author | nipkow |
Fri, 27 Mar 2020 12:28:05 +0100 | |
changeset 71593 | 71579bd59cd4 |
parent 71544 | 66bc4b668d6e |
child 71695 | 65489718f4dc |
permissions | -rw-r--r-- |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
1 |
(* Title: HOL/Hilbert_Choice.thy |
32988 | 2 |
Author: Lawrence C Paulson, Tobias Nipkow |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
3 |
Author: Viorel Preoteasa (Results about complete distributive lattices) |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
4 |
Copyright 2001 University of Cambridge |
12023 | 5 |
*) |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
6 |
|
60758 | 7 |
section \<open>Hilbert's Epsilon-Operator and the Axiom of Choice\<close> |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
8 |
|
15131 | 9 |
theory Hilbert_Choice |
63612 | 10 |
imports Wellfounded |
69913 | 11 |
keywords "specification" :: thy_goal_defn |
15131 | 12 |
begin |
12298 | 13 |
|
60758 | 14 |
subsection \<open>Hilbert's epsilon\<close> |
12298 | 15 |
|
63612 | 16 |
axiomatization Eps :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" |
17 |
where someI: "P x \<Longrightarrow> P (Eps P)" |
|
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
18 |
|
14872
3f2144aebd76
improved symbolic syntax of Eps: \<some> for mode "epsilon";
wenzelm
parents:
14760
diff
changeset
|
19 |
syntax (epsilon) |
63612 | 20 |
"_Eps" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a" ("(3\<some>_./ _)" [0, 10] 10) |
62521 | 21 |
syntax (input) |
63612 | 22 |
"_Eps" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a" ("(3@ _./ _)" [0, 10] 10) |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
23 |
syntax |
63612 | 24 |
"_Eps" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a" ("(3SOME _./ _)" [0, 10] 10) |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
25 |
translations |
63612 | 26 |
"SOME x. P" \<rightleftharpoons> "CONST Eps (\<lambda>x. P)" |
13763
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13585
diff
changeset
|
27 |
|
60758 | 28 |
print_translation \<open> |
69593 | 29 |
[(\<^const_syntax>\<open>Eps\<close>, fn _ => fn [Abs abs] => |
42284 | 30 |
let val (x, t) = Syntax_Trans.atomic_abs_tr' abs |
69593 | 31 |
in Syntax.const \<^syntax_const>\<open>_Eps\<close> $ x $ t end)] |
61799 | 32 |
\<close> \<comment> \<open>to avoid eta-contraction of body\<close> |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
33 |
|
65815 | 34 |
definition inv_into :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where |
35 |
"inv_into A f = (\<lambda>x. SOME y. y \<in> A \<and> f y = x)" |
|
11454
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset
|
36 |
|
65815 | 37 |
lemma inv_into_def2: "inv_into A f x = (SOME y. y \<in> A \<and> f y = x)" |
38 |
by(simp add: inv_into_def) |
|
39 |
||
40 |
abbreviation inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where |
|
41 |
"inv \<equiv> inv_into UNIV" |
|
14760 | 42 |
|
43 |
||
60758 | 44 |
subsection \<open>Hilbert's Epsilon-operator\<close> |
14760 | 45 |
|
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69913
diff
changeset
|
46 |
lemma Eps_cong: |
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69913
diff
changeset
|
47 |
assumes "\<And>x. P x = Q x" |
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69913
diff
changeset
|
48 |
shows "Eps P = Eps Q" |
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69913
diff
changeset
|
49 |
using ext[of P Q, OF assms] by simp |
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69913
diff
changeset
|
50 |
|
63612 | 51 |
text \<open> |
52 |
Easier to apply than \<open>someI\<close> if the witness comes from an |
|
53 |
existential formula. |
|
54 |
\<close> |
|
55 |
lemma someI_ex [elim?]: "\<exists>x. P x \<Longrightarrow> P (SOME x. P x)" |
|
56 |
apply (erule exE) |
|
57 |
apply (erule someI) |
|
58 |
done |
|
14760 | 59 |
|
71544 | 60 |
lemma some_eq_imp: |
61 |
assumes "Eps P = a" "P b" shows "P a" |
|
62 |
using assms someI_ex by force |
|
63 |
||
63612 | 64 |
text \<open> |
65 |
Easier to apply than \<open>someI\<close> because the conclusion has only one |
|
69593 | 66 |
occurrence of \<^term>\<open>P\<close>. |
63612 | 67 |
\<close> |
68 |
lemma someI2: "P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (SOME x. P x)" |
|
60974
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
69 |
by (blast intro: someI) |
14760 | 70 |
|
63612 | 71 |
text \<open> |
72 |
Easier to apply than \<open>someI2\<close> if the witness comes from an |
|
73 |
existential formula. |
|
74 |
\<close> |
|
75 |
lemma someI2_ex: "\<exists>a. P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (SOME x. P x)" |
|
60974
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
76 |
by (blast intro: someI2) |
14760 | 77 |
|
63612 | 78 |
lemma someI2_bex: "\<exists>a\<in>A. P a \<Longrightarrow> (\<And>x. x \<in> A \<and> P x \<Longrightarrow> Q x) \<Longrightarrow> Q (SOME x. x \<in> A \<and> P x)" |
79 |
by (blast intro: someI2) |
|
80 |
||
81 |
lemma some_equality [intro]: "P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> x = a) \<Longrightarrow> (SOME x. P x) = a" |
|
82 |
by (blast intro: someI2) |
|
14760 | 83 |
|
63629 | 84 |
lemma some1_equality: "\<exists>!x. P x \<Longrightarrow> P a \<Longrightarrow> (SOME x. P x) = a" |
63612 | 85 |
by blast |
14760 | 86 |
|
63612 | 87 |
lemma some_eq_ex: "P (SOME x. P x) \<longleftrightarrow> (\<exists>x. P x)" |
88 |
by (blast intro: someI) |
|
14760 | 89 |
|
59000 | 90 |
lemma some_in_eq: "(SOME x. x \<in> A) \<in> A \<longleftrightarrow> A \<noteq> {}" |
91 |
unfolding ex_in_conv[symmetric] by (rule some_eq_ex) |
|
92 |
||
63612 | 93 |
lemma some_eq_trivial [simp]: "(SOME y. y = x) = x" |
94 |
by (rule some_equality) (rule refl) |
|
14760 | 95 |
|
63612 | 96 |
lemma some_sym_eq_trivial [simp]: "(SOME y. x = y) = x" |
97 |
apply (rule some_equality) |
|
98 |
apply (rule refl) |
|
99 |
apply (erule sym) |
|
100 |
done |
|
14760 | 101 |
|
102 |
||
63612 | 103 |
subsection \<open>Axiom of Choice, Proved Using the Description Operator\<close> |
14760 | 104 |
|
63612 | 105 |
lemma choice: "\<forall>x. \<exists>y. Q x y \<Longrightarrow> \<exists>f. \<forall>x. Q x (f x)" |
106 |
by (fast elim: someI) |
|
14760 | 107 |
|
63612 | 108 |
lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y \<Longrightarrow> \<exists>f. \<forall>x\<in>S. Q x (f x)" |
109 |
by (fast elim: someI) |
|
14760 | 110 |
|
50105 | 111 |
lemma choice_iff: "(\<forall>x. \<exists>y. Q x y) \<longleftrightarrow> (\<exists>f. \<forall>x. Q x (f x))" |
63612 | 112 |
by (fast elim: someI) |
50105 | 113 |
|
114 |
lemma choice_iff': "(\<forall>x. P x \<longrightarrow> (\<exists>y. Q x y)) \<longleftrightarrow> (\<exists>f. \<forall>x. P x \<longrightarrow> Q x (f x))" |
|
63612 | 115 |
by (fast elim: someI) |
50105 | 116 |
|
117 |
lemma bchoice_iff: "(\<forall>x\<in>S. \<exists>y. Q x y) \<longleftrightarrow> (\<exists>f. \<forall>x\<in>S. Q x (f x))" |
|
63612 | 118 |
by (fast elim: someI) |
50105 | 119 |
|
120 |
lemma bchoice_iff': "(\<forall>x\<in>S. P x \<longrightarrow> (\<exists>y. Q x y)) \<longleftrightarrow> (\<exists>f. \<forall>x\<in>S. P x \<longrightarrow> Q x (f x))" |
|
63612 | 121 |
by (fast elim: someI) |
14760 | 122 |
|
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56740
diff
changeset
|
123 |
lemma dependent_nat_choice: |
63612 | 124 |
assumes 1: "\<exists>x. P 0 x" |
125 |
and 2: "\<And>x n. P n x \<Longrightarrow> \<exists>y. P (Suc n) y \<and> Q n x y" |
|
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57275
diff
changeset
|
126 |
shows "\<exists>f. \<forall>n. P n (f n) \<and> Q n (f n) (f (Suc n))" |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56740
diff
changeset
|
127 |
proof (intro exI allI conjI) |
63040 | 128 |
fix n |
129 |
define f where "f = rec_nat (SOME x. P 0 x) (\<lambda>n x. SOME y. P (Suc n) y \<and> Q n x y)" |
|
63612 | 130 |
then have "P 0 (f 0)" "\<And>n. P n (f n) \<Longrightarrow> P (Suc n) (f (Suc n)) \<and> Q n (f n) (f (Suc n))" |
131 |
using someI_ex[OF 1] someI_ex[OF 2] by simp_all |
|
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57275
diff
changeset
|
132 |
then show "P n (f n)" "Q n (f n) (f (Suc n))" |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56740
diff
changeset
|
133 |
by (induct n) auto |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56740
diff
changeset
|
134 |
qed |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56740
diff
changeset
|
135 |
|
68975
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68802
diff
changeset
|
136 |
lemma finite_subset_Union: |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68802
diff
changeset
|
137 |
assumes "finite A" "A \<subseteq> \<Union>\<B>" |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68802
diff
changeset
|
138 |
obtains \<F> where "finite \<F>" "\<F> \<subseteq> \<B>" "A \<subseteq> \<Union>\<F>" |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68802
diff
changeset
|
139 |
proof - |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68802
diff
changeset
|
140 |
have "\<forall>x\<in>A. \<exists>B\<in>\<B>. x\<in>B" |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68802
diff
changeset
|
141 |
using assms by blast |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68802
diff
changeset
|
142 |
then obtain f where f: "\<And>x. x \<in> A \<Longrightarrow> f x \<in> \<B> \<and> x \<in> f x" |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68802
diff
changeset
|
143 |
by (auto simp add: bchoice_iff Bex_def) |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68802
diff
changeset
|
144 |
show thesis |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68802
diff
changeset
|
145 |
proof |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68802
diff
changeset
|
146 |
show "finite (f ` A)" |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68802
diff
changeset
|
147 |
using assms by auto |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68802
diff
changeset
|
148 |
qed (use f in auto) |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68802
diff
changeset
|
149 |
qed |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68802
diff
changeset
|
150 |
|
58074 | 151 |
|
60758 | 152 |
subsection \<open>Function Inverse\<close> |
14760 | 153 |
|
63612 | 154 |
lemma inv_def: "inv f = (\<lambda>y. SOME x. f x = y)" |
155 |
by (simp add: inv_into_def) |
|
33014 | 156 |
|
63612 | 157 |
lemma inv_into_into: "x \<in> f ` A \<Longrightarrow> inv_into A f x \<in> A" |
158 |
by (simp add: inv_into_def) (fast intro: someI2) |
|
14760 | 159 |
|
63612 | 160 |
lemma inv_identity [simp]: "inv (\<lambda>a. a) = (\<lambda>a. a)" |
63365 | 161 |
by (simp add: inv_def) |
162 |
||
63612 | 163 |
lemma inv_id [simp]: "inv id = id" |
63365 | 164 |
by (simp add: id_def) |
14760 | 165 |
|
63612 | 166 |
lemma inv_into_f_f [simp]: "inj_on f A \<Longrightarrow> x \<in> A \<Longrightarrow> inv_into A f (f x) = x" |
167 |
by (simp add: inv_into_def inj_on_def) (blast intro: someI2) |
|
14760 | 168 |
|
63612 | 169 |
lemma inv_f_f: "inj f \<Longrightarrow> inv f (f x) = x" |
170 |
by simp |
|
32988 | 171 |
|
67613 | 172 |
lemma f_inv_into_f: "y \<in> f`A \<Longrightarrow> f (inv_into A f y) = y" |
63612 | 173 |
by (simp add: inv_into_def) (fast intro: someI2) |
32988 | 174 |
|
63612 | 175 |
lemma inv_into_f_eq: "inj_on f A \<Longrightarrow> x \<in> A \<Longrightarrow> f x = y \<Longrightarrow> inv_into A f y = x" |
176 |
by (erule subst) (fast intro: inv_into_f_f) |
|
32988 | 177 |
|
63612 | 178 |
lemma inv_f_eq: "inj f \<Longrightarrow> f x = y \<Longrightarrow> inv f y = x" |
179 |
by (simp add:inv_into_f_eq) |
|
32988 | 180 |
|
63612 | 181 |
lemma inj_imp_inv_eq: "inj f \<Longrightarrow> \<forall>x. f (g x) = x \<Longrightarrow> inv f = g" |
44921 | 182 |
by (blast intro: inv_into_f_eq) |
14760 | 183 |
|
63612 | 184 |
text \<open>But is it useful?\<close> |
14760 | 185 |
lemma inj_transfer: |
63612 | 186 |
assumes inj: "inj f" |
187 |
and minor: "\<And>y. y \<in> range f \<Longrightarrow> P (inv f y)" |
|
14760 | 188 |
shows "P x" |
189 |
proof - |
|
190 |
have "f x \<in> range f" by auto |
|
63612 | 191 |
then have "P(inv f (f x))" by (rule minor) |
192 |
then show "P x" by (simp add: inv_into_f_f [OF inj]) |
|
14760 | 193 |
qed |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
194 |
|
63612 | 195 |
lemma inj_iff: "inj f \<longleftrightarrow> inv f \<circ> f = id" |
196 |
by (simp add: o_def fun_eq_iff) (blast intro: inj_on_inverseI inv_into_f_f) |
|
14760 | 197 |
|
63612 | 198 |
lemma inv_o_cancel[simp]: "inj f \<Longrightarrow> inv f \<circ> f = id" |
199 |
by (simp add: inj_iff) |
|
200 |
||
201 |
lemma o_inv_o_cancel[simp]: "inj f \<Longrightarrow> g \<circ> inv f \<circ> f = g" |
|
202 |
by (simp add: comp_assoc) |
|
23433 | 203 |
|
63612 | 204 |
lemma inv_into_image_cancel[simp]: "inj_on f A \<Longrightarrow> S \<subseteq> A \<Longrightarrow> inv_into A f ` f ` S = S" |
205 |
by (fastforce simp: image_def) |
|
23433 | 206 |
|
63612 | 207 |
lemma inj_imp_surj_inv: "inj f \<Longrightarrow> surj (inv f)" |
208 |
by (blast intro!: surjI inv_into_f_f) |
|
32988 | 209 |
|
63612 | 210 |
lemma surj_f_inv_f: "surj f \<Longrightarrow> f (inv f y) = y" |
211 |
by (simp add: f_inv_into_f) |
|
14760 | 212 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67613
diff
changeset
|
213 |
lemma bij_inv_eq_iff: "bij p \<Longrightarrow> x = inv p y \<longleftrightarrow> p x = y" |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67613
diff
changeset
|
214 |
using surj_f_inv_f[of p] by (auto simp add: bij_def) |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67613
diff
changeset
|
215 |
|
33057 | 216 |
lemma inv_into_injective: |
217 |
assumes eq: "inv_into A f x = inv_into A f y" |
|
63612 | 218 |
and x: "x \<in> f`A" |
219 |
and y: "y \<in> f`A" |
|
220 |
shows "x = y" |
|
14760 | 221 |
proof - |
63612 | 222 |
from eq have "f (inv_into A f x) = f (inv_into A f y)" |
223 |
by simp |
|
224 |
with x y show ?thesis |
|
225 |
by (simp add: f_inv_into_f) |
|
14760 | 226 |
qed |
227 |
||
63612 | 228 |
lemma inj_on_inv_into: "B \<subseteq> f`A \<Longrightarrow> inj_on (inv_into A f) B" |
229 |
by (blast intro: inj_onI dest: inv_into_injective injD) |
|
32988 | 230 |
|
63612 | 231 |
lemma bij_betw_inv_into: "bij_betw f A B \<Longrightarrow> bij_betw (inv_into A f) B A" |
232 |
by (auto simp add: bij_betw_def inj_on_inv_into) |
|
14760 | 233 |
|
63612 | 234 |
lemma surj_imp_inj_inv: "surj f \<Longrightarrow> inj (inv f)" |
235 |
by (simp add: inj_on_inv_into) |
|
14760 | 236 |
|
63612 | 237 |
lemma surj_iff: "surj f \<longleftrightarrow> f \<circ> inv f = id" |
238 |
by (auto intro!: surjI simp: surj_f_inv_f fun_eq_iff[where 'b='a]) |
|
40702 | 239 |
|
240 |
lemma surj_iff_all: "surj f \<longleftrightarrow> (\<forall>x. f (inv f x) = x)" |
|
63612 | 241 |
by (simp add: o_def surj_iff fun_eq_iff) |
14760 | 242 |
|
63612 | 243 |
lemma surj_imp_inv_eq: "surj f \<Longrightarrow> \<forall>x. g (f x) = x \<Longrightarrow> inv f = g" |
244 |
apply (rule ext) |
|
245 |
apply (drule_tac x = "inv f x" in spec) |
|
246 |
apply (simp add: surj_f_inv_f) |
|
247 |
done |
|
14760 | 248 |
|
63612 | 249 |
lemma bij_imp_bij_inv: "bij f \<Longrightarrow> bij (inv f)" |
250 |
by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv) |
|
12372 | 251 |
|
63612 | 252 |
lemma inv_equality: "(\<And>x. g (f x) = x) \<Longrightarrow> (\<And>y. f (g y) = y) \<Longrightarrow> inv f = g" |
253 |
by (rule ext) (auto simp add: inv_into_def) |
|
254 |
||
255 |
lemma inv_inv_eq: "bij f \<Longrightarrow> inv (inv f) = f" |
|
256 |
by (rule inv_equality) (auto simp add: bij_def surj_f_inv_f) |
|
14760 | 257 |
|
63612 | 258 |
text \<open> |
259 |
\<open>bij (inv f)\<close> implies little about \<open>f\<close>. Consider \<open>f :: bool \<Rightarrow> bool\<close> such |
|
260 |
that \<open>f True = f False = True\<close>. Then it ia consistent with axiom \<open>someI\<close> |
|
261 |
that \<open>inv f\<close> could be any function at all, including the identity function. |
|
262 |
If \<open>inv f = id\<close> then \<open>inv f\<close> is a bijection, but \<open>inj f\<close>, \<open>surj f\<close> and \<open>inv |
|
263 |
(inv f) = f\<close> all fail. |
|
264 |
\<close> |
|
14760 | 265 |
|
33057 | 266 |
lemma inv_into_comp: |
63612 | 267 |
"inj_on f (g ` A) \<Longrightarrow> inj_on g A \<Longrightarrow> x \<in> f ` g ` A \<Longrightarrow> |
268 |
inv_into A (f \<circ> g) x = (inv_into A g \<circ> inv_into (g ` A) f) x" |
|
269 |
apply (rule inv_into_f_eq) |
|
270 |
apply (fast intro: comp_inj_on) |
|
271 |
apply (simp add: inv_into_into) |
|
272 |
apply (simp add: f_inv_into_f inv_into_into) |
|
273 |
done |
|
32988 | 274 |
|
63612 | 275 |
lemma o_inv_distrib: "bij f \<Longrightarrow> bij g \<Longrightarrow> inv (f \<circ> g) = inv g \<circ> inv f" |
276 |
by (rule inv_equality) (auto simp add: bij_def surj_f_inv_f) |
|
14760 | 277 |
|
63807 | 278 |
lemma image_f_inv_f: "surj f \<Longrightarrow> f ` (inv f ` A) = A" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
61859
diff
changeset
|
279 |
by (simp add: surj_f_inv_f image_comp comp_def) |
14760 | 280 |
|
63612 | 281 |
lemma image_inv_f_f: "inj f \<Longrightarrow> inv f ` (f ` A) = A" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
61859
diff
changeset
|
282 |
by simp |
14760 | 283 |
|
63612 | 284 |
lemma bij_image_Collect_eq: "bij f \<Longrightarrow> f ` Collect P = {y. P (inv f y)}" |
285 |
apply auto |
|
286 |
apply (force simp add: bij_is_inj) |
|
287 |
apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric]) |
|
288 |
done |
|
14760 | 289 |
|
63612 | 290 |
lemma bij_vimage_eq_inv_image: "bij f \<Longrightarrow> f -` A = inv f ` A" |
291 |
apply (auto simp add: bij_is_surj [THEN surj_f_inv_f]) |
|
292 |
apply (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric]) |
|
293 |
done |
|
14760 | 294 |
|
68610 | 295 |
lemma inv_fn_o_fn_is_id: |
296 |
fixes f::"'a \<Rightarrow> 'a" |
|
297 |
assumes "bij f" |
|
298 |
shows "((inv f)^^n) o (f^^n) = (\<lambda>x. x)" |
|
299 |
proof - |
|
300 |
have "((inv f)^^n)((f^^n) x) = x" for x n |
|
301 |
proof (induction n) |
|
302 |
case (Suc n) |
|
303 |
have *: "(inv f) (f y) = y" for y |
|
304 |
by (simp add: assms bij_is_inj) |
|
305 |
have "(inv f ^^ Suc n) ((f ^^ Suc n) x) = (inv f^^n) (inv f (f ((f^^n) x)))" |
|
306 |
by (simp add: funpow_swap1) |
|
307 |
also have "... = (inv f^^n) ((f^^n) x)" |
|
308 |
using * by auto |
|
309 |
also have "... = x" using Suc.IH by auto |
|
310 |
finally show ?case by simp |
|
311 |
qed (auto) |
|
312 |
then show ?thesis unfolding o_def by blast |
|
313 |
qed |
|
314 |
||
315 |
lemma fn_o_inv_fn_is_id: |
|
316 |
fixes f::"'a \<Rightarrow> 'a" |
|
317 |
assumes "bij f" |
|
318 |
shows "(f^^n) o ((inv f)^^n) = (\<lambda>x. x)" |
|
319 |
proof - |
|
320 |
have "(f^^n) (((inv f)^^n) x) = x" for x n |
|
321 |
proof (induction n) |
|
322 |
case (Suc n) |
|
323 |
have *: "f(inv f y) = y" for y |
|
324 |
using bij_inv_eq_iff[OF assms] by auto |
|
325 |
have "(f ^^ Suc n) ((inv f ^^ Suc n) x) = (f^^n) (f (inv f ((inv f^^n) x)))" |
|
326 |
by (simp add: funpow_swap1) |
|
327 |
also have "... = (f^^n) ((inv f^^n) x)" |
|
328 |
using * by auto |
|
329 |
also have "... = x" using Suc.IH by auto |
|
330 |
finally show ?case by simp |
|
331 |
qed (auto) |
|
332 |
then show ?thesis unfolding o_def by blast |
|
333 |
qed |
|
334 |
||
335 |
lemma inv_fn: |
|
336 |
fixes f::"'a \<Rightarrow> 'a" |
|
337 |
assumes "bij f" |
|
338 |
shows "inv (f^^n) = ((inv f)^^n)" |
|
339 |
proof - |
|
340 |
have "inv (f^^n) x = ((inv f)^^n) x" for x |
|
341 |
apply (rule inv_into_f_eq, auto simp add: inj_fn[OF bij_is_inj[OF assms]]) |
|
342 |
using fn_o_inv_fn_is_id[OF assms, of n, THEN fun_cong] by (simp) |
|
343 |
then show ?thesis by auto |
|
344 |
qed |
|
345 |
||
346 |
lemma mono_inv: |
|
347 |
fixes f::"'a::linorder \<Rightarrow> 'b::linorder" |
|
348 |
assumes "mono f" "bij f" |
|
349 |
shows "mono (inv f)" |
|
350 |
proof |
|
351 |
fix x y::'b assume "x \<le> y" |
|
352 |
from \<open>bij f\<close> obtain a b where x: "x = f a" and y: "y = f b" by(fastforce simp: bij_def surj_def) |
|
353 |
show "inv f x \<le> inv f y" |
|
354 |
proof (rule le_cases) |
|
355 |
assume "a \<le> b" |
|
356 |
thus ?thesis using \<open>bij f\<close> x y by(simp add: bij_def inv_f_f) |
|
357 |
next |
|
358 |
assume "b \<le> a" |
|
359 |
hence "f b \<le> f a" by(rule monoD[OF \<open>mono f\<close>]) |
|
360 |
hence "y \<le> x" using x y by simp |
|
361 |
hence "x = y" using \<open>x \<le> y\<close> by auto |
|
362 |
thus ?thesis by simp |
|
363 |
qed |
|
364 |
qed |
|
365 |
||
366 |
lemma mono_bij_Inf: |
|
367 |
fixes f :: "'a::complete_linorder \<Rightarrow> 'b::complete_linorder" |
|
368 |
assumes "mono f" "bij f" |
|
369 |
shows "f (Inf A) = Inf (f`A)" |
|
370 |
proof - |
|
371 |
have "surj f" using \<open>bij f\<close> by (auto simp: bij_betw_def) |
|
372 |
have *: "(inv f) (Inf (f`A)) \<le> Inf ((inv f)`(f`A))" |
|
373 |
using mono_Inf[OF mono_inv[OF assms], of "f`A"] by simp |
|
374 |
have "Inf (f`A) \<le> f (Inf ((inv f)`(f`A)))" |
|
375 |
using monoD[OF \<open>mono f\<close> *] by(simp add: surj_f_inv_f[OF \<open>surj f\<close>]) |
|
376 |
also have "... = f(Inf A)" |
|
377 |
using assms by (simp add: bij_is_inj) |
|
378 |
finally show ?thesis using mono_Inf[OF assms(1), of A] by auto |
|
379 |
qed |
|
380 |
||
31380 | 381 |
lemma finite_fun_UNIVD1: |
382 |
assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)" |
|
63612 | 383 |
and card: "card (UNIV :: 'b set) \<noteq> Suc 0" |
31380 | 384 |
shows "finite (UNIV :: 'a set)" |
385 |
proof - |
|
63630 | 386 |
let ?UNIV_b = "UNIV :: 'b set" |
387 |
from fin have "finite ?UNIV_b" |
|
63612 | 388 |
by (rule finite_fun_UNIVD2) |
63630 | 389 |
with card have "card ?UNIV_b \<ge> Suc (Suc 0)" |
390 |
by (cases "card ?UNIV_b") (auto simp: card_eq_0_iff) |
|
391 |
then have "card ?UNIV_b = Suc (Suc (card ?UNIV_b - Suc (Suc 0)))" |
|
392 |
by simp |
|
63629 | 393 |
then obtain b1 b2 :: 'b where b1b2: "b1 \<noteq> b2" |
394 |
by (auto simp: card_Suc_eq) |
|
63630 | 395 |
from fin have fin': "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" |
63612 | 396 |
by (rule finite_imageI) |
63630 | 397 |
have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)" |
31380 | 398 |
proof (rule UNIV_eq_I) |
399 |
fix x :: 'a |
|
63612 | 400 |
from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" |
401 |
by (simp add: inv_into_def) |
|
402 |
then show "x \<in> range (\<lambda>f::'a \<Rightarrow> 'b. inv f b1)" |
|
403 |
by blast |
|
31380 | 404 |
qed |
63630 | 405 |
with fin' show ?thesis |
63612 | 406 |
by simp |
31380 | 407 |
qed |
14760 | 408 |
|
60758 | 409 |
text \<open> |
54578
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset
|
410 |
Every infinite set contains a countable subset. More precisely we |
61799 | 411 |
show that a set \<open>S\<close> is infinite if and only if there exists an |
412 |
injective function from the naturals into \<open>S\<close>. |
|
54578
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset
|
413 |
|
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset
|
414 |
The ``only if'' direction is harder because it requires the |
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset
|
415 |
construction of a sequence of pairwise different elements of an |
61799 | 416 |
infinite set \<open>S\<close>. The idea is to construct a sequence of |
417 |
non-empty and infinite subsets of \<open>S\<close> obtained by successively |
|
418 |
removing elements of \<open>S\<close>. |
|
60758 | 419 |
\<close> |
54578
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset
|
420 |
|
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset
|
421 |
lemma infinite_countable_subset: |
63629 | 422 |
assumes inf: "\<not> finite S" |
423 |
shows "\<exists>f::nat \<Rightarrow> 'a. inj f \<and> range f \<subseteq> S" |
|
61799 | 424 |
\<comment> \<open>Courtesy of Stephan Merz\<close> |
54578
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset
|
425 |
proof - |
63040 | 426 |
define Sseq where "Sseq = rec_nat S (\<lambda>n T. T - {SOME e. e \<in> T})" |
427 |
define pick where "pick n = (SOME e. e \<in> Sseq n)" for n |
|
63540 | 428 |
have *: "Sseq n \<subseteq> S" "\<not> finite (Sseq n)" for n |
63612 | 429 |
by (induct n) (auto simp: Sseq_def inf) |
63540 | 430 |
then have **: "\<And>n. pick n \<in> Sseq n" |
55811 | 431 |
unfolding pick_def by (subst (asm) finite.simps) (auto simp add: ex_in_conv intro: someI_ex) |
63540 | 432 |
with * have "range pick \<subseteq> S" by auto |
63612 | 433 |
moreover have "pick n \<noteq> pick (n + Suc m)" for m n |
434 |
proof - |
|
63540 | 435 |
have "pick n \<notin> Sseq (n + Suc m)" |
436 |
by (induct m) (auto simp add: Sseq_def pick_def) |
|
63612 | 437 |
with ** show ?thesis by auto |
438 |
qed |
|
439 |
then have "inj pick" |
|
440 |
by (intro linorder_injI) (auto simp add: less_iff_Suc_add) |
|
54578
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset
|
441 |
ultimately show ?thesis by blast |
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset
|
442 |
qed |
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset
|
443 |
|
63629 | 444 |
lemma infinite_iff_countable_subset: "\<not> finite S \<longleftrightarrow> (\<exists>f::nat \<Rightarrow> 'a. inj f \<and> range f \<subseteq> S)" |
61799 | 445 |
\<comment> \<open>Courtesy of Stephan Merz\<close> |
55811 | 446 |
using finite_imageD finite_subset infinite_UNIV_char_0 infinite_countable_subset by auto |
54578
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset
|
447 |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
448 |
lemma image_inv_into_cancel: |
63612 | 449 |
assumes surj: "f`A = A'" |
450 |
and sub: "B' \<subseteq> A'" |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
451 |
shows "f `((inv_into A f)`B') = B'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
452 |
using assms |
63612 | 453 |
proof (auto simp: f_inv_into_f) |
454 |
let ?f' = "inv_into A f" |
|
455 |
fix a' |
|
456 |
assume *: "a' \<in> B'" |
|
457 |
with sub have "a' \<in> A'" by auto |
|
458 |
with surj have "a' = f (?f' a')" |
|
459 |
by (auto simp: f_inv_into_f) |
|
460 |
with * show "a' \<in> f ` (?f' ` B')" by blast |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
461 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
462 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
463 |
lemma inv_into_inv_into_eq: |
63612 | 464 |
assumes "bij_betw f A A'" |
465 |
and a: "a \<in> A" |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
466 |
shows "inv_into A' (inv_into A f) a = f a" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
467 |
proof - |
63612 | 468 |
let ?f' = "inv_into A f" |
469 |
let ?f'' = "inv_into A' ?f'" |
|
470 |
from assms have *: "bij_betw ?f' A' A" |
|
471 |
by (auto simp: bij_betw_inv_into) |
|
472 |
with a obtain a' where a': "a' \<in> A'" "?f' a' = a" |
|
473 |
unfolding bij_betw_def by force |
|
474 |
with a * have "?f'' a = a'" |
|
475 |
by (auto simp: f_inv_into_f bij_betw_def) |
|
476 |
moreover from assms a' have "f a = a'" |
|
477 |
by (auto simp: bij_betw_def) |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
478 |
ultimately show "?f'' a = f a" by simp |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
479 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
480 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
481 |
lemma inj_on_iff_surj: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
482 |
assumes "A \<noteq> {}" |
63629 | 483 |
shows "(\<exists>f. inj_on f A \<and> f ` A \<subseteq> A') \<longleftrightarrow> (\<exists>g. g ` A' = A)" |
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
484 |
proof safe |
63612 | 485 |
fix f |
486 |
assume inj: "inj_on f A" and incl: "f ` A \<subseteq> A'" |
|
487 |
let ?phi = "\<lambda>a' a. a \<in> A \<and> f a = a'" |
|
488 |
let ?csi = "\<lambda>a. a \<in> A" |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
489 |
let ?g = "\<lambda>a'. if a' \<in> f ` A then (SOME a. ?phi a' a) else (SOME a. ?csi a)" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
490 |
have "?g ` A' = A" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
491 |
proof |
63612 | 492 |
show "?g ` A' \<subseteq> A" |
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
493 |
proof clarify |
63612 | 494 |
fix a' |
495 |
assume *: "a' \<in> A'" |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
496 |
show "?g a' \<in> A" |
63612 | 497 |
proof (cases "a' \<in> f ` A") |
498 |
case True |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
499 |
then obtain a where "?phi a' a" by blast |
63612 | 500 |
then have "?phi a' (SOME a. ?phi a' a)" |
501 |
using someI[of "?phi a'" a] by blast |
|
502 |
with True show ?thesis by auto |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
503 |
next |
63612 | 504 |
case False |
505 |
with assms have "?csi (SOME a. ?csi a)" |
|
506 |
using someI_ex[of ?csi] by blast |
|
507 |
with False show ?thesis by auto |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
508 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
509 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
510 |
next |
63612 | 511 |
show "A \<subseteq> ?g ` A'" |
512 |
proof - |
|
513 |
have "?g (f a) = a \<and> f a \<in> A'" if a: "a \<in> A" for a |
|
514 |
proof - |
|
515 |
let ?b = "SOME aa. ?phi (f a) aa" |
|
516 |
from a have "?phi (f a) a" by auto |
|
517 |
then have *: "?phi (f a) ?b" |
|
518 |
using someI[of "?phi(f a)" a] by blast |
|
519 |
then have "?g (f a) = ?b" using a by auto |
|
520 |
moreover from inj * a have "a = ?b" |
|
521 |
by (auto simp add: inj_on_def) |
|
522 |
ultimately have "?g(f a) = a" by simp |
|
523 |
with incl a show ?thesis by auto |
|
524 |
qed |
|
525 |
then show ?thesis by force |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
526 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
527 |
qed |
63612 | 528 |
then show "\<exists>g. g ` A' = A" by blast |
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
529 |
next |
63612 | 530 |
fix g |
531 |
let ?f = "inv_into A' g" |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
532 |
have "inj_on ?f (g ` A')" |
63612 | 533 |
by (auto simp: inj_on_inv_into) |
534 |
moreover have "?f (g a') \<in> A'" if a': "a' \<in> A'" for a' |
|
535 |
proof - |
|
536 |
let ?phi = "\<lambda> b'. b' \<in> A' \<and> g b' = g a'" |
|
537 |
from a' have "?phi a'" by auto |
|
538 |
then have "?phi (SOME b'. ?phi b')" |
|
539 |
using someI[of ?phi] by blast |
|
540 |
then show ?thesis by (auto simp: inv_into_def) |
|
541 |
qed |
|
542 |
ultimately show "\<exists>f. inj_on f (g ` A') \<and> f ` g ` A' \<subseteq> A'" |
|
543 |
by auto |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
544 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
545 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
546 |
lemma Ex_inj_on_UNION_Sigma: |
63629 | 547 |
"\<exists>f. (inj_on f (\<Union>i \<in> I. A i) \<and> f ` (\<Union>i \<in> I. A i) \<subseteq> (SIGMA i : I. A i))" |
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
548 |
proof |
63612 | 549 |
let ?phi = "\<lambda>a i. i \<in> I \<and> a \<in> A i" |
550 |
let ?sm = "\<lambda>a. SOME i. ?phi a i" |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
551 |
let ?f = "\<lambda>a. (?sm a, a)" |
63612 | 552 |
have "inj_on ?f (\<Union>i \<in> I. A i)" |
553 |
by (auto simp: inj_on_def) |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
554 |
moreover |
63612 | 555 |
have "?sm a \<in> I \<and> a \<in> A(?sm a)" if "i \<in> I" and "a \<in> A i" for i a |
556 |
using that someI[of "?phi a" i] by auto |
|
63629 | 557 |
then have "?f ` (\<Union>i \<in> I. A i) \<subseteq> (SIGMA i : I. A i)" |
63612 | 558 |
by auto |
63629 | 559 |
ultimately show "inj_on ?f (\<Union>i \<in> I. A i) \<and> ?f ` (\<Union>i \<in> I. A i) \<subseteq> (SIGMA i : I. A i)" |
63612 | 560 |
by auto |
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
561 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
562 |
|
56608 | 563 |
lemma inv_unique_comp: |
564 |
assumes fg: "f \<circ> g = id" |
|
565 |
and gf: "g \<circ> f = id" |
|
566 |
shows "inv f = g" |
|
567 |
using fg gf inv_equality[of g f] by (auto simp add: fun_eq_iff) |
|
568 |
||
70179
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
569 |
lemma subset_image_inj: |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
570 |
"S \<subseteq> f ` T \<longleftrightarrow> (\<exists>U. U \<subseteq> T \<and> inj_on f U \<and> S = f ` U)" |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
571 |
proof safe |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
572 |
show "\<exists>U\<subseteq>T. inj_on f U \<and> S = f ` U" |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
573 |
if "S \<subseteq> f ` T" |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
574 |
proof - |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
575 |
from that [unfolded subset_image_iff subset_iff] |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
576 |
obtain g where g: "\<And>x. x \<in> S \<Longrightarrow> g x \<in> T \<and> x = f (g x)" |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
577 |
by (auto simp add: image_iff Bex_def choice_iff') |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
578 |
show ?thesis |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
579 |
proof (intro exI conjI) |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
580 |
show "g ` S \<subseteq> T" |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
581 |
by (simp add: g image_subsetI) |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
582 |
show "inj_on f (g ` S)" |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
583 |
using g by (auto simp: inj_on_def) |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
584 |
show "S = f ` (g ` S)" |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
585 |
using g image_subset_iff by auto |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
586 |
qed |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
587 |
qed |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
588 |
qed blast |
269dcea7426c
moved subset_image_inj into Hilbert_Choice
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
589 |
|
56608 | 590 |
|
60758 | 591 |
subsection \<open>Other Consequences of Hilbert's Epsilon\<close> |
14760 | 592 |
|
69593 | 593 |
text \<open>Hilbert's Epsilon and the \<^term>\<open>split\<close> Operator\<close> |
14760 | 594 |
|
63612 | 595 |
text \<open>Looping simprule!\<close> |
596 |
lemma split_paired_Eps: "(SOME x. P x) = (SOME (a, b). P (a, b))" |
|
26347 | 597 |
by simp |
14760 | 598 |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61076
diff
changeset
|
599 |
lemma Eps_case_prod: "Eps (case_prod P) = (SOME xy. P (fst xy) (snd xy))" |
26347 | 600 |
by (simp add: split_def) |
14760 | 601 |
|
63612 | 602 |
lemma Eps_case_prod_eq [simp]: "(SOME (x', y'). x = x' \<and> y = y') = (x, y)" |
26347 | 603 |
by blast |
14760 | 604 |
|
605 |
||
63612 | 606 |
text \<open>A relation is wellfounded iff it has no infinite descending chain.\<close> |
63981 | 607 |
lemma wf_iff_no_infinite_down_chain: "wf r \<longleftrightarrow> (\<nexists>f. \<forall>i. (f (Suc i), f i) \<in> r)" |
608 |
(is "_ \<longleftrightarrow> \<not> ?ex") |
|
609 |
proof |
|
610 |
assume "wf r" |
|
611 |
show "\<not> ?ex" |
|
612 |
proof |
|
613 |
assume ?ex |
|
614 |
then obtain f where f: "(f (Suc i), f i) \<in> r" for i |
|
615 |
by blast |
|
616 |
from \<open>wf r\<close> have minimal: "x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q" for x Q |
|
617 |
by (auto simp: wf_eq_minimal) |
|
618 |
let ?Q = "{w. \<exists>i. w = f i}" |
|
619 |
fix n |
|
620 |
have "f n \<in> ?Q" by blast |
|
621 |
from minimal [OF this] obtain j where "(y, f j) \<in> r \<Longrightarrow> y \<notin> ?Q" for y by blast |
|
622 |
with this [OF \<open>(f (Suc j), f j) \<in> r\<close>] have "f (Suc j) \<notin> ?Q" by simp |
|
623 |
then show False by blast |
|
624 |
qed |
|
625 |
next |
|
626 |
assume "\<not> ?ex" |
|
627 |
then show "wf r" |
|
628 |
proof (rule contrapos_np) |
|
629 |
assume "\<not> wf r" |
|
630 |
then obtain Q x where x: "x \<in> Q" and rec: "z \<in> Q \<Longrightarrow> \<exists>y. (y, z) \<in> r \<and> y \<in> Q" for z |
|
631 |
by (auto simp add: wf_eq_minimal) |
|
632 |
obtain descend :: "nat \<Rightarrow> 'a" |
|
633 |
where descend_0: "descend 0 = x" |
|
634 |
and descend_Suc: "descend (Suc n) = (SOME y. y \<in> Q \<and> (y, descend n) \<in> r)" for n |
|
635 |
by (rule that [of "rec_nat x (\<lambda>_ rec. (SOME y. y \<in> Q \<and> (y, rec) \<in> r))"]) simp_all |
|
636 |
have descend_Q: "descend n \<in> Q" for n |
|
637 |
proof (induct n) |
|
638 |
case 0 |
|
639 |
with x show ?case by (simp only: descend_0) |
|
640 |
next |
|
641 |
case Suc |
|
642 |
then show ?case by (simp only: descend_Suc) (rule someI2_ex; use rec in blast) |
|
643 |
qed |
|
644 |
have "(descend (Suc i), descend i) \<in> r" for i |
|
645 |
by (simp only: descend_Suc) (rule someI2_ex; use descend_Q rec in blast) |
|
646 |
then show "\<exists>f. \<forall>i. (f (Suc i), f i) \<in> r" by blast |
|
647 |
qed |
|
648 |
qed |
|
14760 | 649 |
|
27760 | 650 |
lemma wf_no_infinite_down_chainE: |
63612 | 651 |
assumes "wf r" |
652 |
obtains k where "(f (Suc k), f k) \<notin> r" |
|
653 |
using assms wf_iff_no_infinite_down_chain[of r] by blast |
|
27760 | 654 |
|
655 |
||
63612 | 656 |
text \<open>A dynamically-scoped fact for TFL\<close> |
657 |
lemma tfl_some: "\<forall>P x. P x \<longrightarrow> P (Eps P)" |
|
12298 | 658 |
by (blast intro: someI) |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
659 |
|
12298 | 660 |
|
60758 | 661 |
subsection \<open>An aside: bounded accessible part\<close> |
49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
662 |
|
60758 | 663 |
text \<open>Finite monotone eventually stable sequences\<close> |
49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
664 |
|
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
665 |
lemma finite_mono_remains_stable_implies_strict_prefix: |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
666 |
fixes f :: "nat \<Rightarrow> 'a::order" |
63612 | 667 |
assumes S: "finite (range f)" "mono f" |
668 |
and eq: "\<forall>n. f n = f (Suc n) \<longrightarrow> f (Suc n) = f (Suc (Suc n))" |
|
49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
669 |
shows "\<exists>N. (\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m < f n) \<and> (\<forall>n\<ge>N. f N = f n)" |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
670 |
using assms |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
671 |
proof - |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
672 |
have "\<exists>n. f n = f (Suc n)" |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
673 |
proof (rule ccontr) |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
674 |
assume "\<not> ?thesis" |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
675 |
then have "\<And>n. f n \<noteq> f (Suc n)" by auto |
63612 | 676 |
with \<open>mono f\<close> have "\<And>n. f n < f (Suc n)" |
677 |
by (auto simp: le_less mono_iff_le_Suc) |
|
678 |
with lift_Suc_mono_less_iff[of f] have *: "\<And>n m. n < m \<Longrightarrow> f n < f m" |
|
679 |
by auto |
|
55811 | 680 |
have "inj f" |
681 |
proof (intro injI) |
|
682 |
fix x y |
|
683 |
assume "f x = f y" |
|
63612 | 684 |
then show "x = y" |
685 |
by (cases x y rule: linorder_cases) (auto dest: *) |
|
55811 | 686 |
qed |
60758 | 687 |
with \<open>finite (range f)\<close> have "finite (UNIV::nat set)" |
49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
688 |
by (rule finite_imageD) |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
689 |
then show False by simp |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
690 |
qed |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
691 |
then obtain n where n: "f n = f (Suc n)" .. |
63040 | 692 |
define N where "N = (LEAST n. f n = f (Suc n))" |
49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
693 |
have N: "f N = f (Suc N)" |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
694 |
unfolding N_def using n by (rule LeastI) |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
695 |
show ?thesis |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
696 |
proof (intro exI[of _ N] conjI allI impI) |
63612 | 697 |
fix n |
698 |
assume "N \<le> n" |
|
49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
699 |
then have "\<And>m. N \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m = f N" |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
700 |
proof (induct rule: dec_induct) |
63612 | 701 |
case base |
702 |
then show ?case by simp |
|
703 |
next |
|
704 |
case (step n) |
|
705 |
then show ?case |
|
706 |
using eq [rule_format, of "n - 1"] N |
|
49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
707 |
by (cases n) (auto simp add: le_Suc_eq) |
63612 | 708 |
qed |
60758 | 709 |
from this[of n] \<open>N \<le> n\<close> show "f N = f n" by auto |
49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
710 |
next |
63612 | 711 |
fix n m :: nat |
712 |
assume "m < n" "n \<le> N" |
|
49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
713 |
then show "f m < f n" |
62683 | 714 |
proof (induct rule: less_Suc_induct) |
49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
715 |
case (1 i) |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
716 |
then have "i < N" by simp |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
717 |
then have "f i \<noteq> f (Suc i)" |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
718 |
unfolding N_def by (rule not_less_Least) |
60758 | 719 |
with \<open>mono f\<close> show ?case by (simp add: mono_iff_le_Suc less_le) |
63612 | 720 |
next |
721 |
case 2 |
|
722 |
then show ?case by simp |
|
723 |
qed |
|
49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
724 |
qed |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
725 |
qed |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
726 |
|
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
727 |
lemma finite_mono_strict_prefix_implies_finite_fixpoint: |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
728 |
fixes f :: "nat \<Rightarrow> 'a set" |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
729 |
assumes S: "\<And>i. f i \<subseteq> S" "finite S" |
63612 | 730 |
and ex: "\<exists>N. (\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m \<subset> f n) \<and> (\<forall>n\<ge>N. f N = f n)" |
49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
731 |
shows "f (card S) = (\<Union>n. f n)" |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
732 |
proof - |
63612 | 733 |
from ex obtain N where inj: "\<And>n m. n \<le> N \<Longrightarrow> m \<le> N \<Longrightarrow> m < n \<Longrightarrow> f m \<subset> f n" |
734 |
and eq: "\<forall>n\<ge>N. f N = f n" |
|
735 |
by atomize auto |
|
736 |
have "i \<le> N \<Longrightarrow> i \<le> card (f i)" for i |
|
737 |
proof (induct i) |
|
738 |
case 0 |
|
739 |
then show ?case by simp |
|
740 |
next |
|
741 |
case (Suc i) |
|
742 |
with inj [of "Suc i" i] have "(f i) \<subset> (f (Suc i))" by auto |
|
743 |
moreover have "finite (f (Suc i))" using S by (rule finite_subset) |
|
744 |
ultimately have "card (f i) < card (f (Suc i))" by (intro psubset_card_mono) |
|
745 |
with Suc inj show ?case by auto |
|
746 |
qed |
|
49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
747 |
then have "N \<le> card (f N)" by simp |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
748 |
also have "\<dots> \<le> card S" using S by (intro card_mono) |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
749 |
finally have "f (card S) = f N" using eq by auto |
63612 | 750 |
then show ?thesis |
751 |
using eq inj [of N] |
|
49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
752 |
apply auto |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
753 |
apply (case_tac "n < N") |
63612 | 754 |
apply (auto simp: not_less) |
49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
755 |
done |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
756 |
qed |
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
757 |
|
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset
|
758 |
|
60758 | 759 |
subsection \<open>More on injections, bijections, and inverses\<close> |
55020 | 760 |
|
63374 | 761 |
locale bijection = |
762 |
fixes f :: "'a \<Rightarrow> 'a" |
|
763 |
assumes bij: "bij f" |
|
764 |
begin |
|
765 |
||
63612 | 766 |
lemma bij_inv: "bij (inv f)" |
63374 | 767 |
using bij by (rule bij_imp_bij_inv) |
768 |
||
63612 | 769 |
lemma surj [simp]: "surj f" |
63374 | 770 |
using bij by (rule bij_is_surj) |
771 |
||
63612 | 772 |
lemma inj: "inj f" |
63374 | 773 |
using bij by (rule bij_is_inj) |
774 |
||
63612 | 775 |
lemma surj_inv [simp]: "surj (inv f)" |
63374 | 776 |
using inj by (rule inj_imp_surj_inv) |
777 |
||
63612 | 778 |
lemma inj_inv: "inj (inv f)" |
63374 | 779 |
using surj by (rule surj_imp_inj_inv) |
780 |
||
63612 | 781 |
lemma eqI: "f a = f b \<Longrightarrow> a = b" |
63374 | 782 |
using inj by (rule injD) |
783 |
||
63612 | 784 |
lemma eq_iff [simp]: "f a = f b \<longleftrightarrow> a = b" |
63374 | 785 |
by (auto intro: eqI) |
786 |
||
63612 | 787 |
lemma eq_invI: "inv f a = inv f b \<Longrightarrow> a = b" |
63374 | 788 |
using inj_inv by (rule injD) |
789 |
||
63612 | 790 |
lemma eq_inv_iff [simp]: "inv f a = inv f b \<longleftrightarrow> a = b" |
63374 | 791 |
by (auto intro: eq_invI) |
792 |
||
63612 | 793 |
lemma inv_left [simp]: "inv f (f a) = a" |
63374 | 794 |
using inj by (simp add: inv_f_eq) |
795 |
||
63612 | 796 |
lemma inv_comp_left [simp]: "inv f \<circ> f = id" |
63374 | 797 |
by (simp add: fun_eq_iff) |
798 |
||
63612 | 799 |
lemma inv_right [simp]: "f (inv f a) = a" |
63374 | 800 |
using surj by (simp add: surj_f_inv_f) |
801 |
||
63612 | 802 |
lemma inv_comp_right [simp]: "f \<circ> inv f = id" |
63374 | 803 |
by (simp add: fun_eq_iff) |
804 |
||
63612 | 805 |
lemma inv_left_eq_iff [simp]: "inv f a = b \<longleftrightarrow> f b = a" |
63374 | 806 |
by auto |
807 |
||
63612 | 808 |
lemma inv_right_eq_iff [simp]: "b = inv f a \<longleftrightarrow> f b = a" |
63374 | 809 |
by auto |
810 |
||
811 |
end |
|
812 |
||
55020 | 813 |
lemma infinite_imp_bij_betw: |
63612 | 814 |
assumes infinite: "\<not> finite A" |
815 |
shows "\<exists>h. bij_betw h A (A - {a})" |
|
816 |
proof (cases "a \<in> A") |
|
817 |
case False |
|
818 |
then have "A - {a} = A" by blast |
|
819 |
then show ?thesis |
|
820 |
using bij_betw_id[of A] by auto |
|
55020 | 821 |
next |
63612 | 822 |
case True |
823 |
with infinite have "\<not> finite (A - {a})" by auto |
|
824 |
with infinite_iff_countable_subset[of "A - {a}"] |
|
825 |
obtain f :: "nat \<Rightarrow> 'a" where 1: "inj f" and 2: "f ` UNIV \<subseteq> A - {a}" by blast |
|
826 |
define g where "g n = (if n = 0 then a else f (Suc n))" for n |
|
827 |
define A' where "A' = g ` UNIV" |
|
828 |
have *: "\<forall>y. f y \<noteq> a" using 2 by blast |
|
829 |
have 3: "inj_on g UNIV \<and> g ` UNIV \<subseteq> A \<and> a \<in> g ` UNIV" |
|
830 |
apply (auto simp add: True g_def [abs_def]) |
|
831 |
apply (unfold inj_on_def) |
|
832 |
apply (intro ballI impI) |
|
833 |
apply (case_tac "x = 0") |
|
834 |
apply (auto simp add: 2) |
|
835 |
proof - |
|
836 |
fix y |
|
837 |
assume "a = (if y = 0 then a else f (Suc y))" |
|
838 |
then show "y = 0" by (cases "y = 0") (use * in auto) |
|
55020 | 839 |
next |
840 |
fix x y |
|
841 |
assume "f (Suc x) = (if y = 0 then a else f (Suc y))" |
|
63612 | 842 |
with 1 * show "x = y" by (cases "y = 0") (auto simp: inj_on_def) |
55020 | 843 |
next |
63612 | 844 |
fix n |
845 |
from 2 show "f (Suc n) \<in> A" by blast |
|
55020 | 846 |
qed |
63612 | 847 |
then have 4: "bij_betw g UNIV A' \<and> a \<in> A' \<and> A' \<subseteq> A" |
848 |
using inj_on_imp_bij_betw[of g] by (auto simp: A'_def) |
|
849 |
then have 5: "bij_betw (inv g) A' UNIV" |
|
850 |
by (auto simp add: bij_betw_inv_into) |
|
851 |
from 3 obtain n where n: "g n = a" by auto |
|
852 |
have 6: "bij_betw g (UNIV - {n}) (A' - {a})" |
|
853 |
by (rule bij_betw_subset) (use 3 4 n in \<open>auto simp: image_set_diff A'_def\<close>) |
|
854 |
define v where "v m = (if m < n then m else Suc m)" for m |
|
55020 | 855 |
have 7: "bij_betw v UNIV (UNIV - {n})" |
63612 | 856 |
proof (unfold bij_betw_def inj_on_def, intro conjI, clarify) |
857 |
fix m1 m2 |
|
858 |
assume "v m1 = v m2" |
|
859 |
then show "m1 = m2" |
|
860 |
apply (cases "m1 < n") |
|
861 |
apply (cases "m2 < n") |
|
862 |
apply (auto simp: inj_on_def v_def [abs_def]) |
|
863 |
apply (cases "m2 < n") |
|
864 |
apply auto |
|
865 |
done |
|
55020 | 866 |
next |
867 |
show "v ` UNIV = UNIV - {n}" |
|
63612 | 868 |
proof (auto simp: v_def [abs_def]) |
869 |
fix m |
|
870 |
assume "m \<noteq> n" |
|
871 |
assume *: "m \<notin> Suc ` {m'. \<not> m' < n}" |
|
872 |
have False if "n \<le> m" |
|
873 |
proof - |
|
874 |
from \<open>m \<noteq> n\<close> that have **: "Suc n \<le> m" by auto |
|
875 |
from Suc_le_D [OF this] obtain m' where m': "m = Suc m'" .. |
|
876 |
with ** have "n \<le> m'" by auto |
|
877 |
with m' * show ?thesis by auto |
|
878 |
qed |
|
879 |
then show "m < n" by force |
|
55020 | 880 |
qed |
881 |
qed |
|
63612 | 882 |
define h' where "h' = g \<circ> v \<circ> (inv g)" |
883 |
with 5 6 7 have 8: "bij_betw h' A' (A' - {a})" |
|
884 |
by (auto simp add: bij_betw_trans) |
|
885 |
define h where "h b = (if b \<in> A' then h' b else b)" for b |
|
886 |
then have "\<forall>b \<in> A'. h b = h' b" by simp |
|
887 |
with 8 have "bij_betw h A' (A' - {a})" |
|
888 |
using bij_betw_cong[of A' h] by auto |
|
55020 | 889 |
moreover |
63612 | 890 |
have "\<forall>b \<in> A - A'. h b = b" by (auto simp: h_def) |
891 |
then have "bij_betw h (A - A') (A - A')" |
|
892 |
using bij_betw_cong[of "A - A'" h id] bij_betw_id[of "A - A'"] by auto |
|
55020 | 893 |
moreover |
63612 | 894 |
from 4 have "(A' \<inter> (A - A') = {} \<and> A' \<union> (A - A') = A) \<and> |
895 |
((A' - {a}) \<inter> (A - A') = {} \<and> (A' - {a}) \<union> (A - A') = A - {a})" |
|
896 |
by blast |
|
55020 | 897 |
ultimately have "bij_betw h A (A - {a})" |
63612 | 898 |
using bij_betw_combine[of h A' "A' - {a}" "A - A'" "A - A'"] by simp |
899 |
then show ?thesis by blast |
|
55020 | 900 |
qed |
901 |
||
902 |
lemma infinite_imp_bij_betw2: |
|
63612 | 903 |
assumes "\<not> finite A" |
904 |
shows "\<exists>h. bij_betw h A (A \<union> {a})" |
|
905 |
proof (cases "a \<in> A") |
|
906 |
case True |
|
907 |
then have "A \<union> {a} = A" by blast |
|
908 |
then show ?thesis using bij_betw_id[of A] by auto |
|
55020 | 909 |
next |
63612 | 910 |
case False |
55020 | 911 |
let ?A' = "A \<union> {a}" |
63612 | 912 |
from False have "A = ?A' - {a}" by blast |
913 |
moreover from assms have "\<not> finite ?A'" by auto |
|
55020 | 914 |
ultimately obtain f where "bij_betw f ?A' A" |
63612 | 915 |
using infinite_imp_bij_betw[of ?A' a] by auto |
916 |
then have "bij_betw (inv_into ?A' f) A ?A'" by (rule bij_betw_inv_into) |
|
917 |
then show ?thesis by auto |
|
55020 | 918 |
qed |
919 |
||
63612 | 920 |
lemma bij_betw_inv_into_left: "bij_betw f A A' \<Longrightarrow> a \<in> A \<Longrightarrow> inv_into A f (f a) = a" |
921 |
unfolding bij_betw_def by clarify (rule inv_into_f_f) |
|
55020 | 922 |
|
63612 | 923 |
lemma bij_betw_inv_into_right: "bij_betw f A A' \<Longrightarrow> a' \<in> A' \<Longrightarrow> f (inv_into A f a') = a'" |
924 |
unfolding bij_betw_def using f_inv_into_f by force |
|
55020 | 925 |
|
926 |
lemma bij_betw_inv_into_subset: |
|
63612 | 927 |
"bij_betw f A A' \<Longrightarrow> B \<subseteq> A \<Longrightarrow> f ` B = B' \<Longrightarrow> bij_betw (inv_into A f) B' B" |
928 |
by (auto simp: bij_betw_def intro: inj_on_inv_into) |
|
55020 | 929 |
|
930 |
||
60758 | 931 |
subsection \<open>Specification package -- Hilbertized version\<close> |
17893
aef5a6d11c2a
added lemma exE_some (from specification_package.ML);
wenzelm
parents:
17702
diff
changeset
|
932 |
|
63612 | 933 |
lemma exE_some: "Ex P \<Longrightarrow> c \<equiv> Eps P \<Longrightarrow> P c" |
17893
aef5a6d11c2a
added lemma exE_some (from specification_package.ML);
wenzelm
parents:
17702
diff
changeset
|
934 |
by (simp only: someI_ex) |
aef5a6d11c2a
added lemma exE_some (from specification_package.ML);
wenzelm
parents:
17702
diff
changeset
|
935 |
|
69605 | 936 |
ML_file \<open>Tools/choice_specification.ML\<close> |
14115 | 937 |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
938 |
subsection \<open>Complete Distributive Lattices -- Properties depending on Hilbert Choice\<close> |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
939 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
940 |
context complete_distrib_lattice |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
941 |
begin |
69479 | 942 |
|
943 |
lemma Sup_Inf: "\<Squnion> (Inf ` A) = \<Sqinter> (Sup ` {f ` A |f. \<forall>B\<in>A. f B \<in> B})" |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
944 |
proof (rule antisym) |
69479 | 945 |
show "\<Squnion> (Inf ` A) \<le> \<Sqinter> (Sup ` {f ` A |f. \<forall>B\<in>A. f B \<in> B})" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
946 |
apply (rule Sup_least, rule INF_greatest) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
947 |
using Inf_lower2 Sup_upper by auto |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
948 |
next |
69479 | 949 |
show "\<Sqinter> (Sup ` {f ` A |f. \<forall>B\<in>A. f B \<in> B}) \<le> \<Squnion> (Inf ` A)" |
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
950 |
proof (simp add: Inf_Sup, rule SUP_least, simp, safe) |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
951 |
fix f |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
952 |
assume "\<forall>Y. (\<exists>f. Y = f ` A \<and> (\<forall>Y\<in>A. f Y \<in> Y)) \<longrightarrow> f Y \<in> Y" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
953 |
from this have B: "\<And> F . (\<forall> Y \<in> A . F Y \<in> Y) \<Longrightarrow> \<exists> Z \<in> A . f (F ` A) = F Z" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
954 |
by auto |
69275 | 955 |
show "\<Sqinter>(f ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}) \<le> \<Squnion>(Inf ` A)" |
956 |
proof (cases "\<exists> Z \<in> A . \<Sqinter>(f ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}) \<le> Inf Z") |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
957 |
case True |
69275 | 958 |
from this obtain Z where [simp]: "Z \<in> A" and A: "\<Sqinter>(f ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}) \<le> Inf Z" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
959 |
by blast |
69275 | 960 |
have B: "... \<le> \<Squnion>(Inf ` A)" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
961 |
by (simp add: SUP_upper) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
962 |
from A and B show ?thesis |
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
963 |
by simp |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
964 |
next |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
965 |
case False |
69275 | 966 |
from this have X: "\<And> Z . Z \<in> A \<Longrightarrow> \<exists> x . x \<in> Z \<and> \<not> \<Sqinter>(f ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}) \<le> x" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
967 |
using Inf_greatest by blast |
69275 | 968 |
define F where "F = (\<lambda> Z . SOME x . x \<in> Z \<and> \<not> \<Sqinter>(f ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}) \<le> x)" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
969 |
have C: "\<And> Y . Y \<in> A \<Longrightarrow> F Y \<in> Y" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
970 |
using X by (simp add: F_def, rule someI2_ex, auto) |
69275 | 971 |
have E: "\<And> Y . Y \<in> A \<Longrightarrow> \<not> \<Sqinter>(f ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}) \<le> F Y" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
972 |
using X by (simp add: F_def, rule someI2_ex, auto) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
973 |
from C and B obtain Z where D: "Z \<in> A " and Y: "f (F ` A) = F Z" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
974 |
by blast |
69275 | 975 |
from E and D have W: "\<not> \<Sqinter>(f ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}) \<le> F Z" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
976 |
by simp |
69275 | 977 |
have "\<Sqinter>(f ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}) \<le> f (F ` A)" |
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
978 |
apply (rule INF_lower) |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
979 |
using C by blast |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
980 |
from this and W and Y show ?thesis |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
981 |
by simp |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
982 |
qed |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
983 |
qed |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
984 |
qed |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
985 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
986 |
lemma dual_complete_distrib_lattice: |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
987 |
"class.complete_distrib_lattice Sup Inf sup (\<ge>) (>) inf \<top> \<bottom>" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
988 |
apply (rule class.complete_distrib_lattice.intro) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
989 |
apply (fact dual_complete_lattice) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
990 |
by (simp add: class.complete_distrib_lattice_axioms_def Sup_Inf) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
991 |
|
68802 | 992 |
lemma sup_Inf: "a \<squnion> \<Sqinter>B = \<Sqinter>((\<squnion>) a ` B)" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
993 |
proof (rule antisym) |
68802 | 994 |
show "a \<squnion> \<Sqinter>B \<le> \<Sqinter>((\<squnion>) a ` B)" |
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
995 |
apply (rule INF_greatest) |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
996 |
using Inf_lower sup.mono by fastforce |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
997 |
next |
68802 | 998 |
have "\<Sqinter>((\<squnion>) a ` B) \<le> \<Sqinter>(Sup ` {{f {a}, f B} |f. f {a} = a \<and> f B \<in> B})" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
999 |
by (rule INF_greatest, auto simp add: INF_lower) |
69275 | 1000 |
also have "... = \<Squnion>(Inf ` {{a}, B})" |
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1001 |
by (unfold Sup_Inf, simp) |
68802 | 1002 |
finally show "\<Sqinter>((\<squnion>) a ` B) \<le> a \<squnion> \<Sqinter>B" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1003 |
by simp |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1004 |
qed |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1005 |
|
68802 | 1006 |
lemma inf_Sup: "a \<sqinter> \<Squnion>B = \<Squnion>((\<sqinter>) a ` B)" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1007 |
using dual_complete_distrib_lattice |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1008 |
by (rule complete_distrib_lattice.sup_Inf) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1009 |
|
69479 | 1010 |
lemma INF_SUP: "(\<Sqinter>y. \<Squnion>x. P x y) = (\<Squnion>f. \<Sqinter>x. P (f x) x)" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1011 |
proof (rule antisym) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1012 |
show "(SUP x. INF y. P (x y) y) \<le> (INF y. SUP x. P x y)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1013 |
by (rule SUP_least, rule INF_greatest, rule SUP_upper2, simp_all, rule INF_lower2, simp, blast) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1014 |
next |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1015 |
have "(INF y. SUP x. ((P x y))) \<le> Inf (Sup ` {{P x y | x . True} | y . True })" (is "?A \<le> ?B") |
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1016 |
proof (rule INF_greatest, clarsimp) |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1017 |
fix y |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1018 |
have "?A \<le> (SUP x. P x y)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1019 |
by (rule INF_lower, simp) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1020 |
also have "... \<le> Sup {uu. \<exists>x. uu = P x y}" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1021 |
by (simp add: full_SetCompr_eq) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1022 |
finally show "?A \<le> Sup {uu. \<exists>x. uu = P x y}" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1023 |
by simp |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1024 |
qed |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1025 |
also have "... \<le> (SUP x. INF y. P (x y) y)" |
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1026 |
proof (subst Inf_Sup, rule SUP_least, clarsimp) |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1027 |
fix f |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1028 |
assume A: "\<forall>Y. (\<exists>y. Y = {uu. \<exists>x. uu = P x y}) \<longrightarrow> f Y \<in> Y" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1029 |
|
68802 | 1030 |
have " \<Sqinter>(f ` {uu. \<exists>y. uu = {uu. \<exists>x. uu = P x y}}) \<le> |
1031 |
(\<Sqinter>y. P (SOME x. f {P x y |x. True} = P x y) y)" |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1032 |
proof (rule INF_greatest, clarsimp) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1033 |
fix y |
68802 | 1034 |
have "(INF x\<in>{uu. \<exists>y. uu = {uu. \<exists>x. uu = P x y}}. f x) \<le> f {uu. \<exists>x. uu = P x y}" |
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1035 |
by (rule INF_lower, blast) |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1036 |
also have "... \<le> P (SOME x. f {uu . \<exists>x. uu = P x y} = P x y) y" |
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1037 |
apply (rule someI2_ex) |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1038 |
using A by auto |
68802 | 1039 |
finally show "\<Sqinter>(f ` {uu. \<exists>y. uu = {uu. \<exists>x. uu = P x y}}) \<le> |
1040 |
P (SOME x. f {uu. \<exists>x. uu = P x y} = P x y) y" |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1041 |
by simp |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1042 |
qed |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1043 |
also have "... \<le> (SUP x. INF y. P (x y) y)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1044 |
by (rule SUP_upper, simp) |
68802 | 1045 |
finally show "\<Sqinter>(f ` {uu. \<exists>y. uu = {uu. \<exists>x. uu = P x y}}) \<le> (\<Squnion>x. \<Sqinter>y. P (x y) y)" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1046 |
by simp |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1047 |
qed |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1048 |
finally show "(INF y. SUP x. P x y) \<le> (SUP x. INF y. P (x y) y)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1049 |
by simp |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1050 |
qed |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1051 |
|
69478 | 1052 |
lemma INF_SUP_set: "(\<Sqinter>B\<in>A. \<Squnion>(g ` B)) = (\<Squnion>B\<in>{f ` A |f. \<forall>C\<in>A. f C \<in> C}. \<Sqinter>(g ` B))" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1053 |
proof (rule antisym) |
69478 | 1054 |
have "\<Sqinter> ((g \<circ> f) ` A) \<le> \<Squnion> (g ` B)" if "\<And>B. B \<in> A \<Longrightarrow> f B \<in> B" and "B \<in> A" |
1055 |
for f and B |
|
1056 |
using that by (auto intro: SUP_upper2 INF_lower2) |
|
1057 |
then show "(\<Squnion>x\<in>{f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}. \<Sqinter>a\<in>x. g a) \<le> (\<Sqinter>x\<in>A. \<Squnion>a\<in>x. g a)" |
|
69861
62e47f06d22c
avoid context-sensitive simp rules whose context-free form (image_comp) is not simp by default
haftmann
parents:
69768
diff
changeset
|
1058 |
by (auto intro!: SUP_least INF_greatest simp add: image_comp) |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1059 |
next |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1060 |
show "(\<Sqinter>x\<in>A. \<Squnion>a\<in>x. g a) \<le> (\<Squnion>x\<in>{f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}. \<Sqinter>a\<in>x. g a)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1061 |
proof (cases "{} \<in> A") |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1062 |
case True |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1063 |
then show ?thesis |
69478 | 1064 |
by (rule INF_lower2) simp_all |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1065 |
next |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1066 |
case False |
69478 | 1067 |
have *: "\<And>f B. B \<in> A \<Longrightarrow> f B \<in> B \<Longrightarrow> |
1068 |
(\<Sqinter>B. if B \<in> A then if f B \<in> B then g (f B) else \<bottom> else \<top>) \<le> g (f B)" |
|
1069 |
by (rule INF_lower2, auto) |
|
1070 |
have **: "\<And>f B. B \<in> A \<Longrightarrow> f B \<notin> B \<Longrightarrow> |
|
1071 |
(\<Sqinter>B. if B \<in> A then if f B \<in> B then g (f B) else \<bottom> else \<top>) \<le> g (SOME x. x \<in> B)" |
|
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1072 |
by (rule INF_lower2, auto) |
69478 | 1073 |
have ****: "\<And>f B. B \<in> A \<Longrightarrow> |
1074 |
(\<Sqinter>B. if B \<in> A then if f B \<in> B then g (f B) else \<bottom> else \<top>) |
|
1075 |
\<le> (if f B \<in> B then g (f B) else g (SOME x. x \<in> B))" |
|
1076 |
by (rule INF_lower2) auto |
|
1077 |
have ***: "\<And>x. (\<Sqinter>B. if B \<in> A then if x B \<in> B then g (x B) else \<bottom> else \<top>) |
|
1078 |
\<le> (\<Squnion>x\<in>{f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}. \<Sqinter>x\<in>x. g x)" |
|
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1079 |
proof - |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1080 |
fix x |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1081 |
define F where "F = (\<lambda> (y::'b set) . if x y \<in> y then x y else (SOME x . x \<in>y))" |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1082 |
have B: "(\<forall>Y\<in>A. F Y \<in> Y)" |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1083 |
using False some_in_eq F_def by auto |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1084 |
have A: "F ` A \<in> {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}" |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1085 |
using B by blast |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1086 |
show "(\<Sqinter>xa. if xa \<in> A then if x xa \<in> xa then g (x xa) else \<bottom> else \<top>) \<le> (\<Squnion>x\<in>{f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}. \<Sqinter>x\<in>x. g x)" |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1087 |
using A apply (rule SUP_upper2) |
69478 | 1088 |
apply (rule INF_greatest) |
69768 | 1089 |
using * ** |
1090 |
apply (auto simp add: F_def) |
|
69478 | 1091 |
done |
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1092 |
qed |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1093 |
|
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1094 |
{fix x |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1095 |
have "(\<Sqinter>x\<in>A. \<Squnion>x\<in>x. g x) \<le> (\<Squnion>xa. if x \<in> A then if xa \<in> x then g xa else \<bottom> else \<top>)" |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1096 |
proof (cases "x \<in> A") |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1097 |
case True |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1098 |
then show ?thesis |
69768 | 1099 |
apply (rule INF_lower2) |
1100 |
apply (rule SUP_least) |
|
1101 |
apply (rule SUP_upper2) |
|
1102 |
apply auto |
|
1103 |
done |
|
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1104 |
next |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1105 |
case False |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1106 |
then show ?thesis by simp |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1107 |
qed |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1108 |
} |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1109 |
from this have "(\<Sqinter>x\<in>A. \<Squnion>a\<in>x. g a) \<le> (\<Sqinter>x. \<Squnion>xa. if x \<in> A then if xa \<in> x then g xa else \<bottom> else \<top>)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1110 |
by (rule INF_greatest) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1111 |
also have "... = (\<Squnion>x. \<Sqinter>xa. if xa \<in> A then if x xa \<in> xa then g (x xa) else \<bottom> else \<top>)" |
69768 | 1112 |
by (simp only: INF_SUP) |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1113 |
also have "... \<le> (\<Squnion>x\<in>{f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}. \<Sqinter>a\<in>x. g a)" |
69768 | 1114 |
apply (rule SUP_least) |
1115 |
using *** apply simp |
|
1116 |
done |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1117 |
finally show ?thesis by simp |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1118 |
qed |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1119 |
qed |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1120 |
|
69479 | 1121 |
lemma SUP_INF: "(\<Squnion>y. \<Sqinter>x. P x y) = (\<Sqinter>x. \<Squnion>y. P (x y) y)" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1122 |
using dual_complete_distrib_lattice |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1123 |
by (rule complete_distrib_lattice.INF_SUP) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1124 |
|
69479 | 1125 |
lemma SUP_INF_set: "(\<Squnion>x\<in>A. \<Sqinter> (g ` x)) = (\<Sqinter>x\<in>{f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}. \<Squnion> (g ` x))" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1126 |
using dual_complete_distrib_lattice |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1127 |
by (rule complete_distrib_lattice.INF_SUP_set) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1128 |
|
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
1129 |
end |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1130 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1131 |
(*properties of the former complete_distrib_lattice*) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1132 |
context complete_distrib_lattice |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1133 |
begin |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1134 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1135 |
lemma sup_INF: "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)" |
69861
62e47f06d22c
avoid context-sensitive simp rules whose context-free form (image_comp) is not simp by default
haftmann
parents:
69768
diff
changeset
|
1136 |
by (simp add: sup_Inf image_comp) |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1137 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1138 |
lemma inf_SUP: "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)" |
69861
62e47f06d22c
avoid context-sensitive simp rules whose context-free form (image_comp) is not simp by default
haftmann
parents:
69768
diff
changeset
|
1139 |
by (simp add: inf_Sup image_comp) |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1140 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1141 |
lemma Inf_sup: "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1142 |
by (simp add: sup_Inf sup_commute) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1143 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1144 |
lemma Sup_inf: "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1145 |
by (simp add: inf_Sup inf_commute) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1146 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1147 |
lemma INF_sup: "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1148 |
by (simp add: sup_INF sup_commute) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1149 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1150 |
lemma SUP_inf: "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1151 |
by (simp add: inf_SUP inf_commute) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1152 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1153 |
lemma Inf_sup_eq_top_iff: "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1154 |
by (simp only: Inf_sup INF_top_conv) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1155 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1156 |
lemma Sup_inf_eq_bot_iff: "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1157 |
by (simp only: Sup_inf SUP_bot_conv) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1158 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1159 |
lemma INF_sup_distrib2: "(\<Sqinter>a\<in>A. f a) \<squnion> (\<Sqinter>b\<in>B. g b) = (\<Sqinter>a\<in>A. \<Sqinter>b\<in>B. f a \<squnion> g b)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1160 |
by (subst INF_commute) (simp add: sup_INF INF_sup) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1161 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1162 |
lemma SUP_inf_distrib2: "(\<Squnion>a\<in>A. f a) \<sqinter> (\<Squnion>b\<in>B. g b) = (\<Squnion>a\<in>A. \<Squnion>b\<in>B. f a \<sqinter> g b)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1163 |
by (subst SUP_commute) (simp add: inf_SUP SUP_inf) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1164 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1165 |
end |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1166 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1167 |
context complete_boolean_algebra |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1168 |
begin |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1169 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1170 |
lemma dual_complete_boolean_algebra: |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1171 |
"class.complete_boolean_algebra Sup Inf sup (\<ge>) (>) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1172 |
by (rule class.complete_boolean_algebra.intro, |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1173 |
rule dual_complete_distrib_lattice, |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1174 |
rule dual_boolean_algebra) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1175 |
end |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1176 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1177 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1178 |
|
68802 | 1179 |
instantiation set :: (type) complete_distrib_lattice |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1180 |
begin |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1181 |
instance proof (standard, clarsimp) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1182 |
fix A :: "(('a set) set) set" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1183 |
fix x::'a |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1184 |
define F where "F = (\<lambda> Y . (SOME X . (Y \<in> A \<and> X \<in> Y \<and> x \<in> X)))" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1185 |
assume A: "\<forall>xa\<in>A. \<exists>X\<in>xa. x \<in> X" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1186 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1187 |
from this have B: " (\<forall>xa \<in> F ` A. x \<in> xa)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1188 |
apply (safe, simp add: F_def) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1189 |
by (rule someI2_ex, auto) |
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1190 |
|
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1191 |
have C: "(\<forall>Y\<in>A. F Y \<in> Y)" |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1192 |
apply (simp add: F_def, safe) |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1193 |
apply (rule someI2_ex) |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1194 |
using A by auto |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1195 |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1196 |
have "(\<exists>f. F ` A = f ` A \<and> (\<forall>Y\<in>A. f Y \<in> Y))" |
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1197 |
using C by blast |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1198 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1199 |
from B and this show "\<exists>X. (\<exists>f. X = f ` A \<and> (\<forall>Y\<in>A. f Y \<in> Y)) \<and> (\<forall>xa\<in>X. x \<in> xa)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1200 |
by auto |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1201 |
qed |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1202 |
end |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1203 |
|
68802 | 1204 |
instance set :: (type) complete_boolean_algebra .. |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1205 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1206 |
instantiation "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1207 |
begin |
69861
62e47f06d22c
avoid context-sensitive simp rules whose context-free form (image_comp) is not simp by default
haftmann
parents:
69768
diff
changeset
|
1208 |
instance by standard (simp add: le_fun_def INF_SUP_set image_comp) |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1209 |
end |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1210 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1211 |
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra .. |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1212 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1213 |
context complete_linorder |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1214 |
begin |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1215 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1216 |
subclass complete_distrib_lattice |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1217 |
proof (standard, rule ccontr) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1218 |
fix A |
69275 | 1219 |
assume "\<not> \<Sqinter>(Sup ` A) \<le> \<Squnion>(Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y})" |
1220 |
then have C: "\<Sqinter>(Sup ` A) > \<Squnion>(Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y})" |
|
1221 |
by (simp add: not_le) |
|
1222 |
show False |
|
1223 |
proof (cases "\<exists> z . \<Sqinter>(Sup ` A) > z \<and> z > \<Squnion>(Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y})") |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1224 |
case True |
69275 | 1225 |
from this obtain z where A: "z < \<Sqinter>(Sup ` A)" and X: "z > \<Squnion>(Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y})" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1226 |
by blast |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1227 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1228 |
from A have "\<And> Y . Y \<in> A \<Longrightarrow> z < Sup Y" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1229 |
by (simp add: less_INF_D) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1230 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1231 |
from this have B: "\<And> Y . Y \<in> A \<Longrightarrow> \<exists> k \<in>Y . z < k" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1232 |
using local.less_Sup_iff by blast |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1233 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1234 |
define F where "F = (\<lambda> Y . SOME k . k \<in> Y \<and> z < k)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1235 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1236 |
have D: "\<And> Y . Y \<in> A \<Longrightarrow> z < F Y" |
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1237 |
using B apply (simp add: F_def) |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1238 |
by (rule someI2_ex, auto) |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1239 |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1240 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1241 |
have E: "\<And> Y . Y \<in> A \<Longrightarrow> F Y \<in> Y" |
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1242 |
using B apply (simp add: F_def) |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1243 |
by (rule someI2_ex, auto) |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1244 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1245 |
have "z \<le> Inf (F ` A)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1246 |
by (simp add: D local.INF_greatest local.order.strict_implies_order) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1247 |
|
69275 | 1248 |
also have "... \<le> \<Squnion>(Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y})" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1249 |
apply (rule SUP_upper, safe) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1250 |
using E by blast |
69275 | 1251 |
finally have "z \<le> \<Squnion>(Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y})" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1252 |
by simp |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1253 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1254 |
from X and this show ?thesis |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1255 |
using local.not_less by blast |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1256 |
next |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1257 |
case False |
69275 | 1258 |
from this have A: "\<And> z . \<Sqinter>(Sup ` A) \<le> z \<or> z \<le> \<Squnion>(Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y})" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1259 |
using local.le_less_linear by blast |
69275 | 1260 |
|
1261 |
from C have "\<And> Y . Y \<in> A \<Longrightarrow> \<Squnion>(Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}) < Sup Y" |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1262 |
by (simp add: less_INF_D) |
69275 | 1263 |
|
1264 |
from this have B: "\<And> Y . Y \<in> A \<Longrightarrow> \<exists> k \<in>Y . \<Squnion>(Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}) < k" |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1265 |
using local.less_Sup_iff by blast |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1266 |
|
69275 | 1267 |
define F where "F = (\<lambda> Y . SOME k . k \<in> Y \<and> \<Squnion>(Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}) < k)" |
1268 |
||
1269 |
have D: "\<And> Y . Y \<in> A \<Longrightarrow> \<Squnion>(Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}) < F Y" |
|
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1270 |
using B apply (simp add: F_def) |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1271 |
by (rule someI2_ex, auto) |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1272 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1273 |
have E: "\<And> Y . Y \<in> A \<Longrightarrow> F Y \<in> Y" |
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1274 |
using B apply (simp add: F_def) |
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1275 |
by (rule someI2_ex, auto) |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1276 |
|
69275 | 1277 |
have "\<And> Y . Y \<in> A \<Longrightarrow> \<Sqinter>(Sup ` A) \<le> F Y" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1278 |
using D False local.leI by blast |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1279 |
|
69275 | 1280 |
from this have "\<Sqinter>(Sup ` A) \<le> Inf (F ` A)" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1281 |
by (simp add: local.INF_greatest) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1282 |
|
69275 | 1283 |
also have "Inf (F ` A) \<le> \<Squnion>(Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y})" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1284 |
apply (rule SUP_upper, safe) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1285 |
using E by blast |
69275 | 1286 |
|
1287 |
finally have "\<Sqinter>(Sup ` A) \<le> \<Squnion>(Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y})" |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1288 |
by simp |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1289 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1290 |
from C and this show ?thesis |
67951
655aa11359dc
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
Manuel Eberl <eberlm@in.tum.de>
parents:
67829
diff
changeset
|
1291 |
using not_less by blast |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1292 |
qed |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1293 |
qed |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1294 |
end |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1295 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1296 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1297 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1298 |
end |