite A"
+ shows "\<not> finite B"
+ using assms card_of_ordLeq_finite by auto
lemma card_of_ordIso_finite:
-assumes "|A| =o |B|"
-shows "finite A = finite B"
-using assms unfolding ordIso_def iso_def[abs_def]
-by (auto simp: bij_betw_finite Field_card_of)
+ assumes "|A| =o |B|"
+ shows "finite A = finite B"
+ using assms unfolding ordIso_def iso_def[abs_def]
+ by (auto simp: bij_betw_finite Field_card_of)
lemma card_of_ordIso_finite_Field:
-assumes "Card_order r" and "r =o |A|"
-shows "finite(Field r) = finite A"
-using assms card_of_Field_ordIso card_of_ordIso_finite ordIso_equivalence by blast
+ assumes "Card_order r" and "r =o |A|"
+ shows "finite(Field r) = finite A"
+ using assms card_of_Field_ordIso card_of_ordIso_finite ordIso_equivalence by blast
subsection \<open>Cardinals versus set operations involving infinite sets\<close>
@@ -835,18 +762,18 @@
at page 47 in @{cite "card-book"}. Then everything else follows fairly easily.\<close>
lemma infinite_iff_card_of_nat:
-"\<not> finite A \<longleftrightarrow> ( |UNIV::nat set| \<le>o |A| )"
-unfolding infinite_iff_countable_subset card_of_ordLeq ..
+ "\<not> finite A \<longleftrightarrow> ( |UNIV::nat set| \<le>o |A| )"
+ unfolding infinite_iff_countable_subset card_of_ordLeq ..
text\<open>The next two results correspond to the ZF fact that all infinite cardinals are
limit ordinals:\<close>
lemma Card_order_infinite_not_under:
-assumes CARD: "Card_order r" and INF: "\<not>finite (Field r)"
-shows "\<not> (\<exists>a. Field r = under r a)"
+ assumes CARD: "Card_order r" and INF: "\<not>finite (Field r)"
+ shows "\<not> (\<exists>a. Field r = under r a)"
proof(auto)
have 0: "Well_order r \<and> wo_rel r \<and> Refl r"
- using CARD unfolding wo_rel_def card_order_on_def order_on_defs by auto
+ using CARD unfolding wo_rel_def card_order_on_def order_on_defs by auto
fix a assume *: "Field r = under r a"
show False
proof(cases "a \<in> Field r")
@@ -857,115 +784,113 @@
let ?r' = "Restr r (underS r a)"
assume Case2: "a \<in> Field r"
hence 1: "under r a = underS r a \<union> {a} \<and> a \<notin> underS r a"
- using 0 Refl_under_underS[of r a] underS_notIn[of a r] by blast
+ using 0 Refl_under_underS[of r a] underS_notIn[of a r] by blast
have 2: "wo_rel.ofilter r (underS r a) \<and> underS r a < Field r"
- using 0 wo_rel.underS_ofilter * 1 Case2 by fast
+ using 0 wo_rel.underS_ofilter * 1 Case2 by fast
hence "?r' <o r" using 0 using ofilter_ordLess by blast
moreover
have "Field ?r' = underS r a \<and> Well_order ?r'"
- using 2 0 Field_Restr_ofilter[of r] Well_order_Restr[of r] by blast
+ using 2 0 Field_Restr_ofilter[of r] Well_order_Restr[of r] by blast
ultimately have "|underS r a| <o r" using ordLess_Field[of ?r'] by auto
moreover have "|under r a| =o r" using * CARD card_of_Field_ordIso[of r] by auto
ultimately have "|underS r a| <o |under r a|"
- using ordIso_symmetric ordLess_ordIso_trans by blast
+ using ordIso_symmetric ordLess_ordIso_trans by blast
moreover
{have "\<exists>f. bij_betw f (under r a) (underS r a)"
- using infinite_imp_bij_betw[of "Field r" a] INF * 1 by auto
- hence "|under r a| =o |underS r a|" using card_of_ordIso by blast
+ using infinite_imp_bij_betw[of "Field r" a] INF * 1 by auto
+ hence "|under r a| =o |underS r a|" using card_of_ordIso by blast
}
ultimately show False using not_ordLess_ordIso ordIso_symmetric by blast
qed
qed
lemma infinite_Card_order_limit:
-assumes r: "Card_order r" and "\<not>finite (Field r)"
-and a: "a \<in> Field r"
-shows "\<exists>b \<in> Field r. a \<noteq> b \<and> (a,b) \<in> r"
-proof-
+ assumes r: "Card_order r" and "\<not>finite (Field r)"
+ and a: "a \<in> Field r"
+ shows "\<exists>b \<in> Field r. a \<noteq> b \<and> (a,b) \<in> r"
+proof -
have "Field r \<noteq> under r a"
- using assms Card_order_infinite_not_under by blast
+ using assms Card_order_infinite_not_under by blast
moreover have "under r a \<le> Field r"
- using under_Field .
- ultimately have "under r a < Field r" by blast
- then obtain b where 1: "b \<in> Field r \<and> \<not> (b,a) \<in> r"
- unfolding under_def by blast
+ using under_Field .
+ ultimately obtain b where b: "b \<in> Field r \<and> \<not> (b,a) \<in> r"
+ unfolding under_def by blast
moreover have ba: "b \<noteq> a"
- using 1 r unfolding card_order_on_def well_order_on_def
- linear_order_on_def partial_order_on_def preorder_on_def refl_on_def by auto
+ using b r unfolding card_order_on_def well_order_on_def
+ linear_order_on_def partial_order_on_def preorder_on_def refl_on_def by auto
ultimately have "(a,b) \<in> r"
- using a r unfolding card_order_on_def well_order_on_def linear_order_on_def
- total_on_def by blast
- thus ?thesis using 1 ba by auto
+ using a r unfolding card_order_on_def well_order_on_def linear_order_on_def
+ total_on_def by blast
+ thus ?thesis using b ba by auto
qed
theorem Card_order_Times_same_infinite:
-assumes CO: "Card_order r" and INF: "\<not>finite(Field r)"
-shows "|Field r \<times> Field r| \<le>o r"
-proof-
- obtain phi where phi_def:
- "phi = (\<lambda>r::'a rel. Card_order r \<and> \<not>finite(Field r) \<and>
- \<not> |Field r \<times> Field r| \<le>o r )" by blast
+ assumes CO: "Card_order r" and INF: "\<not>finite(Field r)"
+ shows "|Field r \<times> Field r| \<le>o r"
+proof -
+ define phi where
+ "phi \<equiv> \<lambda>r::'a rel. Card_order r \<and> \<not>finite(Field r) \<and> \<not> |Field r \<times> Field r| \<le>o r"
have temp1: "\<forall>r. phi r \<longrightarrow> Well_order r"
- unfolding phi_def card_order_on_def by auto
+ unfolding phi_def card_order_on_def by auto
have Ft: "\<not>(\<exists>r. phi r)"
proof
assume "\<exists>r. phi r"
hence "{r. phi r} \<noteq> {} \<and> {r. phi r} \<le> {r. Well_order r}"
- using temp1 by auto
+ using temp1 by auto
then obtain r where 1: "phi r" and 2: "\<forall>r'. phi r' \<longrightarrow> r \<le>o r'" and
- 3: "Card_order r \<and> Well_order r"
- using exists_minim_Well_order[of "{r. phi r}"] temp1 phi_def by blast
+ 3: "Card_order r \<and> Well_order r"
+ using exists_minim_Well_order[of "{r. phi r}"] temp1 phi_def by blast
let ?A = "Field r" let ?r' = "bsqr r"
have 4: "Well_order ?r' \<and> Field ?r' = ?A \<times> ?A \<and> |?A| =o r"
- using 3 bsqr_Well_order Field_bsqr card_of_Field_ordIso by blast
+ using 3 bsqr_Well_order Field_bsqr card_of_Field_ordIso by blast
have 5: "Card_order |?A \<times> ?A| \<and> Well_order |?A \<times> ?A|"
- using card_of_Card_order card_of_Well_order by blast
- (* *)
+ using card_of_Card_order card_of_Well_order by blast
+ (* *)
have "r <o |?A \<times> ?A|"
- using 1 3 5 ordLess_or_ordLeq unfolding phi_def by blast
+ using 1 3 5 ordLess_or_ordLeq unfolding phi_def by blast
moreover have "|?A \<times> ?A| \<le>o ?r'"
- using card_of_least[of "?A \<times> ?A"] 4 by auto
+ using card_of_least[of "?A \<times> ?A"] 4 by auto
ultimately have "r <o ?r'" using ordLess_ordLeq_trans by auto
then obtain f where 6: "embed r ?r' f" and 7: "\<not> bij_betw f ?A (?A \<times> ?A)"
- unfolding ordLess_def embedS_def[abs_def]
- by (auto simp add: Field_bsqr)
+ unfolding ordLess_def embedS_def[abs_def]
+ by (auto simp add: Field_bsqr)
let ?B = "f ` ?A"
have "|?A| =o |?B|"
- using 3 6 embed_inj_on inj_on_imp_bij_betw card_of_ordIso by blast
+ using 3 6 embed_inj_on inj_on_imp_bij_betw card_of_ordIso by blast
hence 8: "r =o |?B|" using 4 ordIso_transitive ordIso_symmetric by blast
- (* *)
+ (* *)
have "wo_rel.ofilter ?r' ?B"
- using 6 embed_Field_ofilter 3 4 by blast
+ using 6 embed_Field_ofilter 3 4 by blast
hence "wo_rel.ofilter ?r' ?B \<and> ?B \<noteq> ?A \<times> ?A \<and> ?B \<noteq> Field ?r'"
- using 7 unfolding bij_betw_def using 6 3 embed_inj_on 4 by auto
+ using 7 unfolding bij_betw_def using 6 3 embed_inj_on 4 by auto
hence temp2: "wo_rel.ofilter ?r' ?B \<and> ?B < ?A \<times> ?A"
- using 4 wo_rel_def[of ?r'] wo_rel.ofilter_def[of ?r' ?B] by blast
+ using 4 wo_rel_def[of ?r'] wo_rel.ofilter_def[of ?r' ?B] by blast
have "\<not> (\<exists>a. Field r = under r a)"
- using 1 unfolding phi_def using Card_order_infinite_not_under[of r] by auto
+ using 1 unfolding phi_def using Card_order_infinite_not_under[of r] by auto
then obtain A1 where temp3: "wo_rel.ofilter r A1 \<and> A1 < ?A" and 9: "?B \<le> A1 \<times> A1"
- using temp2 3 bsqr_ofilter[of r ?B] by blast
+ using temp2 3 bsqr_ofilter[of r ?B] by blast
hence "|?B| \<le>o |A1 \<times> A1|" using card_of_mono1 by blast
hence 10: "r \<le>o |A1 \<times> A1|" using 8 ordIso_ordLeq_trans by blast
let ?r1 = "Restr r A1"
have "?r1 <o r" using temp3 ofilter_ordLess 3 by blast
moreover
{have "well_order_on A1 ?r1" using 3 temp3 well_order_on_Restr by blast
- hence "|A1| \<le>o ?r1" using 3 Well_order_Restr card_of_least by blast
+ hence "|A1| \<le>o ?r1" using 3 Well_order_Restr card_of_least by blast
}
ultimately have 11: "|A1| <o r" using ordLeq_ordLess_trans by blast
- (* *)
+ (* *)
have "\<not> finite (Field r)" using 1 unfolding phi_def by simp
hence "\<not> finite ?B" using 8 3 card_of_ordIso_finite_Field[of r ?B] by blast
hence "\<not> finite A1" using 9 finite_cartesian_product finite_subset by blast
moreover have temp4: "Field |A1| = A1 \<and> Well_order |A1| \<and> Card_order |A1|"
- using card_of_Card_order[of A1] card_of_Well_order[of A1]
- by (simp add: Field_card_of)
+ using card_of_Card_order[of A1] card_of_Well_order[of A1]
+ by (simp add: Field_card_of)
moreover have "\<not> r \<le>o | A1 |"
- using temp4 11 3 using not_ordLeq_iff_ordLess by blast
+ using temp4 11 3 using not_ordLeq_iff_ordLess by blast
ultimately have "\<not> finite(Field |A1| ) \<and> Card_order |A1| \<and> \<not> r \<le>o | A1 |"
- by (simp add: card_of_card_order_on)
+ by (simp add: card_of_card_order_on)
hence "|Field |A1| \<times> Field |A1| | \<le>o |A1|"
- using 2 unfolding phi_def by blast
+ using 2 unfolding phi_def by blast
hence "|A1 \<times> A1 | \<le>o |A1|" using temp4 by auto
hence "r \<le>o |A1|" using 10 ordLeq_transitive by blast
thus False using 11 not_ordLess_ordLeq by auto
@@ -974,175 +899,175 @@
qed
corollary card_of_Times_same_infinite:
-assumes "\<not>finite A"
-shows "|A \<times> A| =o |A|"
-proof-
+ assumes "\<not>finite A"
+ shows "|A \<times> A| =o |A|"
+proof -
let ?r = "|A|"
have "Field ?r = A \<and> Card_order ?r"
- using Field_card_of card_of_Card_order[of A] by fastforce
+ using Field_card_of card_of_Card_order[of A] by fastforce
hence "|A \<times> A| \<le>o |A|"
- using Card_order_Times_same_infinite[of ?r] assms by auto
+ using Card_order_Times_same_infinite[of ?r] assms by auto
thus ?thesis using card_of_Times3 ordIso_iff_ordLeq by blast
qed
lemma card_of_Times_infinite:
-assumes INF: "\<not>finite A" and NE: "B \<noteq> {}" and LEQ: "|B| \<le>o |A|"
-shows "|A \<times> B| =o |A| \<and> |B \<times> A| =o |A|"
-proof-
+ assumes INF: "\<not>finite A" and NE: "B \<noteq> {}" and LEQ: "|B| \<le>o |A|"
+ shows "|A \<times> B| =o |A| \<and> |B \<times> A| =o |A|"
+proof -
have "|A| \<le>o |A \<times> B| \<and> |A| \<le>o |B \<times> A|"
- using assms by (simp add: card_of_Times1 card_of_Times2)
+ using assms by (simp add: card_of_Times1 card_of_Times2)
moreover
{have "|A \<times> B| \<le>o |A \<times> A| \<and> |B \<times> A| \<le>o |A \<times> A|"
- using LEQ card_of_Times_mono1 card_of_Times_mono2 by blast
- moreover have "|A \<times> A| =o |A|" using INF card_of_Times_same_infinite by blast
- ultimately have "|A \<times> B| \<le>o |A| \<and> |B \<times> A| \<le>o |A|"
- using ordLeq_ordIso_trans[of "|A \<times> B|"] ordLeq_ordIso_trans[of "|B \<times> A|"] by auto
+ using LEQ card_of_Times_mono1 card_of_Times_mono2 by blast
+ moreover have "|A \<times> A| =o |A|" using INF card_of_Times_same_infinite by blast
+ ultimately have "|A \<times> B| \<le>o |A| \<and> |B \<times> A| \<le>o |A|"
+ using ordLeq_ordIso_trans[of "|A \<times> B|"] ordLeq_ordIso_trans[of "|B \<times> A|"] by auto
}
ultimately show ?thesis by (simp add: ordIso_iff_ordLeq)
qed
corollary Card_order_Times_infinite:
-assumes INF: "\<not>finite(Field r)" and CARD: "Card_order r" and
- NE: "Field p \<noteq> {}" and LEQ: "p \<le>o r"
-shows "| (Field r) \<times> (Field p) | =o r \<and> | (Field p) \<times> (Field r) | =o r"
-proof-
+ assumes INF: "\<not>finite(Field r)" and CARD: "Card_order r" and
+ NE: "Field p \<noteq> {}" and LEQ: "p \<le>o r"
+ shows "| (Field r) \<times> (Field p) | =o r \<and> | (Field p) \<times> (Field r) | =o r"
+proof -
have "|Field r \<times> Field p| =o |Field r| \<and> |Field p \<times> Field r| =o |Field r|"
- using assms by (simp add: card_of_Times_infinite card_of_mono2)
+ using assms by (simp add: card_of_Times_infinite card_of_mono2)
thus ?thesis
- using assms card_of_Field_ordIso[of r]
- ordIso_transitive[of "|Field r \<times> Field p|"]
- ordIso_transitive[of _ "|Field r|"] by blast
+ using assms card_of_Field_ordIso by (blast intro: ordIso_transitive)
qed
lemma card_of_Sigma_ordLeq_infinite:
-assumes INF: "\<not>finite B" and
- LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"
-shows "|SIGMA i : I. A i| \<le>o |B|"
-proof(cases "I = {}", simp add: card_of_empty)
- assume *: "I \<noteq> {}"
+ assumes INF: "\<not>finite B" and
+ LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"
+ shows "|SIGMA i : I. A i| \<le>o |B|"
+proof(cases "I = {}")
+ case False
have "|SIGMA i : I. A i| \<le>o |I \<times> B|"
- using card_of_Sigma_mono1[OF LEQ] by blast
+ using card_of_Sigma_mono1[OF LEQ] by blast
moreover have "|I \<times> B| =o |B|"
- using INF * LEQ_I by (auto simp add: card_of_Times_infinite)
+ using INF False LEQ_I by (auto simp add: card_of_Times_infinite)
ultimately show ?thesis using ordLeq_ordIso_trans by blast
-qed
+qed (simp add: card_of_empty)
lemma card_of_Sigma_ordLeq_infinite_Field:
-assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and
- LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"
-shows "|SIGMA i : I. A i| \<le>o r"
-proof-
+ assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and
+ LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"
+ shows "|SIGMA i : I. A i| \<le>o r"
+proof -
let ?B = "Field r"
- have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso
- ordIso_symmetric by blast
+ have 1: "r =o |?B| \<and> |?B| =o r"
+ using r card_of_Field_ordIso ordIso_symmetric by blast
hence "|I| \<le>o |?B|" "\<forall>i \<in> I. |A i| \<le>o |?B|"
- using LEQ_I LEQ ordLeq_ordIso_trans by blast+
+ using LEQ_I LEQ ordLeq_ordIso_trans by blast+
hence "|SIGMA i : I. A i| \<le>o |?B|" using INF LEQ
- card_of_Sigma_ordLeq_infinite by blast
+ card_of_Sigma_ordLeq_infinite by blast
thus ?thesis using 1 ordLeq_ordIso_trans by blast
qed
lemma card_of_Times_ordLeq_infinite_Field:
-"\<lbrakk>\<not>finite (Field r); |A| \<le>o r; |B| \<le>o r; Card_order r\<rbrakk>
- \<Longrightarrow> |A \<times> B| \<le>o r"
-by(simp add: card_of_Sigma_ordLeq_infinite_Field)
+ "\<lbrakk>\<not>finite (Field r); |A| \<le>o r; |B| \<le>o r; Card_order r\<rbrakk> \<Longrightarrow> |A \<times> B| \<le>o r"
+ by(simp add: card_of_Sigma_ordLeq_infinite_Field)
lemma card_of_Times_infinite_simps:
-"\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A \<times> B| =o |A|"
-"\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |A \<times> B|"
-"\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |B \<times> A| =o |A|"
-"\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |B \<times> A|"
-by (auto simp add: card_of_Times_infinite ordIso_symmetric)
+ "\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A \<times> B| =o |A|"
+ "\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |A \<times> B|"
+ "\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |B \<times> A| =o |A|"
+ "\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |B \<times> A|"
+ by (auto simp add: card_of_Times_infinite ordIso_symmetric)
lemma card_of_UNION_ordLeq_infinite:
-assumes INF: "\<not>finite B" and
- LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"
-shows "|\<Union>i \<in> I. A i| \<le>o |B|"
-proof(cases "I = {}", simp add: card_of_empty)
- assume *: "I \<noteq> {}"
+ assumes INF: "\<not>finite B" and LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"
+ shows "|\<Union>i \<in> I. A i| \<le>o |B|"
+proof(cases "I = {}")
+ case False
have "|\<Union>i \<in> I. A i| \<le>o |SIGMA i : I. A i|"
- using card_of_UNION_Sigma by blast
+ using card_of_UNION_Sigma by blast
moreover have "|SIGMA i : I. A i| \<le>o |B|"
- using assms card_of_Sigma_ordLeq_infinite by blast
+ using assms card_of_Sigma_ordLeq_infinite by blast
ultimately show ?thesis using ordLeq_transitive by blast
-qed
+qed (simp add: card_of_empty)
corollary card_of_UNION_ordLeq_infinite_Field:
-assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and
- LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"
-shows "|\<Union>i \<in> I. A i| \<le>o r"
-proof-
+ assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and
+ LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"
+ shows "|\<Union>i \<in> I. A i| \<le>o r"
+proof -
let ?B = "Field r"
- have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso
- ordIso_symmetric by blast
+ have 1: "r =o |?B| \<and> |?B| =o r"
+ using r card_of_Field_ordIso ordIso_symmetric by blast
hence "|I| \<le>o |?B|" "\<forall>i \<in> I. |A i| \<le>o |?B|"
- using LEQ_I LEQ ordLeq_ordIso_trans by blast+
+ using LEQ_I LEQ ordLeq_ordIso_trans by blast+
hence "|\<Union>i \<in> I. A i| \<le>o |?B|" using INF LEQ
- card_of_UNION_ordLeq_infinite by blast
+ card_of_UNION_ordLeq_infinite by blast
thus ?thesis using 1 ordLeq_ordIso_trans by blast
qed
lemma card_of_Plus_infinite1:
-assumes INF: "\<not>finite A" and LEQ: "|B| \<le>o |A|"
-shows "|A <+> B| =o |A|"
-proof(cases "B = {}", simp add: card_of_Plus_empty1 card_of_Plus_empty2 ordIso_symmetric)
+ assumes INF: "\<not>finite A" and LEQ: "|B| \<le>o |A|"
+ shows "|A <+> B| =o |A|"
+proof(cases "B = {}")
+ case True
+ then show ?thesis
+ by (simp add: card_of_Plus_empty1 card_of_Plus_empty2 ordIso_symmetric)
+next
+ case False
let ?Inl = "Inl::'a \<Rightarrow> 'a + 'b" let ?Inr = "Inr::'b \<Rightarrow> 'a + 'b"
assume *: "B \<noteq> {}"
then obtain b1 where 1: "b1 \<in> B" by blast
show ?thesis
proof(cases "B = {b1}")
- assume Case1: "B = {b1}"
+ case True
have 2: "bij_betw ?Inl A ((?Inl ` A))"
- unfolding bij_betw_def inj_on_def by auto
+ unfolding bij_betw_def inj_on_def by auto
hence 3: "\<not>finite (?Inl ` A)"
- using INF bij_betw_finite[of ?Inl A] by blast
+ using INF bij_betw_finite[of ?Inl A] by blast
let ?A' = "?Inl ` A \<union> {?Inr b1}"
obtain g where "bij_betw g (?Inl ` A) ?A'"
- using 3 infinite_imp_bij_betw2[of "?Inl ` A"] by auto
- moreover have "?A' = A <+> B" using Case1 by blast
+ using 3 infinite_imp_bij_betw2[of "?Inl ` A"] by auto
+ moreover have "?A' = A <+> B" using True by blast
ultimately have "bij_betw g (?Inl ` A) (A <+> B)" by simp
hence "bij_betw (g \<circ> ?Inl) A (A <+> B)"
- using 2 by (auto simp add: bij_betw_trans)
+ using 2 by (auto simp add: bij_betw_trans)
thus ?thesis using card_of_ordIso ordIso_symmetric by blast
next
- assume Case2: "B \<noteq> {b1}"
+ case False
with * 1 obtain b2 where 3: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B" by fastforce
obtain f where "inj_on f B \<and> f ` B \<le> A"
- using LEQ card_of_ordLeq[of B] by fastforce
+ using LEQ card_of_ordLeq[of B] by fastforce
with 3 have "f b1 \<noteq> f b2 \<and> {f b1, f b2} \<le> A"
- unfolding inj_on_def by auto
+ unfolding inj_on_def by auto
with 3 have "|A <+> B| \<le>o |A \<times> B|"
- by (auto simp add: card_of_Plus_Times)
+ by (auto simp add: card_of_Plus_Times)
moreover have "|A \<times> B| =o |A|"
- using assms * by (simp add: card_of_Times_infinite_simps)
+ using assms * by (simp add: card_of_Times_infinite_simps)
ultimately have "|A <+> B| \<le>o |A|" using ordLeq_ordIso_trans by blast
thus ?thesis using card_of_Plus1 ordIso_iff_ordLeq by blast
qed
qed
lemma card_of_Plus_infinite2:
-assumes INF: "\<not>finite A" and LEQ: "|B| \<le>o |A|"
-shows "|B <+> A| =o |A|"
-using assms card_of_Plus_commute card_of_Plus_infinite1
-ordIso_equivalence by blast
+ assumes INF: "\<not>finite A" and LEQ: "|B| \<le>o |A|"
+ shows "|B <+> A| =o |A|"
+ using assms card_of_Plus_commute card_of_Plus_infinite1
+ ordIso_equivalence by blast
lemma card_of_Plus_infinite:
-assumes INF: "\<not>finite A" and LEQ: "|B| \<le>o |A|"
-shows "|A <+> B| =o |A| \<and> |B <+> A| =o |A|"
-using assms by (auto simp: card_of_Plus_infinite1 card_of_Plus_infinite2)
+ assumes INF: "\<not>finite A" and LEQ: "|B| \<le>o |A|"
+ shows "|A <+> B| =o |A| \<and> |B <+> A| =o |A|"
+ using assms by (auto simp: card_of_Plus_infinite1 card_of_Plus_infinite2)
corollary Card_order_Plus_infinite:
-assumes INF: "\<not>finite(Field r)" and CARD: "Card_order r" and
- LEQ: "p \<le>o r"
-shows "| (Field r) <+> (Field p) | =o r \<and> | (Field p) <+> (Field r) | =o r"
-proof-
+ assumes INF: "\<not>finite(Field r)" and CARD: "Card_order r" and
+ LEQ: "p \<le>o r"
+ shows "| (Field r) <+> (Field p) | =o r \<and> | (Field p) <+> (Field r) | =o r"
+proof -
have "| Field r <+> Field p | =o | Field r | \<and>
| Field p <+> Field r | =o | Field r |"
- using assms by (simp add: card_of_Plus_infinite card_of_mono2)
+ using assms by (simp add: card_of_Plus_infinite card_of_mono2)
thus ?thesis
- using assms card_of_Field_ordIso[of r]
- ordIso_transitive[of "|Field r <+> Field p|"]
- ordIso_transitive[of _ "|Field r|"] by blast
+ using assms card_of_Field_ordIso by (blast intro: ordIso_transitive)
+
qed
@@ -1158,110 +1083,111 @@
definition "(natLess::(nat * nat) set) \<equiv> {(x,y). x < y}"
abbreviation natLeq_on :: "nat \<Rightarrow> (nat * nat) set"
-where "natLeq_on n \<equiv> {(x,y). x < n \<and> y < n \<and> x \<le> y}"
+ where "natLeq_on n \<equiv> {(x,y). x < n \<and> y < n \<and> x \<le> y}"
lemma infinite_cartesian_product:
-assumes "\<not>finite A" "\<not>finite B"
-shows "\<not>finite (A \<times> B)"
-proof
- assume "finite (A \<times> B)"
- from assms(1) have "A \<noteq> {}" by auto
- with \<open>finite (A \<times> B)\<close> have "finite B" using finite_cartesian_productD2 by auto
- with assms(2) show False by simp
-qed
+ assumes "\<not>finite A" "\<not>finite B"
+ shows "\<not>finite (A \<times> B)"
+using assms finite_cartesian_productD2 by auto
subsubsection \<open>First as well-orders\<close>
lemma Field_natLeq: "Field natLeq = (UNIV::nat set)"
-by(unfold Field_def natLeq_def, auto)
+ by(unfold Field_def natLeq_def, auto)
lemma natLeq_Refl: "Refl natLeq"
-unfolding refl_on_def Field_def natLeq_def by auto
+ unfolding refl_on_def Field_def natLeq_def by auto
lemma natLeq_trans: "trans natLeq"
-unfolding trans_def natLeq_def by auto
+ unfolding trans_def natLeq_def by auto
lemma natLeq_Preorder: "Preorder natLeq"
-unfolding preorder_on_def
-by (auto simp add: natLeq_Refl natLeq_trans)
+ unfolding preorder_on_def
+ by (auto simp add: natLeq_Refl natLeq_trans)
lemma natLeq_antisym: "antisym natLeq"
-unfolding antisym_def natLeq_def by auto
+ unfolding antisym_def natLeq_def by auto
lemma natLeq_Partial_order: "Partial_order natLeq"
-unfolding partial_order_on_def
-by (auto simp add: natLeq_Preorder natLeq_antisym)
+ unfolding partial_order_on_def
+ by (auto simp add: natLeq_Preorder natLeq_antisym)
lemma natLeq_Total: "Total natLeq"
-unfolding total_on_def natLeq_def by auto
+ unfolding total_on_def natLeq_def by auto
lemma natLeq_Linear_order: "Linear_order natLeq"
-unfolding linear_order_on_def
-by (auto simp add: natLeq_Partial_order natLeq_Total)
+ unfolding linear_order_on_def
+ by (auto simp add: natLeq_Partial_order natLeq_Total)
lemma natLeq_natLess_Id: "natLess = natLeq - Id"
-unfolding natLeq_def natLess_def by auto
+ unfolding natLeq_def natLess_def by auto
lemma natLeq_Well_order: "Well_order natLeq"
-unfolding well_order_on_def
-using natLeq_Linear_order wf_less natLeq_natLess_Id natLeq_def natLess_def by auto
+ unfolding well_order_on_def
+ using natLeq_Linear_order wf_less natLeq_natLess_Id natLeq_def natLess_def by auto
lemma Field_natLeq_on: "Field (natLeq_on n) = {x. x < n}"
-unfolding Field_def by auto
+ unfolding Field_def by auto
lemma natLeq_underS_less: "underS natLeq n = {x. x < n}"
-unfolding underS_def natLeq_def by auto
+ unfolding underS_def natLeq_def by auto
lemma Restr_natLeq: "Restr natLeq {x. x < n} = natLeq_on n"
-unfolding natLeq_def by force
+ unfolding natLeq_def by force
lemma Restr_natLeq2:
-"Restr natLeq (underS natLeq n) = natLeq_on n"
-by (auto simp add: Restr_natLeq natLeq_underS_less)
+ "Restr natLeq (underS natLeq n) = natLeq_on n"
+ by (auto simp add: Restr_natLeq natLeq_underS_less)
lemma natLeq_on_Well_order: "Well_order(natLeq_on n)"
-using Restr_natLeq[of n] natLeq_Well_order
- Well_order_Restr[of natLeq "{x. x < n}"] by auto
+ using Restr_natLeq[of n] natLeq_Well_order
+ Well_order_Restr[of natLeq "{x. x < n}"] by auto
corollary natLeq_on_well_order_on: "well_order_on {x. x < n} (natLeq_on n)"
-using natLeq_on_Well_order Field_natLeq_on by auto
+ using natLeq_on_Well_order Field_natLeq_on by auto
lemma natLeq_on_wo_rel: "wo_rel(natLeq_on n)"
-unfolding wo_rel_def using natLeq_on_Well_order .
+ unfolding wo_rel_def using natLeq_on_Well_order .
subsubsection \<open>Then as cardinals\<close>
lemma natLeq_Card_order: "Card_order natLeq"
-proof(auto simp add: natLeq_Well_order
- Card_order_iff_Restr_underS Restr_natLeq2, simp add: Field_natLeq)
- fix n have "finite(Field (natLeq_on n))" by (auto simp: Field_def)
- moreover have "\<not>finite(UNIV::nat set)" by auto
- ultimately show "natLeq_on n <o |UNIV::nat set|"
- using finite_ordLess_infinite[of "natLeq_on n" "|UNIV::nat set|"]
- Field_card_of[of "UNIV::nat set"]
- card_of_Well_order[of "UNIV::nat set"] natLeq_on_Well_order[of n] by auto
+proof -
+ have "natLeq_on n <o |UNIV::nat set|" for n
+ proof -
+ have "finite(Field (natLeq_on n))" by (auto simp: Field_def)
+ moreover have "\<not>finite(UNIV::nat set)" by auto
+ ultimately show ?thesis
+ using finite_ordLess_infinite[of "natLeq_on n" "|UNIV::nat set|"]
+ card_of_Well_order[of "UNIV::nat set"] natLeq_on_Well_order
+ by (force simp add: Field_card_of)
+ qed
+ then show ?thesis
+ apply (simp add: natLeq_Well_order Card_order_iff_Restr_underS Restr_natLeq2)
+ apply (force simp add: Field_natLeq)
+ done
qed
corollary card_of_Field_natLeq:
-"|Field natLeq| =o natLeq"
-using Field_natLeq natLeq_Card_order Card_order_iff_ordIso_card_of[of natLeq]
- ordIso_symmetric[of natLeq] by blast
+ "|Field natLeq| =o natLeq"
+ using Field_natLeq natLeq_Card_order Card_order_iff_ordIso_card_of[of natLeq]
+ ordIso_symmetric[of natLeq] by blast
corollary card_of_nat:
-"|UNIV::nat set| =o natLeq"
-using Field_natLeq card_of_Field_natLeq by auto
+ "|UNIV::nat set| =o natLeq"
+ using Field_natLeq card_of_Field_natLeq by auto
corollary infinite_iff_natLeq_ordLeq:
-"\<not>finite A = ( natLeq \<le>o |A| )"
-using infinite_iff_card_of_nat[of A] card_of_nat
- ordIso_ordLeq_trans ordLeq_ordIso_trans ordIso_symmetric by blast
+ "\<not>finite A = ( natLeq \<le>o |A| )"
+ using infinite_iff_card_of_nat[of A] card_of_nat
+ ordIso_ordLeq_trans ordLeq_ordIso_trans ordIso_symmetric by blast
corollary finite_iff_ordLess_natLeq:
-"finite A = ( |A| <o natLeq)"
-using infinite_iff_natLeq_ordLeq not_ordLeq_iff_ordLess
- card_of_Well_order natLeq_Well_order by blast
+ "finite A = ( |A| <o natLeq)"
+ using infinite_iff_natLeq_ordLeq not_ordLeq_iff_ordLess
+ card_of_Well_order natLeq_Well_order by blast
subsection \<open>The successor of a cardinal\<close>
@@ -1271,169 +1197,167 @@
not require \<open>r\<close> to be a cardinal, only this case will be meaningful.\<close>
definition isCardSuc :: "'a rel \<Rightarrow> 'a set rel \<Rightarrow> bool"
-where
-"isCardSuc r r' \<equiv>
- Card_order r' \<and> r <o r' \<and>
- (\<forall>(r''::'a set rel). Card_order r'' \<and> r <o r'' \<longrightarrow> r' \<le>o r'')"
+ where
+ "isCardSuc r r' \<equiv>
+ Card_order r' \<and> r <o r' \<and>
+ (\<forall>(r''::'a set rel). Card_order r'' \<and> r <o r'' \<longrightarrow> r' \<le>o r'')"
text\<open>Now we introduce the cardinal-successor operator \<open>cardSuc\<close>,
by picking {\em some} cardinal-order relation fulfilling \<open>isCardSuc\<close>.
Again, the picked item shall be proved unique up to order-isomorphism.\<close>
definition cardSuc :: "'a rel \<Rightarrow> 'a set rel"
-where
-"cardSuc r \<equiv> SOME r'. isCardSuc r r'"
+ where "cardSuc r \<equiv> SOME r'. isCardSuc r r'"
lemma exists_minim_Card_order:
-"\<lbrakk>R \<noteq> {}; \<forall>r \<in> R. Card_order r\<rbrakk> \<Longrightarrow> \<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
-unfolding card_order_on_def using exists_minim_Well_order by blast
+ "\<lbrakk>R \<noteq> {}; \<forall>r \<in> R. Card_order r\<rbrakk> \<Longrightarrow> \<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
+ unfolding card_order_on_def using exists_minim_Well_order by blast
lemma exists_isCardSuc:
-assumes "Card_order r"
-shows "\<exists>r'. isCardSuc r r'"
-proof-
+ assumes "Card_order r"
+ shows "\<exists>r'. isCardSuc r r'"
+proof -
let ?R = "{(r'::'a set rel). Card_order r' \<and> r <o r'}"
have "|Pow(Field r)| \<in> ?R \<and> (\<forall>r \<in> ?R. Card_order r)" using assms
- by (simp add: card_of_Card_order Card_order_Pow)
+ by (simp add: card_of_Card_order Card_order_Pow)
then obtain r where "r \<in> ?R \<and> (\<forall>r' \<in> ?R. r \<le>o r')"
- using exists_minim_Card_order[of ?R] by blast
+ using exists_minim_Card_order[of ?R] by blast
thus ?thesis unfolding isCardSuc_def by auto
qed
lemma cardSuc_isCardSuc:
-assumes "Card_order r"
-shows "isCardSuc r (cardSuc r)"
-unfolding cardSuc_def using assms
-by (simp add: exists_isCardSuc someI_ex)
+ assumes "Card_order r"
+ shows "isCardSuc r (cardSuc r)"
+ unfolding cardSuc_def using assms
+ by (simp add: exists_isCardSuc someI_ex)
lemma cardSuc_Card_order:
-"Card_order r \<Longrightarrow> Card_order(cardSuc r)"
-using cardSuc_isCardSuc unfolding isCardSuc_def by blast
+ "Card_order r \<Longrightarrow> Card_order(cardSuc r)"
+ using cardSuc_isCardSuc unfolding isCardSuc_def by blast
lemma cardSuc_greater:
-"Card_order r \<Longrightarrow> r <o cardSuc r"
-using cardSuc_isCardSuc unfolding isCardSuc_def by blast
+ "Card_order r \<Longrightarrow> r <o cardSuc r"
+ using cardSuc_isCardSuc unfolding isCardSuc_def by blast
lemma cardSuc_ordLeq:
-"Card_order r \<Longrightarrow> r \<le>o cardSuc r"
-using cardSuc_greater ordLeq_iff_ordLess_or_ordIso by blast
+ "Card_order r \<Longrightarrow> r \<le>o cardSuc r"
+ using cardSuc_greater ordLeq_iff_ordLess_or_ordIso by blast
text\<open>The minimality property of \<open>cardSuc\<close> originally present in its definition
is local to the type \<open>'a set rel\<close>, i.e., that of \<open>cardSuc r\<close>:\<close>
lemma cardSuc_least_aux:
-"\<lbrakk>Card_order (r::'a rel); Card_order (r'::'a set rel); r <o r'\<rbrakk> \<Longrightarrow> cardSuc r \<le>o r'"
-using cardSuc_isCardSuc unfolding isCardSuc_def by blast
+ "\<lbrakk>Card_order (r::'a rel); Card_order (r'::'a set rel); r <o r'\<rbrakk> \<Longrightarrow> cardSuc r \<le>o r'"
+ using cardSuc_isCardSuc unfolding isCardSuc_def by blast
text\<open>But from this we can infer general minimality:\<close>
lemma cardSuc_least:
-assumes CARD: "Card_order r" and CARD': "Card_order r'" and LESS: "r <o r'"
-shows "cardSuc r \<le>o r'"
-proof-
+ assumes CARD: "Card_order r" and CARD': "Card_order r'" and LESS: "r <o r'"
+ shows "cardSuc r \<le>o r'"
+proof -
let ?p = "cardSuc r"
have 0: "Well_order ?p \<and> Well_order r'"
- using assms cardSuc_Card_order unfolding card_order_on_def by blast
- {assume "r' <o ?p"
- then obtain r'' where 1: "Field r'' < Field ?p" and 2: "r' =o r'' \<and> r'' <o ?p"
- using internalize_ordLess[of r' ?p] by blast
- (* *)
- have "Card_order r''" using CARD' Card_order_ordIso2 2 by blast
- moreover have "r <o r''" using LESS 2 ordLess_ordIso_trans by blast
- ultimately have "?p \<le>o r''" using cardSuc_least_aux CARD by blast
- hence False using 2 not_ordLess_ordLeq by blast
+ using assms cardSuc_Card_order unfolding card_order_on_def by blast
+ { assume "r' <o ?p"
+ then obtain r'' where 1: "Field r'' < Field ?p" and 2: "r' =o r'' \<and> r'' <o ?p"
+ using internalize_ordLess[of r' ?p] by blast
+ (* *)
+ have "Card_order r''" using CARD' Card_order_ordIso2 2 by blast
+ moreover have "r <o r''" using LESS 2 ordLess_ordIso_trans by blast
+ ultimately have "?p \<le>o r''" using cardSuc_least_aux CARD by blast
+ hence False using 2 not_ordLess_ordLeq by blast
}
thus ?thesis using 0 ordLess_or_ordLeq by blast
qed
lemma cardSuc_ordLess_ordLeq:
-assumes CARD: "Card_order r" and CARD': "Card_order r'"
-shows "(r <o r') = (cardSuc r \<le>o r')"
-proof(auto simp add: assms cardSuc_least)
- assume "cardSuc r \<le>o r'"
- thus "r <o r'" using assms cardSuc_greater ordLess_ordLeq_trans by blast
-qed
+ assumes CARD: "Card_order r" and CARD': "Card_order r'"
+ shows "(r <o r') = (cardSuc r \<le>o r')"
+proof
+ show "cardSuc r \<le>o r' \<Longrightarrow> r <o r'"
+ using assms cardSuc_greater ordLess_ordLeq_trans by blast
+qed (auto simp add: assms cardSuc_least)
lemma cardSuc_ordLeq_ordLess:
-assumes CARD: "Card_order r" and CARD': "Card_order r'"
-shows "(r' <o cardSuc r) = (r' \<le>o r)"
-proof-
+ assumes CARD: "Card_order r" and CARD': "Card_order r'"
+ shows "(r' <o cardSuc r) = (r' \<le>o r)"
+proof -
have "Well_order r \<and> Well_order r'"
- using assms unfolding card_order_on_def by auto
+ using assms unfolding card_order_on_def by auto
moreover have "Well_order(cardSuc r)"
- using assms cardSuc_Card_order card_order_on_def by blast
+ using assms cardSuc_Card_order card_order_on_def by blast
ultimately show ?thesis
- using assms cardSuc_ordLess_ordLeq[of r r']
- not_ordLeq_iff_ordLess[of r r'] not_ordLeq_iff_ordLess[of r' "cardSuc r"] by blast
+ using assms cardSuc_ordLess_ordLeq by (blast dest: not_ordLeq_iff_ordLess)
qed
lemma cardSuc_mono_ordLeq:
-assumes CARD: "Card_order r" and CARD': "Card_order r'"
-shows "(cardSuc r \<le>o cardSuc r') = (r \<le>o r')"
-using assms cardSuc_ordLeq_ordLess cardSuc_ordLess_ordLeq cardSuc_Card_order by blast
+ assumes CARD: "Card_order r" and CARD': "Card_order r'"
+ shows "(cardSuc r \<le>o cardSuc r') = (r \<le>o r')"
+ using assms cardSuc_ordLeq_ordLess cardSuc_ordLess_ordLeq cardSuc_Card_order by blast
lemma cardSuc_invar_ordIso:
-assumes CARD: "Card_order r" and CARD': "Card_order r'"
-shows "(cardSuc r =o cardSuc r') = (r =o r')"
-proof-
+ 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 (simp add: card_order_on_well_order_on cardSuc_Card_order)
+ using assms by (simp add: card_order_on_well_order_on cardSuc_Card_order)
thus ?thesis
- using ordIso_iff_ordLeq[of r r'] ordIso_iff_ordLeq
- using cardSuc_mono_ordLeq[of r r'] cardSuc_mono_ordLeq[of r' r] assms by blast
+ using ordIso_iff_ordLeq[of r r'] ordIso_iff_ordLeq
+ using cardSuc_mono_ordLeq[of r r'] cardSuc_mono_ordLeq[of r' r] assms by blast
qed
lemma card_of_cardSuc_finite:
-"finite(Field(cardSuc |A| )) = finite A"
+ "finite(Field(cardSuc |A| )) = finite A"
proof
assume *: "finite (Field (cardSuc |A| ))"
have 0: "|Field(cardSuc |A| )| =o cardSuc |A|"
- using card_of_Card_order cardSuc_Card_order card_of_Field_ordIso by blast
+ using card_of_Card_order cardSuc_Card_order card_of_Field_ordIso by blast
hence "|A| \<le>o |Field(cardSuc |A| )|"
- using card_of_Card_order[of A] cardSuc_ordLeq[of "|A|"] ordIso_symmetric
- ordLeq_ordIso_trans by blast
+ using card_of_Card_order[of A] cardSuc_ordLeq[of "|A|"] ordIso_symmetric
+ ordLeq_ordIso_trans by blast
thus "finite A" using * card_of_ordLeq_finite by blast
next
assume "finite A"
then have "finite ( Field |Pow A| )" unfolding Field_card_of by simp
- then show "finite (Field (cardSuc |A| ))"
- proof (rule card_of_ordLeq_finite[OF card_of_mono2, rotated])
- show "cardSuc |A| \<le>o |Pow A|"
- by (rule iffD1[OF cardSuc_ordLess_ordLeq card_of_Pow]) (simp_all add: card_of_Card_order)
- qed
+ moreover
+ have "cardSuc |A| \<le>o |Pow A|"
+ by (rule iffD1[OF cardSuc_ordLess_ordLeq card_of_Pow]) (simp_all add: card_of_Card_order)
+ ultimately show "finite (Field (cardSuc |A| ))"
+ by (blast intro: card_of_ordLeq_finite card_of_mono2)
qed
lemma cardSuc_finite:
-assumes "Card_order r"
-shows "finite (Field (cardSuc r)) = finite (Field r)"
-proof-
+ assumes "Card_order r"
+ shows "finite (Field (cardSuc r)) = finite (Field r)"
+proof -
let ?A = "Field r"
have "|?A| =o r" using assms by (simp add: card_of_Field_ordIso)
hence "cardSuc |?A| =o cardSuc r" using assms
- by (simp add: card_of_Card_order cardSuc_invar_ordIso)
+ by (simp add: card_of_Card_order cardSuc_invar_ordIso)
moreover have "|Field (cardSuc |?A| ) | =o cardSuc |?A|"
- by (simp add: card_of_card_order_on Field_card_of card_of_Field_ordIso cardSuc_Card_order)
+ by (simp add: card_of_card_order_on Field_card_of card_of_Field_ordIso cardSuc_Card_order)
moreover
- {have "|Field (cardSuc r) | =o cardSuc r"
- using assms by (simp add: card_of_Field_ordIso cardSuc_Card_order)
- hence "cardSuc r =o |Field (cardSuc r) |"
- using ordIso_symmetric by blast
+ { have "|Field (cardSuc r) | =o cardSuc r"
+ using assms by (simp add: card_of_Field_ordIso cardSuc_Card_order)
+ hence "cardSuc r =o |Field (cardSuc r) |"
+ using ordIso_symmetric by blast
}
ultimately have "|Field (cardSuc |?A| ) | =o |Field (cardSuc r) |"
- using ordIso_transitive by blast
+ using ordIso_transitive by blast
hence "finite (Field (cardSuc |?A| )) = finite (Field (cardSuc r))"
- using card_of_ordIso_finite by blast
+ using card_of_ordIso_finite by blast
thus ?thesis by (simp only: card_of_cardSuc_finite)
qed
lemma Field_cardSuc_not_empty:
-assumes "Card_order r"
-shows "Field (cardSuc r) \<noteq> {}"
+ assumes "Card_order r"
+ shows "Field (cardSuc r) \<noteq> {}"
proof
assume "Field(cardSuc r) = {}"
then have "|Field(cardSuc r)| \<le>o r" using assms Card_order_empty[of r] by auto
then have "cardSuc r \<le>o r" using assms card_of_Field_ordIso
- cardSuc_Card_order ordIso_symmetric ordIso_ordLeq_trans by blast
+ cardSuc_Card_order ordIso_symmetric ordIso_ordLeq_trans by blast
then show False using cardSuc_greater not_ordLess_ordLeq assms by blast
qed
@@ -1487,265 +1411,263 @@
using ordLess_ordIso_trans[OF cardSuc_greater cardSuc_ordIso_card_suc] card_order_on_Card_order by blast
lemma card_of_Plus_ordLess_infinite:
-assumes INF: "\<not>finite C" and
- LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|"
-shows "|A <+> B| <o |C|"
+ assumes INF: "\<not>finite C" and LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|"
+ shows "|A <+> B| <o |C|"
proof(cases "A = {} \<or> B = {}")
- assume Case1: "A = {} \<or> B = {}"
+ case True
hence "|A| =o |A <+> B| \<or> |B| =o |A <+> B|"
- using card_of_Plus_empty1 card_of_Plus_empty2 by blast
+ using card_of_Plus_empty1 card_of_Plus_empty2 by blast
hence "|A <+> B| =o |A| \<or> |A <+> B| =o |B|"
- using ordIso_symmetric[of "|A|"] ordIso_symmetric[of "|B|"] by blast
+ using ordIso_symmetric[of "|A|"] ordIso_symmetric[of "|B|"] by blast
thus ?thesis using LESS1 LESS2
- ordIso_ordLess_trans[of "|A <+> B|" "|A|"]
- ordIso_ordLess_trans[of "|A <+> B|" "|B|"] by blast
+ ordIso_ordLess_trans[of "|A <+> B|" "|A|"]
+ ordIso_ordLess_trans[of "|A <+> B|" "|B|"] by blast
next
- assume Case2: "\<not>(A = {} \<or> B = {})"
- {assume *: "|C| \<le>o |A <+> B|"
- hence "\<not>finite (A <+> B)" using INF card_of_ordLeq_finite by blast
- hence 1: "\<not>finite A \<or> \<not>finite B" using finite_Plus by blast
- {assume Case21: "|A| \<le>o |B|"
- hence "\<not>finite B" using 1 card_of_ordLeq_finite by blast
- hence "|A <+> B| =o |B|" using Case2 Case21
- by (auto simp add: card_of_Plus_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 <+> B| =o |A|" using Case2 Case22
- by (auto simp add: card_of_Plus_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 <+> B|" "|C|"]
- card_of_Well_order[of "A <+> B"] card_of_Well_order[of "C"] by auto
+ case False
+ have False if "|C| \<le>o |A <+> B|"
+ proof -
+ have \<section>: "\<not>finite A \<or> \<not>finite B"
+ using that INF card_of_ordLeq_finite finite_Plus by blast
+ consider "|A| \<le>o |B|" | "|B| \<le>o |A|"
+ using ordLeq_total [OF card_of_Well_order card_of_Well_order] by blast
+ then show False
+ proof cases
+ case 1
+ hence "\<not>finite B" using \<section> card_of_ordLeq_finite by blast
+ hence "|A <+> B| =o |B|" using False 1
+ by (auto simp add: card_of_Plus_infinite)
+ thus False using LESS2 not_ordLess_ordLeq that ordLeq_ordIso_trans by blast
+ next
+ case 2
+ hence "\<not>finite A" using \<section> card_of_ordLeq_finite by blast
+ hence "|A <+> B| =o |A|" using False 2
+ by (auto simp add: card_of_Plus_infinite)
+ thus False using LESS1 not_ordLess_ordLeq that ordLeq_ordIso_trans by blast
+ qed
+ qed
+ thus ?thesis
+ using ordLess_or_ordLeq[of "|A <+> B|" "|C|"]
+ card_of_Well_order[of "A <+> B"] card_of_Well_order[of "C"] by auto
qed
lemma card_of_Plus_ordLess_infinite_Field:
-assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and
- LESS1: "|A| <o r" and LESS2: "|B| <o r"
-shows "|A <+> B| <o r"
-proof-
+ assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and
+ LESS1: "|A| <o r" and LESS2: "|B| <o r"
+ shows "|A <+> 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
+ 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+
+ using LESS1 LESS2 ordLess_ordIso_trans by blast+
hence "|A <+> B| <o |?C|" using INF
- card_of_Plus_ordLess_infinite by blast
+ card_of_Plus_ordLess_infinite by blast
thus ?thesis using 1 ordLess_ordIso_trans by blast
qed
lemma card_of_Plus_ordLeq_infinite_Field:
-assumes r: "\<not>finite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"
-and c: "Card_order r"
-shows "|A <+> B| \<le>o r"
-proof-
+ assumes r: "\<not>finite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"
+ and c: "Card_order r"
+ shows "|A <+> B| \<le>o r"
+proof -
let ?r' = "cardSuc r"
have "Card_order ?r' \<and> \<not>finite (Field ?r')" using assms
- by (simp add: cardSuc_Card_order cardSuc_finite)
+ by (simp add: cardSuc_Card_order cardSuc_finite)
moreover have "|A| <o ?r'" and "|B| <o ?r'" using A B c
- by (auto simp: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)
+ by (auto simp: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)
ultimately have "|A <+> B| <o ?r'"
- using card_of_Plus_ordLess_infinite_Field by blast
+ using card_of_Plus_ordLess_infinite_Field by blast
thus ?thesis using c r
- by (simp add: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)
+ by (simp add: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)
qed
lemma card_of_Un_ordLeq_infinite_Field:
-assumes C: "\<not>finite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"
-and "Card_order r"
-shows "|A Un B| \<le>o r"
-using assms card_of_Plus_ordLeq_infinite_Field card_of_Un_Plus_ordLeq
-ordLeq_transitive by fast
+ assumes C: "\<not>finite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"
+ and "Card_order r"
+ shows "|A Un B| \<le>o r"
+ using assms card_of_Plus_ordLeq_infinite_Field card_of_Un_Plus_ordLeq
+ ordLeq_transitive by fast
lemma card_of_Un_ordLess_infinite:
-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
+ 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:
-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-
+ 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
+ ordIso_symmetric by blast
hence "|A| <o |?C|" "|B| <o |?C|"
- using LESS1 LESS2 ordLess_ordIso_trans by blast+
+ using LESS1 LESS2 ordLess_ordIso_trans by blast+
hence "|A Un B| <o |?C|" using INF
- card_of_Un_ordLess_infinite by blast
+ card_of_Un_ordLess_infinite by blast
thus ?thesis using 1 ordLess_ordIso_trans by blast
qed
subsection \<open>Regular cardinals\<close>
definition cofinal where
-"cofinal A r \<equiv>
- \<forall>a \<in> Field r. \<exists>b \<in> A. a \<noteq> b \<and> (a,b) \<in> r"
+ "cofinal A r \<equiv> \<forall>a \<in> Field r. \<exists>b \<in> A. a \<noteq> b \<and> (a,b) \<in> r"
definition regularCard where
-"regularCard r \<equiv>
- \<forall>K. K \<le> Field r \<and> cofinal K r \<longrightarrow> |K| =o r"
+ "regularCard r \<equiv> \<forall>K. K \<le> Field r \<and> cofinal K r \<longrightarrow> |K| =o r"
definition relChain where
-"relChain r As \<equiv>
- \<forall>i j. (i,j) \<in> r \<longrightarrow> As i \<le> As j"
+ "relChain r As \<equiv> \<forall>i j. (i,j) \<in> r \<longrightarrow> As i \<le> As j"
lemma regularCard_UNION:
-assumes r: "Card_order r" "regularCard r"
-and As: "relChain r As"
-and Bsub: "B \<le> (\<Union>i \<in> Field r. As i)"
-and cardB: "|B| <o r"
-shows "\<exists>i \<in> Field r. B \<le> As i"
-proof-
+ assumes r: "Card_order r" "regularCard r"
+ and As: "relChain r As"
+ and Bsub: "B \<le> (\<Union>i \<in> Field r. As i)"
+ and cardB: "|B| <o r"
+ shows "\<exists>i \<in> Field r. B \<le> As i"
+proof -
let ?phi = "\<lambda>b j. j \<in> Field r \<and> b \<in> As j"
have "\<forall>b\<in>B. \<exists>j. ?phi b j" using Bsub by blast
then obtain f where f: "\<And>b. b \<in> B \<Longrightarrow> ?phi b (f b)"
- using bchoice[of B ?phi] by blast
+ using bchoice[of B ?phi] by blast
let ?K = "f ` B"
{assume 1: "\<And>i. i \<in> Field r \<Longrightarrow> \<not> B \<le> As i"
- have 2: "cofinal ?K r"
- unfolding cofinal_def proof auto
- fix i assume i: "i \<in> Field r"
- with 1 obtain b where b: "b \<in> B \<and> b \<notin> As i" by blast
- hence "i \<noteq> f b \<and> \<not> (f b,i) \<in> r"
- using As f unfolding relChain_def by auto
- hence "i \<noteq> f b \<and> (i, f b) \<in> r" using r
- unfolding card_order_on_def well_order_on_def linear_order_on_def
- total_on_def using i f b by auto
- with b show "\<exists>b\<in>B. i \<noteq> f b \<and> (i, f b) \<in> r" by blast
- qed
- moreover have "?K \<le> Field r" using f by blast
- ultimately have "|?K| =o r" using 2 r unfolding regularCard_def by blast
- moreover
- {
- have "|?K| <=o |B|" using card_of_image .
- hence "|?K| <o r" using cardB ordLeq_ordLess_trans by blast
- }
- ultimately have False using not_ordLess_ordIso by blast
+ have 2: "cofinal ?K r"
+ unfolding cofinal_def
+ proof (intro strip)
+ fix i assume i: "i \<in> Field r"
+ with 1 obtain b where b: "b \<in> B \<and> b \<notin> As i" by blast
+ hence "i \<noteq> f b \<and> \<not> (f b,i) \<in> r"
+ using As f unfolding relChain_def by auto
+ hence "i \<noteq> f b \<and> (i, f b) \<in> r" using r
+ unfolding card_order_on_def well_order_on_def linear_order_on_def
+ total_on_def using i f b by auto
+ with b show "\<exists>b \<in> f`B. i \<noteq> b \<and> (i,b) \<in> r" by blast
+ qed
+ moreover have "?K \<le> Field r" using f by blast
+ ultimately have "|?K| =o r" using 2 r unfolding regularCard_def by blast
+ moreover
+ have "|?K| <o r" using cardB ordLeq_ordLess_trans card_of_image by blast
+ ultimately have False using not_ordLess_ordIso by blast
}
thus ?thesis by blast
qed
lemma infinite_cardSuc_regularCard:
-assumes r_inf: "\<not>finite (Field r)" and r_card: "Card_order r"
-shows "regularCard (cardSuc r)"
-proof-
+ assumes r_inf: "\<not>finite (Field r)" and r_card: "Card_order r"
+ shows "regularCard (cardSuc r)"
+proof -
let ?r' = "cardSuc r"
- have r': "Card_order ?r'"
- "!! p. Card_order p \<longrightarrow> (p \<le>o r) = (p <o ?r')"
- using r_card by (auto simp: cardSuc_Card_order cardSuc_ordLeq_ordLess)
+ have r': "Card_order ?r'" "\<And>p. Card_order p \<longrightarrow> (p \<le>o r) = (p <o ?r')"
+ using r_card by (auto simp: cardSuc_Card_order cardSuc_ordLeq_ordLess)
show ?thesis
- unfolding regularCard_def proof auto
+ unfolding regularCard_def proof auto
fix K assume 1: "K \<le> Field ?r'" and 2: "cofinal K ?r'"
hence "|K| \<le>o |Field ?r'|" by (simp only: card_of_mono1)
also have 22: "|Field ?r'| =o ?r'"
- using r' by (simp add: card_of_Field_ordIso[of ?r'])
+ using r' by (simp add: card_of_Field_ordIso[of ?r'])
finally have "|K| \<le>o ?r'" .
moreover
- {let ?L = "\<Union> j \<in> K. underS ?r' j"
- let ?J = "Field r"
- have rJ: "r =o |?J|"
- using r_card card_of_Field_ordIso ordIso_symmetric by blast
- assume "|K| <o ?r'"
- hence "|K| <=o r" using r' card_of_Card_order[of K] by blast
- hence "|K| \<le>o |?J|" using rJ ordLeq_ordIso_trans by blast
- moreover
- {have "\<forall>j\<in>K. |underS ?r' j| <o ?r'"
- using r' 1 by (auto simp: card_of_underS)
- hence "\<forall>j\<in>K. |underS ?r' j| \<le>o r"
- using r' card_of_Card_order by blast
- hence "\<forall>j\<in>K. |underS ?r' j| \<le>o |?J|"
- using rJ ordLeq_ordIso_trans by blast
- }
- ultimately have "|?L| \<le>o |?J|"
- using r_inf card_of_UNION_ordLeq_infinite by blast
- hence "|?L| \<le>o r" using rJ ordIso_symmetric ordLeq_ordIso_trans by blast
- hence "|?L| <o ?r'" using r' card_of_Card_order by blast
- moreover
- {
- have "Field ?r' \<le> ?L"
- using 2 unfolding underS_def cofinal_def by auto
- hence "|Field ?r'| \<le>o |?L|" by (simp add: card_of_mono1)
- hence "?r' \<le>o |?L|"
- using 22 ordIso_ordLeq_trans ordIso_symmetric by blast
- }
- ultimately have "|?L| <o |?L|" using ordLess_ordLeq_trans by blast
- hence False using ordLess_irreflexive by blast
+ { let ?L = "\<Union> j \<in> K. underS ?r' j"
+ let ?J = "Field r"
+ have rJ: "r =o |?J|"
+ using r_card card_of_Field_ordIso ordIso_symmetric by blast
+ assume "|K| <o ?r'"
+ hence "|K| \<le>o r" using r' card_of_Card_order[of K] by blast
+ hence "|K| \<le>o |?J|" using rJ ordLeq_ordIso_trans by blast
+ moreover
+ {have "\<forall>j\<in>K. |underS ?r' j| <o ?r'"
+ using r' 1 by (auto simp: card_of_underS)
+ hence "\<forall>j\<in>K. |underS ?r' j| \<le>o r"
+ using r' card_of_Card_order by blast
+ hence "\<forall>j\<in>K. |underS ?r' j| \<le>o |?J|"
+ using rJ ordLeq_ordIso_trans by blast
+ }
+ ultimately have "|?L| \<le>o |?J|"
+ using r_inf card_of_UNION_ordLeq_infinite by blast
+ hence "|?L| \<le>o r" using rJ ordIso_symmetric ordLeq_ordIso_trans by blast
+ hence "|?L| <o ?r'" using r' card_of_Card_order by blast
+ moreover
+ {
+ have "Field ?r' \<le> ?L"
+ using 2 unfolding underS_def cofinal_def by auto
+ hence "|Field ?r'| \<le>o |?L|" by (simp add: card_of_mono1)
+ hence "?r' \<le>o |?L|"
+ using 22 ordIso_ordLeq_trans ordIso_symmetric by blast
+ }
+ ultimately have "|?L| <o |?L|" using ordLess_ordLeq_trans by blast
+ hence False using ordLess_irreflexive by blast
}
ultimately show "|K| =o ?r'"
- unfolding ordLeq_iff_ordLess_or_ordIso by blast
+ unfolding ordLeq_iff_ordLess_or_ordIso by blast
qed
qed
lemma cardSuc_UNION:
-assumes r: "Card_order r" and "\<not>finite (Field r)"
-and As: "relChain (cardSuc r) As"
-and Bsub: "B \<le> (\<Union> i \<in> Field (cardSuc r). As i)"
-and cardB: "|B| <=o r"
-shows "\<exists>i \<in> Field (cardSuc r). B \<le> As i"
-proof-
+ assumes r: "Card_order r" and "\<not>finite (Field r)"
+ and As: "relChain (cardSuc r) As"
+ and Bsub: "B \<le> (\<Union> i \<in> Field (cardSuc r). As i)"
+ and cardB: "|B| \<le>o r"
+ shows "\<exists>i \<in> Field (cardSuc r). B \<le> As i"
+proof -
let ?r' = "cardSuc r"
have "Card_order ?r' \<and> |B| <o ?r'"
- using r cardB cardSuc_ordLeq_ordLess cardSuc_Card_order
- card_of_Card_order by blast
+ using r cardB cardSuc_ordLeq_ordLess cardSuc_Card_order
+ card_of_Card_order by blast
moreover have "regularCard ?r'"
- using assms by(simp add: infinite_cardSuc_regularCard)
+ using assms by(simp add: infinite_cardSuc_regularCard)
ultimately show ?thesis
- using As Bsub cardB regularCard_UNION by blast
+ using As Bsub cardB regularCard_UNION by blast
qed
subsection \<open>Others\<close>
lemma card_of_Func_Times:
-"|Func (A \<times> B) C| =o |Func A (Func B C)|"
-unfolding card_of_ordIso[symmetric]
-using bij_betw_curr by blast
+ "|Func (A \<times> B) C| =o |Func A (Func B C)|"
+ unfolding card_of_ordIso[symmetric]
+ using bij_betw_curr by blast
lemma card_of_Pow_Func:
-"|Pow A| =o |Func A (UNIV::bool set)|"
-proof-
- define F where [abs_def]: "F A' a =
+ "|Pow A| =o |Func A (UNIV::bool set)|"
+proof -
+ define F where [abs_def]: "F A' a \<equiv>
(if a \<in> A then (if a \<in> A' then True else False) else undefined)" for A' a
have "bij_betw F (Pow A) (Func A (UNIV::bool set))"
- unfolding bij_betw_def inj_on_def proof (intro ballI impI conjI)
+ unfolding bij_betw_def inj_on_def proof (intro ballI impI conjI)
fix A1 A2 assume "A1 \<in> Pow A" "A2 \<in> Pow A" "F A1 = F A2"
thus "A1 = A2" unfolding F_def Pow_def fun_eq_iff by (auto split: if_split_asm)
next
show "F ` Pow A = Func A UNIV"
proof safe
fix f assume f: "f \<in> Func A (UNIV::bool set)"
- show "f \<in> F ` Pow A" unfolding image_def mem_Collect_eq proof(intro bexI)
- let ?A1 = "{a \<in> A. f a = True}"
- show "f = F ?A1"
- unfolding F_def apply(rule ext)
- using f unfolding Func_def mem_Collect_eq by auto
+ show "f \<in> F ` Pow A"
+ unfolding image_iff
+ proof
+ show "f = F {a \<in> A. f a = True}"
+ using f unfolding Func_def F_def by force
qed auto
- qed(unfold Func_def mem_Collect_eq F_def, auto)
+ qed(unfold Func_def F_def, auto)
qed
thus ?thesis unfolding card_of_ordIso[symmetric] by blast
qed
lemma card_of_Func_UNIV:
-"|Func (UNIV::'a set) (B::'b set)| =o |{f::'a \<Rightarrow> 'b. range f \<subseteq> B}|"
-apply(rule ordIso_symmetric) proof(intro card_of_ordIsoI)
+ "|Func (UNIV::'a set) (B::'b set)| =o |{f::'a \<Rightarrow> 'b. range f \<subseteq> B}|"
+proof -
let ?F = "\<lambda> f (a::'a). ((f a)::'b)"
- show "bij_betw ?F {f. range f \<subseteq> B} (Func UNIV B)"
- unfolding bij_betw_def inj_on_def proof safe
+ have "bij_betw ?F {f. range f \<subseteq> B} (Func UNIV B)"
+ unfolding bij_betw_def inj_on_def
+ proof safe
fix h :: "'a \<Rightarrow> 'b" assume h: "h \<in> Func UNIV B"
- hence "\<forall> a. \<exists> b. h a = b" unfolding Func_def by auto
then obtain f where f: "\<forall> a. h a = f a" by blast
hence "range f \<subseteq> B" using h unfolding Func_def by auto
thus "h \<in> (\<lambda>f a. f a) ` {f. range f \<subseteq> B}" using f by auto
qed(unfold Func_def fun_eq_iff, auto)
+ then show ?thesis
+ by (blast intro: ordIso_symmetric card_of_ordIsoI)
qed
lemma Func_Times_Range:
@@ -1774,8 +1696,8 @@
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).
+ 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"
@@ -1793,14 +1715,13 @@
{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)
+ have "bij_betw snd {(a, u) |u. u \<in> F a} (F a)"
+ unfolding bij_betw_def inj_on_def by (auto simp: image_def)
+ then have "|L| =o |F a|" unfolding L_def card_of_ordIso[symmetric] by blast
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
- apply simp
- apply (drule not_ordLess_ordIso)
- by auto
+ by (force simp add: dest: 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"
@@ -1816,11 +1737,13 @@
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
+ 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 infinite_Card_order_limit by fast
hence "j \<in> Field r" using card_order_on_def cr well_order_on_domain by fast
- then obtain a where a: "a \<in> A" and j: "(j, g a) \<in> r" using 1 unfolding under_def by auto
+ 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
@@ -1836,40 +1759,40 @@
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
+ 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 card_of_Field_ordIso card_of_mono1 cr ordLeq_ordIso_trans by blast
moreover
{assume Kr: "|K| <o r"
- have x: "\<And>a. a \<in> Field r \<Longrightarrow> \<exists>b. b \<in> K \<and> a \<noteq> b \<and> (a, b) \<in> r" using cof unfolding cofinal_def by blast
- then obtain f where "\<And>a. a \<in> Field r \<Longrightarrow> f a = (SOME b. b \<in> K \<and> a \<noteq> b \<and> (a, b) \<in> r)" by simp
- then have "\<forall>a\<in>Field r. f a \<in> K \<and> a \<noteq> f a \<and> (a, f a) \<in> r" using someI_ex[OF x] by simp
- 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 Card_order_iff_ordLeq_card_of card_of_mono1 ordLeq_transitive by blast
- 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 card_of_underS[OF cr] subsetD by auto
- hence "|SIGMA a: K. underS r a| <o r" using st Kr unfolding stable_def by auto
- }
- finally have "r <o r" .
- hence False using ordLess_irreflexive by blast
+ have x: "\<And>a. a \<in> Field r \<Longrightarrow> \<exists>b. b \<in> K \<and> a \<noteq> b \<and> (a, b) \<in> r" using cof unfolding cofinal_def by blast
+ then obtain f where "\<And>a. a \<in> Field r \<Longrightarrow> f a = (SOME b. b \<in> K \<and> a \<noteq> b \<and> (a, b) \<in> r)" by simp
+ then have "\<forall>a\<in>Field r. f a \<in> K \<and> a \<noteq> f a \<and> (a, f a) \<in> r" using someI_ex[OF x] by simp
+ 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 Card_order_iff_ordLeq_card_of card_of_mono1 ordLeq_transitive by blast
+ 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 card_of_underS[OF cr] subsetD by auto
+ hence "|SIGMA a: K. underS r a| <o r" using st Kr unfolding stable_def by auto
+ }
+ finally have "r <o r" .
+ hence False using ordLess_irreflexive by blast
}
ultimately show "|K| =o r" using ordLeq_iff_ordLess_or_ordIso by blast
qed
lemma internalize_card_of_ordLess:
-"( |A| <o r) = (\<exists>B < Field r. |A| =o |B| \<and> |B| <o r)"
+ "( |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
+ 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
+ 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"
@@ -1877,62 +1800,61 @@
qed
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)
+ 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-
+ 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
+ define u where "u \<equiv> \<lambda>(i::'i,a::'a). (f i,a)"
have "bij_betw u ?LEFT ?RIGHT"
- using assms unfolding u_def bij_betw_def inj_on_def by auto
+ 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-
+ 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
+ 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
+ using BIJ card_of_Sigma_cong2 by blast
ultimately show ?thesis using ordIso_transitive by blast
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-
+ 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
+ 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
- (* *)
+ 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
+ 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
+ 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
+ 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
+ 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
+ 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
@@ -1941,57 +1863,51 @@
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)
+ 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)
+ 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
+ using stable_regularCard[OF natLeq_Card_order _ stable_natLeq] Field_natLeq by simp
lemma stable_ordIso1:
-assumes ST: "stable r" and ISO: "r' =o r"
-shows "stable r'"
+ 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
+ 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
+ using ISO ordIso_symmetric ordLess_ordIso_trans by blast
qed
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
+ assumes "stable r" and "|A| <o r" and "\<And> a. a \<in> A \<Longrightarrow> |F a| <o r"
+ shows "|\<Union>a \<in> A. F a| <o r"
+ using assms card_of_UNION_Sigma stable_elim ordLeq_ordLess_trans by blast
corollary card_of_UNION_ordLess_infinite:
-assumes INF: "stable |B|" and
- LEQ_I: "|I| <o |B|" and LEQ: "\<forall>i \<in> I. |A i| <o |B|"
-shows "|\<Union>i \<in> I. A i| <o |B|"
+ assumes "stable |B|" and "|I| <o |B|" and "\<forall>i \<in> I. |A i| <o |B|"
+ shows "|\<Union>i \<in> I. A i| <o |B|"
using assms stable_UNION by blast
corollary card_of_UNION_ordLess_infinite_Field:
-assumes ST: "stable r" and r: "Card_order r" and
- LEQ_I: "|I| <o r" and LEQ: "\<forall>i \<in> I. |A i| <o r"
-shows "|\<Union>i \<in> I. A i| <o r"
-proof-
+ assumes ST: "stable r" and r: "Card_order r" and
+ LEQ_I: "|I| <o r" and LEQ: "\<forall>i \<in> I. |A i| <o r"
+ shows "|\<Union>i \<in> I. A i| <o r"
+proof -
let ?B = "Field r"
have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso
- ordIso_symmetric by blast
+ ordIso_symmetric by blast
hence "|I| <o |?B|" "\<forall>i \<in> I. |A i| <o |?B|"
using LEQ_I LEQ ordLess_ordIso_trans by blast+
moreover have "stable |?B|" using stable_ordIso1 ST 1 by blast
ultimately have "|\<Union>i \<in> I. A i| <o |?B|" using LEQ
- card_of_UNION_ordLess_infinite by blast
+ card_of_UNION_ordLess_infinite by blast
thus ?thesis using 1 ordLess_ordIso_trans by blast
qed