author | hoelzl |
Wed, 18 Apr 2012 14:29:19 +0200 | |
changeset 47596 | c031e65c8ddc |
parent 46950 | d0181abdbdac |
child 47988 | e4b69e10b990 |
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 |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
3 |
Copyright 2001 University of Cambridge |
12023 | 4 |
*) |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
5 |
|
14760 | 6 |
header {* Hilbert's Epsilon-Operator and the Axiom of Choice *} |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
7 |
|
15131 | 8 |
theory Hilbert_Choice |
29655
ac31940cfb69
Plain, Main form meeting points in import hierarchy
haftmann
parents:
27760
diff
changeset
|
9 |
imports Nat Wellfounded Plain |
46950
d0181abdbdac
declare command keywords via theory header, including strict checking outside Pure;
wenzelm
parents:
45607
diff
changeset
|
10 |
keywords "specification" "ax_specification" :: thy_goal |
39943 | 11 |
uses ("Tools/choice_specification.ML") |
15131 | 12 |
begin |
12298 | 13 |
|
14 |
subsection {* Hilbert's epsilon *} |
|
15 |
||
31454 | 16 |
axiomatization Eps :: "('a => bool) => 'a" where |
22690
0b08f218f260
replaced axioms/finalconsts by proper axiomatization;
wenzelm
parents:
21999
diff
changeset
|
17 |
someI: "P x ==> 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) |
3f2144aebd76
improved symbolic syntax of Eps: \<some> for mode "epsilon";
wenzelm
parents:
14760
diff
changeset
|
20 |
"_Eps" :: "[pttrn, bool] => 'a" ("(3\<some>_./ _)" [0, 10] 10) |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
21 |
syntax (HOL) |
12298 | 22 |
"_Eps" :: "[pttrn, bool] => 'a" ("(3@ _./ _)" [0, 10] 10) |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
23 |
syntax |
12298 | 24 |
"_Eps" :: "[pttrn, bool] => 'a" ("(3SOME _./ _)" [0, 10] 10) |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
25 |
translations |
22690
0b08f218f260
replaced axioms/finalconsts by proper axiomatization;
wenzelm
parents:
21999
diff
changeset
|
26 |
"SOME x. P" == "CONST Eps (%x. P)" |
13763
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13585
diff
changeset
|
27 |
|
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13585
diff
changeset
|
28 |
print_translation {* |
35115 | 29 |
[(@{const_syntax Eps}, fn [Abs abs] => |
42284 | 30 |
let val (x, t) = Syntax_Trans.atomic_abs_tr' abs |
35115 | 31 |
in Syntax.const @{syntax_const "_Eps"} $ x $ t end)] |
32 |
*} -- {* to avoid eta-contraction of body *} |
|
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
33 |
|
33057 | 34 |
definition inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where |
35 |
"inv_into A f == %x. SOME y. y : A & f y = x" |
|
11454
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset
|
36 |
|
32988 | 37 |
abbreviation inv :: "('a => 'b) => ('b => 'a)" where |
33057 | 38 |
"inv == inv_into UNIV" |
14760 | 39 |
|
40 |
||
41 |
subsection {*Hilbert's Epsilon-operator*} |
|
42 |
||
43 |
text{*Easier to apply than @{text someI} if the witness comes from an |
|
44 |
existential formula*} |
|
45 |
lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)" |
|
46 |
apply (erule exE) |
|
47 |
apply (erule someI) |
|
48 |
done |
|
49 |
||
50 |
text{*Easier to apply than @{text someI} because the conclusion has only one |
|
51 |
occurrence of @{term P}.*} |
|
52 |
lemma someI2: "[| P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)" |
|
53 |
by (blast intro: someI) |
|
54 |
||
55 |
text{*Easier to apply than @{text someI2} if the witness comes from an |
|
56 |
existential formula*} |
|
57 |
lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)" |
|
58 |
by (blast intro: someI2) |
|
59 |
||
60 |
lemma some_equality [intro]: |
|
61 |
"[| P a; !!x. P x ==> x=a |] ==> (SOME x. P x) = a" |
|
62 |
by (blast intro: someI2) |
|
63 |
||
64 |
lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a" |
|
35216 | 65 |
by blast |
14760 | 66 |
|
67 |
lemma some_eq_ex: "P (SOME x. P x) = (\<exists>x. P x)" |
|
68 |
by (blast intro: someI) |
|
69 |
||
70 |
lemma some_eq_trivial [simp]: "(SOME y. y=x) = x" |
|
71 |
apply (rule some_equality) |
|
72 |
apply (rule refl, assumption) |
|
73 |
done |
|
74 |
||
75 |
lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x" |
|
76 |
apply (rule some_equality) |
|
77 |
apply (rule refl) |
|
78 |
apply (erule sym) |
|
79 |
done |
|
80 |
||
81 |
||
82 |
subsection{*Axiom of Choice, Proved Using the Description Operator*} |
|
83 |
||
39950 | 84 |
lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)" |
14760 | 85 |
by (fast elim: someI) |
86 |
||
87 |
lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)" |
|
88 |
by (fast elim: someI) |
|
89 |
||
90 |
||
91 |
subsection {*Function Inverse*} |
|
92 |
||
33014 | 93 |
lemma inv_def: "inv f = (%y. SOME x. f x = y)" |
33057 | 94 |
by(simp add: inv_into_def) |
33014 | 95 |
|
33057 | 96 |
lemma inv_into_into: "x : f ` A ==> inv_into A f x : A" |
97 |
apply (simp add: inv_into_def) |
|
32988 | 98 |
apply (fast intro: someI2) |
99 |
done |
|
14760 | 100 |
|
32988 | 101 |
lemma inv_id [simp]: "inv id = id" |
33057 | 102 |
by (simp add: inv_into_def id_def) |
14760 | 103 |
|
33057 | 104 |
lemma inv_into_f_f [simp]: |
105 |
"[| inj_on f A; x : A |] ==> inv_into A f (f x) = x" |
|
106 |
apply (simp add: inv_into_def inj_on_def) |
|
32988 | 107 |
apply (blast intro: someI2) |
14760 | 108 |
done |
109 |
||
32988 | 110 |
lemma inv_f_f: "inj f ==> inv f (f x) = x" |
35216 | 111 |
by simp |
32988 | 112 |
|
33057 | 113 |
lemma f_inv_into_f: "y : f`A ==> f (inv_into A f y) = y" |
114 |
apply (simp add: inv_into_def) |
|
32988 | 115 |
apply (fast intro: someI2) |
116 |
done |
|
117 |
||
33057 | 118 |
lemma inv_into_f_eq: "[| inj_on f A; x : A; f x = y |] ==> inv_into A f y = x" |
32988 | 119 |
apply (erule subst) |
33057 | 120 |
apply (fast intro: inv_into_f_f) |
32988 | 121 |
done |
122 |
||
123 |
lemma inv_f_eq: "[| inj f; f x = y |] ==> inv f y = x" |
|
33057 | 124 |
by (simp add:inv_into_f_eq) |
32988 | 125 |
|
126 |
lemma inj_imp_inv_eq: "[| inj f; ALL x. f(g x) = x |] ==> inv f = g" |
|
44921 | 127 |
by (blast intro: inv_into_f_eq) |
14760 | 128 |
|
129 |
text{*But is it useful?*} |
|
130 |
lemma inj_transfer: |
|
131 |
assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)" |
|
132 |
shows "P x" |
|
133 |
proof - |
|
134 |
have "f x \<in> range f" by auto |
|
135 |
hence "P(inv f (f x))" by (rule minor) |
|
33057 | 136 |
thus "P x" by (simp add: inv_into_f_f [OF injf]) |
14760 | 137 |
qed |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
138 |
|
14760 | 139 |
lemma inj_iff: "(inj f) = (inv f o f = id)" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
140 |
apply (simp add: o_def fun_eq_iff) |
33057 | 141 |
apply (blast intro: inj_on_inverseI inv_into_f_f) |
14760 | 142 |
done |
143 |
||
23433 | 144 |
lemma inv_o_cancel[simp]: "inj f ==> inv f o f = id" |
145 |
by (simp add: inj_iff) |
|
146 |
||
147 |
lemma o_inv_o_cancel[simp]: "inj f ==> g o inv f o f = g" |
|
148 |
by (simp add: o_assoc[symmetric]) |
|
149 |
||
33057 | 150 |
lemma inv_into_image_cancel[simp]: |
151 |
"inj_on f A ==> S <= A ==> inv_into A f ` f ` S = S" |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
42284
diff
changeset
|
152 |
by(fastforce simp: image_def) |
32988 | 153 |
|
14760 | 154 |
lemma inj_imp_surj_inv: "inj f ==> surj (inv f)" |
40702 | 155 |
by (blast intro!: surjI inv_into_f_f) |
14760 | 156 |
|
157 |
lemma surj_f_inv_f: "surj f ==> f(inv f y) = y" |
|
40702 | 158 |
by (simp add: f_inv_into_f) |
14760 | 159 |
|
33057 | 160 |
lemma inv_into_injective: |
161 |
assumes eq: "inv_into A f x = inv_into A f y" |
|
32988 | 162 |
and x: "x: f`A" |
163 |
and y: "y: f`A" |
|
14760 | 164 |
shows "x=y" |
165 |
proof - |
|
33057 | 166 |
have "f (inv_into A f x) = f (inv_into A f y)" using eq by simp |
167 |
thus ?thesis by (simp add: f_inv_into_f x y) |
|
14760 | 168 |
qed |
169 |
||
33057 | 170 |
lemma inj_on_inv_into: "B <= f`A ==> inj_on (inv_into A f) B" |
171 |
by (blast intro: inj_onI dest: inv_into_injective injD) |
|
32988 | 172 |
|
33057 | 173 |
lemma bij_betw_inv_into: "bij_betw f A B ==> bij_betw (inv_into A f) B A" |
174 |
by (auto simp add: bij_betw_def inj_on_inv_into) |
|
14760 | 175 |
|
176 |
lemma surj_imp_inj_inv: "surj f ==> inj (inv f)" |
|
40702 | 177 |
by (simp add: inj_on_inv_into) |
14760 | 178 |
|
179 |
lemma surj_iff: "(surj f) = (f o inv f = id)" |
|
40702 | 180 |
by (auto intro!: surjI simp: surj_f_inv_f fun_eq_iff[where 'b='a]) |
181 |
||
182 |
lemma surj_iff_all: "surj f \<longleftrightarrow> (\<forall>x. f (inv f x) = x)" |
|
183 |
unfolding surj_iff by (simp add: o_def fun_eq_iff) |
|
14760 | 184 |
|
185 |
lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g" |
|
186 |
apply (rule ext) |
|
187 |
apply (drule_tac x = "inv f x" in spec) |
|
188 |
apply (simp add: surj_f_inv_f) |
|
189 |
done |
|
190 |
||
191 |
lemma bij_imp_bij_inv: "bij f ==> bij (inv f)" |
|
192 |
by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv) |
|
12372 | 193 |
|
14760 | 194 |
lemma inv_equality: "[| !!x. g (f x) = x; !!y. f (g y) = y |] ==> inv f = g" |
195 |
apply (rule ext) |
|
33057 | 196 |
apply (auto simp add: inv_into_def) |
14760 | 197 |
done |
198 |
||
199 |
lemma inv_inv_eq: "bij f ==> inv (inv f) = f" |
|
200 |
apply (rule inv_equality) |
|
201 |
apply (auto simp add: bij_def surj_f_inv_f) |
|
202 |
done |
|
203 |
||
204 |
(** bij(inv f) implies little about f. Consider f::bool=>bool such that |
|
205 |
f(True)=f(False)=True. Then it's consistent with axiom someI that |
|
206 |
inv f could be any function at all, including the identity function. |
|
207 |
If inv f=id then inv f is a bijection, but inj f, surj(f) and |
|
208 |
inv(inv f)=f all fail. |
|
209 |
**) |
|
210 |
||
33057 | 211 |
lemma inv_into_comp: |
32988 | 212 |
"[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==> |
33057 | 213 |
inv_into A (f o g) x = (inv_into A g o inv_into (g ` A) f) x" |
214 |
apply (rule inv_into_f_eq) |
|
32988 | 215 |
apply (fast intro: comp_inj_on) |
33057 | 216 |
apply (simp add: inv_into_into) |
217 |
apply (simp add: f_inv_into_f inv_into_into) |
|
32988 | 218 |
done |
219 |
||
14760 | 220 |
lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f" |
221 |
apply (rule inv_equality) |
|
222 |
apply (auto simp add: bij_def surj_f_inv_f) |
|
223 |
done |
|
224 |
||
225 |
lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A" |
|
226 |
by (simp add: image_eq_UN surj_f_inv_f) |
|
227 |
||
228 |
lemma image_inv_f_f: "inj f ==> (inv f) ` (f ` A) = A" |
|
229 |
by (simp add: image_eq_UN) |
|
230 |
||
231 |
lemma inv_image_comp: "inj f ==> inv f ` (f`X) = X" |
|
232 |
by (auto simp add: image_def) |
|
233 |
||
234 |
lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}" |
|
235 |
apply auto |
|
236 |
apply (force simp add: bij_is_inj) |
|
237 |
apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric]) |
|
238 |
done |
|
239 |
||
240 |
lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A" |
|
241 |
apply (auto simp add: bij_is_surj [THEN surj_f_inv_f]) |
|
33057 | 242 |
apply (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric]) |
14760 | 243 |
done |
244 |
||
31380 | 245 |
lemma finite_fun_UNIVD1: |
246 |
assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)" |
|
247 |
and card: "card (UNIV :: 'b set) \<noteq> Suc 0" |
|
248 |
shows "finite (UNIV :: 'a set)" |
|
249 |
proof - |
|
250 |
from fin have finb: "finite (UNIV :: 'b set)" by (rule finite_fun_UNIVD2) |
|
251 |
with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)" |
|
252 |
by (cases "card (UNIV :: 'b set)") (auto simp add: card_eq_0_iff) |
|
253 |
then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - Suc (Suc 0)" by auto |
|
254 |
then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by (auto simp add: card_Suc_eq) |
|
255 |
from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by (rule finite_imageI) |
|
256 |
moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)" |
|
257 |
proof (rule UNIV_eq_I) |
|
258 |
fix x :: 'a |
|
33057 | 259 |
from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_into_def) |
31380 | 260 |
thus "x \<in> range (\<lambda>f\<Colon>'a \<Rightarrow> 'b. inv f b1)" by blast |
261 |
qed |
|
262 |
ultimately show "finite (UNIV :: 'a set)" by simp |
|
263 |
qed |
|
14760 | 264 |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
265 |
lemma image_inv_into_cancel: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
266 |
assumes SURJ: "f`A=A'" and SUB: "B' \<le> A'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
267 |
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
|
268 |
using assms |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
269 |
proof (auto simp add: f_inv_into_f) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
270 |
let ?f' = "(inv_into A f)" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
271 |
fix a' assume *: "a' \<in> B'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
272 |
then have "a' \<in> A'" using SUB by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
273 |
then have "a' = f (?f' a')" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
274 |
using SURJ by (auto simp add: f_inv_into_f) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
275 |
then show "a' \<in> f ` (?f' ` B')" using * by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
276 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
277 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
278 |
lemma inv_into_inv_into_eq: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
279 |
assumes "bij_betw f A A'" "a \<in> A" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
280 |
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
|
281 |
proof - |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
282 |
let ?f' = "inv_into A f" let ?f'' = "inv_into A' ?f'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
283 |
have 1: "bij_betw ?f' A' A" using assms |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
284 |
by (auto simp add: bij_betw_inv_into) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
285 |
obtain a' where 2: "a' \<in> A'" and 3: "?f' a' = a" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
286 |
using 1 `a \<in> A` unfolding bij_betw_def by force |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
287 |
hence "?f'' a = a'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
288 |
using `a \<in> A` 1 3 by (auto simp add: f_inv_into_f bij_betw_def) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
289 |
moreover have "f a = a'" using assms 2 3 |
44921 | 290 |
by (auto simp add: 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
|
291 |
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
|
292 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
293 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
294 |
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
|
295 |
assumes "A \<noteq> {}" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
296 |
shows "(\<exists>f. inj_on f A \<and> f ` A \<le> A') \<longleftrightarrow> (\<exists>g. 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
|
297 |
proof safe |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
298 |
fix f assume INJ: "inj_on f A" and INCL: "f ` A \<le> A'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
299 |
let ?phi = "\<lambda>a' a. a \<in> A \<and> f a = a'" let ?csi = "\<lambda>a. a \<in> A" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
300 |
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
|
301 |
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
|
302 |
proof |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
303 |
show "?g ` A' \<le> A" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
304 |
proof clarify |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
305 |
fix a' assume *: "a' \<in> A'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
306 |
show "?g a' \<in> A" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
307 |
proof cases |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
308 |
assume Case1: "a' \<in> f ` A" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
309 |
then obtain a where "?phi a' a" by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
310 |
hence "?phi a' (SOME a. ?phi a' a)" using someI[of "?phi a'" a] by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
311 |
with Case1 show ?thesis by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
312 |
next |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
313 |
assume Case2: "a' \<notin> f ` A" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
314 |
hence "?csi (SOME a. ?csi a)" using assms someI_ex[of ?csi] by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
315 |
with Case2 show ?thesis by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
316 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
317 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
318 |
next |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
319 |
show "A \<le> ?g ` A'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
320 |
proof- |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
321 |
{fix a assume *: "a \<in> A" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
322 |
let ?b = "SOME aa. ?phi (f a) aa" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
323 |
have "?phi (f a) a" using * by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
324 |
hence 1: "?phi (f a) ?b" using someI[of "?phi(f a)" a] by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
325 |
hence "?g(f a) = ?b" using * by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
326 |
moreover have "a = ?b" using 1 INJ * by (auto simp add: inj_on_def) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
327 |
ultimately have "?g(f a) = 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
|
328 |
with INCL * have "?g(f a) = a \<and> f a \<in> A'" by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
329 |
} |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
330 |
thus ?thesis by force |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
331 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
332 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
333 |
thus "\<exists>g. g ` A' = A" by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
334 |
next |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
335 |
fix g let ?f = "inv_into A' g" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
336 |
have "inj_on ?f (g ` A')" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
337 |
by (auto simp add: inj_on_inv_into) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
338 |
moreover |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
339 |
{fix a' assume *: "a' \<in> A'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
340 |
let ?phi = "\<lambda> b'. b' \<in> A' \<and> g b' = g a'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
341 |
have "?phi a'" using * by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
342 |
hence "?phi(SOME b'. ?phi b')" using someI[of ?phi] by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
343 |
hence "?f(g a') \<in> A'" unfolding inv_into_def by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
344 |
} |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
345 |
ultimately show "\<exists>f. inj_on f (g ` A') \<and> f ` g ` A' \<subseteq> A'" by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
346 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
347 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
348 |
lemma Ex_inj_on_UNION_Sigma: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
349 |
"\<exists>f. (inj_on f (\<Union> i \<in> I. A i) \<and> f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i))" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
350 |
proof |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
351 |
let ?phi = "\<lambda> a i. i \<in> I \<and> a \<in> A i" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
352 |
let ?sm = "\<lambda> a. SOME i. ?phi a i" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
353 |
let ?f = "\<lambda>a. (?sm a, a)" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
354 |
have "inj_on ?f (\<Union> i \<in> I. A i)" unfolding inj_on_def by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
355 |
moreover |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
356 |
{ { fix i a assume "i \<in> I" and "a \<in> A i" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
357 |
hence "?sm a \<in> I \<and> a \<in> A(?sm a)" using someI[of "?phi a" i] by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
358 |
} |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
359 |
hence "?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)" by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
360 |
} |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
361 |
ultimately |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
362 |
show "inj_on ?f (\<Union> i \<in> I. A i) \<and> ?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
363 |
by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
364 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
365 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
366 |
subsection {* The Cantor-Bernstein Theorem *} |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
367 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
368 |
lemma Cantor_Bernstein_aux: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
369 |
shows "\<exists>A' h. A' \<le> A \<and> |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
370 |
(\<forall>a \<in> A'. a \<notin> g`(B - f ` A')) \<and> |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
371 |
(\<forall>a \<in> A'. h a = f a) \<and> |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
372 |
(\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a))" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
373 |
proof- |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
374 |
obtain H where H_def: "H = (\<lambda> A'. A - (g`(B - (f ` A'))))" by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
375 |
have 0: "mono H" unfolding mono_def H_def by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
376 |
then obtain A' where 1: "H A' = A'" using lfp_unfold by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
377 |
hence 2: "A' = A - (g`(B - (f ` A')))" unfolding H_def by simp |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
378 |
hence 3: "A' \<le> A" by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
379 |
have 4: "\<forall>a \<in> A'. a \<notin> g`(B - f ` A')" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
380 |
using 2 by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
381 |
have 5: "\<forall>a \<in> A - A'. \<exists>b \<in> B - (f ` A'). a = g b" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
382 |
using 2 by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
383 |
(* *) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
384 |
obtain h where h_def: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
385 |
"h = (\<lambda> a. if a \<in> A' then f a else (SOME b. b \<in> B - (f ` A') \<and> a = g b))" by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
386 |
hence "\<forall>a \<in> A'. h a = f a" by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
387 |
moreover |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
388 |
have "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
389 |
proof |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
390 |
fix a assume *: "a \<in> A - A'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
391 |
let ?phi = "\<lambda> b. b \<in> B - (f ` A') \<and> a = g b" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
392 |
have "h a = (SOME b. ?phi b)" using h_def * by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
393 |
moreover have "\<exists>b. ?phi b" using 5 * by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
394 |
ultimately show "?phi (h a)" using someI_ex[of ?phi] by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
395 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
396 |
ultimately show ?thesis using 3 4 by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
397 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
398 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
399 |
theorem Cantor_Bernstein: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
400 |
assumes INJ1: "inj_on f A" and SUB1: "f ` A \<le> B" and |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
401 |
INJ2: "inj_on g B" and SUB2: "g ` B \<le> A" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
402 |
shows "\<exists>h. bij_betw h A B" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
403 |
proof- |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
404 |
obtain A' and h where 0: "A' \<le> A" and |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
405 |
1: "\<forall>a \<in> A'. a \<notin> g`(B - f ` A')" and |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
406 |
2: "\<forall>a \<in> A'. h a = f a" and |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
407 |
3: "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
408 |
using Cantor_Bernstein_aux[of A g B f] by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
409 |
have "inj_on h A" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
410 |
proof (intro inj_onI) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
411 |
fix a1 a2 |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
412 |
assume 4: "a1 \<in> A" and 5: "a2 \<in> A" and 6: "h a1 = h a2" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
413 |
show "a1 = a2" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
414 |
proof(cases "a1 \<in> A'") |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
415 |
assume Case1: "a1 \<in> A'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
416 |
show ?thesis |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
417 |
proof(cases "a2 \<in> A'") |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
418 |
assume Case11: "a2 \<in> A'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
419 |
hence "f a1 = f a2" using Case1 2 6 by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
420 |
thus ?thesis using INJ1 Case1 Case11 0 |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
421 |
unfolding inj_on_def by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
422 |
next |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
423 |
assume Case12: "a2 \<notin> A'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
424 |
hence False using 3 5 2 6 Case1 by force |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
425 |
thus ?thesis by simp |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
426 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
427 |
next |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
428 |
assume Case2: "a1 \<notin> A'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
429 |
show ?thesis |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
430 |
proof(cases "a2 \<in> A'") |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
431 |
assume Case21: "a2 \<in> A'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
432 |
hence False using 3 4 2 6 Case2 by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
433 |
thus ?thesis by simp |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
434 |
next |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
435 |
assume Case22: "a2 \<notin> A'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
436 |
hence "a1 = g(h a1) \<and> a2 = g(h a2)" using Case2 4 5 3 by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
437 |
thus ?thesis using 6 by simp |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
438 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
439 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
440 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
441 |
(* *) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
442 |
moreover |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
443 |
have "h ` A = B" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
444 |
proof safe |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
445 |
fix a assume "a \<in> A" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
446 |
thus "h a \<in> B" using SUB1 2 3 by (case_tac "a \<in> A'", auto) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
447 |
next |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
448 |
fix b assume *: "b \<in> B" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
449 |
show "b \<in> h ` A" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
450 |
proof(cases "b \<in> f ` A'") |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
451 |
assume Case1: "b \<in> f ` A'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
452 |
then obtain a where "a \<in> A' \<and> b = f a" by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
453 |
thus ?thesis using 2 0 by force |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
454 |
next |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
455 |
assume Case2: "b \<notin> f ` A'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
456 |
hence "g b \<notin> A'" using 1 * by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
457 |
hence 4: "g b \<in> A - A'" using * SUB2 by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
458 |
hence "h(g b) \<in> B \<and> g(h(g b)) = g b" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
459 |
using 3 by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
460 |
hence "h(g b) = b" using * INJ2 unfolding inj_on_def by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
461 |
thus ?thesis using 4 by force |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
462 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
463 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
464 |
(* *) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
465 |
ultimately show ?thesis unfolding bij_betw_def by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
466 |
qed |
14760 | 467 |
|
468 |
subsection {*Other Consequences of Hilbert's Epsilon*} |
|
469 |
||
470 |
text {*Hilbert's Epsilon and the @{term split} Operator*} |
|
471 |
||
472 |
text{*Looping simprule*} |
|
473 |
lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))" |
|
26347 | 474 |
by simp |
14760 | 475 |
|
476 |
lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))" |
|
26347 | 477 |
by (simp add: split_def) |
14760 | 478 |
|
479 |
lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)" |
|
26347 | 480 |
by blast |
14760 | 481 |
|
482 |
||
483 |
text{*A relation is wellfounded iff it has no infinite descending chain*} |
|
484 |
lemma wf_iff_no_infinite_down_chain: |
|
485 |
"wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))" |
|
486 |
apply (simp only: wf_eq_minimal) |
|
487 |
apply (rule iffI) |
|
488 |
apply (rule notI) |
|
489 |
apply (erule exE) |
|
490 |
apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast) |
|
491 |
apply (erule contrapos_np, simp, clarify) |
|
492 |
apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q") |
|
493 |
apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI) |
|
494 |
apply (rule allI, simp) |
|
495 |
apply (rule someI2_ex, blast, blast) |
|
496 |
apply (rule allI) |
|
497 |
apply (induct_tac "n", simp_all) |
|
498 |
apply (rule someI2_ex, blast+) |
|
499 |
done |
|
500 |
||
27760 | 501 |
lemma wf_no_infinite_down_chainE: |
502 |
assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r" |
|
503 |
using `wf r` wf_iff_no_infinite_down_chain[of r] by blast |
|
504 |
||
505 |
||
14760 | 506 |
text{*A dynamically-scoped fact for TFL *} |
12298 | 507 |
lemma tfl_some: "\<forall>P x. P x --> P (Eps P)" |
508 |
by (blast intro: someI) |
|
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
509 |
|
12298 | 510 |
|
511 |
subsection {* Least value operator *} |
|
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
512 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35216
diff
changeset
|
513 |
definition |
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35216
diff
changeset
|
514 |
LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where |
14760 | 515 |
"LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)" |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
516 |
|
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
517 |
syntax |
12298 | 518 |
"_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a" ("LEAST _ WRT _. _" [0, 4, 10] 10) |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
519 |
translations |
35115 | 520 |
"LEAST x WRT m. P" == "CONST LeastM m (%x. P)" |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
521 |
|
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
522 |
lemma LeastMI2: |
12298 | 523 |
"P x ==> (!!y. P y ==> m x <= m y) |
524 |
==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x) |
|
525 |
==> Q (LeastM m P)" |
|
14760 | 526 |
apply (simp add: LeastM_def) |
14208 | 527 |
apply (rule someI2_ex, blast, blast) |
12298 | 528 |
done |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
529 |
|
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
530 |
lemma LeastM_equality: |
12298 | 531 |
"P k ==> (!!x. P x ==> m k <= m x) |
532 |
==> m (LEAST x WRT m. P x) = (m k::'a::order)" |
|
14208 | 533 |
apply (rule LeastMI2, assumption, blast) |
12298 | 534 |
apply (blast intro!: order_antisym) |
535 |
done |
|
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
536 |
|
11454
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset
|
537 |
lemma wf_linord_ex_has_least: |
14760 | 538 |
"wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k |
539 |
==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)" |
|
12298 | 540 |
apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]]) |
14208 | 541 |
apply (drule_tac x = "m`Collect P" in spec, force) |
12298 | 542 |
done |
11454
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset
|
543 |
|
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset
|
544 |
lemma ex_has_least_nat: |
14760 | 545 |
"P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))" |
12298 | 546 |
apply (simp only: pred_nat_trancl_eq_le [symmetric]) |
547 |
apply (rule wf_pred_nat [THEN wf_linord_ex_has_least]) |
|
16796 | 548 |
apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption) |
12298 | 549 |
done |
11454
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset
|
550 |
|
12298 | 551 |
lemma LeastM_nat_lemma: |
14760 | 552 |
"P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))" |
553 |
apply (simp add: LeastM_def) |
|
12298 | 554 |
apply (rule someI_ex) |
555 |
apply (erule ex_has_least_nat) |
|
556 |
done |
|
11454
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset
|
557 |
|
45607 | 558 |
lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1] |
11454
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset
|
559 |
|
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset
|
560 |
lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)" |
14208 | 561 |
by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption) |
11454
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset
|
562 |
|
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
563 |
|
12298 | 564 |
subsection {* Greatest value operator *} |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
565 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35216
diff
changeset
|
566 |
definition |
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35216
diff
changeset
|
567 |
GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where |
14760 | 568 |
"GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)" |
12298 | 569 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35216
diff
changeset
|
570 |
definition |
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35216
diff
changeset
|
571 |
Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where |
12298 | 572 |
"Greatest == GreatestM (%x. x)" |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
573 |
|
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
574 |
syntax |
35115 | 575 |
"_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a" |
12298 | 576 |
("GREATEST _ WRT _. _" [0, 4, 10] 10) |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
577 |
translations |
35115 | 578 |
"GREATEST x WRT m. P" == "CONST GreatestM m (%x. P)" |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
579 |
|
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
580 |
lemma GreatestMI2: |
12298 | 581 |
"P x ==> (!!y. P y ==> m y <= m x) |
582 |
==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x) |
|
583 |
==> Q (GreatestM m P)" |
|
14760 | 584 |
apply (simp add: GreatestM_def) |
14208 | 585 |
apply (rule someI2_ex, blast, blast) |
12298 | 586 |
done |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
587 |
|
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
588 |
lemma GreatestM_equality: |
12298 | 589 |
"P k ==> (!!x. P x ==> m x <= m k) |
590 |
==> m (GREATEST x WRT m. P x) = (m k::'a::order)" |
|
14208 | 591 |
apply (rule_tac m = m in GreatestMI2, assumption, blast) |
12298 | 592 |
apply (blast intro!: order_antisym) |
593 |
done |
|
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
594 |
|
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
595 |
lemma Greatest_equality: |
12298 | 596 |
"P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k" |
14760 | 597 |
apply (simp add: Greatest_def) |
14208 | 598 |
apply (erule GreatestM_equality, blast) |
12298 | 599 |
done |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
600 |
|
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
601 |
lemma ex_has_greatest_nat_lemma: |
14760 | 602 |
"P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x)) |
603 |
==> \<exists>y. P y & ~ (m y < m k + n)" |
|
15251 | 604 |
apply (induct n, force) |
12298 | 605 |
apply (force simp add: le_Suc_eq) |
606 |
done |
|
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
607 |
|
12298 | 608 |
lemma ex_has_greatest_nat: |
14760 | 609 |
"P k ==> \<forall>y. P y --> m y < b |
610 |
==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)" |
|
12298 | 611 |
apply (rule ccontr) |
612 |
apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma) |
|
14208 | 613 |
apply (subgoal_tac [3] "m k <= b", auto) |
12298 | 614 |
done |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
615 |
|
12298 | 616 |
lemma GreatestM_nat_lemma: |
14760 | 617 |
"P k ==> \<forall>y. P y --> m y < b |
618 |
==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))" |
|
619 |
apply (simp add: GreatestM_def) |
|
12298 | 620 |
apply (rule someI_ex) |
14208 | 621 |
apply (erule ex_has_greatest_nat, assumption) |
12298 | 622 |
done |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
623 |
|
45607 | 624 |
lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1] |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
625 |
|
12298 | 626 |
lemma GreatestM_nat_le: |
14760 | 627 |
"P x ==> \<forall>y. P y --> m y < b |
12298 | 628 |
==> (m x::nat) <= m (GreatestM m P)" |
21020 | 629 |
apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P]) |
12298 | 630 |
done |
631 |
||
632 |
||
633 |
text {* \medskip Specialization to @{text GREATEST}. *} |
|
634 |
||
14760 | 635 |
lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)" |
636 |
apply (simp add: Greatest_def) |
|
14208 | 637 |
apply (rule GreatestM_natI, auto) |
12298 | 638 |
done |
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
639 |
|
12298 | 640 |
lemma Greatest_le: |
14760 | 641 |
"P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)" |
642 |
apply (simp add: Greatest_def) |
|
14208 | 643 |
apply (rule GreatestM_nat_le, auto) |
12298 | 644 |
done |
645 |
||
646 |
||
17893
aef5a6d11c2a
added lemma exE_some (from specification_package.ML);
wenzelm
parents:
17702
diff
changeset
|
647 |
subsection {* Specification package -- Hilbertized version *} |
aef5a6d11c2a
added lemma exE_some (from specification_package.ML);
wenzelm
parents:
17702
diff
changeset
|
648 |
|
aef5a6d11c2a
added lemma exE_some (from specification_package.ML);
wenzelm
parents:
17702
diff
changeset
|
649 |
lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c" |
aef5a6d11c2a
added lemma exE_some (from specification_package.ML);
wenzelm
parents:
17702
diff
changeset
|
650 |
by (simp only: someI_ex) |
aef5a6d11c2a
added lemma exE_some (from specification_package.ML);
wenzelm
parents:
17702
diff
changeset
|
651 |
|
31723
f5cafe803b55
discontinued ancient tradition to suffix certain ML module names with "_package"
haftmann
parents:
31454
diff
changeset
|
652 |
use "Tools/choice_specification.ML" |
14115 | 653 |
|
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset
|
654 |
end |