--- a/src/HOL/Zorn.thy Sun Jul 31 19:09:21 2016 +0200
+++ b/src/HOL/Zorn.thy Sun Jul 31 22:56:18 2016 +0200
@@ -1,7 +1,7 @@
-(* Title: HOL/Zorn.thy
- Author: Jacques D. Fleuriot
- Author: Tobias Nipkow, TUM
- Author: Christian Sternagel, JAIST
+(* Title: HOL/Zorn.thy
+ Author: Jacques D. Fleuriot
+ Author: Tobias Nipkow, TUM
+ Author: Christian Sternagel, JAIST
Zorn's Lemma (ported from Larry Paulson's Zorn.thy in ZF).
The well-ordering theorem.
@@ -10,7 +10,7 @@
section \<open>Zorn's Lemma\<close>
theory Zorn
-imports Order_Relation Hilbert_Choice
+ imports Order_Relation Hilbert_Choice
begin
subsection \<open>Zorn's Lemma for the Subset Relation\<close>
@@ -20,36 +20,38 @@
text \<open>Let \<open>P\<close> be a binary predicate on the set \<open>A\<close>.\<close>
locale pred_on =
fixes A :: "'a set"
- and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubset>" 50)
+ and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubset>" 50)
begin
-abbreviation Peq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubseteq>" 50) where
- "x \<sqsubseteq> y \<equiv> P\<^sup>=\<^sup>= x y"
+abbreviation Peq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubseteq>" 50)
+ where "x \<sqsubseteq> y \<equiv> P\<^sup>=\<^sup>= x y"
+
+text \<open>A chain is a totally ordered subset of \<open>A\<close>.\<close>
+definition chain :: "'a set \<Rightarrow> bool"
+ where "chain C \<longleftrightarrow> C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. x \<sqsubseteq> y \<or> y \<sqsubseteq> x)"
-text \<open>A chain is a totally ordered subset of @{term A}.\<close>
-definition chain :: "'a set \<Rightarrow> bool" where
- "chain C \<longleftrightarrow> C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. x \<sqsubseteq> y \<or> y \<sqsubseteq> x)"
-
-text \<open>We call a chain that is a proper superset of some set @{term X},
-but not necessarily a chain itself, a superchain of @{term X}.\<close>
-abbreviation superchain :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infix "<c" 50) where
- "X <c C \<equiv> chain C \<and> X \<subset> C"
+text \<open>
+ We call a chain that is a proper superset of some set \<open>X\<close>,
+ but not necessarily a chain itself, a superchain of \<open>X\<close>.
+\<close>
+abbreviation superchain :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infix "<c" 50)
+ where "X <c C \<equiv> chain C \<and> X \<subset> C"
text \<open>A maximal chain is a chain that does not have a superchain.\<close>
-definition maxchain :: "'a set \<Rightarrow> bool" where
- "maxchain C \<longleftrightarrow> chain C \<and> \<not> (\<exists>S. C <c S)"
+definition maxchain :: "'a set \<Rightarrow> bool"
+ where "maxchain C \<longleftrightarrow> chain C \<and> (\<nexists>S. C <c S)"
-text \<open>We define the successor of a set to be an arbitrary
-superchain, if such exists, or the set itself, otherwise.\<close>
-definition suc :: "'a set \<Rightarrow> 'a set" where
- "suc C = (if \<not> chain C \<or> maxchain C then C else (SOME D. C <c D))"
+text \<open>
+ We define the successor of a set to be an arbitrary
+ superchain, if such exists, or the set itself, otherwise.
+\<close>
+definition suc :: "'a set \<Rightarrow> 'a set"
+ where "suc C = (if \<not> chain C \<or> maxchain C then C else (SOME D. C <c D))"
-lemma chainI [Pure.intro?]:
- "\<lbrakk>C \<subseteq> A; \<And>x y. \<lbrakk>x \<in> C; y \<in> C\<rbrakk> \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x\<rbrakk> \<Longrightarrow> chain C"
+lemma chainI [Pure.intro?]: "C \<subseteq> A \<Longrightarrow> (\<And>x y. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x) \<Longrightarrow> chain C"
unfolding chain_def by blast
-lemma chain_total:
- "chain C \<Longrightarrow> x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
+lemma chain_total: "chain C \<Longrightarrow> x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
by (simp add: chain_def)
lemma not_chain_suc [simp]: "\<not> chain X \<Longrightarrow> suc X = X"
@@ -64,62 +66,67 @@
lemma chain_empty [simp]: "chain {}"
by (auto simp: chain_def)
-lemma not_maxchain_Some:
- "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> C <c (SOME D. C <c D)"
+lemma not_maxchain_Some: "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> C <c (SOME D. C <c D)"
by (rule someI_ex) (auto simp: maxchain_def)
-lemma suc_not_equals:
- "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> suc C \<noteq> C"
+lemma suc_not_equals: "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> suc C \<noteq> C"
using not_maxchain_Some by (auto simp: suc_def)
lemma subset_suc:
- assumes "X \<subseteq> Y" shows "X \<subseteq> suc Y"
+ assumes "X \<subseteq> Y"
+ shows "X \<subseteq> suc Y"
using assms by (rule subset_trans) (rule suc_subset)
-text \<open>We build a set @{term \<C>} that is closed under applications
-of @{term suc} and contains the union of all its subsets.\<close>
-inductive_set suc_Union_closed ("\<C>") where
- suc: "X \<in> \<C> \<Longrightarrow> suc X \<in> \<C>" |
- Union [unfolded Pow_iff]: "X \<in> Pow \<C> \<Longrightarrow> \<Union>X \<in> \<C>"
-
-text \<open>Since the empty set as well as the set itself is a subset of
-every set, @{term \<C>} contains at least @{term "{} \<in> \<C>"} and
-@{term "\<Union>\<C> \<in> \<C>"}.\<close>
-lemma
- suc_Union_closed_empty: "{} \<in> \<C>" and
- suc_Union_closed_Union: "\<Union>\<C> \<in> \<C>"
- using Union [of "{}"] and Union [of "\<C>"] by simp+
-text \<open>Thus closure under @{term suc} will hit a maximal chain
-eventually, as is shown below.\<close>
+text \<open>
+ We build a set @{term \<C>} that is closed under applications
+ of @{term suc} and contains the union of all its subsets.
+\<close>
+inductive_set suc_Union_closed ("\<C>")
+ where
+ suc: "X \<in> \<C> \<Longrightarrow> suc X \<in> \<C>"
+ | Union [unfolded Pow_iff]: "X \<in> Pow \<C> \<Longrightarrow> \<Union>X \<in> \<C>"
-lemma suc_Union_closed_induct [consumes 1, case_names suc Union,
- induct pred: suc_Union_closed]:
- assumes "X \<in> \<C>"
- and "\<And>X. \<lbrakk>X \<in> \<C>; Q X\<rbrakk> \<Longrightarrow> Q (suc X)"
- and "\<And>X. \<lbrakk>X \<subseteq> \<C>; \<forall>x\<in>X. Q x\<rbrakk> \<Longrightarrow> Q (\<Union>X)"
- shows "Q X"
- using assms by (induct) blast+
+text \<open>
+ Since the empty set as well as the set itself is a subset of
+ every set, @{term \<C>} contains at least @{term "{} \<in> \<C>"} and
+ @{term "\<Union>\<C> \<in> \<C>"}.
+\<close>
+lemma suc_Union_closed_empty: "{} \<in> \<C>"
+ and suc_Union_closed_Union: "\<Union>\<C> \<in> \<C>"
+ using Union [of "{}"] and Union [of "\<C>"] by simp_all
+
+text \<open>Thus closure under @{term suc} will hit a maximal chain
+ eventually, as is shown below.\<close>
-lemma suc_Union_closed_cases [consumes 1, case_names suc Union,
- cases pred: suc_Union_closed]:
+lemma suc_Union_closed_induct [consumes 1, case_names suc Union, induct pred: suc_Union_closed]:
assumes "X \<in> \<C>"
- and "\<And>Y. \<lbrakk>X = suc Y; Y \<in> \<C>\<rbrakk> \<Longrightarrow> Q"
- and "\<And>Y. \<lbrakk>X = \<Union>Y; Y \<subseteq> \<C>\<rbrakk> \<Longrightarrow> Q"
+ and "\<And>X. X \<in> \<C> \<Longrightarrow> Q X \<Longrightarrow> Q (suc X)"
+ and "\<And>X. X \<subseteq> \<C> \<Longrightarrow> \<forall>x\<in>X. Q x \<Longrightarrow> Q (\<Union>X)"
+ shows "Q X"
+ using assms by induct blast+
+
+lemma suc_Union_closed_cases [consumes 1, case_names suc Union, cases pred: suc_Union_closed]:
+ assumes "X \<in> \<C>"
+ and "\<And>Y. X = suc Y \<Longrightarrow> Y \<in> \<C> \<Longrightarrow> Q"
+ and "\<And>Y. X = \<Union>Y \<Longrightarrow> Y \<subseteq> \<C> \<Longrightarrow> Q"
shows "Q"
- using assms by (cases) simp+
+ using assms by cases simp_all
text \<open>On chains, @{term suc} yields a chain.\<close>
lemma chain_suc:
- assumes "chain X" shows "chain (suc X)"
+ assumes "chain X"
+ shows "chain (suc X)"
using assms
- by (cases "\<not> chain X \<or> maxchain X")
- (force simp: suc_def dest: not_maxchain_Some)+
+ by (cases "\<not> chain X \<or> maxchain X") (force simp: suc_def dest: not_maxchain_Some)+
lemma chain_sucD:
- assumes "chain X" shows "suc X \<subseteq> A \<and> chain (suc X)"
+ assumes "chain X"
+ shows "suc X \<subseteq> A \<and> chain (suc X)"
proof -
- from \<open>chain X\<close> have *: "chain (suc X)" by (rule chain_suc)
- then have "suc X \<subseteq> A" unfolding chain_def by blast
+ from \<open>chain X\<close> have *: "chain (suc X)"
+ by (rule chain_suc)
+ then have "suc X \<subseteq> A"
+ unfolding chain_def by blast
with * show ?thesis by blast
qed
@@ -128,27 +135,31 @@
and *: "\<And>Z. Z \<in> \<C> \<Longrightarrow> Z \<subseteq> Y \<Longrightarrow> Z = Y \<or> suc Z \<subseteq> Y"
shows "X \<subseteq> Y \<or> suc Y \<subseteq> X"
using \<open>X \<in> \<C>\<close>
-proof (induct)
+proof induct
case (suc X)
with * show ?case by (blast del: subsetI intro: subset_suc)
-qed blast
+next
+ case Union
+ then show ?case by blast
+qed
lemma suc_Union_closed_subsetD:
assumes "Y \<subseteq> X" and "X \<in> \<C>" and "Y \<in> \<C>"
shows "X = Y \<or> suc Y \<subseteq> X"
- using assms(2-, 1)
+ using assms(2,3,1)
proof (induct arbitrary: Y)
case (suc X)
- note * = \<open>\<And>Y. \<lbrakk>Y \<in> \<C>; Y \<subseteq> X\<rbrakk> \<Longrightarrow> X = Y \<or> suc Y \<subseteq> X\<close>
+ note * = \<open>\<And>Y. Y \<in> \<C> \<Longrightarrow> Y \<subseteq> X \<Longrightarrow> X = Y \<or> suc Y \<subseteq> X\<close>
with suc_Union_closed_total' [OF \<open>Y \<in> \<C>\<close> \<open>X \<in> \<C>\<close>]
- have "Y \<subseteq> X \<or> suc X \<subseteq> Y" by blast
+ have "Y \<subseteq> X \<or> suc X \<subseteq> Y" by blast
then show ?case
proof
assume "Y \<subseteq> X"
with * and \<open>Y \<in> \<C>\<close> have "X = Y \<or> suc Y \<subseteq> X" by blast
then show ?thesis
proof
- assume "X = Y" then show ?thesis by simp
+ assume "X = Y"
+ then show ?thesis by simp
next
assume "suc Y \<subseteq> X"
then have "suc Y \<subseteq> suc X" by (rule subset_suc)
@@ -164,21 +175,22 @@
proof (rule ccontr)
assume "\<not> ?thesis"
with \<open>Y \<subseteq> \<Union>X\<close> obtain x y z
- where "\<not> suc Y \<subseteq> \<Union>X"
- and "x \<in> X" and "y \<in> x" and "y \<notin> Y"
- and "z \<in> suc Y" and "\<forall>x\<in>X. z \<notin> x" by blast
+ where "\<not> suc Y \<subseteq> \<Union>X"
+ and "x \<in> X" and "y \<in> x" and "y \<notin> Y"
+ and "z \<in> suc Y" and "\<forall>x\<in>X. z \<notin> x" by blast
with \<open>X \<subseteq> \<C>\<close> have "x \<in> \<C>" by blast
- from Union and \<open>x \<in> X\<close>
- have *: "\<And>y. \<lbrakk>y \<in> \<C>; y \<subseteq> x\<rbrakk> \<Longrightarrow> x = y \<or> suc y \<subseteq> x" by blast
- with suc_Union_closed_total' [OF \<open>Y \<in> \<C>\<close> \<open>x \<in> \<C>\<close>]
- have "Y \<subseteq> x \<or> suc x \<subseteq> Y" by blast
+ from Union and \<open>x \<in> X\<close> have *: "\<And>y. y \<in> \<C> \<Longrightarrow> y \<subseteq> x \<Longrightarrow> x = y \<or> suc y \<subseteq> x"
+ by blast
+ with suc_Union_closed_total' [OF \<open>Y \<in> \<C>\<close> \<open>x \<in> \<C>\<close>] have "Y \<subseteq> x \<or> suc x \<subseteq> Y"
+ by blast
then show False
proof
assume "Y \<subseteq> x"
with * [OF \<open>Y \<in> \<C>\<close>] have "x = Y \<or> suc Y \<subseteq> x" by blast
then show False
proof
- assume "x = Y" with \<open>y \<in> x\<close> and \<open>y \<notin> Y\<close> show False by blast
+ assume "x = Y"
+ with \<open>y \<in> x\<close> and \<open>y \<notin> Y\<close> show False by blast
next
assume "suc Y \<subseteq> x"
with \<open>x \<in> X\<close> have "suc Y \<subseteq> \<Union>X" by blast
@@ -199,75 +211,87 @@
proof (cases "\<forall>Z\<in>\<C>. Z \<subseteq> Y \<longrightarrow> Z = Y \<or> suc Z \<subseteq> Y")
case True
with suc_Union_closed_total' [OF assms]
- have "X \<subseteq> Y \<or> suc Y \<subseteq> X" by blast
- then show ?thesis using suc_subset [of Y] by blast
+ have "X \<subseteq> Y \<or> suc Y \<subseteq> X" by blast
+ with suc_subset [of Y] show ?thesis by blast
next
case False
- then obtain Z
- where "Z \<in> \<C>" and "Z \<subseteq> Y" and "Z \<noteq> Y" and "\<not> suc Z \<subseteq> Y" by blast
- with suc_Union_closed_subsetD and \<open>Y \<in> \<C>\<close> show ?thesis by blast
+ then obtain Z where "Z \<in> \<C>" and "Z \<subseteq> Y" and "Z \<noteq> Y" and "\<not> suc Z \<subseteq> Y"
+ by blast
+ with suc_Union_closed_subsetD and \<open>Y \<in> \<C>\<close> show ?thesis
+ by blast
qed
text \<open>Once we hit a fixed point w.r.t. @{term suc}, all other elements
-of @{term \<C>} are subsets of this fixed point.\<close>
+ of @{term \<C>} are subsets of this fixed point.\<close>
lemma suc_Union_closed_suc:
assumes "X \<in> \<C>" and "Y \<in> \<C>" and "suc Y = Y"
shows "X \<subseteq> Y"
-using \<open>X \<in> \<C>\<close>
-proof (induct)
+ using \<open>X \<in> \<C>\<close>
+proof induct
case (suc X)
- with \<open>Y \<in> \<C>\<close> and suc_Union_closed_subsetD
- have "X = Y \<or> suc X \<subseteq> Y" by blast
- then show ?case by (auto simp: \<open>suc Y = Y\<close>)
-qed blast
+ with \<open>Y \<in> \<C>\<close> and suc_Union_closed_subsetD have "X = Y \<or> suc X \<subseteq> Y"
+ by blast
+ then show ?case
+ by (auto simp: \<open>suc Y = Y\<close>)
+next
+ case Union
+ then show ?case by blast
+qed
lemma eq_suc_Union:
assumes "X \<in> \<C>"
shows "suc X = X \<longleftrightarrow> X = \<Union>\<C>"
+ (is "?lhs \<longleftrightarrow> ?rhs")
proof
- assume "suc X = X"
- with suc_Union_closed_suc [OF suc_Union_closed_Union \<open>X \<in> \<C>\<close>]
- have "\<Union>\<C> \<subseteq> X" .
- with \<open>X \<in> \<C>\<close> show "X = \<Union>\<C>" by blast
+ assume ?lhs
+ then have "\<Union>\<C> \<subseteq> X"
+ by (rule suc_Union_closed_suc [OF suc_Union_closed_Union \<open>X \<in> \<C>\<close>])
+ with \<open>X \<in> \<C>\<close> show ?rhs
+ by blast
next
from \<open>X \<in> \<C>\<close> have "suc X \<in> \<C>" by (rule suc)
then have "suc X \<subseteq> \<Union>\<C>" by blast
- moreover assume "X = \<Union>\<C>"
+ moreover assume ?rhs
ultimately have "suc X \<subseteq> X" by simp
moreover have "X \<subseteq> suc X" by (rule suc_subset)
- ultimately show "suc X = X" ..
+ ultimately show ?lhs ..
qed
lemma suc_in_carrier:
assumes "X \<subseteq> A"
shows "suc X \<subseteq> A"
using assms
- by (cases "\<not> chain X \<or> maxchain X")
- (auto dest: chain_sucD)
+ by (cases "\<not> chain X \<or> maxchain X") (auto dest: chain_sucD)
lemma suc_Union_closed_in_carrier:
assumes "X \<in> \<C>"
shows "X \<subseteq> A"
using assms
- by (induct) (auto dest: suc_in_carrier)
+ by induct (auto dest: suc_in_carrier)
text \<open>All elements of @{term \<C>} are chains.\<close>
lemma suc_Union_closed_chain:
assumes "X \<in> \<C>"
shows "chain X"
-using assms
-proof (induct)
- case (suc X) then show ?case using not_maxchain_Some by (simp add: suc_def)
+ using assms
+proof induct
+ case (suc X)
+ then show ?case
+ using not_maxchain_Some by (simp add: suc_def)
next
case (Union X)
- then have "\<Union>X \<subseteq> A" by (auto dest: suc_Union_closed_in_carrier)
+ then have "\<Union>X \<subseteq> A"
+ by (auto dest: suc_Union_closed_in_carrier)
moreover have "\<forall>x\<in>\<Union>X. \<forall>y\<in>\<Union>X. x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
proof (intro ballI)
fix x y
assume "x \<in> \<Union>X" and "y \<in> \<Union>X"
- then obtain u v where "x \<in> u" and "u \<in> X" and "y \<in> v" and "v \<in> X" by blast
- with Union have "u \<in> \<C>" and "v \<in> \<C>" and "chain u" and "chain v" by blast+
- with suc_Union_closed_total have "u \<subseteq> v \<or> v \<subseteq> u" by blast
+ then obtain u v where "x \<in> u" and "u \<in> X" and "y \<in> v" and "v \<in> X"
+ by blast
+ with Union have "u \<in> \<C>" and "v \<in> \<C>" and "chain u" and "chain v"
+ by blast+
+ with suc_Union_closed_total have "u \<subseteq> v \<or> v \<subseteq> u"
+ by blast
then show "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
proof
assume "u \<subseteq> v"
@@ -290,18 +314,17 @@
subsubsection \<open>Hausdorff's Maximum Principle\<close>
-text \<open>There exists a maximal totally ordered subset of @{term A}. (Note that we do not
-require @{term A} to be partially ordered.)\<close>
+text \<open>There exists a maximal totally ordered subset of \<open>A\<close>. (Note that we do not
+ require \<open>A\<close> to be partially ordered.)\<close>
theorem Hausdorff: "\<exists>C. maxchain C"
proof -
let ?M = "\<Union>\<C>"
have "maxchain ?M"
proof (rule ccontr)
- assume "\<not> maxchain ?M"
+ assume "\<not> ?thesis"
then have "suc ?M \<noteq> ?M"
- using suc_not_equals and
- suc_Union_closed_chain [OF suc_Union_closed_Union] by simp
+ using suc_not_equals and suc_Union_closed_chain [OF suc_Union_closed_Union] by simp
moreover have "suc ?M = ?M"
using eq_suc_Union [OF suc_Union_closed_Union] by simp
ultimately show False by contradiction
@@ -310,34 +333,35 @@
qed
text \<open>Make notation @{term \<C>} available again.\<close>
-no_notation suc_Union_closed ("\<C>")
+no_notation suc_Union_closed ("\<C>")
-lemma chain_extend:
- "chain C \<Longrightarrow> z \<in> A \<Longrightarrow> \<forall>x\<in>C. x \<sqsubseteq> z \<Longrightarrow> chain ({z} \<union> C)"
+lemma chain_extend: "chain C \<Longrightarrow> z \<in> A \<Longrightarrow> \<forall>x\<in>C. x \<sqsubseteq> z \<Longrightarrow> chain ({z} \<union> C)"
unfolding chain_def by blast
-lemma maxchain_imp_chain:
- "maxchain C \<Longrightarrow> chain C"
+lemma maxchain_imp_chain: "maxchain C \<Longrightarrow> chain C"
by (simp add: maxchain_def)
end
text \<open>Hide constant @{const pred_on.suc_Union_closed}, which was just needed
-for the proof of Hausforff's maximum principle.\<close>
+ for the proof of Hausforff's maximum principle.\<close>
hide_const pred_on.suc_Union_closed
lemma chain_mono:
- assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; P x y\<rbrakk> \<Longrightarrow> Q x y"
+ assumes "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> P x y \<Longrightarrow> Q x y"
and "pred_on.chain A P C"
shows "pred_on.chain A Q C"
using assms unfolding pred_on.chain_def by blast
+
subsubsection \<open>Results for the proper subset relation\<close>
interpretation subset: pred_on "A" "op \<subset>" for A .
lemma subset_maxchain_max:
- assumes "subset.maxchain A C" and "X \<in> A" and "\<Union>C \<subseteq> X"
+ assumes "subset.maxchain A C"
+ and "X \<in> A"
+ and "\<Union>C \<subseteq> X"
shows "\<Union>C = X"
proof (rule ccontr)
let ?C = "{X} \<union> C"
@@ -352,6 +376,7 @@
ultimately show False using * by blast
qed
+
subsubsection \<open>Zorn's lemma\<close>
text \<open>If every chain has an upper bound, then there is a maximal set.\<close>
@@ -360,19 +385,23 @@
shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
proof -
from subset.Hausdorff [of A] obtain M where "subset.maxchain A M" ..
- then have "subset.chain A M" by (rule subset.maxchain_imp_chain)
- with assms obtain Y where "Y \<in> A" and "\<forall>X\<in>M. X \<subseteq> Y" by blast
+ then have "subset.chain A M"
+ by (rule subset.maxchain_imp_chain)
+ with assms obtain Y where "Y \<in> A" and "\<forall>X\<in>M. X \<subseteq> Y"
+ by blast
moreover have "\<forall>X\<in>A. Y \<subseteq> X \<longrightarrow> Y = X"
proof (intro ballI impI)
fix X
assume "X \<in> A" and "Y \<subseteq> X"
show "Y = X"
proof (rule ccontr)
- assume "Y \<noteq> X"
+ assume "\<not> ?thesis"
with \<open>Y \<subseteq> X\<close> have "\<not> X \<subseteq> Y" by blast
from subset.chain_extend [OF \<open>subset.chain A M\<close> \<open>X \<in> A\<close>] and \<open>\<forall>X\<in>M. X \<subseteq> Y\<close>
- have "subset.chain A ({X} \<union> M)" using \<open>Y \<subseteq> X\<close> by auto
- moreover have "M \<subset> {X} \<union> M" using \<open>\<forall>X\<in>M. X \<subseteq> Y\<close> and \<open>\<not> X \<subseteq> Y\<close> by auto
+ have "subset.chain A ({X} \<union> M)"
+ using \<open>Y \<subseteq> X\<close> by auto
+ moreover have "M \<subset> {X} \<union> M"
+ using \<open>\<forall>X\<in>M. X \<subseteq> Y\<close> and \<open>\<not> X \<subseteq> Y\<close> by auto
ultimately show False
using \<open>subset.maxchain A M\<close> by (auto simp: subset.maxchain_def)
qed
@@ -380,13 +409,14 @@
ultimately show ?thesis by blast
qed
-text\<open>Alternative version of Zorn's lemma for the subset relation.\<close>
+text \<open>Alternative version of Zorn's lemma for the subset relation.\<close>
lemma subset_Zorn':
assumes "\<And>C. subset.chain A C \<Longrightarrow> \<Union>C \<in> A"
shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
proof -
from subset.Hausdorff [of A] obtain M where "subset.maxchain A M" ..
- then have "subset.chain A M" by (rule subset.maxchain_imp_chain)
+ then have "subset.chain A M"
+ by (rule subset.maxchain_imp_chain)
with assms have "\<Union>M \<in> A" .
moreover have "\<forall>Z\<in>A. \<Union>M \<subseteq> Z \<longrightarrow> \<Union>M = Z"
proof (intro ballI impI)
@@ -403,19 +433,17 @@
text \<open>Relate old to new definitions.\<close>
-(* Define globally? In Set.thy? *)
-definition chain_subset :: "'a set set \<Rightarrow> bool" ("chain\<^sub>\<subseteq>") where
- "chain\<^sub>\<subseteq> C \<longleftrightarrow> (\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A)"
+definition chain_subset :: "'a set set \<Rightarrow> bool" ("chain\<^sub>\<subseteq>") (* Define globally? In Set.thy? *)
+ where "chain\<^sub>\<subseteq> C \<longleftrightarrow> (\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A)"
-definition chains :: "'a set set \<Rightarrow> 'a set set set" where
- "chains A = {C. C \<subseteq> A \<and> chain\<^sub>\<subseteq> C}"
+definition chains :: "'a set set \<Rightarrow> 'a set set set"
+ where "chains A = {C. C \<subseteq> A \<and> chain\<^sub>\<subseteq> C}"
-(* Define globally? In Relation.thy? *)
-definition Chains :: "('a \<times> 'a) set \<Rightarrow> 'a set set" where
- "Chains r = {C. \<forall>a\<in>C. \<forall>b\<in>C. (a, b) \<in> r \<or> (b, a) \<in> r}"
+definition Chains :: "('a \<times> 'a) set \<Rightarrow> 'a set set" (* Define globally? In Relation.thy? *)
+ where "Chains r = {C. \<forall>a\<in>C. \<forall>b\<in>C. (a, b) \<in> r \<or> (b, a) \<in> r}"
-lemma chains_extend:
- "[| c \<in> chains S; z \<in> S; \<forall>x \<in> c. x \<subseteq> (z:: 'a set) |] ==> {z} Un c \<in> chains S"
+lemma chains_extend: "c \<in> chains S \<Longrightarrow> z \<in> S \<Longrightarrow> \<forall>x \<in> c. x \<subseteq> z \<Longrightarrow> {z} \<union> c \<in> chains S"
+ for z :: "'a set"
unfolding chains_def chain_subset_def by blast
lemma mono_Chains: "r \<subseteq> s \<Longrightarrow> Chains r \<subseteq> Chains s"
@@ -427,8 +455,7 @@
lemma chains_alt_def: "chains A = {C. subset.chain A C}"
by (simp add: chains_def chain_subset_alt_def subset.chain_def)
-lemma Chains_subset:
- "Chains r \<subseteq> {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}"
+lemma Chains_subset: "Chains r \<subseteq> {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}"
by (force simp add: Chains_def pred_on.chain_def)
lemma Chains_subset':
@@ -442,20 +469,18 @@
shows "Chains r = {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}"
using assms Chains_subset Chains_subset' by blast
-lemma Zorn_Lemma:
- "\<forall>C\<in>chains A. \<Union>C \<in> A \<Longrightarrow> \<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
+lemma Zorn_Lemma: "\<forall>C\<in>chains A. \<Union>C \<in> A \<Longrightarrow> \<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
using subset_Zorn' [of A] by (force simp: chains_alt_def)
-lemma Zorn_Lemma2:
- "\<forall>C\<in>chains A. \<exists>U\<in>A. \<forall>X\<in>C. X \<subseteq> U \<Longrightarrow> \<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
+lemma Zorn_Lemma2: "\<forall>C\<in>chains A. \<exists>U\<in>A. \<forall>X\<in>C. X \<subseteq> U \<Longrightarrow> \<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
using subset_Zorn [of A] by (auto simp: chains_alt_def)
-text\<open>Various other lemmas\<close>
+text \<open>Various other lemmas\<close>
-lemma chainsD: "[| c \<in> chains S; x \<in> c; y \<in> c |] ==> x \<subseteq> y | y \<subseteq> x"
+lemma chainsD: "c \<in> chains S \<Longrightarrow> x \<in> c \<Longrightarrow> y \<in> c \<Longrightarrow> x \<subseteq> y \<or> y \<subseteq> x"
unfolding chains_def chain_subset_def by blast
-lemma chainsD2: "!!(c :: 'a set set). c \<in> chains S ==> c \<subseteq> S"
+lemma chainsD2: "c \<in> chains S \<Longrightarrow> c \<subseteq> S"
unfolding chains_def by blast
lemma Zorns_po_lemma:
@@ -463,42 +488,49 @@
and u: "\<forall>C\<in>Chains r. \<exists>u\<in>Field r. \<forall>a\<in>C. (a, u) \<in> r"
shows "\<exists>m\<in>Field r. \<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
proof -
- have "Preorder r" using po by (simp add: partial_order_on_def)
-\<comment>\<open>Mirror r in the set of subsets below (wrt r) elements of A\<close>
- let ?B = "%x. r\<inverse> `` {x}" let ?S = "?B ` Field r"
- {
- fix C assume 1: "C \<subseteq> ?S" and 2: "\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A"
+ have "Preorder r"
+ using po by (simp add: partial_order_on_def)
+ txt \<open>Mirror \<open>r\<close> in the set of subsets below (wrt \<open>r\<close>) elements of \<open>A\<close>.\<close>
+ let ?B = "\<lambda>x. r\<inverse> `` {x}"
+ let ?S = "?B ` Field r"
+ have "\<exists>u\<in>Field r. \<forall>A\<in>C. A \<subseteq> r\<inverse> `` {u}" (is "\<exists>u\<in>Field r. ?P u")
+ if 1: "C \<subseteq> ?S" and 2: "\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A" for C
+ proof -
let ?A = "{x\<in>Field r. \<exists>M\<in>C. M = ?B x}"
- have "C = ?B ` ?A" using 1 by (auto simp: image_def)
+ from 1 have "C = ?B ` ?A" by (auto simp: image_def)
have "?A \<in> Chains r"
proof (simp add: Chains_def, intro allI impI, elim conjE)
fix a b
assume "a \<in> Field r" and "?B a \<in> C" and "b \<in> Field r" and "?B b \<in> C"
- hence "?B a \<subseteq> ?B b \<or> ?B b \<subseteq> ?B a" using 2 by auto
- thus "(a, b) \<in> r \<or> (b, a) \<in> r"
+ with 2 have "?B a \<subseteq> ?B b \<or> ?B b \<subseteq> ?B a" by auto
+ then show "(a, b) \<in> r \<or> (b, a) \<in> r"
using \<open>Preorder r\<close> and \<open>a \<in> Field r\<close> and \<open>b \<in> Field r\<close>
by (simp add:subset_Image1_Image1_iff)
qed
- then obtain u where uA: "u \<in> Field r" "\<forall>a\<in>?A. (a, u) \<in> r" using u by auto
- have "\<forall>A\<in>C. A \<subseteq> r\<inverse> `` {u}" (is "?P u")
+ with u obtain u where uA: "u \<in> Field r" "\<forall>a\<in>?A. (a, u) \<in> r" by auto
+ have "?P u"
proof auto
fix a B assume aB: "B \<in> C" "a \<in> B"
with 1 obtain x where "x \<in> Field r" and "B = r\<inverse> `` {x}" by auto
- thus "(a, u) \<in> r" using uA and aB and \<open>Preorder r\<close>
+ then show "(a, u) \<in> r"
+ using uA and aB and \<open>Preorder r\<close>
unfolding preorder_on_def refl_on_def by simp (fast dest: transD)
qed
- then have "\<exists>u\<in>Field r. ?P u" using \<open>u \<in> Field r\<close> by blast
- }
+ then show ?thesis
+ using \<open>u \<in> Field r\<close> by blast
+ qed
then have "\<forall>C\<in>chains ?S. \<exists>U\<in>?S. \<forall>A\<in>C. A \<subseteq> U"
by (auto simp: chains_def chain_subset_def)
- from Zorn_Lemma2 [OF this]
- obtain m B where "m \<in> Field r" and "B = r\<inverse> `` {m}"
- and "\<forall>x\<in>Field r. B \<subseteq> r\<inverse> `` {x} \<longrightarrow> r\<inverse> `` {x} = B"
+ from Zorn_Lemma2 [OF this] obtain m B
+ where "m \<in> Field r"
+ and "B = r\<inverse> `` {m}"
+ and "\<forall>x\<in>Field r. B \<subseteq> r\<inverse> `` {x} \<longrightarrow> r\<inverse> `` {x} = B"
by auto
- hence "\<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
+ then have "\<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
using po and \<open>Preorder r\<close> and \<open>m \<in> Field r\<close>
by (auto simp: subset_Image1_Image1_iff Partial_order_eq_Image1_Image1_iff)
- thus ?thesis using \<open>m \<in> Field r\<close> by blast
+ then show ?thesis
+ using \<open>m \<in> Field r\<close> by blast
qed
@@ -509,13 +541,12 @@
Definition correct/most general?
Naming?
*)
-definition init_seg_of :: "(('a \<times> 'a) set \<times> ('a \<times> 'a) set) set" where
- "init_seg_of = {(r, s). r \<subseteq> s \<and> (\<forall>a b c. (a, b) \<in> s \<and> (b, c) \<in> r \<longrightarrow> (a, b) \<in> r)}"
+definition init_seg_of :: "(('a \<times> 'a) set \<times> ('a \<times> 'a) set) set"
+ where "init_seg_of = {(r, s). r \<subseteq> s \<and> (\<forall>a b c. (a, b) \<in> s \<and> (b, c) \<in> r \<longrightarrow> (a, b) \<in> r)}"
-abbreviation
- initialSegmentOf :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" (infix "initial'_segment'_of" 55)
-where
- "r initial_segment_of s \<equiv> (r, s) \<in> init_seg_of"
+abbreviation initial_segment_of_syntax :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool"
+ (infix "initial'_segment'_of" 55)
+ where "r initial_segment_of s \<equiv> (r, s) \<in> init_seg_of"
lemma refl_on_init_seg_of [simp]: "r initial_segment_of r"
by (simp add: init_seg_of_def)
@@ -524,85 +555,97 @@
"r initial_segment_of s \<Longrightarrow> s initial_segment_of t \<Longrightarrow> r initial_segment_of t"
by (simp (no_asm_use) add: init_seg_of_def) blast
-lemma antisym_init_seg_of:
- "r initial_segment_of s \<Longrightarrow> s initial_segment_of r \<Longrightarrow> r = s"
+lemma antisym_init_seg_of: "r initial_segment_of s \<Longrightarrow> s initial_segment_of r \<Longrightarrow> r = s"
unfolding init_seg_of_def by safe
-lemma Chains_init_seg_of_Union:
- "R \<in> Chains init_seg_of \<Longrightarrow> r\<in>R \<Longrightarrow> r initial_segment_of \<Union>R"
+lemma Chains_init_seg_of_Union: "R \<in> Chains init_seg_of \<Longrightarrow> r\<in>R \<Longrightarrow> r initial_segment_of \<Union>R"
by (auto simp: init_seg_of_def Ball_def Chains_def) blast
lemma chain_subset_trans_Union:
assumes "chain\<^sub>\<subseteq> R" "\<forall>r\<in>R. trans r"
shows "trans (\<Union>R)"
proof (intro transI, elim UnionE)
- fix S1 S2 :: "'a rel" and x y z :: 'a
+ fix S1 S2 :: "'a rel" and x y z :: 'a
assume "S1 \<in> R" "S2 \<in> R"
- with assms(1) have "S1 \<subseteq> S2 \<or> S2 \<subseteq> S1" unfolding chain_subset_def by blast
+ with assms(1) have "S1 \<subseteq> S2 \<or> S2 \<subseteq> S1"
+ unfolding chain_subset_def by blast
moreover assume "(x, y) \<in> S1" "(y, z) \<in> S2"
- ultimately have "((x, y) \<in> S1 \<and> (y, z) \<in> S1) \<or> ((x, y) \<in> S2 \<and> (y, z) \<in> S2)" by blast
- with \<open>S1 \<in> R\<close> \<open>S2 \<in> R\<close> assms(2) show "(x, z) \<in> \<Union>R" by (auto elim: transE)
+ ultimately have "((x, y) \<in> S1 \<and> (y, z) \<in> S1) \<or> ((x, y) \<in> S2 \<and> (y, z) \<in> S2)"
+ by blast
+ with \<open>S1 \<in> R\<close> \<open>S2 \<in> R\<close> assms(2) show "(x, z) \<in> \<Union>R"
+ by (auto elim: transE)
qed
lemma chain_subset_antisym_Union:
assumes "chain\<^sub>\<subseteq> R" "\<forall>r\<in>R. antisym r"
shows "antisym (\<Union>R)"
proof (intro antisymI, elim UnionE)
- fix S1 S2 :: "'a rel" and x y :: 'a
+ fix S1 S2 :: "'a rel" and x y :: 'a
assume "S1 \<in> R" "S2 \<in> R"
- with assms(1) have "S1 \<subseteq> S2 \<or> S2 \<subseteq> S1" unfolding chain_subset_def by blast
+ with assms(1) have "S1 \<subseteq> S2 \<or> S2 \<subseteq> S1"
+ unfolding chain_subset_def by blast
moreover assume "(x, y) \<in> S1" "(y, x) \<in> S2"
- ultimately have "((x, y) \<in> S1 \<and> (y, x) \<in> S1) \<or> ((x, y) \<in> S2 \<and> (y, x) \<in> S2)" by blast
- with \<open>S1 \<in> R\<close> \<open>S2 \<in> R\<close> assms(2) show "x = y" unfolding antisym_def by auto
+ ultimately have "((x, y) \<in> S1 \<and> (y, x) \<in> S1) \<or> ((x, y) \<in> S2 \<and> (y, x) \<in> S2)"
+ by blast
+ with \<open>S1 \<in> R\<close> \<open>S2 \<in> R\<close> assms(2) show "x = y"
+ unfolding antisym_def by auto
qed
lemma chain_subset_Total_Union:
assumes "chain\<^sub>\<subseteq> R" and "\<forall>r\<in>R. Total r"
shows "Total (\<Union>R)"
proof (simp add: total_on_def Ball_def, auto del: disjCI)
- fix r s a b assume A: "r \<in> R" "s \<in> R" "a \<in> Field r" "b \<in> Field s" "a \<noteq> b"
+ fix r s a b
+ assume A: "r \<in> R" "s \<in> R" "a \<in> Field r" "b \<in> Field s" "a \<noteq> b"
from \<open>chain\<^sub>\<subseteq> R\<close> and \<open>r \<in> R\<close> and \<open>s \<in> R\<close> have "r \<subseteq> s \<or> s \<subseteq> r"
by (auto simp add: chain_subset_def)
- thus "(\<exists>r\<in>R. (a, b) \<in> r) \<or> (\<exists>r\<in>R. (b, a) \<in> r)"
+ then show "(\<exists>r\<in>R. (a, b) \<in> r) \<or> (\<exists>r\<in>R. (b, a) \<in> r)"
proof
- assume "r \<subseteq> s" hence "(a, b) \<in> s \<or> (b, a) \<in> s" using assms(2) A mono_Field[of r s]
+ assume "r \<subseteq> s"
+ then have "(a, b) \<in> s \<or> (b, a) \<in> s"
+ using assms(2) A mono_Field[of r s]
by (auto simp add: total_on_def)
- thus ?thesis using \<open>s \<in> R\<close> by blast
+ then show ?thesis
+ using \<open>s \<in> R\<close> by blast
next
- assume "s \<subseteq> r" hence "(a, b) \<in> r \<or> (b, a) \<in> r" using assms(2) A mono_Field[of s r]
+ assume "s \<subseteq> r"
+ then have "(a, b) \<in> r \<or> (b, a) \<in> r"
+ using assms(2) A mono_Field[of s r]
by (fastforce simp add: total_on_def)
- thus ?thesis using \<open>r \<in> R\<close> by blast
+ then show ?thesis
+ using \<open>r \<in> R\<close> by blast
qed
qed
lemma wf_Union_wf_init_segs:
- assumes "R \<in> Chains init_seg_of" and "\<forall>r\<in>R. wf r"
+ assumes "R \<in> Chains init_seg_of"
+ and "\<forall>r\<in>R. wf r"
shows "wf (\<Union>R)"
-proof(simp add: wf_iff_no_infinite_down_chain, rule ccontr, auto)
- fix f assume 1: "\<forall>i. \<exists>r\<in>R. (f (Suc i), f i) \<in> r"
+proof (simp add: wf_iff_no_infinite_down_chain, rule ccontr, auto)
+ fix f
+ assume 1: "\<forall>i. \<exists>r\<in>R. (f (Suc i), f i) \<in> r"
then obtain r where "r \<in> R" and "(f (Suc 0), f 0) \<in> r" by auto
- { fix i have "(f (Suc i), f i) \<in> r"
- proof (induct i)
- case 0 show ?case by fact
- next
- case (Suc i)
- then obtain s where s: "s \<in> R" "(f (Suc (Suc i)), f(Suc i)) \<in> s"
- using 1 by auto
- then have "s initial_segment_of r \<or> r initial_segment_of s"
- using assms(1) \<open>r \<in> R\<close> by (simp add: Chains_def)
- with Suc s show ?case by (simp add: init_seg_of_def) blast
- qed
- }
- thus False using assms(2) and \<open>r \<in> R\<close>
+ have "(f (Suc i), f i) \<in> r" for i
+ proof (induct i)
+ case 0
+ show ?case by fact
+ next
+ case (Suc i)
+ then obtain s where s: "s \<in> R" "(f (Suc (Suc i)), f(Suc i)) \<in> s"
+ using 1 by auto
+ then have "s initial_segment_of r \<or> r initial_segment_of s"
+ using assms(1) \<open>r \<in> R\<close> by (simp add: Chains_def)
+ with Suc s show ?case by (simp add: init_seg_of_def) blast
+ qed
+ then show False
+ using assms(2) and \<open>r \<in> R\<close>
by (simp add: wf_iff_no_infinite_down_chain) blast
qed
-lemma initial_segment_of_Diff:
- "p initial_segment_of q \<Longrightarrow> p - s initial_segment_of q - s"
+lemma initial_segment_of_Diff: "p initial_segment_of q \<Longrightarrow> p - s initial_segment_of q - s"
unfolding init_seg_of_def by blast
-lemma Chains_inits_DiffI:
- "R \<in> Chains init_seg_of \<Longrightarrow> {r - s |r. r \<in> R} \<in> Chains init_seg_of"
+lemma Chains_inits_DiffI: "R \<in> Chains init_seg_of \<Longrightarrow> {r - s |r. r \<in> R} \<in> Chains init_seg_of"
unfolding Chains_def by (blast intro: initial_segment_of_Diff)
theorem well_ordering: "\<exists>r::'a rel. Well_order r \<and> Field r = UNIV"
@@ -610,24 +653,28 @@
\<comment> \<open>The initial segment relation on well-orders:\<close>
let ?WO = "{r::'a rel. Well_order r}"
define I where "I = init_seg_of \<inter> ?WO \<times> ?WO"
- have I_init: "I \<subseteq> init_seg_of" by (auto simp: I_def)
- hence subch: "\<And>R. R \<in> Chains I \<Longrightarrow> chain\<^sub>\<subseteq> R"
+ then have I_init: "I \<subseteq> init_seg_of" by simp
+ then have subch: "\<And>R. R \<in> Chains I \<Longrightarrow> chain\<^sub>\<subseteq> R"
unfolding init_seg_of_def chain_subset_def Chains_def by blast
have Chains_wo: "\<And>R r. R \<in> Chains I \<Longrightarrow> r \<in> R \<Longrightarrow> Well_order r"
by (simp add: Chains_def I_def) blast
- have FI: "Field I = ?WO" by (auto simp add: I_def init_seg_of_def Field_def)
- hence 0: "Partial_order I"
+ have FI: "Field I = ?WO"
+ by (auto simp add: I_def init_seg_of_def Field_def)
+ then have 0: "Partial_order I"
by (auto simp: partial_order_on_def preorder_on_def antisym_def antisym_init_seg_of refl_on_def
- trans_def I_def elim!: trans_init_seg_of)
-\<comment> \<open>I-chains have upper bounds in ?WO wrt I: their Union\<close>
- { fix R assume "R \<in> Chains I"
- hence Ris: "R \<in> Chains init_seg_of" using mono_Chains [OF I_init] by blast
- have subch: "chain\<^sub>\<subseteq> R" using \<open>R : Chains I\<close> I_init
- by (auto simp: init_seg_of_def chain_subset_def Chains_def)
+ trans_def I_def elim!: trans_init_seg_of)
+\<comment> \<open>\<open>I\<close>-chains have upper bounds in \<open>?WO\<close> wrt \<open>I\<close>: their Union\<close>
+ have "\<Union>R \<in> ?WO \<and> (\<forall>r\<in>R. (r, \<Union>R) \<in> I)" if "R \<in> Chains I" for R
+ proof -
+ from that have Ris: "R \<in> Chains init_seg_of"
+ using mono_Chains [OF I_init] by blast
+ have subch: "chain\<^sub>\<subseteq> R"
+ using \<open>R : Chains I\<close> I_init by (auto simp: init_seg_of_def chain_subset_def Chains_def)
have "\<forall>r\<in>R. Refl r" and "\<forall>r\<in>R. trans r" and "\<forall>r\<in>R. antisym r"
and "\<forall>r\<in>R. Total r" and "\<forall>r\<in>R. wf (r - Id)"
using Chains_wo [OF \<open>R \<in> Chains I\<close>] by (simp_all add: order_on_defs)
- have "Refl (\<Union>R)" using \<open>\<forall>r\<in>R. Refl r\<close> unfolding refl_on_def by fastforce
+ have "Refl (\<Union>R)"
+ using \<open>\<forall>r\<in>R. Refl r\<close> unfolding refl_on_def by fastforce
moreover have "trans (\<Union>R)"
by (rule chain_subset_trans_Union [OF subch \<open>\<forall>r\<in>R. trans r\<close>])
moreover have "antisym (\<Union>R)"
@@ -640,21 +687,25 @@
with \<open>\<forall>r\<in>R. wf (r - Id)\<close> and wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]]
show ?thesis by fastforce
qed
- ultimately have "Well_order (\<Union>R)" by(simp add:order_on_defs)
- moreover have "\<forall>r \<in> R. r initial_segment_of \<Union>R" using Ris
- by(simp add: Chains_init_seg_of_Union)
- ultimately have "\<Union>R \<in> ?WO \<and> (\<forall>r\<in>R. (r, \<Union>R) \<in> I)"
+ ultimately have "Well_order (\<Union>R)"
+ by (simp add:order_on_defs)
+ moreover have "\<forall>r \<in> R. r initial_segment_of \<Union>R"
+ using Ris by (simp add: Chains_init_seg_of_Union)
+ ultimately show ?thesis
using mono_Chains [OF I_init] Chains_wo[of R] and \<open>R \<in> Chains I\<close>
unfolding I_def by blast
- }
- hence 1: "\<forall>R \<in> Chains I. \<exists>u\<in>Field I. \<forall>r\<in>R. (r, u) \<in> I" by (subst FI) blast
-\<comment>\<open>Zorn's Lemma yields a maximal well-order m:\<close>
- then obtain m::"'a rel" where "Well_order m" and
- max: "\<forall>r. Well_order r \<and> (m, r) \<in> I \<longrightarrow> r = m"
+ qed
+ then have 1: "\<forall>R \<in> Chains I. \<exists>u\<in>Field I. \<forall>r\<in>R. (r, u) \<in> I"
+ by (subst FI) blast
+\<comment>\<open>Zorn's Lemma yields a maximal well-order \<open>m\<close>:\<close>
+ then obtain m :: "'a rel"
+ where "Well_order m"
+ and max: "\<forall>r. Well_order r \<and> (m, r) \<in> I \<longrightarrow> r = m"
using Zorns_po_lemma[OF 0 1] unfolding FI by fastforce
-\<comment>\<open>Now show by contradiction that m covers the whole type:\<close>
- { fix x::'a assume "x \<notin> Field m"
-\<comment>\<open>We assume that x is not covered and extend m at the top with x\<close>
+\<comment>\<open>Now show by contradiction that \<open>m\<close> covers the whole type:\<close>
+ have False if "x \<notin> Field m" for x :: 'a
+ proof -
+\<comment>\<open>Assuming that \<open>x\<close> is not covered and extend \<open>m\<close> at the top with \<open>x\<close>\<close>
have "m \<noteq> {}"
proof
assume "m = {}"
@@ -663,10 +714,10 @@
ultimately show False using max
by (auto simp: I_def init_seg_of_def simp del: Field_insert)
qed
- hence "Field m \<noteq> {}" by(auto simp:Field_def)
- moreover have "wf (m - Id)" using \<open>Well_order m\<close>
- by (simp add: well_order_on_def)
-\<comment>\<open>The extension of m by x:\<close>
+ then have "Field m \<noteq> {}" by (auto simp: Field_def)
+ moreover have "wf (m - Id)"
+ using \<open>Well_order m\<close> by (simp add: well_order_on_def)
+\<comment>\<open>The extension of \<open>m\<close> by \<open>x\<close>:\<close>
let ?s = "{(a, x) | a. a \<in> Field m}"
let ?m = "insert (x, x) m \<union> ?s"
have Fm: "Field ?m = insert x (Field m)"
@@ -674,49 +725,58 @@
have "Refl m" and "trans m" and "antisym m" and "Total m" and "wf (m - Id)"
using \<open>Well_order m\<close> by (simp_all add: order_on_defs)
\<comment>\<open>We show that the extension is a well-order\<close>
- have "Refl ?m" using \<open>Refl m\<close> Fm unfolding refl_on_def by blast
+ have "Refl ?m"
+ using \<open>Refl m\<close> Fm unfolding refl_on_def by blast
moreover have "trans ?m" using \<open>trans m\<close> and \<open>x \<notin> Field m\<close>
unfolding trans_def Field_def by blast
- moreover have "antisym ?m" using \<open>antisym m\<close> and \<open>x \<notin> Field m\<close>
- unfolding antisym_def Field_def by blast
- moreover have "Total ?m" using \<open>Total m\<close> and Fm by (auto simp: total_on_def)
+ moreover have "antisym ?m"
+ using \<open>antisym m\<close> and \<open>x \<notin> Field m\<close> unfolding antisym_def Field_def by blast
+ moreover have "Total ?m"
+ using \<open>Total m\<close> and Fm by (auto simp: total_on_def)
moreover have "wf (?m - Id)"
proof -
- have "wf ?s" using \<open>x \<notin> Field m\<close>
- by (auto simp: wf_eq_minimal Field_def Bex_def)
- thus ?thesis using \<open>wf (m - Id)\<close> and \<open>x \<notin> Field m\<close>
- wf_subset [OF \<open>wf ?s\<close> Diff_subset]
+ have "wf ?s"
+ using \<open>x \<notin> Field m\<close> by (auto simp: wf_eq_minimal Field_def Bex_def)
+ then show ?thesis
+ using \<open>wf (m - Id)\<close> and \<open>x \<notin> Field m\<close> wf_subset [OF \<open>wf ?s\<close> Diff_subset]
by (auto simp: Un_Diff Field_def intro: wf_Un)
qed
- ultimately have "Well_order ?m" by (simp add: order_on_defs)
-\<comment>\<open>We show that the extension is above m\<close>
- moreover have "(m, ?m) \<in> I" using \<open>Well_order ?m\<close> and \<open>Well_order m\<close> and \<open>x \<notin> Field m\<close>
+ ultimately have "Well_order ?m"
+ by (simp add: order_on_defs)
+\<comment>\<open>We show that the extension is above \<open>m\<close>\<close>
+ moreover have "(m, ?m) \<in> I"
+ using \<open>Well_order ?m\<close> and \<open>Well_order m\<close> and \<open>x \<notin> Field m\<close>
by (fastforce simp: I_def init_seg_of_def Field_def)
ultimately
-\<comment>\<open>This contradicts maximality of m:\<close>
- have False using max and \<open>x \<notin> Field m\<close> unfolding Field_def by blast
- }
- hence "Field m = UNIV" by auto
+\<comment>\<open>This contradicts maximality of \<open>m\<close>:\<close>
+ show False
+ using max and \<open>x \<notin> Field m\<close> unfolding Field_def by blast
+ qed
+ then have "Field m = UNIV" by auto
with \<open>Well_order m\<close> show ?thesis by blast
qed
corollary well_order_on: "\<exists>r::'a rel. well_order_on A r"
proof -
- obtain r::"'a rel" where wo: "Well_order r" and univ: "Field r = UNIV"
+ obtain r :: "'a rel" where wo: "Well_order r" and univ: "Field r = UNIV"
using well_ordering [where 'a = "'a"] by blast
let ?r = "{(x, y). x \<in> A \<and> y \<in> A \<and> (x, y) \<in> r}"
- have 1: "Field ?r = A" using wo univ
- by (fastforce simp: Field_def order_on_defs refl_on_def)
- have "Refl r" and "trans r" and "antisym r" and "Total r" and "wf (r - Id)"
- using \<open>Well_order r\<close> by (simp_all add: order_on_defs)
- have "Refl ?r" using \<open>Refl r\<close> by (auto simp: refl_on_def 1 univ)
- moreover have "trans ?r" using \<open>trans r\<close>
+ have 1: "Field ?r = A"
+ using wo univ by (fastforce simp: Field_def order_on_defs refl_on_def)
+ from \<open>Well_order r\<close> have "Refl r" "trans r" "antisym r" "Total r" "wf (r - Id)"
+ by (simp_all add: order_on_defs)
+ from \<open>Refl r\<close> have "Refl ?r"
+ by (auto simp: refl_on_def 1 univ)
+ moreover from \<open>trans r\<close> have "trans ?r"
unfolding trans_def by blast
- moreover have "antisym ?r" using \<open>antisym r\<close>
+ moreover from \<open>antisym r\<close> have "antisym ?r"
unfolding antisym_def by blast
- moreover have "Total ?r" using \<open>Total r\<close> by (simp add:total_on_def 1 univ)
- moreover have "wf (?r - Id)" by (rule wf_subset [OF \<open>wf (r - Id)\<close>]) blast
- ultimately have "Well_order ?r" by (simp add: order_on_defs)
+ moreover from \<open>Total r\<close> have "Total ?r"
+ by (simp add:total_on_def 1 univ)
+ moreover have "wf (?r - Id)"
+ by (rule wf_subset [OF \<open>wf (r - Id)\<close>]) blast
+ ultimately have "Well_order ?r"
+ by (simp add: order_on_defs)
with 1 show ?thesis by auto
qed
@@ -727,15 +787,16 @@
lemma dependent_wf_choice:
fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
- assumes "wf R" and adm: "\<And>f g x r. (\<And>z. (z, x) \<in> R \<Longrightarrow> f z = g z) \<Longrightarrow> P f x r = P g x r"
- assumes P: "\<And>x f. (\<And>y. (y, x) \<in> R \<Longrightarrow> P f y (f y)) \<Longrightarrow> \<exists>r. P f x r"
+ assumes "wf R"
+ and adm: "\<And>f g x r. (\<And>z. (z, x) \<in> R \<Longrightarrow> f z = g z) \<Longrightarrow> P f x r = P g x r"
+ and P: "\<And>x f. (\<And>y. (y, x) \<in> R \<Longrightarrow> P f y (f y)) \<Longrightarrow> \<exists>r. P f x r"
shows "\<exists>f. \<forall>x. P f x (f x)"
proof (intro exI allI)
- fix x
+ fix x
define f where "f \<equiv> wfrec R (\<lambda>f x. SOME r. P f x r)"
from \<open>wf R\<close> show "P f x (f x)"
proof (induct x)
- fix x assume "\<And>y. (y, x) \<in> R \<Longrightarrow> P f y (f y)"
+ case (less x)
show "P f x (f x)"
proof (subst (2) wfrec_def_adm[OF f_def \<open>wf R\<close>])
show "adm_wf R (\<lambda>f x. SOME r. P f x r)"
@@ -748,7 +809,7 @@
lemma (in wellorder) dependent_wellorder_choice:
assumes "\<And>r f g x. (\<And>y. y < x \<Longrightarrow> f y = g y) \<Longrightarrow> P f x r = P g x r"
- assumes P: "\<And>x f. (\<And>y. y < x \<Longrightarrow> P f y (f y)) \<Longrightarrow> \<exists>r. P f x r"
+ and P: "\<And>x f. (\<And>y. y < x \<Longrightarrow> P f y (f y)) \<Longrightarrow> \<exists>r. P f x r"
shows "\<exists>f. \<forall>x. P f x (f x)"
using wf by (rule dependent_wf_choice) (auto intro!: assms)