src/HOL/Hilbert_Choice.thy
author wenzelm
Thu May 24 17:25:53 2012 +0200 (2012-05-24)
changeset 47988 e4b69e10b990
parent 46950 d0181abdbdac
child 48891 c0eafbd55de3
permissions -rw-r--r--
tuned proofs;
paulson@11451
     1
(*  Title:      HOL/Hilbert_Choice.thy
nipkow@32988
     2
    Author:     Lawrence C Paulson, Tobias Nipkow
paulson@11451
     3
    Copyright   2001  University of Cambridge
wenzelm@12023
     4
*)
paulson@11451
     5
paulson@14760
     6
header {* Hilbert's Epsilon-Operator and the Axiom of Choice *}
paulson@11451
     7
nipkow@15131
     8
theory Hilbert_Choice
haftmann@29655
     9
imports Nat Wellfounded Plain
wenzelm@46950
    10
keywords "specification" "ax_specification" :: thy_goal
blanchet@39943
    11
uses ("Tools/choice_specification.ML")
nipkow@15131
    12
begin
wenzelm@12298
    13
wenzelm@12298
    14
subsection {* Hilbert's epsilon *}
wenzelm@12298
    15
haftmann@31454
    16
axiomatization Eps :: "('a => bool) => 'a" where
wenzelm@22690
    17
  someI: "P x ==> P (Eps P)"
paulson@11451
    18
wenzelm@14872
    19
syntax (epsilon)
wenzelm@14872
    20
  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3\<some>_./ _)" [0, 10] 10)
paulson@11451
    21
syntax (HOL)
wenzelm@12298
    22
  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3@ _./ _)" [0, 10] 10)
paulson@11451
    23
syntax
wenzelm@12298
    24
  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3SOME _./ _)" [0, 10] 10)
paulson@11451
    25
translations
wenzelm@22690
    26
  "SOME x. P" == "CONST Eps (%x. P)"
nipkow@13763
    27
nipkow@13763
    28
print_translation {*
wenzelm@35115
    29
  [(@{const_syntax Eps}, fn [Abs abs] =>
wenzelm@42284
    30
      let val (x, t) = Syntax_Trans.atomic_abs_tr' abs
wenzelm@35115
    31
      in Syntax.const @{syntax_const "_Eps"} $ x $ t end)]
wenzelm@35115
    32
*} -- {* to avoid eta-contraction of body *}
paulson@11451
    33
nipkow@33057
    34
definition inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
nipkow@33057
    35
"inv_into A f == %x. SOME y. y : A & f y = x"
paulson@11454
    36
nipkow@32988
    37
abbreviation inv :: "('a => 'b) => ('b => 'a)" where
nipkow@33057
    38
"inv == inv_into UNIV"
paulson@14760
    39
paulson@14760
    40
paulson@14760
    41
subsection {*Hilbert's Epsilon-operator*}
paulson@14760
    42
paulson@14760
    43
text{*Easier to apply than @{text someI} if the witness comes from an
paulson@14760
    44
existential formula*}
paulson@14760
    45
lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)"
paulson@14760
    46
apply (erule exE)
paulson@14760
    47
apply (erule someI)
paulson@14760
    48
done
paulson@14760
    49
paulson@14760
    50
text{*Easier to apply than @{text someI} because the conclusion has only one
paulson@14760
    51
occurrence of @{term P}.*}
paulson@14760
    52
lemma someI2: "[| P a;  !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
paulson@14760
    53
by (blast intro: someI)
paulson@14760
    54
paulson@14760
    55
text{*Easier to apply than @{text someI2} if the witness comes from an
paulson@14760
    56
existential formula*}
paulson@14760
    57
lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
paulson@14760
    58
by (blast intro: someI2)
paulson@14760
    59
paulson@14760
    60
lemma some_equality [intro]:
paulson@14760
    61
     "[| P a;  !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
paulson@14760
    62
by (blast intro: someI2)
paulson@14760
    63
paulson@14760
    64
lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
huffman@35216
    65
by blast
paulson@14760
    66
paulson@14760
    67
lemma some_eq_ex: "P (SOME x. P x) =  (\<exists>x. P x)"
paulson@14760
    68
by (blast intro: someI)
paulson@14760
    69
paulson@14760
    70
lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
paulson@14760
    71
apply (rule some_equality)
paulson@14760
    72
apply (rule refl, assumption)
paulson@14760
    73
done
paulson@14760
    74
paulson@14760
    75
lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"
paulson@14760
    76
apply (rule some_equality)
paulson@14760
    77
apply (rule refl)
paulson@14760
    78
apply (erule sym)
paulson@14760
    79
done
paulson@14760
    80
paulson@14760
    81
paulson@14760
    82
subsection{*Axiom of Choice, Proved Using the Description Operator*}
paulson@14760
    83
blanchet@39950
    84
lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)"
paulson@14760
    85
by (fast elim: someI)
paulson@14760
    86
paulson@14760
    87
lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)"
paulson@14760
    88
by (fast elim: someI)
paulson@14760
    89
paulson@14760
    90
paulson@14760
    91
subsection {*Function Inverse*}
paulson@14760
    92
nipkow@33014
    93
lemma inv_def: "inv f = (%y. SOME x. f x = y)"
nipkow@33057
    94
by(simp add: inv_into_def)
nipkow@33014
    95
nipkow@33057
    96
lemma inv_into_into: "x : f ` A ==> inv_into A f x : A"
nipkow@33057
    97
apply (simp add: inv_into_def)
nipkow@32988
    98
apply (fast intro: someI2)
nipkow@32988
    99
done
paulson@14760
   100
nipkow@32988
   101
lemma inv_id [simp]: "inv id = id"
nipkow@33057
   102
by (simp add: inv_into_def id_def)
paulson@14760
   103
nipkow@33057
   104
lemma inv_into_f_f [simp]:
nipkow@33057
   105
  "[| inj_on f A;  x : A |] ==> inv_into A f (f x) = x"
nipkow@33057
   106
apply (simp add: inv_into_def inj_on_def)
nipkow@32988
   107
apply (blast intro: someI2)
paulson@14760
   108
done
paulson@14760
   109
nipkow@32988
   110
lemma inv_f_f: "inj f ==> inv f (f x) = x"
huffman@35216
   111
by simp
nipkow@32988
   112
nipkow@33057
   113
lemma f_inv_into_f: "y : f`A  ==> f (inv_into A f y) = y"
nipkow@33057
   114
apply (simp add: inv_into_def)
nipkow@32988
   115
apply (fast intro: someI2)
nipkow@32988
   116
done
nipkow@32988
   117
nipkow@33057
   118
lemma inv_into_f_eq: "[| inj_on f A; x : A; f x = y |] ==> inv_into A f y = x"
nipkow@32988
   119
apply (erule subst)
nipkow@33057
   120
apply (fast intro: inv_into_f_f)
nipkow@32988
   121
done
nipkow@32988
   122
nipkow@32988
   123
lemma inv_f_eq: "[| inj f; f x = y |] ==> inv f y = x"
nipkow@33057
   124
by (simp add:inv_into_f_eq)
nipkow@32988
   125
nipkow@32988
   126
lemma inj_imp_inv_eq: "[| inj f; ALL x. f(g x) = x |] ==> inv f = g"
huffman@44921
   127
  by (blast intro: inv_into_f_eq)
paulson@14760
   128
paulson@14760
   129
text{*But is it useful?*}
paulson@14760
   130
lemma inj_transfer:
paulson@14760
   131
  assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)"
paulson@14760
   132
  shows "P x"
paulson@14760
   133
proof -
paulson@14760
   134
  have "f x \<in> range f" by auto
paulson@14760
   135
  hence "P(inv f (f x))" by (rule minor)
nipkow@33057
   136
  thus "P x" by (simp add: inv_into_f_f [OF injf])
paulson@14760
   137
qed
paulson@11451
   138
paulson@14760
   139
lemma inj_iff: "(inj f) = (inv f o f = id)"
nipkow@39302
   140
apply (simp add: o_def fun_eq_iff)
nipkow@33057
   141
apply (blast intro: inj_on_inverseI inv_into_f_f)
paulson@14760
   142
done
paulson@14760
   143
nipkow@23433
   144
lemma inv_o_cancel[simp]: "inj f ==> inv f o f = id"
nipkow@23433
   145
by (simp add: inj_iff)
nipkow@23433
   146
nipkow@23433
   147
lemma o_inv_o_cancel[simp]: "inj f ==> g o inv f o f = g"
nipkow@23433
   148
by (simp add: o_assoc[symmetric])
nipkow@23433
   149
nipkow@33057
   150
lemma inv_into_image_cancel[simp]:
nipkow@33057
   151
  "inj_on f A ==> S <= A ==> inv_into A f ` f ` S = S"
nipkow@44890
   152
by(fastforce simp: image_def)
nipkow@32988
   153
paulson@14760
   154
lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
hoelzl@40702
   155
by (blast intro!: surjI inv_into_f_f)
paulson@14760
   156
paulson@14760
   157
lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
hoelzl@40702
   158
by (simp add: f_inv_into_f)
paulson@14760
   159
nipkow@33057
   160
lemma inv_into_injective:
nipkow@33057
   161
  assumes eq: "inv_into A f x = inv_into A f y"
nipkow@32988
   162
      and x: "x: f`A"
nipkow@32988
   163
      and y: "y: f`A"
paulson@14760
   164
  shows "x=y"
paulson@14760
   165
proof -
nipkow@33057
   166
  have "f (inv_into A f x) = f (inv_into A f y)" using eq by simp
nipkow@33057
   167
  thus ?thesis by (simp add: f_inv_into_f x y)
paulson@14760
   168
qed
paulson@14760
   169
nipkow@33057
   170
lemma inj_on_inv_into: "B <= f`A ==> inj_on (inv_into A f) B"
nipkow@33057
   171
by (blast intro: inj_onI dest: inv_into_injective injD)
nipkow@32988
   172
nipkow@33057
   173
lemma bij_betw_inv_into: "bij_betw f A B ==> bij_betw (inv_into A f) B A"
nipkow@33057
   174
by (auto simp add: bij_betw_def inj_on_inv_into)
paulson@14760
   175
paulson@14760
   176
lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
hoelzl@40702
   177
by (simp add: inj_on_inv_into)
paulson@14760
   178
paulson@14760
   179
lemma surj_iff: "(surj f) = (f o inv f = id)"
hoelzl@40702
   180
by (auto intro!: surjI simp: surj_f_inv_f fun_eq_iff[where 'b='a])
hoelzl@40702
   181
hoelzl@40702
   182
lemma surj_iff_all: "surj f \<longleftrightarrow> (\<forall>x. f (inv f x) = x)"
hoelzl@40702
   183
  unfolding surj_iff by (simp add: o_def fun_eq_iff)
paulson@14760
   184
paulson@14760
   185
lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
paulson@14760
   186
apply (rule ext)
paulson@14760
   187
apply (drule_tac x = "inv f x" in spec)
paulson@14760
   188
apply (simp add: surj_f_inv_f)
paulson@14760
   189
done
paulson@14760
   190
paulson@14760
   191
lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
paulson@14760
   192
by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
wenzelm@12372
   193
paulson@14760
   194
lemma inv_equality: "[| !!x. g (f x) = x;  !!y. f (g y) = y |] ==> inv f = g"
paulson@14760
   195
apply (rule ext)
nipkow@33057
   196
apply (auto simp add: inv_into_def)
paulson@14760
   197
done
paulson@14760
   198
paulson@14760
   199
lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
paulson@14760
   200
apply (rule inv_equality)
paulson@14760
   201
apply (auto simp add: bij_def surj_f_inv_f)
paulson@14760
   202
done
paulson@14760
   203
paulson@14760
   204
(** bij(inv f) implies little about f.  Consider f::bool=>bool such that
paulson@14760
   205
    f(True)=f(False)=True.  Then it's consistent with axiom someI that
paulson@14760
   206
    inv f could be any function at all, including the identity function.
paulson@14760
   207
    If inv f=id then inv f is a bijection, but inj f, surj(f) and
paulson@14760
   208
    inv(inv f)=f all fail.
paulson@14760
   209
**)
paulson@14760
   210
nipkow@33057
   211
lemma inv_into_comp:
nipkow@32988
   212
  "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
nipkow@33057
   213
  inv_into A (f o g) x = (inv_into A g o inv_into (g ` A) f) x"
nipkow@33057
   214
apply (rule inv_into_f_eq)
nipkow@32988
   215
  apply (fast intro: comp_inj_on)
nipkow@33057
   216
 apply (simp add: inv_into_into)
nipkow@33057
   217
apply (simp add: f_inv_into_f inv_into_into)
nipkow@32988
   218
done
nipkow@32988
   219
paulson@14760
   220
lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
paulson@14760
   221
apply (rule inv_equality)
paulson@14760
   222
apply (auto simp add: bij_def surj_f_inv_f)
paulson@14760
   223
done
paulson@14760
   224
paulson@14760
   225
lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
paulson@14760
   226
by (simp add: image_eq_UN surj_f_inv_f)
paulson@14760
   227
paulson@14760
   228
lemma image_inv_f_f: "inj f ==> (inv f) ` (f ` A) = A"
paulson@14760
   229
by (simp add: image_eq_UN)
paulson@14760
   230
paulson@14760
   231
lemma inv_image_comp: "inj f ==> inv f ` (f`X) = X"
paulson@14760
   232
by (auto simp add: image_def)
paulson@14760
   233
paulson@14760
   234
lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
paulson@14760
   235
apply auto
paulson@14760
   236
apply (force simp add: bij_is_inj)
paulson@14760
   237
apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
paulson@14760
   238
done
paulson@14760
   239
paulson@14760
   240
lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A" 
paulson@14760
   241
apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
nipkow@33057
   242
apply (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric])
paulson@14760
   243
done
paulson@14760
   244
haftmann@31380
   245
lemma finite_fun_UNIVD1:
haftmann@31380
   246
  assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
haftmann@31380
   247
  and card: "card (UNIV :: 'b set) \<noteq> Suc 0"
haftmann@31380
   248
  shows "finite (UNIV :: 'a set)"
haftmann@31380
   249
proof -
haftmann@31380
   250
  from fin have finb: "finite (UNIV :: 'b set)" by (rule finite_fun_UNIVD2)
haftmann@31380
   251
  with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)"
haftmann@31380
   252
    by (cases "card (UNIV :: 'b set)") (auto simp add: card_eq_0_iff)
haftmann@31380
   253
  then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - Suc (Suc 0)" by auto
haftmann@31380
   254
  then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by (auto simp add: card_Suc_eq)
haftmann@31380
   255
  from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by (rule finite_imageI)
haftmann@31380
   256
  moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)"
haftmann@31380
   257
  proof (rule UNIV_eq_I)
haftmann@31380
   258
    fix x :: 'a
nipkow@33057
   259
    from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_into_def)
haftmann@31380
   260
    thus "x \<in> range (\<lambda>f\<Colon>'a \<Rightarrow> 'b. inv f b1)" by blast
haftmann@31380
   261
  qed
haftmann@31380
   262
  ultimately show "finite (UNIV :: 'a set)" by simp
haftmann@31380
   263
qed
paulson@14760
   264
hoelzl@40703
   265
lemma image_inv_into_cancel:
hoelzl@40703
   266
  assumes SURJ: "f`A=A'" and SUB: "B' \<le> A'"
hoelzl@40703
   267
  shows "f `((inv_into A f)`B') = B'"
hoelzl@40703
   268
  using assms
hoelzl@40703
   269
proof (auto simp add: f_inv_into_f)
hoelzl@40703
   270
  let ?f' = "(inv_into A f)"
hoelzl@40703
   271
  fix a' assume *: "a' \<in> B'"
hoelzl@40703
   272
  then have "a' \<in> A'" using SUB by auto
hoelzl@40703
   273
  then have "a' = f (?f' a')"
hoelzl@40703
   274
    using SURJ by (auto simp add: f_inv_into_f)
hoelzl@40703
   275
  then show "a' \<in> f ` (?f' ` B')" using * by blast
hoelzl@40703
   276
qed
hoelzl@40703
   277
hoelzl@40703
   278
lemma inv_into_inv_into_eq:
hoelzl@40703
   279
  assumes "bij_betw f A A'" "a \<in> A"
hoelzl@40703
   280
  shows "inv_into A' (inv_into A f) a = f a"
hoelzl@40703
   281
proof -
hoelzl@40703
   282
  let ?f' = "inv_into A f"   let ?f'' = "inv_into A' ?f'"
hoelzl@40703
   283
  have 1: "bij_betw ?f' A' A" using assms
hoelzl@40703
   284
  by (auto simp add: bij_betw_inv_into)
hoelzl@40703
   285
  obtain a' where 2: "a' \<in> A'" and 3: "?f' a' = a"
hoelzl@40703
   286
    using 1 `a \<in> A` unfolding bij_betw_def by force
hoelzl@40703
   287
  hence "?f'' a = a'"
hoelzl@40703
   288
    using `a \<in> A` 1 3 by (auto simp add: f_inv_into_f bij_betw_def)
hoelzl@40703
   289
  moreover have "f a = a'" using assms 2 3
huffman@44921
   290
    by (auto simp add: bij_betw_def)
hoelzl@40703
   291
  ultimately show "?f'' a = f a" by simp
hoelzl@40703
   292
qed
hoelzl@40703
   293
hoelzl@40703
   294
lemma inj_on_iff_surj:
hoelzl@40703
   295
  assumes "A \<noteq> {}"
hoelzl@40703
   296
  shows "(\<exists>f. inj_on f A \<and> f ` A \<le> A') \<longleftrightarrow> (\<exists>g. g ` A' = A)"
hoelzl@40703
   297
proof safe
hoelzl@40703
   298
  fix f assume INJ: "inj_on f A" and INCL: "f ` A \<le> A'"
hoelzl@40703
   299
  let ?phi = "\<lambda>a' a. a \<in> A \<and> f a = a'"  let ?csi = "\<lambda>a. a \<in> A"
hoelzl@40703
   300
  let ?g = "\<lambda>a'. if a' \<in> f ` A then (SOME a. ?phi a' a) else (SOME a. ?csi a)"
hoelzl@40703
   301
  have "?g ` A' = A"
hoelzl@40703
   302
  proof
hoelzl@40703
   303
    show "?g ` A' \<le> A"
hoelzl@40703
   304
    proof clarify
hoelzl@40703
   305
      fix a' assume *: "a' \<in> A'"
hoelzl@40703
   306
      show "?g a' \<in> A"
hoelzl@40703
   307
      proof cases
hoelzl@40703
   308
        assume Case1: "a' \<in> f ` A"
hoelzl@40703
   309
        then obtain a where "?phi a' a" by blast
hoelzl@40703
   310
        hence "?phi a' (SOME a. ?phi a' a)" using someI[of "?phi a'" a] by blast
hoelzl@40703
   311
        with Case1 show ?thesis by auto
hoelzl@40703
   312
      next
hoelzl@40703
   313
        assume Case2: "a' \<notin> f ` A"
hoelzl@40703
   314
        hence "?csi (SOME a. ?csi a)" using assms someI_ex[of ?csi] by blast
hoelzl@40703
   315
        with Case2 show ?thesis by auto
hoelzl@40703
   316
      qed
hoelzl@40703
   317
    qed
hoelzl@40703
   318
  next
hoelzl@40703
   319
    show "A \<le> ?g ` A'"
hoelzl@40703
   320
    proof-
hoelzl@40703
   321
      {fix a assume *: "a \<in> A"
hoelzl@40703
   322
       let ?b = "SOME aa. ?phi (f a) aa"
hoelzl@40703
   323
       have "?phi (f a) a" using * by auto
hoelzl@40703
   324
       hence 1: "?phi (f a) ?b" using someI[of "?phi(f a)" a] by blast
hoelzl@40703
   325
       hence "?g(f a) = ?b" using * by auto
hoelzl@40703
   326
       moreover have "a = ?b" using 1 INJ * by (auto simp add: inj_on_def)
hoelzl@40703
   327
       ultimately have "?g(f a) = a" by simp
hoelzl@40703
   328
       with INCL * have "?g(f a) = a \<and> f a \<in> A'" by auto
hoelzl@40703
   329
      }
hoelzl@40703
   330
      thus ?thesis by force
hoelzl@40703
   331
    qed
hoelzl@40703
   332
  qed
hoelzl@40703
   333
  thus "\<exists>g. g ` A' = A" by blast
hoelzl@40703
   334
next
hoelzl@40703
   335
  fix g  let ?f = "inv_into A' g"
hoelzl@40703
   336
  have "inj_on ?f (g ` A')"
hoelzl@40703
   337
    by (auto simp add: inj_on_inv_into)
hoelzl@40703
   338
  moreover
hoelzl@40703
   339
  {fix a' assume *: "a' \<in> A'"
hoelzl@40703
   340
   let ?phi = "\<lambda> b'. b' \<in> A' \<and> g b' = g a'"
hoelzl@40703
   341
   have "?phi a'" using * by auto
hoelzl@40703
   342
   hence "?phi(SOME b'. ?phi b')" using someI[of ?phi] by blast
hoelzl@40703
   343
   hence "?f(g a') \<in> A'" unfolding inv_into_def by auto
hoelzl@40703
   344
  }
hoelzl@40703
   345
  ultimately show "\<exists>f. inj_on f (g ` A') \<and> f ` g ` A' \<subseteq> A'" by auto
hoelzl@40703
   346
qed
hoelzl@40703
   347
hoelzl@40703
   348
lemma Ex_inj_on_UNION_Sigma:
hoelzl@40703
   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))"
hoelzl@40703
   350
proof
hoelzl@40703
   351
  let ?phi = "\<lambda> a i. i \<in> I \<and> a \<in> A i"
hoelzl@40703
   352
  let ?sm = "\<lambda> a. SOME i. ?phi a i"
hoelzl@40703
   353
  let ?f = "\<lambda>a. (?sm a, a)"
hoelzl@40703
   354
  have "inj_on ?f (\<Union> i \<in> I. A i)" unfolding inj_on_def by auto
hoelzl@40703
   355
  moreover
hoelzl@40703
   356
  { { fix i a assume "i \<in> I" and "a \<in> A i"
hoelzl@40703
   357
      hence "?sm a \<in> I \<and> a \<in> A(?sm a)" using someI[of "?phi a" i] by auto
hoelzl@40703
   358
    }
hoelzl@40703
   359
    hence "?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)" by auto
hoelzl@40703
   360
  }
hoelzl@40703
   361
  ultimately
hoelzl@40703
   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)"
hoelzl@40703
   363
  by auto
hoelzl@40703
   364
qed
hoelzl@40703
   365
hoelzl@40703
   366
subsection {* The Cantor-Bernstein Theorem *}
hoelzl@40703
   367
hoelzl@40703
   368
lemma Cantor_Bernstein_aux:
hoelzl@40703
   369
  shows "\<exists>A' h. A' \<le> A \<and>
hoelzl@40703
   370
                (\<forall>a \<in> A'. a \<notin> g`(B - f ` A')) \<and>
hoelzl@40703
   371
                (\<forall>a \<in> A'. h a = f a) \<and>
hoelzl@40703
   372
                (\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a))"
hoelzl@40703
   373
proof-
hoelzl@40703
   374
  obtain H where H_def: "H = (\<lambda> A'. A - (g`(B - (f ` A'))))" by blast
hoelzl@40703
   375
  have 0: "mono H" unfolding mono_def H_def by blast
hoelzl@40703
   376
  then obtain A' where 1: "H A' = A'" using lfp_unfold by blast
hoelzl@40703
   377
  hence 2: "A' = A - (g`(B - (f ` A')))" unfolding H_def by simp
hoelzl@40703
   378
  hence 3: "A' \<le> A" by blast
hoelzl@40703
   379
  have 4: "\<forall>a \<in> A'.  a \<notin> g`(B - f ` A')"
hoelzl@40703
   380
  using 2 by blast
hoelzl@40703
   381
  have 5: "\<forall>a \<in> A - A'. \<exists>b \<in> B - (f ` A'). a = g b"
hoelzl@40703
   382
  using 2 by blast
hoelzl@40703
   383
  (*  *)
hoelzl@40703
   384
  obtain h where h_def:
hoelzl@40703
   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
hoelzl@40703
   386
  hence "\<forall>a \<in> A'. h a = f a" by auto
hoelzl@40703
   387
  moreover
hoelzl@40703
   388
  have "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
hoelzl@40703
   389
  proof
hoelzl@40703
   390
    fix a assume *: "a \<in> A - A'"
hoelzl@40703
   391
    let ?phi = "\<lambda> b. b \<in> B - (f ` A') \<and> a = g b"
hoelzl@40703
   392
    have "h a = (SOME b. ?phi b)" using h_def * by auto
hoelzl@40703
   393
    moreover have "\<exists>b. ?phi b" using 5 *  by auto
hoelzl@40703
   394
    ultimately show  "?phi (h a)" using someI_ex[of ?phi] by auto
hoelzl@40703
   395
  qed
hoelzl@40703
   396
  ultimately show ?thesis using 3 4 by blast
hoelzl@40703
   397
qed
hoelzl@40703
   398
hoelzl@40703
   399
theorem Cantor_Bernstein:
hoelzl@40703
   400
  assumes INJ1: "inj_on f A" and SUB1: "f ` A \<le> B" and
hoelzl@40703
   401
          INJ2: "inj_on g B" and SUB2: "g ` B \<le> A"
hoelzl@40703
   402
  shows "\<exists>h. bij_betw h A B"
hoelzl@40703
   403
proof-
hoelzl@40703
   404
  obtain A' and h where 0: "A' \<le> A" and
hoelzl@40703
   405
  1: "\<forall>a \<in> A'. a \<notin> g`(B - f ` A')" and
hoelzl@40703
   406
  2: "\<forall>a \<in> A'. h a = f a" and
hoelzl@40703
   407
  3: "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
hoelzl@40703
   408
  using Cantor_Bernstein_aux[of A g B f] by blast
hoelzl@40703
   409
  have "inj_on h A"
hoelzl@40703
   410
  proof (intro inj_onI)
hoelzl@40703
   411
    fix a1 a2
hoelzl@40703
   412
    assume 4: "a1 \<in> A" and 5: "a2 \<in> A" and 6: "h a1 = h a2"
hoelzl@40703
   413
    show "a1 = a2"
hoelzl@40703
   414
    proof(cases "a1 \<in> A'")
hoelzl@40703
   415
      assume Case1: "a1 \<in> A'"
hoelzl@40703
   416
      show ?thesis
hoelzl@40703
   417
      proof(cases "a2 \<in> A'")
hoelzl@40703
   418
        assume Case11: "a2 \<in> A'"
hoelzl@40703
   419
        hence "f a1 = f a2" using Case1 2 6 by auto
hoelzl@40703
   420
        thus ?thesis using INJ1 Case1 Case11 0
hoelzl@40703
   421
        unfolding inj_on_def by blast
hoelzl@40703
   422
      next
hoelzl@40703
   423
        assume Case12: "a2 \<notin> A'"
hoelzl@40703
   424
        hence False using 3 5 2 6 Case1 by force
hoelzl@40703
   425
        thus ?thesis by simp
hoelzl@40703
   426
      qed
hoelzl@40703
   427
    next
hoelzl@40703
   428
    assume Case2: "a1 \<notin> A'"
hoelzl@40703
   429
      show ?thesis
hoelzl@40703
   430
      proof(cases "a2 \<in> A'")
hoelzl@40703
   431
        assume Case21: "a2 \<in> A'"
hoelzl@40703
   432
        hence False using 3 4 2 6 Case2 by auto
hoelzl@40703
   433
        thus ?thesis by simp
hoelzl@40703
   434
      next
hoelzl@40703
   435
        assume Case22: "a2 \<notin> A'"
hoelzl@40703
   436
        hence "a1 = g(h a1) \<and> a2 = g(h a2)" using Case2 4 5 3 by auto
hoelzl@40703
   437
        thus ?thesis using 6 by simp
hoelzl@40703
   438
      qed
hoelzl@40703
   439
    qed
hoelzl@40703
   440
  qed
hoelzl@40703
   441
  (*  *)
hoelzl@40703
   442
  moreover
hoelzl@40703
   443
  have "h ` A = B"
hoelzl@40703
   444
  proof safe
hoelzl@40703
   445
    fix a assume "a \<in> A"
wenzelm@47988
   446
    thus "h a \<in> B" using SUB1 2 3 by (cases "a \<in> A'") auto
hoelzl@40703
   447
  next
hoelzl@40703
   448
    fix b assume *: "b \<in> B"
hoelzl@40703
   449
    show "b \<in> h ` A"
hoelzl@40703
   450
    proof(cases "b \<in> f ` A'")
hoelzl@40703
   451
      assume Case1: "b \<in> f ` A'"
hoelzl@40703
   452
      then obtain a where "a \<in> A' \<and> b = f a" by blast
hoelzl@40703
   453
      thus ?thesis using 2 0 by force
hoelzl@40703
   454
    next
hoelzl@40703
   455
      assume Case2: "b \<notin> f ` A'"
hoelzl@40703
   456
      hence "g b \<notin> A'" using 1 * by auto
hoelzl@40703
   457
      hence 4: "g b \<in> A - A'" using * SUB2 by auto
hoelzl@40703
   458
      hence "h(g b) \<in> B \<and> g(h(g b)) = g b"
hoelzl@40703
   459
      using 3 by auto
hoelzl@40703
   460
      hence "h(g b) = b" using * INJ2 unfolding inj_on_def by auto
hoelzl@40703
   461
      thus ?thesis using 4 by force
hoelzl@40703
   462
    qed
hoelzl@40703
   463
  qed
hoelzl@40703
   464
  (*  *)
hoelzl@40703
   465
  ultimately show ?thesis unfolding bij_betw_def by auto
hoelzl@40703
   466
qed
paulson@14760
   467
paulson@14760
   468
subsection {*Other Consequences of Hilbert's Epsilon*}
paulson@14760
   469
paulson@14760
   470
text {*Hilbert's Epsilon and the @{term split} Operator*}
paulson@14760
   471
paulson@14760
   472
text{*Looping simprule*}
paulson@14760
   473
lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
haftmann@26347
   474
  by simp
paulson@14760
   475
paulson@14760
   476
lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"
haftmann@26347
   477
  by (simp add: split_def)
paulson@14760
   478
paulson@14760
   479
lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
haftmann@26347
   480
  by blast
paulson@14760
   481
paulson@14760
   482
paulson@14760
   483
text{*A relation is wellfounded iff it has no infinite descending chain*}
paulson@14760
   484
lemma wf_iff_no_infinite_down_chain:
paulson@14760
   485
  "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
paulson@14760
   486
apply (simp only: wf_eq_minimal)
paulson@14760
   487
apply (rule iffI)
paulson@14760
   488
 apply (rule notI)
paulson@14760
   489
 apply (erule exE)
paulson@14760
   490
 apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
paulson@14760
   491
apply (erule contrapos_np, simp, clarify)
paulson@14760
   492
apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
paulson@14760
   493
 apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI)
paulson@14760
   494
 apply (rule allI, simp)
paulson@14760
   495
 apply (rule someI2_ex, blast, blast)
paulson@14760
   496
apply (rule allI)
paulson@14760
   497
apply (induct_tac "n", simp_all)
paulson@14760
   498
apply (rule someI2_ex, blast+)
paulson@14760
   499
done
paulson@14760
   500
nipkow@27760
   501
lemma wf_no_infinite_down_chainE:
nipkow@27760
   502
  assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r"
nipkow@27760
   503
using `wf r` wf_iff_no_infinite_down_chain[of r] by blast
nipkow@27760
   504
nipkow@27760
   505
paulson@14760
   506
text{*A dynamically-scoped fact for TFL *}
wenzelm@12298
   507
lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
wenzelm@12298
   508
  by (blast intro: someI)
paulson@11451
   509
wenzelm@12298
   510
wenzelm@12298
   511
subsection {* Least value operator *}
paulson@11451
   512
haftmann@35416
   513
definition
haftmann@35416
   514
  LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where
paulson@14760
   515
  "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
paulson@11451
   516
paulson@11451
   517
syntax
wenzelm@12298
   518
  "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
paulson@11451
   519
translations
wenzelm@35115
   520
  "LEAST x WRT m. P" == "CONST LeastM m (%x. P)"
paulson@11451
   521
paulson@11451
   522
lemma LeastMI2:
wenzelm@12298
   523
  "P x ==> (!!y. P y ==> m x <= m y)
wenzelm@12298
   524
    ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
wenzelm@12298
   525
    ==> Q (LeastM m P)"
paulson@14760
   526
  apply (simp add: LeastM_def)
paulson@14208
   527
  apply (rule someI2_ex, blast, blast)
wenzelm@12298
   528
  done
paulson@11451
   529
paulson@11451
   530
lemma LeastM_equality:
wenzelm@12298
   531
  "P k ==> (!!x. P x ==> m k <= m x)
wenzelm@12298
   532
    ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
paulson@14208
   533
  apply (rule LeastMI2, assumption, blast)
wenzelm@12298
   534
  apply (blast intro!: order_antisym)
wenzelm@12298
   535
  done
paulson@11451
   536
paulson@11454
   537
lemma wf_linord_ex_has_least:
paulson@14760
   538
  "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
paulson@14760
   539
    ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
wenzelm@12298
   540
  apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
paulson@14208
   541
  apply (drule_tac x = "m`Collect P" in spec, force)
wenzelm@12298
   542
  done
paulson@11454
   543
paulson@11454
   544
lemma ex_has_least_nat:
paulson@14760
   545
    "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
wenzelm@12298
   546
  apply (simp only: pred_nat_trancl_eq_le [symmetric])
wenzelm@12298
   547
  apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
paulson@16796
   548
   apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
wenzelm@12298
   549
  done
paulson@11454
   550
wenzelm@12298
   551
lemma LeastM_nat_lemma:
paulson@14760
   552
    "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
paulson@14760
   553
  apply (simp add: LeastM_def)
wenzelm@12298
   554
  apply (rule someI_ex)
wenzelm@12298
   555
  apply (erule ex_has_least_nat)
wenzelm@12298
   556
  done
paulson@11454
   557
wenzelm@45607
   558
lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1]
paulson@11454
   559
paulson@11454
   560
lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
paulson@14208
   561
by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
paulson@11454
   562
paulson@11451
   563
wenzelm@12298
   564
subsection {* Greatest value operator *}
paulson@11451
   565
haftmann@35416
   566
definition
haftmann@35416
   567
  GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where
paulson@14760
   568
  "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
wenzelm@12298
   569
haftmann@35416
   570
definition
haftmann@35416
   571
  Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where
wenzelm@12298
   572
  "Greatest == GreatestM (%x. x)"
paulson@11451
   573
paulson@11451
   574
syntax
wenzelm@35115
   575
  "_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"
wenzelm@12298
   576
      ("GREATEST _ WRT _. _" [0, 4, 10] 10)
paulson@11451
   577
translations
wenzelm@35115
   578
  "GREATEST x WRT m. P" == "CONST GreatestM m (%x. P)"
paulson@11451
   579
paulson@11451
   580
lemma GreatestMI2:
wenzelm@12298
   581
  "P x ==> (!!y. P y ==> m y <= m x)
wenzelm@12298
   582
    ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
wenzelm@12298
   583
    ==> Q (GreatestM m P)"
paulson@14760
   584
  apply (simp add: GreatestM_def)
paulson@14208
   585
  apply (rule someI2_ex, blast, blast)
wenzelm@12298
   586
  done
paulson@11451
   587
paulson@11451
   588
lemma GreatestM_equality:
wenzelm@12298
   589
 "P k ==> (!!x. P x ==> m x <= m k)
wenzelm@12298
   590
    ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
paulson@14208
   591
  apply (rule_tac m = m in GreatestMI2, assumption, blast)
wenzelm@12298
   592
  apply (blast intro!: order_antisym)
wenzelm@12298
   593
  done
paulson@11451
   594
paulson@11451
   595
lemma Greatest_equality:
wenzelm@12298
   596
  "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
paulson@14760
   597
  apply (simp add: Greatest_def)
paulson@14208
   598
  apply (erule GreatestM_equality, blast)
wenzelm@12298
   599
  done
paulson@11451
   600
paulson@11451
   601
lemma ex_has_greatest_nat_lemma:
paulson@14760
   602
  "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
paulson@14760
   603
    ==> \<exists>y. P y & ~ (m y < m k + n)"
paulson@15251
   604
  apply (induct n, force)
wenzelm@12298
   605
  apply (force simp add: le_Suc_eq)
wenzelm@12298
   606
  done
paulson@11451
   607
wenzelm@12298
   608
lemma ex_has_greatest_nat:
paulson@14760
   609
  "P k ==> \<forall>y. P y --> m y < b
paulson@14760
   610
    ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
wenzelm@12298
   611
  apply (rule ccontr)
wenzelm@12298
   612
  apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
paulson@14208
   613
    apply (subgoal_tac [3] "m k <= b", auto)
wenzelm@12298
   614
  done
paulson@11451
   615
wenzelm@12298
   616
lemma GreatestM_nat_lemma:
paulson@14760
   617
  "P k ==> \<forall>y. P y --> m y < b
paulson@14760
   618
    ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
paulson@14760
   619
  apply (simp add: GreatestM_def)
wenzelm@12298
   620
  apply (rule someI_ex)
paulson@14208
   621
  apply (erule ex_has_greatest_nat, assumption)
wenzelm@12298
   622
  done
paulson@11451
   623
wenzelm@45607
   624
lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1]
paulson@11451
   625
wenzelm@12298
   626
lemma GreatestM_nat_le:
paulson@14760
   627
  "P x ==> \<forall>y. P y --> m y < b
wenzelm@12298
   628
    ==> (m x::nat) <= m (GreatestM m P)"
berghofe@21020
   629
  apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
wenzelm@12298
   630
  done
wenzelm@12298
   631
wenzelm@12298
   632
wenzelm@12298
   633
text {* \medskip Specialization to @{text GREATEST}. *}
wenzelm@12298
   634
paulson@14760
   635
lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
paulson@14760
   636
  apply (simp add: Greatest_def)
paulson@14208
   637
  apply (rule GreatestM_natI, auto)
wenzelm@12298
   638
  done
paulson@11451
   639
wenzelm@12298
   640
lemma Greatest_le:
paulson@14760
   641
    "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
paulson@14760
   642
  apply (simp add: Greatest_def)
paulson@14208
   643
  apply (rule GreatestM_nat_le, auto)
wenzelm@12298
   644
  done
wenzelm@12298
   645
wenzelm@12298
   646
wenzelm@17893
   647
subsection {* Specification package -- Hilbertized version *}
wenzelm@17893
   648
wenzelm@17893
   649
lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
wenzelm@17893
   650
  by (simp only: someI_ex)
wenzelm@17893
   651
haftmann@31723
   652
use "Tools/choice_specification.ML"
skalberg@14115
   653
paulson@11451
   654
end