1 (* Title: HOL/Zorn.thy |
1 (* Title: HOL/Zorn.thy |
2 Author: Jacques D. Fleuriot |
2 Author: Jacques D. Fleuriot |
3 Author: Tobias Nipkow, TUM |
3 Author: Tobias Nipkow, TUM |
4 Author: Christian Sternagel, JAIST |
4 Author: Christian Sternagel, JAIST |
5 |
5 |
6 Zorn's Lemma (ported from Larry Paulson's Zorn.thy in ZF). |
6 Zorn's Lemma (ported from Larry Paulson's Zorn.thy in ZF). |
7 The well-ordering theorem. |
7 The well-ordering theorem. |
8 *) |
8 *) |
9 |
9 |
10 section \<open>Zorn's Lemma\<close> |
10 section \<open>Zorn's Lemma\<close> |
11 |
11 |
12 theory Zorn |
12 theory Zorn |
13 imports Order_Relation Hilbert_Choice |
13 imports Order_Relation Hilbert_Choice |
14 begin |
14 begin |
15 |
15 |
16 subsection \<open>Zorn's Lemma for the Subset Relation\<close> |
16 subsection \<open>Zorn's Lemma for the Subset Relation\<close> |
17 |
17 |
18 subsubsection \<open>Results that do not require an order\<close> |
18 subsubsection \<open>Results that do not require an order\<close> |
19 |
19 |
20 text \<open>Let \<open>P\<close> be a binary predicate on the set \<open>A\<close>.\<close> |
20 text \<open>Let \<open>P\<close> be a binary predicate on the set \<open>A\<close>.\<close> |
21 locale pred_on = |
21 locale pred_on = |
22 fixes A :: "'a set" |
22 fixes A :: "'a set" |
23 and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubset>" 50) |
23 and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubset>" 50) |
24 begin |
24 begin |
25 |
25 |
26 abbreviation Peq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubseteq>" 50) where |
26 abbreviation Peq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubseteq>" 50) |
27 "x \<sqsubseteq> y \<equiv> P\<^sup>=\<^sup>= x y" |
27 where "x \<sqsubseteq> y \<equiv> P\<^sup>=\<^sup>= x y" |
28 |
28 |
29 text \<open>A chain is a totally ordered subset of @{term A}.\<close> |
29 text \<open>A chain is a totally ordered subset of \<open>A\<close>.\<close> |
30 definition chain :: "'a set \<Rightarrow> bool" where |
30 definition chain :: "'a set \<Rightarrow> bool" |
31 "chain C \<longleftrightarrow> C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. x \<sqsubseteq> y \<or> y \<sqsubseteq> x)" |
31 where "chain C \<longleftrightarrow> C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. x \<sqsubseteq> y \<or> y \<sqsubseteq> x)" |
32 |
32 |
33 text \<open>We call a chain that is a proper superset of some set @{term X}, |
33 text \<open> |
34 but not necessarily a chain itself, a superchain of @{term X}.\<close> |
34 We call a chain that is a proper superset of some set \<open>X\<close>, |
35 abbreviation superchain :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infix "<c" 50) where |
35 but not necessarily a chain itself, a superchain of \<open>X\<close>. |
36 "X <c C \<equiv> chain C \<and> X \<subset> C" |
36 \<close> |
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37 abbreviation superchain :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infix "<c" 50) |
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38 where "X <c C \<equiv> chain C \<and> X \<subset> C" |
37 |
39 |
38 text \<open>A maximal chain is a chain that does not have a superchain.\<close> |
40 text \<open>A maximal chain is a chain that does not have a superchain.\<close> |
39 definition maxchain :: "'a set \<Rightarrow> bool" where |
41 definition maxchain :: "'a set \<Rightarrow> bool" |
40 "maxchain C \<longleftrightarrow> chain C \<and> \<not> (\<exists>S. C <c S)" |
42 where "maxchain C \<longleftrightarrow> chain C \<and> (\<nexists>S. C <c S)" |
41 |
43 |
42 text \<open>We define the successor of a set to be an arbitrary |
44 text \<open> |
43 superchain, if such exists, or the set itself, otherwise.\<close> |
45 We define the successor of a set to be an arbitrary |
44 definition suc :: "'a set \<Rightarrow> 'a set" where |
46 superchain, if such exists, or the set itself, otherwise. |
45 "suc C = (if \<not> chain C \<or> maxchain C then C else (SOME D. C <c D))" |
47 \<close> |
46 |
48 definition suc :: "'a set \<Rightarrow> 'a set" |
47 lemma chainI [Pure.intro?]: |
49 where "suc C = (if \<not> chain C \<or> maxchain C then C else (SOME D. C <c D))" |
48 "\<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" |
50 |
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51 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" |
49 unfolding chain_def by blast |
52 unfolding chain_def by blast |
50 |
53 |
51 lemma chain_total: |
54 lemma chain_total: "chain C \<Longrightarrow> x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x" |
52 "chain C \<Longrightarrow> x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x" |
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53 by (simp add: chain_def) |
55 by (simp add: chain_def) |
54 |
56 |
55 lemma not_chain_suc [simp]: "\<not> chain X \<Longrightarrow> suc X = X" |
57 lemma not_chain_suc [simp]: "\<not> chain X \<Longrightarrow> suc X = X" |
56 by (simp add: suc_def) |
58 by (simp add: suc_def) |
57 |
59 |
62 by (auto simp: suc_def maxchain_def intro: someI2) |
64 by (auto simp: suc_def maxchain_def intro: someI2) |
63 |
65 |
64 lemma chain_empty [simp]: "chain {}" |
66 lemma chain_empty [simp]: "chain {}" |
65 by (auto simp: chain_def) |
67 by (auto simp: chain_def) |
66 |
68 |
67 lemma not_maxchain_Some: |
69 lemma not_maxchain_Some: "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> C <c (SOME D. C <c D)" |
68 "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> C <c (SOME D. C <c D)" |
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69 by (rule someI_ex) (auto simp: maxchain_def) |
70 by (rule someI_ex) (auto simp: maxchain_def) |
70 |
71 |
71 lemma suc_not_equals: |
72 lemma suc_not_equals: "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> suc C \<noteq> C" |
72 "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> suc C \<noteq> C" |
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73 using not_maxchain_Some by (auto simp: suc_def) |
73 using not_maxchain_Some by (auto simp: suc_def) |
74 |
74 |
75 lemma subset_suc: |
75 lemma subset_suc: |
76 assumes "X \<subseteq> Y" shows "X \<subseteq> suc Y" |
76 assumes "X \<subseteq> Y" |
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77 shows "X \<subseteq> suc Y" |
77 using assms by (rule subset_trans) (rule suc_subset) |
78 using assms by (rule subset_trans) (rule suc_subset) |
78 |
79 |
79 text \<open>We build a set @{term \<C>} that is closed under applications |
80 text \<open> |
80 of @{term suc} and contains the union of all its subsets.\<close> |
81 We build a set @{term \<C>} that is closed under applications |
81 inductive_set suc_Union_closed ("\<C>") where |
82 of @{term suc} and contains the union of all its subsets. |
82 suc: "X \<in> \<C> \<Longrightarrow> suc X \<in> \<C>" | |
83 \<close> |
83 Union [unfolded Pow_iff]: "X \<in> Pow \<C> \<Longrightarrow> \<Union>X \<in> \<C>" |
84 inductive_set suc_Union_closed ("\<C>") |
84 |
85 where |
85 text \<open>Since the empty set as well as the set itself is a subset of |
86 suc: "X \<in> \<C> \<Longrightarrow> suc X \<in> \<C>" |
86 every set, @{term \<C>} contains at least @{term "{} \<in> \<C>"} and |
87 | Union [unfolded Pow_iff]: "X \<in> Pow \<C> \<Longrightarrow> \<Union>X \<in> \<C>" |
87 @{term "\<Union>\<C> \<in> \<C>"}.\<close> |
88 |
88 lemma |
89 text \<open> |
89 suc_Union_closed_empty: "{} \<in> \<C>" and |
90 Since the empty set as well as the set itself is a subset of |
90 suc_Union_closed_Union: "\<Union>\<C> \<in> \<C>" |
91 every set, @{term \<C>} contains at least @{term "{} \<in> \<C>"} and |
91 using Union [of "{}"] and Union [of "\<C>"] by simp+ |
92 @{term "\<Union>\<C> \<in> \<C>"}. |
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93 \<close> |
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94 lemma suc_Union_closed_empty: "{} \<in> \<C>" |
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95 and suc_Union_closed_Union: "\<Union>\<C> \<in> \<C>" |
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96 using Union [of "{}"] and Union [of "\<C>"] by simp_all |
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97 |
92 text \<open>Thus closure under @{term suc} will hit a maximal chain |
98 text \<open>Thus closure under @{term suc} will hit a maximal chain |
93 eventually, as is shown below.\<close> |
99 eventually, as is shown below.\<close> |
94 |
100 |
95 lemma suc_Union_closed_induct [consumes 1, case_names suc Union, |
101 lemma suc_Union_closed_induct [consumes 1, case_names suc Union, induct pred: suc_Union_closed]: |
96 induct pred: suc_Union_closed]: |
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97 assumes "X \<in> \<C>" |
102 assumes "X \<in> \<C>" |
98 and "\<And>X. \<lbrakk>X \<in> \<C>; Q X\<rbrakk> \<Longrightarrow> Q (suc X)" |
103 and "\<And>X. X \<in> \<C> \<Longrightarrow> Q X \<Longrightarrow> Q (suc X)" |
99 and "\<And>X. \<lbrakk>X \<subseteq> \<C>; \<forall>x\<in>X. Q x\<rbrakk> \<Longrightarrow> Q (\<Union>X)" |
104 and "\<And>X. X \<subseteq> \<C> \<Longrightarrow> \<forall>x\<in>X. Q x \<Longrightarrow> Q (\<Union>X)" |
100 shows "Q X" |
105 shows "Q X" |
101 using assms by (induct) blast+ |
106 using assms by induct blast+ |
102 |
107 |
103 lemma suc_Union_closed_cases [consumes 1, case_names suc Union, |
108 lemma suc_Union_closed_cases [consumes 1, case_names suc Union, cases pred: suc_Union_closed]: |
104 cases pred: suc_Union_closed]: |
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105 assumes "X \<in> \<C>" |
109 assumes "X \<in> \<C>" |
106 and "\<And>Y. \<lbrakk>X = suc Y; Y \<in> \<C>\<rbrakk> \<Longrightarrow> Q" |
110 and "\<And>Y. X = suc Y \<Longrightarrow> Y \<in> \<C> \<Longrightarrow> Q" |
107 and "\<And>Y. \<lbrakk>X = \<Union>Y; Y \<subseteq> \<C>\<rbrakk> \<Longrightarrow> Q" |
111 and "\<And>Y. X = \<Union>Y \<Longrightarrow> Y \<subseteq> \<C> \<Longrightarrow> Q" |
108 shows "Q" |
112 shows "Q" |
109 using assms by (cases) simp+ |
113 using assms by cases simp_all |
110 |
114 |
111 text \<open>On chains, @{term suc} yields a chain.\<close> |
115 text \<open>On chains, @{term suc} yields a chain.\<close> |
112 lemma chain_suc: |
116 lemma chain_suc: |
113 assumes "chain X" shows "chain (suc X)" |
117 assumes "chain X" |
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118 shows "chain (suc X)" |
114 using assms |
119 using assms |
115 by (cases "\<not> chain X \<or> maxchain X") |
120 by (cases "\<not> chain X \<or> maxchain X") (force simp: suc_def dest: not_maxchain_Some)+ |
116 (force simp: suc_def dest: not_maxchain_Some)+ |
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117 |
121 |
118 lemma chain_sucD: |
122 lemma chain_sucD: |
119 assumes "chain X" shows "suc X \<subseteq> A \<and> chain (suc X)" |
123 assumes "chain X" |
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124 shows "suc X \<subseteq> A \<and> chain (suc X)" |
120 proof - |
125 proof - |
121 from \<open>chain X\<close> have *: "chain (suc X)" by (rule chain_suc) |
126 from \<open>chain X\<close> have *: "chain (suc X)" |
122 then have "suc X \<subseteq> A" unfolding chain_def by blast |
127 by (rule chain_suc) |
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128 then have "suc X \<subseteq> A" |
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129 unfolding chain_def by blast |
123 with * show ?thesis by blast |
130 with * show ?thesis by blast |
124 qed |
131 qed |
125 |
132 |
126 lemma suc_Union_closed_total': |
133 lemma suc_Union_closed_total': |
127 assumes "X \<in> \<C>" and "Y \<in> \<C>" |
134 assumes "X \<in> \<C>" and "Y \<in> \<C>" |
128 and *: "\<And>Z. Z \<in> \<C> \<Longrightarrow> Z \<subseteq> Y \<Longrightarrow> Z = Y \<or> suc Z \<subseteq> Y" |
135 and *: "\<And>Z. Z \<in> \<C> \<Longrightarrow> Z \<subseteq> Y \<Longrightarrow> Z = Y \<or> suc Z \<subseteq> Y" |
129 shows "X \<subseteq> Y \<or> suc Y \<subseteq> X" |
136 shows "X \<subseteq> Y \<or> suc Y \<subseteq> X" |
130 using \<open>X \<in> \<C>\<close> |
137 using \<open>X \<in> \<C>\<close> |
131 proof (induct) |
138 proof induct |
132 case (suc X) |
139 case (suc X) |
133 with * show ?case by (blast del: subsetI intro: subset_suc) |
140 with * show ?case by (blast del: subsetI intro: subset_suc) |
134 qed blast |
141 next |
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142 case Union |
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143 then show ?case by blast |
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144 qed |
135 |
145 |
136 lemma suc_Union_closed_subsetD: |
146 lemma suc_Union_closed_subsetD: |
137 assumes "Y \<subseteq> X" and "X \<in> \<C>" and "Y \<in> \<C>" |
147 assumes "Y \<subseteq> X" and "X \<in> \<C>" and "Y \<in> \<C>" |
138 shows "X = Y \<or> suc Y \<subseteq> X" |
148 shows "X = Y \<or> suc Y \<subseteq> X" |
139 using assms(2-, 1) |
149 using assms(2,3,1) |
140 proof (induct arbitrary: Y) |
150 proof (induct arbitrary: Y) |
141 case (suc X) |
151 case (suc X) |
142 note * = \<open>\<And>Y. \<lbrakk>Y \<in> \<C>; Y \<subseteq> X\<rbrakk> \<Longrightarrow> X = Y \<or> suc Y \<subseteq> X\<close> |
152 note * = \<open>\<And>Y. Y \<in> \<C> \<Longrightarrow> Y \<subseteq> X \<Longrightarrow> X = Y \<or> suc Y \<subseteq> X\<close> |
143 with suc_Union_closed_total' [OF \<open>Y \<in> \<C>\<close> \<open>X \<in> \<C>\<close>] |
153 with suc_Union_closed_total' [OF \<open>Y \<in> \<C>\<close> \<open>X \<in> \<C>\<close>] |
144 have "Y \<subseteq> X \<or> suc X \<subseteq> Y" by blast |
154 have "Y \<subseteq> X \<or> suc X \<subseteq> Y" by blast |
145 then show ?case |
155 then show ?case |
146 proof |
156 proof |
147 assume "Y \<subseteq> X" |
157 assume "Y \<subseteq> X" |
148 with * and \<open>Y \<in> \<C>\<close> have "X = Y \<or> suc Y \<subseteq> X" by blast |
158 with * and \<open>Y \<in> \<C>\<close> have "X = Y \<or> suc Y \<subseteq> X" by blast |
149 then show ?thesis |
159 then show ?thesis |
150 proof |
160 proof |
151 assume "X = Y" then show ?thesis by simp |
161 assume "X = Y" |
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162 then show ?thesis by simp |
152 next |
163 next |
153 assume "suc Y \<subseteq> X" |
164 assume "suc Y \<subseteq> X" |
154 then have "suc Y \<subseteq> suc X" by (rule subset_suc) |
165 then have "suc Y \<subseteq> suc X" by (rule subset_suc) |
155 then show ?thesis by simp |
166 then show ?thesis by simp |
156 qed |
167 qed |
197 assumes "X \<in> \<C>" and "Y \<in> \<C>" |
209 assumes "X \<in> \<C>" and "Y \<in> \<C>" |
198 shows "X \<subseteq> Y \<or> Y \<subseteq> X" |
210 shows "X \<subseteq> Y \<or> Y \<subseteq> X" |
199 proof (cases "\<forall>Z\<in>\<C>. Z \<subseteq> Y \<longrightarrow> Z = Y \<or> suc Z \<subseteq> Y") |
211 proof (cases "\<forall>Z\<in>\<C>. Z \<subseteq> Y \<longrightarrow> Z = Y \<or> suc Z \<subseteq> Y") |
200 case True |
212 case True |
201 with suc_Union_closed_total' [OF assms] |
213 with suc_Union_closed_total' [OF assms] |
202 have "X \<subseteq> Y \<or> suc Y \<subseteq> X" by blast |
214 have "X \<subseteq> Y \<or> suc Y \<subseteq> X" by blast |
203 then show ?thesis using suc_subset [of Y] by blast |
215 with suc_subset [of Y] show ?thesis by blast |
204 next |
216 next |
205 case False |
217 case False |
206 then obtain Z |
218 then obtain Z where "Z \<in> \<C>" and "Z \<subseteq> Y" and "Z \<noteq> Y" and "\<not> suc Z \<subseteq> Y" |
207 where "Z \<in> \<C>" and "Z \<subseteq> Y" and "Z \<noteq> Y" and "\<not> suc Z \<subseteq> Y" by blast |
219 by blast |
208 with suc_Union_closed_subsetD and \<open>Y \<in> \<C>\<close> show ?thesis by blast |
220 with suc_Union_closed_subsetD and \<open>Y \<in> \<C>\<close> show ?thesis |
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221 by blast |
209 qed |
222 qed |
210 |
223 |
211 text \<open>Once we hit a fixed point w.r.t. @{term suc}, all other elements |
224 text \<open>Once we hit a fixed point w.r.t. @{term suc}, all other elements |
212 of @{term \<C>} are subsets of this fixed point.\<close> |
225 of @{term \<C>} are subsets of this fixed point.\<close> |
213 lemma suc_Union_closed_suc: |
226 lemma suc_Union_closed_suc: |
214 assumes "X \<in> \<C>" and "Y \<in> \<C>" and "suc Y = Y" |
227 assumes "X \<in> \<C>" and "Y \<in> \<C>" and "suc Y = Y" |
215 shows "X \<subseteq> Y" |
228 shows "X \<subseteq> Y" |
216 using \<open>X \<in> \<C>\<close> |
229 using \<open>X \<in> \<C>\<close> |
217 proof (induct) |
230 proof induct |
218 case (suc X) |
231 case (suc X) |
219 with \<open>Y \<in> \<C>\<close> and suc_Union_closed_subsetD |
232 with \<open>Y \<in> \<C>\<close> and suc_Union_closed_subsetD have "X = Y \<or> suc X \<subseteq> Y" |
220 have "X = Y \<or> suc X \<subseteq> Y" by blast |
233 by blast |
221 then show ?case by (auto simp: \<open>suc Y = Y\<close>) |
234 then show ?case |
222 qed blast |
235 by (auto simp: \<open>suc Y = Y\<close>) |
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236 next |
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237 case Union |
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238 then show ?case by blast |
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239 qed |
223 |
240 |
224 lemma eq_suc_Union: |
241 lemma eq_suc_Union: |
225 assumes "X \<in> \<C>" |
242 assumes "X \<in> \<C>" |
226 shows "suc X = X \<longleftrightarrow> X = \<Union>\<C>" |
243 shows "suc X = X \<longleftrightarrow> X = \<Union>\<C>" |
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244 (is "?lhs \<longleftrightarrow> ?rhs") |
227 proof |
245 proof |
228 assume "suc X = X" |
246 assume ?lhs |
229 with suc_Union_closed_suc [OF suc_Union_closed_Union \<open>X \<in> \<C>\<close>] |
247 then have "\<Union>\<C> \<subseteq> X" |
230 have "\<Union>\<C> \<subseteq> X" . |
248 by (rule suc_Union_closed_suc [OF suc_Union_closed_Union \<open>X \<in> \<C>\<close>]) |
231 with \<open>X \<in> \<C>\<close> show "X = \<Union>\<C>" by blast |
249 with \<open>X \<in> \<C>\<close> show ?rhs |
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250 by blast |
232 next |
251 next |
233 from \<open>X \<in> \<C>\<close> have "suc X \<in> \<C>" by (rule suc) |
252 from \<open>X \<in> \<C>\<close> have "suc X \<in> \<C>" by (rule suc) |
234 then have "suc X \<subseteq> \<Union>\<C>" by blast |
253 then have "suc X \<subseteq> \<Union>\<C>" by blast |
235 moreover assume "X = \<Union>\<C>" |
254 moreover assume ?rhs |
236 ultimately have "suc X \<subseteq> X" by simp |
255 ultimately have "suc X \<subseteq> X" by simp |
237 moreover have "X \<subseteq> suc X" by (rule suc_subset) |
256 moreover have "X \<subseteq> suc X" by (rule suc_subset) |
238 ultimately show "suc X = X" .. |
257 ultimately show ?lhs .. |
239 qed |
258 qed |
240 |
259 |
241 lemma suc_in_carrier: |
260 lemma suc_in_carrier: |
242 assumes "X \<subseteq> A" |
261 assumes "X \<subseteq> A" |
243 shows "suc X \<subseteq> A" |
262 shows "suc X \<subseteq> A" |
244 using assms |
263 using assms |
245 by (cases "\<not> chain X \<or> maxchain X") |
264 by (cases "\<not> chain X \<or> maxchain X") (auto dest: chain_sucD) |
246 (auto dest: chain_sucD) |
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247 |
265 |
248 lemma suc_Union_closed_in_carrier: |
266 lemma suc_Union_closed_in_carrier: |
249 assumes "X \<in> \<C>" |
267 assumes "X \<in> \<C>" |
250 shows "X \<subseteq> A" |
268 shows "X \<subseteq> A" |
251 using assms |
269 using assms |
252 by (induct) (auto dest: suc_in_carrier) |
270 by induct (auto dest: suc_in_carrier) |
253 |
271 |
254 text \<open>All elements of @{term \<C>} are chains.\<close> |
272 text \<open>All elements of @{term \<C>} are chains.\<close> |
255 lemma suc_Union_closed_chain: |
273 lemma suc_Union_closed_chain: |
256 assumes "X \<in> \<C>" |
274 assumes "X \<in> \<C>" |
257 shows "chain X" |
275 shows "chain X" |
258 using assms |
276 using assms |
259 proof (induct) |
277 proof induct |
260 case (suc X) then show ?case using not_maxchain_Some by (simp add: suc_def) |
278 case (suc X) |
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279 then show ?case |
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280 using not_maxchain_Some by (simp add: suc_def) |
261 next |
281 next |
262 case (Union X) |
282 case (Union X) |
263 then have "\<Union>X \<subseteq> A" by (auto dest: suc_Union_closed_in_carrier) |
283 then have "\<Union>X \<subseteq> A" |
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284 by (auto dest: suc_Union_closed_in_carrier) |
264 moreover have "\<forall>x\<in>\<Union>X. \<forall>y\<in>\<Union>X. x \<sqsubseteq> y \<or> y \<sqsubseteq> x" |
285 moreover have "\<forall>x\<in>\<Union>X. \<forall>y\<in>\<Union>X. x \<sqsubseteq> y \<or> y \<sqsubseteq> x" |
265 proof (intro ballI) |
286 proof (intro ballI) |
266 fix x y |
287 fix x y |
267 assume "x \<in> \<Union>X" and "y \<in> \<Union>X" |
288 assume "x \<in> \<Union>X" and "y \<in> \<Union>X" |
268 then obtain u v where "x \<in> u" and "u \<in> X" and "y \<in> v" and "v \<in> X" by blast |
289 then obtain u v where "x \<in> u" and "u \<in> X" and "y \<in> v" and "v \<in> X" |
269 with Union have "u \<in> \<C>" and "v \<in> \<C>" and "chain u" and "chain v" by blast+ |
290 by blast |
270 with suc_Union_closed_total have "u \<subseteq> v \<or> v \<subseteq> u" by blast |
291 with Union have "u \<in> \<C>" and "v \<in> \<C>" and "chain u" and "chain v" |
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292 by blast+ |
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293 with suc_Union_closed_total have "u \<subseteq> v \<or> v \<subseteq> u" |
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294 by blast |
271 then show "x \<sqsubseteq> y \<or> y \<sqsubseteq> x" |
295 then show "x \<sqsubseteq> y \<or> y \<sqsubseteq> x" |
272 proof |
296 proof |
273 assume "u \<subseteq> v" |
297 assume "u \<subseteq> v" |
274 from \<open>chain v\<close> show ?thesis |
298 from \<open>chain v\<close> show ?thesis |
275 proof (rule chain_total) |
299 proof (rule chain_total) |
440 lemma Chains_alt_def: |
467 lemma Chains_alt_def: |
441 assumes "refl r" |
468 assumes "refl r" |
442 shows "Chains r = {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}" |
469 shows "Chains r = {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}" |
443 using assms Chains_subset Chains_subset' by blast |
470 using assms Chains_subset Chains_subset' by blast |
444 |
471 |
445 lemma Zorn_Lemma: |
472 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" |
446 "\<forall>C\<in>chains A. \<Union>C \<in> A \<Longrightarrow> \<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M" |
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447 using subset_Zorn' [of A] by (force simp: chains_alt_def) |
473 using subset_Zorn' [of A] by (force simp: chains_alt_def) |
448 |
474 |
449 lemma Zorn_Lemma2: |
475 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" |
450 "\<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" |
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451 using subset_Zorn [of A] by (auto simp: chains_alt_def) |
476 using subset_Zorn [of A] by (auto simp: chains_alt_def) |
452 |
477 |
453 text\<open>Various other lemmas\<close> |
478 text \<open>Various other lemmas\<close> |
454 |
479 |
455 lemma chainsD: "[| c \<in> chains S; x \<in> c; y \<in> c |] ==> x \<subseteq> y | y \<subseteq> x" |
480 lemma chainsD: "c \<in> chains S \<Longrightarrow> x \<in> c \<Longrightarrow> y \<in> c \<Longrightarrow> x \<subseteq> y \<or> y \<subseteq> x" |
456 unfolding chains_def chain_subset_def by blast |
481 unfolding chains_def chain_subset_def by blast |
457 |
482 |
458 lemma chainsD2: "!!(c :: 'a set set). c \<in> chains S ==> c \<subseteq> S" |
483 lemma chainsD2: "c \<in> chains S \<Longrightarrow> c \<subseteq> S" |
459 unfolding chains_def by blast |
484 unfolding chains_def by blast |
460 |
485 |
461 lemma Zorns_po_lemma: |
486 lemma Zorns_po_lemma: |
462 assumes po: "Partial_order r" |
487 assumes po: "Partial_order r" |
463 and u: "\<forall>C\<in>Chains r. \<exists>u\<in>Field r. \<forall>a\<in>C. (a, u) \<in> r" |
488 and u: "\<forall>C\<in>Chains r. \<exists>u\<in>Field r. \<forall>a\<in>C. (a, u) \<in> r" |
464 shows "\<exists>m\<in>Field r. \<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m" |
489 shows "\<exists>m\<in>Field r. \<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m" |
465 proof - |
490 proof - |
466 have "Preorder r" using po by (simp add: partial_order_on_def) |
491 have "Preorder r" |
467 \<comment>\<open>Mirror r in the set of subsets below (wrt r) elements of A\<close> |
492 using po by (simp add: partial_order_on_def) |
468 let ?B = "%x. r\<inverse> `` {x}" let ?S = "?B ` Field r" |
493 txt \<open>Mirror \<open>r\<close> in the set of subsets below (wrt \<open>r\<close>) elements of \<open>A\<close>.\<close> |
469 { |
494 let ?B = "\<lambda>x. r\<inverse> `` {x}" |
470 fix C assume 1: "C \<subseteq> ?S" and 2: "\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A" |
495 let ?S = "?B ` Field r" |
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496 have "\<exists>u\<in>Field r. \<forall>A\<in>C. A \<subseteq> r\<inverse> `` {u}" (is "\<exists>u\<in>Field r. ?P u") |
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497 if 1: "C \<subseteq> ?S" and 2: "\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A" for C |
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498 proof - |
471 let ?A = "{x\<in>Field r. \<exists>M\<in>C. M = ?B x}" |
499 let ?A = "{x\<in>Field r. \<exists>M\<in>C. M = ?B x}" |
472 have "C = ?B ` ?A" using 1 by (auto simp: image_def) |
500 from 1 have "C = ?B ` ?A" by (auto simp: image_def) |
473 have "?A \<in> Chains r" |
501 have "?A \<in> Chains r" |
474 proof (simp add: Chains_def, intro allI impI, elim conjE) |
502 proof (simp add: Chains_def, intro allI impI, elim conjE) |
475 fix a b |
503 fix a b |
476 assume "a \<in> Field r" and "?B a \<in> C" and "b \<in> Field r" and "?B b \<in> C" |
504 assume "a \<in> Field r" and "?B a \<in> C" and "b \<in> Field r" and "?B b \<in> C" |
477 hence "?B a \<subseteq> ?B b \<or> ?B b \<subseteq> ?B a" using 2 by auto |
505 with 2 have "?B a \<subseteq> ?B b \<or> ?B b \<subseteq> ?B a" by auto |
478 thus "(a, b) \<in> r \<or> (b, a) \<in> r" |
506 then show "(a, b) \<in> r \<or> (b, a) \<in> r" |
479 using \<open>Preorder r\<close> and \<open>a \<in> Field r\<close> and \<open>b \<in> Field r\<close> |
507 using \<open>Preorder r\<close> and \<open>a \<in> Field r\<close> and \<open>b \<in> Field r\<close> |
480 by (simp add:subset_Image1_Image1_iff) |
508 by (simp add:subset_Image1_Image1_iff) |
481 qed |
509 qed |
482 then obtain u where uA: "u \<in> Field r" "\<forall>a\<in>?A. (a, u) \<in> r" using u by auto |
510 with u obtain u where uA: "u \<in> Field r" "\<forall>a\<in>?A. (a, u) \<in> r" by auto |
483 have "\<forall>A\<in>C. A \<subseteq> r\<inverse> `` {u}" (is "?P u") |
511 have "?P u" |
484 proof auto |
512 proof auto |
485 fix a B assume aB: "B \<in> C" "a \<in> B" |
513 fix a B assume aB: "B \<in> C" "a \<in> B" |
486 with 1 obtain x where "x \<in> Field r" and "B = r\<inverse> `` {x}" by auto |
514 with 1 obtain x where "x \<in> Field r" and "B = r\<inverse> `` {x}" by auto |
487 thus "(a, u) \<in> r" using uA and aB and \<open>Preorder r\<close> |
515 then show "(a, u) \<in> r" |
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516 using uA and aB and \<open>Preorder r\<close> |
488 unfolding preorder_on_def refl_on_def by simp (fast dest: transD) |
517 unfolding preorder_on_def refl_on_def by simp (fast dest: transD) |
489 qed |
518 qed |
490 then have "\<exists>u\<in>Field r. ?P u" using \<open>u \<in> Field r\<close> by blast |
519 then show ?thesis |
491 } |
520 using \<open>u \<in> Field r\<close> by blast |
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521 qed |
492 then have "\<forall>C\<in>chains ?S. \<exists>U\<in>?S. \<forall>A\<in>C. A \<subseteq> U" |
522 then have "\<forall>C\<in>chains ?S. \<exists>U\<in>?S. \<forall>A\<in>C. A \<subseteq> U" |
493 by (auto simp: chains_def chain_subset_def) |
523 by (auto simp: chains_def chain_subset_def) |
494 from Zorn_Lemma2 [OF this] |
524 from Zorn_Lemma2 [OF this] obtain m B |
495 obtain m B where "m \<in> Field r" and "B = r\<inverse> `` {m}" |
525 where "m \<in> Field r" |
496 and "\<forall>x\<in>Field r. B \<subseteq> r\<inverse> `` {x} \<longrightarrow> r\<inverse> `` {x} = B" |
526 and "B = r\<inverse> `` {m}" |
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527 and "\<forall>x\<in>Field r. B \<subseteq> r\<inverse> `` {x} \<longrightarrow> r\<inverse> `` {x} = B" |
497 by auto |
528 by auto |
498 hence "\<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m" |
529 then have "\<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m" |
499 using po and \<open>Preorder r\<close> and \<open>m \<in> Field r\<close> |
530 using po and \<open>Preorder r\<close> and \<open>m \<in> Field r\<close> |
500 by (auto simp: subset_Image1_Image1_iff Partial_order_eq_Image1_Image1_iff) |
531 by (auto simp: subset_Image1_Image1_iff Partial_order_eq_Image1_Image1_iff) |
501 thus ?thesis using \<open>m \<in> Field r\<close> by blast |
532 then show ?thesis |
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533 using \<open>m \<in> Field r\<close> by blast |
502 qed |
534 qed |
503 |
535 |
504 |
536 |
505 subsection \<open>The Well Ordering Theorem\<close> |
537 subsection \<open>The Well Ordering Theorem\<close> |
506 |
538 |
507 (* The initial segment of a relation appears generally useful. |
539 (* The initial segment of a relation appears generally useful. |
508 Move to Relation.thy? |
540 Move to Relation.thy? |
509 Definition correct/most general? |
541 Definition correct/most general? |
510 Naming? |
542 Naming? |
511 *) |
543 *) |
512 definition init_seg_of :: "(('a \<times> 'a) set \<times> ('a \<times> 'a) set) set" where |
544 definition init_seg_of :: "(('a \<times> 'a) set \<times> ('a \<times> 'a) set) set" |
513 "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)}" |
545 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)}" |
514 |
546 |
515 abbreviation |
547 abbreviation initial_segment_of_syntax :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" |
516 initialSegmentOf :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" (infix "initial'_segment'_of" 55) |
548 (infix "initial'_segment'_of" 55) |
517 where |
549 where "r initial_segment_of s \<equiv> (r, s) \<in> init_seg_of" |
518 "r initial_segment_of s \<equiv> (r, s) \<in> init_seg_of" |
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519 |
550 |
520 lemma refl_on_init_seg_of [simp]: "r initial_segment_of r" |
551 lemma refl_on_init_seg_of [simp]: "r initial_segment_of r" |
521 by (simp add: init_seg_of_def) |
552 by (simp add: init_seg_of_def) |
522 |
553 |
523 lemma trans_init_seg_of: |
554 lemma trans_init_seg_of: |
524 "r initial_segment_of s \<Longrightarrow> s initial_segment_of t \<Longrightarrow> r initial_segment_of t" |
555 "r initial_segment_of s \<Longrightarrow> s initial_segment_of t \<Longrightarrow> r initial_segment_of t" |
525 by (simp (no_asm_use) add: init_seg_of_def) blast |
556 by (simp (no_asm_use) add: init_seg_of_def) blast |
526 |
557 |
527 lemma antisym_init_seg_of: |
558 lemma antisym_init_seg_of: "r initial_segment_of s \<Longrightarrow> s initial_segment_of r \<Longrightarrow> r = s" |
528 "r initial_segment_of s \<Longrightarrow> s initial_segment_of r \<Longrightarrow> r = s" |
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529 unfolding init_seg_of_def by safe |
559 unfolding init_seg_of_def by safe |
530 |
560 |
531 lemma Chains_init_seg_of_Union: |
561 lemma Chains_init_seg_of_Union: "R \<in> Chains init_seg_of \<Longrightarrow> r\<in>R \<Longrightarrow> r initial_segment_of \<Union>R" |
532 "R \<in> Chains init_seg_of \<Longrightarrow> r\<in>R \<Longrightarrow> r initial_segment_of \<Union>R" |
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533 by (auto simp: init_seg_of_def Ball_def Chains_def) blast |
562 by (auto simp: init_seg_of_def Ball_def Chains_def) blast |
534 |
563 |
535 lemma chain_subset_trans_Union: |
564 lemma chain_subset_trans_Union: |
536 assumes "chain\<^sub>\<subseteq> R" "\<forall>r\<in>R. trans r" |
565 assumes "chain\<^sub>\<subseteq> R" "\<forall>r\<in>R. trans r" |
537 shows "trans (\<Union>R)" |
566 shows "trans (\<Union>R)" |
538 proof (intro transI, elim UnionE) |
567 proof (intro transI, elim UnionE) |
539 fix S1 S2 :: "'a rel" and x y z :: 'a |
568 fix S1 S2 :: "'a rel" and x y z :: 'a |
540 assume "S1 \<in> R" "S2 \<in> R" |
569 assume "S1 \<in> R" "S2 \<in> R" |
541 with assms(1) have "S1 \<subseteq> S2 \<or> S2 \<subseteq> S1" unfolding chain_subset_def by blast |
570 with assms(1) have "S1 \<subseteq> S2 \<or> S2 \<subseteq> S1" |
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571 unfolding chain_subset_def by blast |
542 moreover assume "(x, y) \<in> S1" "(y, z) \<in> S2" |
572 moreover assume "(x, y) \<in> S1" "(y, z) \<in> S2" |
543 ultimately have "((x, y) \<in> S1 \<and> (y, z) \<in> S1) \<or> ((x, y) \<in> S2 \<and> (y, z) \<in> S2)" by blast |
573 ultimately have "((x, y) \<in> S1 \<and> (y, z) \<in> S1) \<or> ((x, y) \<in> S2 \<and> (y, z) \<in> S2)" |
544 with \<open>S1 \<in> R\<close> \<open>S2 \<in> R\<close> assms(2) show "(x, z) \<in> \<Union>R" by (auto elim: transE) |
574 by blast |
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575 with \<open>S1 \<in> R\<close> \<open>S2 \<in> R\<close> assms(2) show "(x, z) \<in> \<Union>R" |
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576 by (auto elim: transE) |
545 qed |
577 qed |
546 |
578 |
547 lemma chain_subset_antisym_Union: |
579 lemma chain_subset_antisym_Union: |
548 assumes "chain\<^sub>\<subseteq> R" "\<forall>r\<in>R. antisym r" |
580 assumes "chain\<^sub>\<subseteq> R" "\<forall>r\<in>R. antisym r" |
549 shows "antisym (\<Union>R)" |
581 shows "antisym (\<Union>R)" |
550 proof (intro antisymI, elim UnionE) |
582 proof (intro antisymI, elim UnionE) |
551 fix S1 S2 :: "'a rel" and x y :: 'a |
583 fix S1 S2 :: "'a rel" and x y :: 'a |
552 assume "S1 \<in> R" "S2 \<in> R" |
584 assume "S1 \<in> R" "S2 \<in> R" |
553 with assms(1) have "S1 \<subseteq> S2 \<or> S2 \<subseteq> S1" unfolding chain_subset_def by blast |
585 with assms(1) have "S1 \<subseteq> S2 \<or> S2 \<subseteq> S1" |
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586 unfolding chain_subset_def by blast |
554 moreover assume "(x, y) \<in> S1" "(y, x) \<in> S2" |
587 moreover assume "(x, y) \<in> S1" "(y, x) \<in> S2" |
555 ultimately have "((x, y) \<in> S1 \<and> (y, x) \<in> S1) \<or> ((x, y) \<in> S2 \<and> (y, x) \<in> S2)" by blast |
588 ultimately have "((x, y) \<in> S1 \<and> (y, x) \<in> S1) \<or> ((x, y) \<in> S2 \<and> (y, x) \<in> S2)" |
556 with \<open>S1 \<in> R\<close> \<open>S2 \<in> R\<close> assms(2) show "x = y" unfolding antisym_def by auto |
589 by blast |
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590 with \<open>S1 \<in> R\<close> \<open>S2 \<in> R\<close> assms(2) show "x = y" |
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591 unfolding antisym_def by auto |
557 qed |
592 qed |
558 |
593 |
559 lemma chain_subset_Total_Union: |
594 lemma chain_subset_Total_Union: |
560 assumes "chain\<^sub>\<subseteq> R" and "\<forall>r\<in>R. Total r" |
595 assumes "chain\<^sub>\<subseteq> R" and "\<forall>r\<in>R. Total r" |
561 shows "Total (\<Union>R)" |
596 shows "Total (\<Union>R)" |
562 proof (simp add: total_on_def Ball_def, auto del: disjCI) |
597 proof (simp add: total_on_def Ball_def, auto del: disjCI) |
563 fix r s a b assume A: "r \<in> R" "s \<in> R" "a \<in> Field r" "b \<in> Field s" "a \<noteq> b" |
598 fix r s a b |
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599 assume A: "r \<in> R" "s \<in> R" "a \<in> Field r" "b \<in> Field s" "a \<noteq> b" |
564 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" |
600 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" |
565 by (auto simp add: chain_subset_def) |
601 by (auto simp add: chain_subset_def) |
566 thus "(\<exists>r\<in>R. (a, b) \<in> r) \<or> (\<exists>r\<in>R. (b, a) \<in> r)" |
602 then show "(\<exists>r\<in>R. (a, b) \<in> r) \<or> (\<exists>r\<in>R. (b, a) \<in> r)" |
567 proof |
603 proof |
568 assume "r \<subseteq> s" hence "(a, b) \<in> s \<or> (b, a) \<in> s" using assms(2) A mono_Field[of r s] |
604 assume "r \<subseteq> s" |
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605 then have "(a, b) \<in> s \<or> (b, a) \<in> s" |
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606 using assms(2) A mono_Field[of r s] |
569 by (auto simp add: total_on_def) |
607 by (auto simp add: total_on_def) |
570 thus ?thesis using \<open>s \<in> R\<close> by blast |
608 then show ?thesis |
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609 using \<open>s \<in> R\<close> by blast |
571 next |
610 next |
572 assume "s \<subseteq> r" hence "(a, b) \<in> r \<or> (b, a) \<in> r" using assms(2) A mono_Field[of s r] |
611 assume "s \<subseteq> r" |
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612 then have "(a, b) \<in> r \<or> (b, a) \<in> r" |
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613 using assms(2) A mono_Field[of s r] |
573 by (fastforce simp add: total_on_def) |
614 by (fastforce simp add: total_on_def) |
574 thus ?thesis using \<open>r \<in> R\<close> by blast |
615 then show ?thesis |
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616 using \<open>r \<in> R\<close> by blast |
575 qed |
617 qed |
576 qed |
618 qed |
577 |
619 |
578 lemma wf_Union_wf_init_segs: |
620 lemma wf_Union_wf_init_segs: |
579 assumes "R \<in> Chains init_seg_of" and "\<forall>r\<in>R. wf r" |
621 assumes "R \<in> Chains init_seg_of" |
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622 and "\<forall>r\<in>R. wf r" |
580 shows "wf (\<Union>R)" |
623 shows "wf (\<Union>R)" |
581 proof(simp add: wf_iff_no_infinite_down_chain, rule ccontr, auto) |
624 proof (simp add: wf_iff_no_infinite_down_chain, rule ccontr, auto) |
582 fix f assume 1: "\<forall>i. \<exists>r\<in>R. (f (Suc i), f i) \<in> r" |
625 fix f |
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626 assume 1: "\<forall>i. \<exists>r\<in>R. (f (Suc i), f i) \<in> r" |
583 then obtain r where "r \<in> R" and "(f (Suc 0), f 0) \<in> r" by auto |
627 then obtain r where "r \<in> R" and "(f (Suc 0), f 0) \<in> r" by auto |
584 { fix i have "(f (Suc i), f i) \<in> r" |
628 have "(f (Suc i), f i) \<in> r" for i |
585 proof (induct i) |
629 proof (induct i) |
586 case 0 show ?case by fact |
630 case 0 |
587 next |
631 show ?case by fact |
588 case (Suc i) |
632 next |
589 then obtain s where s: "s \<in> R" "(f (Suc (Suc i)), f(Suc i)) \<in> s" |
633 case (Suc i) |
590 using 1 by auto |
634 then obtain s where s: "s \<in> R" "(f (Suc (Suc i)), f(Suc i)) \<in> s" |
591 then have "s initial_segment_of r \<or> r initial_segment_of s" |
635 using 1 by auto |
592 using assms(1) \<open>r \<in> R\<close> by (simp add: Chains_def) |
636 then have "s initial_segment_of r \<or> r initial_segment_of s" |
593 with Suc s show ?case by (simp add: init_seg_of_def) blast |
637 using assms(1) \<open>r \<in> R\<close> by (simp add: Chains_def) |
594 qed |
638 with Suc s show ?case by (simp add: init_seg_of_def) blast |
595 } |
639 qed |
596 thus False using assms(2) and \<open>r \<in> R\<close> |
640 then show False |
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641 using assms(2) and \<open>r \<in> R\<close> |
597 by (simp add: wf_iff_no_infinite_down_chain) blast |
642 by (simp add: wf_iff_no_infinite_down_chain) blast |
598 qed |
643 qed |
599 |
644 |
600 lemma initial_segment_of_Diff: |
645 lemma initial_segment_of_Diff: "p initial_segment_of q \<Longrightarrow> p - s initial_segment_of q - s" |
601 "p initial_segment_of q \<Longrightarrow> p - s initial_segment_of q - s" |
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602 unfolding init_seg_of_def by blast |
646 unfolding init_seg_of_def by blast |
603 |
647 |
604 lemma Chains_inits_DiffI: |
648 lemma Chains_inits_DiffI: "R \<in> Chains init_seg_of \<Longrightarrow> {r - s |r. r \<in> R} \<in> Chains init_seg_of" |
605 "R \<in> Chains init_seg_of \<Longrightarrow> {r - s |r. r \<in> R} \<in> Chains init_seg_of" |
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606 unfolding Chains_def by (blast intro: initial_segment_of_Diff) |
649 unfolding Chains_def by (blast intro: initial_segment_of_Diff) |
607 |
650 |
608 theorem well_ordering: "\<exists>r::'a rel. Well_order r \<and> Field r = UNIV" |
651 theorem well_ordering: "\<exists>r::'a rel. Well_order r \<and> Field r = UNIV" |
609 proof - |
652 proof - |
610 \<comment> \<open>The initial segment relation on well-orders:\<close> |
653 \<comment> \<open>The initial segment relation on well-orders:\<close> |
611 let ?WO = "{r::'a rel. Well_order r}" |
654 let ?WO = "{r::'a rel. Well_order r}" |
612 define I where "I = init_seg_of \<inter> ?WO \<times> ?WO" |
655 define I where "I = init_seg_of \<inter> ?WO \<times> ?WO" |
613 have I_init: "I \<subseteq> init_seg_of" by (auto simp: I_def) |
656 then have I_init: "I \<subseteq> init_seg_of" by simp |
614 hence subch: "\<And>R. R \<in> Chains I \<Longrightarrow> chain\<^sub>\<subseteq> R" |
657 then have subch: "\<And>R. R \<in> Chains I \<Longrightarrow> chain\<^sub>\<subseteq> R" |
615 unfolding init_seg_of_def chain_subset_def Chains_def by blast |
658 unfolding init_seg_of_def chain_subset_def Chains_def by blast |
616 have Chains_wo: "\<And>R r. R \<in> Chains I \<Longrightarrow> r \<in> R \<Longrightarrow> Well_order r" |
659 have Chains_wo: "\<And>R r. R \<in> Chains I \<Longrightarrow> r \<in> R \<Longrightarrow> Well_order r" |
617 by (simp add: Chains_def I_def) blast |
660 by (simp add: Chains_def I_def) blast |
618 have FI: "Field I = ?WO" by (auto simp add: I_def init_seg_of_def Field_def) |
661 have FI: "Field I = ?WO" |
619 hence 0: "Partial_order I" |
662 by (auto simp add: I_def init_seg_of_def Field_def) |
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663 then have 0: "Partial_order I" |
620 by (auto simp: partial_order_on_def preorder_on_def antisym_def antisym_init_seg_of refl_on_def |
664 by (auto simp: partial_order_on_def preorder_on_def antisym_def antisym_init_seg_of refl_on_def |
621 trans_def I_def elim!: trans_init_seg_of) |
665 trans_def I_def elim!: trans_init_seg_of) |
622 \<comment> \<open>I-chains have upper bounds in ?WO wrt I: their Union\<close> |
666 \<comment> \<open>\<open>I\<close>-chains have upper bounds in \<open>?WO\<close> wrt \<open>I\<close>: their Union\<close> |
623 { fix R assume "R \<in> Chains I" |
667 have "\<Union>R \<in> ?WO \<and> (\<forall>r\<in>R. (r, \<Union>R) \<in> I)" if "R \<in> Chains I" for R |
624 hence Ris: "R \<in> Chains init_seg_of" using mono_Chains [OF I_init] by blast |
668 proof - |
625 have subch: "chain\<^sub>\<subseteq> R" using \<open>R : Chains I\<close> I_init |
669 from that have Ris: "R \<in> Chains init_seg_of" |
626 by (auto simp: init_seg_of_def chain_subset_def Chains_def) |
670 using mono_Chains [OF I_init] by blast |
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671 have subch: "chain\<^sub>\<subseteq> R" |
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672 using \<open>R : Chains I\<close> I_init by (auto simp: init_seg_of_def chain_subset_def Chains_def) |
627 have "\<forall>r\<in>R. Refl r" and "\<forall>r\<in>R. trans r" and "\<forall>r\<in>R. antisym r" |
673 have "\<forall>r\<in>R. Refl r" and "\<forall>r\<in>R. trans r" and "\<forall>r\<in>R. antisym r" |
628 and "\<forall>r\<in>R. Total r" and "\<forall>r\<in>R. wf (r - Id)" |
674 and "\<forall>r\<in>R. Total r" and "\<forall>r\<in>R. wf (r - Id)" |
629 using Chains_wo [OF \<open>R \<in> Chains I\<close>] by (simp_all add: order_on_defs) |
675 using Chains_wo [OF \<open>R \<in> Chains I\<close>] by (simp_all add: order_on_defs) |
630 have "Refl (\<Union>R)" using \<open>\<forall>r\<in>R. Refl r\<close> unfolding refl_on_def by fastforce |
676 have "Refl (\<Union>R)" |
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677 using \<open>\<forall>r\<in>R. Refl r\<close> unfolding refl_on_def by fastforce |
631 moreover have "trans (\<Union>R)" |
678 moreover have "trans (\<Union>R)" |
632 by (rule chain_subset_trans_Union [OF subch \<open>\<forall>r\<in>R. trans r\<close>]) |
679 by (rule chain_subset_trans_Union [OF subch \<open>\<forall>r\<in>R. trans r\<close>]) |
633 moreover have "antisym (\<Union>R)" |
680 moreover have "antisym (\<Union>R)" |
634 by (rule chain_subset_antisym_Union [OF subch \<open>\<forall>r\<in>R. antisym r\<close>]) |
681 by (rule chain_subset_antisym_Union [OF subch \<open>\<forall>r\<in>R. antisym r\<close>]) |
635 moreover have "Total (\<Union>R)" |
682 moreover have "Total (\<Union>R)" |
638 proof - |
685 proof - |
639 have "(\<Union>R) - Id = \<Union>{r - Id | r. r \<in> R}" by blast |
686 have "(\<Union>R) - Id = \<Union>{r - Id | r. r \<in> R}" by blast |
640 with \<open>\<forall>r\<in>R. wf (r - Id)\<close> and wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]] |
687 with \<open>\<forall>r\<in>R. wf (r - Id)\<close> and wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]] |
641 show ?thesis by fastforce |
688 show ?thesis by fastforce |
642 qed |
689 qed |
643 ultimately have "Well_order (\<Union>R)" by(simp add:order_on_defs) |
690 ultimately have "Well_order (\<Union>R)" |
644 moreover have "\<forall>r \<in> R. r initial_segment_of \<Union>R" using Ris |
691 by (simp add:order_on_defs) |
645 by(simp add: Chains_init_seg_of_Union) |
692 moreover have "\<forall>r \<in> R. r initial_segment_of \<Union>R" |
646 ultimately have "\<Union>R \<in> ?WO \<and> (\<forall>r\<in>R. (r, \<Union>R) \<in> I)" |
693 using Ris by (simp add: Chains_init_seg_of_Union) |
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694 ultimately show ?thesis |
647 using mono_Chains [OF I_init] Chains_wo[of R] and \<open>R \<in> Chains I\<close> |
695 using mono_Chains [OF I_init] Chains_wo[of R] and \<open>R \<in> Chains I\<close> |
648 unfolding I_def by blast |
696 unfolding I_def by blast |
649 } |
697 qed |
650 hence 1: "\<forall>R \<in> Chains I. \<exists>u\<in>Field I. \<forall>r\<in>R. (r, u) \<in> I" by (subst FI) blast |
698 then have 1: "\<forall>R \<in> Chains I. \<exists>u\<in>Field I. \<forall>r\<in>R. (r, u) \<in> I" |
651 \<comment>\<open>Zorn's Lemma yields a maximal well-order m:\<close> |
699 by (subst FI) blast |
652 then obtain m::"'a rel" where "Well_order m" and |
700 \<comment>\<open>Zorn's Lemma yields a maximal well-order \<open>m\<close>:\<close> |
653 max: "\<forall>r. Well_order r \<and> (m, r) \<in> I \<longrightarrow> r = m" |
701 then obtain m :: "'a rel" |
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702 where "Well_order m" |
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703 and max: "\<forall>r. Well_order r \<and> (m, r) \<in> I \<longrightarrow> r = m" |
654 using Zorns_po_lemma[OF 0 1] unfolding FI by fastforce |
704 using Zorns_po_lemma[OF 0 1] unfolding FI by fastforce |
655 \<comment>\<open>Now show by contradiction that m covers the whole type:\<close> |
705 \<comment>\<open>Now show by contradiction that \<open>m\<close> covers the whole type:\<close> |
656 { fix x::'a assume "x \<notin> Field m" |
706 have False if "x \<notin> Field m" for x :: 'a |
657 \<comment>\<open>We assume that x is not covered and extend m at the top with x\<close> |
707 proof - |
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708 \<comment>\<open>Assuming that \<open>x\<close> is not covered and extend \<open>m\<close> at the top with \<open>x\<close>\<close> |
658 have "m \<noteq> {}" |
709 have "m \<noteq> {}" |
659 proof |
710 proof |
660 assume "m = {}" |
711 assume "m = {}" |
661 moreover have "Well_order {(x, x)}" |
712 moreover have "Well_order {(x, x)}" |
662 by (simp add: order_on_defs refl_on_def trans_def antisym_def total_on_def Field_def) |
713 by (simp add: order_on_defs refl_on_def trans_def antisym_def total_on_def Field_def) |
663 ultimately show False using max |
714 ultimately show False using max |
664 by (auto simp: I_def init_seg_of_def simp del: Field_insert) |
715 by (auto simp: I_def init_seg_of_def simp del: Field_insert) |
665 qed |
716 qed |
666 hence "Field m \<noteq> {}" by(auto simp:Field_def) |
717 then have "Field m \<noteq> {}" by (auto simp: Field_def) |
667 moreover have "wf (m - Id)" using \<open>Well_order m\<close> |
718 moreover have "wf (m - Id)" |
668 by (simp add: well_order_on_def) |
719 using \<open>Well_order m\<close> by (simp add: well_order_on_def) |
669 \<comment>\<open>The extension of m by x:\<close> |
720 \<comment>\<open>The extension of \<open>m\<close> by \<open>x\<close>:\<close> |
670 let ?s = "{(a, x) | a. a \<in> Field m}" |
721 let ?s = "{(a, x) | a. a \<in> Field m}" |
671 let ?m = "insert (x, x) m \<union> ?s" |
722 let ?m = "insert (x, x) m \<union> ?s" |
672 have Fm: "Field ?m = insert x (Field m)" |
723 have Fm: "Field ?m = insert x (Field m)" |
673 by (auto simp: Field_def) |
724 by (auto simp: Field_def) |
674 have "Refl m" and "trans m" and "antisym m" and "Total m" and "wf (m - Id)" |
725 have "Refl m" and "trans m" and "antisym m" and "Total m" and "wf (m - Id)" |
675 using \<open>Well_order m\<close> by (simp_all add: order_on_defs) |
726 using \<open>Well_order m\<close> by (simp_all add: order_on_defs) |
676 \<comment>\<open>We show that the extension is a well-order\<close> |
727 \<comment>\<open>We show that the extension is a well-order\<close> |
677 have "Refl ?m" using \<open>Refl m\<close> Fm unfolding refl_on_def by blast |
728 have "Refl ?m" |
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729 using \<open>Refl m\<close> Fm unfolding refl_on_def by blast |
678 moreover have "trans ?m" using \<open>trans m\<close> and \<open>x \<notin> Field m\<close> |
730 moreover have "trans ?m" using \<open>trans m\<close> and \<open>x \<notin> Field m\<close> |
679 unfolding trans_def Field_def by blast |
731 unfolding trans_def Field_def by blast |
680 moreover have "antisym ?m" using \<open>antisym m\<close> and \<open>x \<notin> Field m\<close> |
732 moreover have "antisym ?m" |
681 unfolding antisym_def Field_def by blast |
733 using \<open>antisym m\<close> and \<open>x \<notin> Field m\<close> unfolding antisym_def Field_def by blast |
682 moreover have "Total ?m" using \<open>Total m\<close> and Fm by (auto simp: total_on_def) |
734 moreover have "Total ?m" |
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735 using \<open>Total m\<close> and Fm by (auto simp: total_on_def) |
683 moreover have "wf (?m - Id)" |
736 moreover have "wf (?m - Id)" |
684 proof - |
737 proof - |
685 have "wf ?s" using \<open>x \<notin> Field m\<close> |
738 have "wf ?s" |
686 by (auto simp: wf_eq_minimal Field_def Bex_def) |
739 using \<open>x \<notin> Field m\<close> by (auto simp: wf_eq_minimal Field_def Bex_def) |
687 thus ?thesis using \<open>wf (m - Id)\<close> and \<open>x \<notin> Field m\<close> |
740 then show ?thesis |
688 wf_subset [OF \<open>wf ?s\<close> Diff_subset] |
741 using \<open>wf (m - Id)\<close> and \<open>x \<notin> Field m\<close> wf_subset [OF \<open>wf ?s\<close> Diff_subset] |
689 by (auto simp: Un_Diff Field_def intro: wf_Un) |
742 by (auto simp: Un_Diff Field_def intro: wf_Un) |
690 qed |
743 qed |
691 ultimately have "Well_order ?m" by (simp add: order_on_defs) |
744 ultimately have "Well_order ?m" |
692 \<comment>\<open>We show that the extension is above m\<close> |
745 by (simp add: order_on_defs) |
693 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> |
746 \<comment>\<open>We show that the extension is above \<open>m\<close>\<close> |
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747 moreover have "(m, ?m) \<in> I" |
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748 using \<open>Well_order ?m\<close> and \<open>Well_order m\<close> and \<open>x \<notin> Field m\<close> |
694 by (fastforce simp: I_def init_seg_of_def Field_def) |
749 by (fastforce simp: I_def init_seg_of_def Field_def) |
695 ultimately |
750 ultimately |
696 \<comment>\<open>This contradicts maximality of m:\<close> |
751 \<comment>\<open>This contradicts maximality of \<open>m\<close>:\<close> |
697 have False using max and \<open>x \<notin> Field m\<close> unfolding Field_def by blast |
752 show False |
698 } |
753 using max and \<open>x \<notin> Field m\<close> unfolding Field_def by blast |
699 hence "Field m = UNIV" by auto |
754 qed |
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755 then have "Field m = UNIV" by auto |
700 with \<open>Well_order m\<close> show ?thesis by blast |
756 with \<open>Well_order m\<close> show ?thesis by blast |
701 qed |
757 qed |
702 |
758 |
703 corollary well_order_on: "\<exists>r::'a rel. well_order_on A r" |
759 corollary well_order_on: "\<exists>r::'a rel. well_order_on A r" |
704 proof - |
760 proof - |
705 obtain r::"'a rel" where wo: "Well_order r" and univ: "Field r = UNIV" |
761 obtain r :: "'a rel" where wo: "Well_order r" and univ: "Field r = UNIV" |
706 using well_ordering [where 'a = "'a"] by blast |
762 using well_ordering [where 'a = "'a"] by blast |
707 let ?r = "{(x, y). x \<in> A \<and> y \<in> A \<and> (x, y) \<in> r}" |
763 let ?r = "{(x, y). x \<in> A \<and> y \<in> A \<and> (x, y) \<in> r}" |
708 have 1: "Field ?r = A" using wo univ |
764 have 1: "Field ?r = A" |
709 by (fastforce simp: Field_def order_on_defs refl_on_def) |
765 using wo univ by (fastforce simp: Field_def order_on_defs refl_on_def) |
710 have "Refl r" and "trans r" and "antisym r" and "Total r" and "wf (r - Id)" |
766 from \<open>Well_order r\<close> have "Refl r" "trans r" "antisym r" "Total r" "wf (r - Id)" |
711 using \<open>Well_order r\<close> by (simp_all add: order_on_defs) |
767 by (simp_all add: order_on_defs) |
712 have "Refl ?r" using \<open>Refl r\<close> by (auto simp: refl_on_def 1 univ) |
768 from \<open>Refl r\<close> have "Refl ?r" |
713 moreover have "trans ?r" using \<open>trans r\<close> |
769 by (auto simp: refl_on_def 1 univ) |
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770 moreover from \<open>trans r\<close> have "trans ?r" |
714 unfolding trans_def by blast |
771 unfolding trans_def by blast |
715 moreover have "antisym ?r" using \<open>antisym r\<close> |
772 moreover from \<open>antisym r\<close> have "antisym ?r" |
716 unfolding antisym_def by blast |
773 unfolding antisym_def by blast |
717 moreover have "Total ?r" using \<open>Total r\<close> by (simp add:total_on_def 1 univ) |
774 moreover from \<open>Total r\<close> have "Total ?r" |
718 moreover have "wf (?r - Id)" by (rule wf_subset [OF \<open>wf (r - Id)\<close>]) blast |
775 by (simp add:total_on_def 1 univ) |
719 ultimately have "Well_order ?r" by (simp add: order_on_defs) |
776 moreover have "wf (?r - Id)" |
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777 by (rule wf_subset [OF \<open>wf (r - Id)\<close>]) blast |
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778 ultimately have "Well_order ?r" |
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779 by (simp add: order_on_defs) |
720 with 1 show ?thesis by auto |
780 with 1 show ?thesis by auto |
721 qed |
781 qed |
722 |
782 |
723 (* Move this to Hilbert Choice and wfrec to Wellfounded*) |
783 (* Move this to Hilbert Choice and wfrec to Wellfounded*) |
724 |
784 |
725 lemma wfrec_def_adm: "f \<equiv> wfrec R F \<Longrightarrow> wf R \<Longrightarrow> adm_wf R F \<Longrightarrow> f = F f" |
785 lemma wfrec_def_adm: "f \<equiv> wfrec R F \<Longrightarrow> wf R \<Longrightarrow> adm_wf R F \<Longrightarrow> f = F f" |
726 using wfrec_fixpoint by simp |
786 using wfrec_fixpoint by simp |
727 |
787 |
728 lemma dependent_wf_choice: |
788 lemma dependent_wf_choice: |
729 fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" |
789 fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" |
730 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" |
790 assumes "wf R" |
731 assumes P: "\<And>x f. (\<And>y. (y, x) \<in> R \<Longrightarrow> P f y (f y)) \<Longrightarrow> \<exists>r. P f x r" |
791 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" |
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792 and P: "\<And>x f. (\<And>y. (y, x) \<in> R \<Longrightarrow> P f y (f y)) \<Longrightarrow> \<exists>r. P f x r" |
732 shows "\<exists>f. \<forall>x. P f x (f x)" |
793 shows "\<exists>f. \<forall>x. P f x (f x)" |
733 proof (intro exI allI) |
794 proof (intro exI allI) |
734 fix x |
795 fix x |
735 define f where "f \<equiv> wfrec R (\<lambda>f x. SOME r. P f x r)" |
796 define f where "f \<equiv> wfrec R (\<lambda>f x. SOME r. P f x r)" |
736 from \<open>wf R\<close> show "P f x (f x)" |
797 from \<open>wf R\<close> show "P f x (f x)" |
737 proof (induct x) |
798 proof (induct x) |
738 fix x assume "\<And>y. (y, x) \<in> R \<Longrightarrow> P f y (f y)" |
799 case (less x) |
739 show "P f x (f x)" |
800 show "P f x (f x)" |
740 proof (subst (2) wfrec_def_adm[OF f_def \<open>wf R\<close>]) |
801 proof (subst (2) wfrec_def_adm[OF f_def \<open>wf R\<close>]) |
741 show "adm_wf R (\<lambda>f x. SOME r. P f x r)" |
802 show "adm_wf R (\<lambda>f x. SOME r. P f x r)" |
742 by (auto simp add: adm_wf_def intro!: arg_cong[where f=Eps] ext adm) |
803 by (auto simp add: adm_wf_def intro!: arg_cong[where f=Eps] ext adm) |
743 show "P f x (Eps (P f x))" |
804 show "P f x (Eps (P f x))" |