src/HOL/Cardinals/Cardinal_Order_Relation.thy
author blanchet
Mon Nov 18 18:04:45 2013 +0100 (2013-11-18)
changeset 54481 5c9819d7713b
parent 54475 b4d644be252c
child 54578 9387251b6a46
permissions -rw-r--r--
compile
blanchet@49310
     1
(*  Title:      HOL/Cardinals/Cardinal_Order_Relation.thy
blanchet@48975
     2
    Author:     Andrei Popescu, TU Muenchen
blanchet@48975
     3
    Copyright   2012
blanchet@48975
     4
blanchet@48975
     5
Cardinal-order relations.
blanchet@48975
     6
*)
blanchet@48975
     7
blanchet@48975
     8
header {* Cardinal-Order Relations *}
blanchet@48975
     9
blanchet@48975
    10
theory Cardinal_Order_Relation
blanchet@54481
    11
imports Cardinal_Order_Relation_FP Constructions_on_Wellorders
blanchet@48975
    12
begin
blanchet@48975
    13
blanchet@48975
    14
declare
blanchet@48975
    15
  card_order_on_well_order_on[simp]
blanchet@48975
    16
  card_of_card_order_on[simp]
blanchet@48975
    17
  card_of_well_order_on[simp]
blanchet@48975
    18
  Field_card_of[simp]
blanchet@48975
    19
  card_of_Card_order[simp]
blanchet@48975
    20
  card_of_Well_order[simp]
blanchet@48975
    21
  card_of_least[simp]
blanchet@48975
    22
  card_of_unique[simp]
blanchet@48975
    23
  card_of_mono1[simp]
blanchet@48975
    24
  card_of_mono2[simp]
blanchet@48975
    25
  card_of_cong[simp]
blanchet@48975
    26
  card_of_Field_ordLess[simp]
blanchet@48975
    27
  card_of_Field_ordIso[simp]
blanchet@48975
    28
  card_of_underS[simp]
blanchet@48975
    29
  ordLess_Field[simp]
blanchet@48975
    30
  card_of_empty[simp]
blanchet@48975
    31
  card_of_empty1[simp]
blanchet@48975
    32
  card_of_image[simp]
blanchet@48975
    33
  card_of_singl_ordLeq[simp]
blanchet@48975
    34
  Card_order_singl_ordLeq[simp]
blanchet@48975
    35
  card_of_Pow[simp]
blanchet@48975
    36
  Card_order_Pow[simp]
blanchet@48975
    37
  card_of_Plus1[simp]
blanchet@48975
    38
  Card_order_Plus1[simp]
blanchet@48975
    39
  card_of_Plus2[simp]
blanchet@48975
    40
  Card_order_Plus2[simp]
blanchet@48975
    41
  card_of_Plus_mono1[simp]
blanchet@48975
    42
  card_of_Plus_mono2[simp]
blanchet@48975
    43
  card_of_Plus_mono[simp]
blanchet@48975
    44
  card_of_Plus_cong2[simp]
blanchet@48975
    45
  card_of_Plus_cong[simp]
blanchet@48975
    46
  card_of_Un_Plus_ordLeq[simp]
blanchet@48975
    47
  card_of_Times1[simp]
blanchet@48975
    48
  card_of_Times2[simp]
blanchet@48975
    49
  card_of_Times3[simp]
blanchet@48975
    50
  card_of_Times_mono1[simp]
blanchet@48975
    51
  card_of_Times_mono2[simp]
blanchet@48975
    52
  card_of_ordIso_finite[simp]
blanchet@48975
    53
  card_of_Times_same_infinite[simp]
blanchet@48975
    54
  card_of_Times_infinite_simps[simp]
blanchet@48975
    55
  card_of_Plus_infinite1[simp]
blanchet@48975
    56
  card_of_Plus_infinite2[simp]
blanchet@48975
    57
  card_of_Plus_ordLess_infinite[simp]
blanchet@48975
    58
  card_of_Plus_ordLess_infinite_Field[simp]
blanchet@48975
    59
  infinite_cartesian_product[simp]
blanchet@48975
    60
  cardSuc_Card_order[simp]
blanchet@48975
    61
  cardSuc_greater[simp]
blanchet@48975
    62
  cardSuc_ordLeq[simp]
blanchet@48975
    63
  cardSuc_ordLeq_ordLess[simp]
blanchet@48975
    64
  cardSuc_mono_ordLeq[simp]
blanchet@48975
    65
  cardSuc_invar_ordIso[simp]
blanchet@48975
    66
  card_of_cardSuc_finite[simp]
blanchet@48975
    67
  cardSuc_finite[simp]
blanchet@48975
    68
  card_of_Plus_ordLeq_infinite_Field[simp]
blanchet@48975
    69
  curr_in[intro, simp]
blanchet@48975
    70
  Func_empty[simp]
blanchet@48975
    71
  Func_is_emp[simp]
blanchet@48975
    72
blanchet@48975
    73
blanchet@48975
    74
subsection {* Cardinal of a set *}
blanchet@48975
    75
blanchet@48975
    76
lemma card_of_inj_rel: assumes INJ: "!! x y y'. \<lbrakk>(x,y) : R; (x,y') : R\<rbrakk> \<Longrightarrow> y = y'"
blanchet@48975
    77
shows "|{y. EX x. (x,y) : R}| <=o |{x. EX y. (x,y) : R}|"
blanchet@48975
    78
proof-
blanchet@48975
    79
  let ?Y = "{y. EX x. (x,y) : R}"  let ?X = "{x. EX y. (x,y) : R}"
blanchet@48975
    80
  let ?f = "% y. SOME x. (x,y) : R"
traytel@51764
    81
  have "?f ` ?Y <= ?X" by (auto dest: someI)
blanchet@48975
    82
  moreover have "inj_on ?f ?Y"
blanchet@48975
    83
  unfolding inj_on_def proof(auto)
blanchet@48975
    84
    fix y1 x1 y2 x2
blanchet@48975
    85
    assume *: "(x1, y1) \<in> R" "(x2, y2) \<in> R" and **: "?f y1 = ?f y2"
blanchet@48975
    86
    hence "(?f y1,y1) : R" using someI[of "% x. (x,y1) : R"] by auto
blanchet@48975
    87
    moreover have "(?f y2,y2) : R" using * someI[of "% x. (x,y2) : R"] by auto
blanchet@48975
    88
    ultimately show "y1 = y2" using ** INJ by auto
blanchet@48975
    89
  qed
blanchet@48975
    90
  ultimately show "|?Y| <=o |?X|" using card_of_ordLeq by blast
blanchet@48975
    91
qed
blanchet@48975
    92
blanchet@48975
    93
lemma card_of_unique2: "\<lbrakk>card_order_on B r; bij_betw f A B\<rbrakk> \<Longrightarrow> r =o |A|"
blanchet@48975
    94
using card_of_ordIso card_of_unique ordIso_equivalence by blast
blanchet@48975
    95
blanchet@48975
    96
lemma internalize_card_of_ordLess:
blanchet@48975
    97
"( |A| <o r) = (\<exists>B < Field r. |A| =o |B| \<and> |B| <o r)"
blanchet@48975
    98
proof
blanchet@48975
    99
  assume "|A| <o r"
blanchet@48975
   100
  then obtain p where 1: "Field p < Field r \<and> |A| =o p \<and> p <o r"
blanchet@48975
   101
  using internalize_ordLess[of "|A|" r] by blast
blanchet@48975
   102
  hence "Card_order p" using card_of_Card_order Card_order_ordIso2 by blast
blanchet@48975
   103
  hence "|Field p| =o p" using card_of_Field_ordIso by blast
blanchet@48975
   104
  hence "|A| =o |Field p| \<and> |Field p| <o r"
blanchet@48975
   105
  using 1 ordIso_equivalence ordIso_ordLess_trans by blast
blanchet@48975
   106
  thus "\<exists>B < Field r. |A| =o |B| \<and> |B| <o r" using 1 by blast
blanchet@48975
   107
next
blanchet@48975
   108
  assume "\<exists>B < Field r. |A| =o |B| \<and> |B| <o r"
blanchet@48975
   109
  thus "|A| <o r" using ordIso_ordLess_trans by blast
blanchet@48975
   110
qed
blanchet@48975
   111
blanchet@48975
   112
lemma internalize_card_of_ordLess2:
blanchet@48975
   113
"( |A| <o |C| ) = (\<exists>B < C. |A| =o |B| \<and> |B| <o |C| )"
blanchet@48975
   114
using internalize_card_of_ordLess[of "A" "|C|"] Field_card_of[of C] by auto
blanchet@48975
   115
blanchet@48975
   116
lemma Card_order_omax:
blanchet@48975
   117
assumes "finite R" and "R \<noteq> {}" and "\<forall>r\<in>R. Card_order r"
blanchet@48975
   118
shows "Card_order (omax R)"
blanchet@48975
   119
proof-
blanchet@48975
   120
  have "\<forall>r\<in>R. Well_order r"
blanchet@48975
   121
  using assms unfolding card_order_on_def by simp
blanchet@48975
   122
  thus ?thesis using assms apply - apply(drule omax_in) by auto
blanchet@48975
   123
qed
blanchet@48975
   124
blanchet@48975
   125
lemma Card_order_omax2:
blanchet@48975
   126
assumes "finite I" and "I \<noteq> {}"
blanchet@48975
   127
shows "Card_order (omax {|A i| | i. i \<in> I})"
blanchet@48975
   128
proof-
blanchet@48975
   129
  let ?R = "{|A i| | i. i \<in> I}"
blanchet@48975
   130
  have "finite ?R" and "?R \<noteq> {}" using assms by auto
blanchet@48975
   131
  moreover have "\<forall>r\<in>?R. Card_order r"
blanchet@48975
   132
  using card_of_Card_order by auto
blanchet@48975
   133
  ultimately show ?thesis by(rule Card_order_omax)
blanchet@48975
   134
qed
blanchet@48975
   135
blanchet@48975
   136
blanchet@48975
   137
subsection {* Cardinals versus set operations on arbitrary sets *}
blanchet@48975
   138
blanchet@54481
   139
lemma card_of_set_type[simp]: "|UNIV::'a set| <o |UNIV::'a set set|"
blanchet@54475
   140
using card_of_Pow[of "UNIV::'a set"] by simp
blanchet@54475
   141
blanchet@54475
   142
lemma card_of_Un1[simp]:
blanchet@54475
   143
shows "|A| \<le>o |A \<union> B| "
blanchet@54475
   144
using inj_on_id[of A] card_of_ordLeq[of A _] by fastforce
blanchet@54475
   145
blanchet@54475
   146
lemma card_of_diff[simp]:
blanchet@54475
   147
shows "|A - B| \<le>o |A|"
blanchet@54475
   148
using inj_on_id[of "A - B"] card_of_ordLeq[of "A - B" _] by fastforce
blanchet@54475
   149
blanchet@48975
   150
lemma subset_ordLeq_strict:
blanchet@48975
   151
assumes "A \<le> B" and "|A| <o |B|"
blanchet@48975
   152
shows "A < B"
blanchet@48975
   153
proof-
blanchet@48975
   154
  {assume "\<not>(A < B)"
blanchet@48975
   155
   hence "A = B" using assms(1) by blast
blanchet@48975
   156
   hence False using assms(2) not_ordLess_ordIso card_of_refl by blast
blanchet@48975
   157
  }
blanchet@48975
   158
  thus ?thesis by blast
blanchet@48975
   159
qed
blanchet@48975
   160
blanchet@48975
   161
corollary subset_ordLeq_diff:
blanchet@48975
   162
assumes "A \<le> B" and "|A| <o |B|"
blanchet@48975
   163
shows "B - A \<noteq> {}"
blanchet@48975
   164
using assms subset_ordLeq_strict by blast
blanchet@48975
   165
blanchet@48975
   166
lemma card_of_empty4:
blanchet@48975
   167
"|{}::'b set| <o |A::'a set| = (A \<noteq> {})"
blanchet@48975
   168
proof(intro iffI notI)
blanchet@48975
   169
  assume *: "|{}::'b set| <o |A|" and "A = {}"
blanchet@48975
   170
  hence "|A| =o |{}::'b set|"
blanchet@48975
   171
  using card_of_ordIso unfolding bij_betw_def inj_on_def by blast
blanchet@48975
   172
  hence "|{}::'b set| =o |A|" using ordIso_symmetric by blast
blanchet@48975
   173
  with * show False using not_ordLess_ordIso[of "|{}::'b set|" "|A|"] by blast
blanchet@48975
   174
next
blanchet@48975
   175
  assume "A \<noteq> {}"
blanchet@48975
   176
  hence "(\<not> (\<exists>f. inj_on f A \<and> f ` A \<subseteq> {}))"
blanchet@48975
   177
  unfolding inj_on_def by blast
blanchet@48975
   178
  thus "| {} | <o | A |"
blanchet@48975
   179
  using card_of_ordLess by blast
blanchet@48975
   180
qed
blanchet@48975
   181
blanchet@48975
   182
lemma card_of_empty5:
blanchet@48975
   183
"|A| <o |B| \<Longrightarrow> B \<noteq> {}"
blanchet@48975
   184
using card_of_empty not_ordLess_ordLeq by blast
blanchet@48975
   185
blanchet@48975
   186
lemma Well_order_card_of_empty:
blanchet@48975
   187
"Well_order r \<Longrightarrow> |{}| \<le>o r" by simp
blanchet@48975
   188
blanchet@48975
   189
lemma card_of_UNIV[simp]:
blanchet@48975
   190
"|A :: 'a set| \<le>o |UNIV :: 'a set|"
blanchet@48975
   191
using card_of_mono1[of A] by simp
blanchet@48975
   192
blanchet@48975
   193
lemma card_of_UNIV2[simp]:
blanchet@48975
   194
"Card_order r \<Longrightarrow> (r :: 'a rel) \<le>o |UNIV :: 'a set|"
blanchet@48975
   195
using card_of_UNIV[of "Field r"] card_of_Field_ordIso
blanchet@48975
   196
      ordIso_symmetric ordIso_ordLeq_trans by blast
blanchet@48975
   197
blanchet@48975
   198
lemma card_of_Pow_mono[simp]:
blanchet@48975
   199
assumes "|A| \<le>o |B|"
blanchet@48975
   200
shows "|Pow A| \<le>o |Pow B|"
blanchet@48975
   201
proof-
blanchet@48975
   202
  obtain f where "inj_on f A \<and> f ` A \<le> B"
blanchet@48975
   203
  using assms card_of_ordLeq[of A B] by auto
blanchet@48975
   204
  hence "inj_on (image f) (Pow A) \<and> (image f) ` (Pow A) \<le> (Pow B)"
blanchet@48975
   205
  by (auto simp add: inj_on_image_Pow image_Pow_mono)
blanchet@48975
   206
  thus ?thesis using card_of_ordLeq[of "Pow A"] by metis
blanchet@48975
   207
qed
blanchet@48975
   208
blanchet@48975
   209
lemma ordIso_Pow_mono[simp]:
blanchet@48975
   210
assumes "r \<le>o r'"
blanchet@48975
   211
shows "|Pow(Field r)| \<le>o |Pow(Field r')|"
blanchet@48975
   212
using assms card_of_mono2 card_of_Pow_mono by blast
blanchet@48975
   213
blanchet@48975
   214
lemma card_of_Pow_cong[simp]:
blanchet@48975
   215
assumes "|A| =o |B|"
blanchet@48975
   216
shows "|Pow A| =o |Pow B|"
blanchet@48975
   217
proof-
blanchet@48975
   218
  obtain f where "bij_betw f A B"
blanchet@48975
   219
  using assms card_of_ordIso[of A B] by auto
blanchet@48975
   220
  hence "bij_betw (image f) (Pow A) (Pow B)"
blanchet@48975
   221
  by (auto simp add: bij_betw_image_Pow)
blanchet@48975
   222
  thus ?thesis using card_of_ordIso[of "Pow A"] by auto
blanchet@48975
   223
qed
blanchet@48975
   224
blanchet@48975
   225
lemma ordIso_Pow_cong[simp]:
blanchet@48975
   226
assumes "r =o r'"
blanchet@48975
   227
shows "|Pow(Field r)| =o |Pow(Field r')|"
blanchet@48975
   228
using assms card_of_cong card_of_Pow_cong by blast
blanchet@48975
   229
blanchet@48975
   230
corollary Card_order_Plus_empty1:
blanchet@48975
   231
"Card_order r \<Longrightarrow> r =o |(Field r) <+> {}|"
blanchet@48975
   232
using card_of_Plus_empty1 card_of_Field_ordIso ordIso_equivalence by blast
blanchet@48975
   233
blanchet@48975
   234
corollary Card_order_Plus_empty2:
blanchet@48975
   235
"Card_order r \<Longrightarrow> r =o |{} <+> (Field r)|"
blanchet@48975
   236
using card_of_Plus_empty2 card_of_Field_ordIso ordIso_equivalence by blast
blanchet@48975
   237
blanchet@48975
   238
lemma Card_order_Un1:
blanchet@48975
   239
shows "Card_order r \<Longrightarrow> |Field r| \<le>o |(Field r) \<union> B| "
blanchet@48975
   240
using card_of_Un1 card_of_Field_ordIso ordIso_symmetric ordIso_ordLeq_trans by auto
blanchet@48975
   241
blanchet@48975
   242
lemma card_of_Un2[simp]:
blanchet@48975
   243
shows "|A| \<le>o |B \<union> A|"
blanchet@48975
   244
using inj_on_id[of A] card_of_ordLeq[of A _] by fastforce
blanchet@48975
   245
blanchet@48975
   246
lemma Card_order_Un2:
blanchet@48975
   247
shows "Card_order r \<Longrightarrow> |Field r| \<le>o |A \<union> (Field r)| "
blanchet@48975
   248
using card_of_Un2 card_of_Field_ordIso ordIso_symmetric ordIso_ordLeq_trans by auto
blanchet@48975
   249
blanchet@48975
   250
lemma Un_Plus_bij_betw:
blanchet@48975
   251
assumes "A Int B = {}"
blanchet@48975
   252
shows "\<exists>f. bij_betw f (A \<union> B) (A <+> B)"
blanchet@48975
   253
proof-
blanchet@48975
   254
  let ?f = "\<lambda> c. if c \<in> A then Inl c else Inr c"
blanchet@48975
   255
  have "bij_betw ?f (A \<union> B) (A <+> B)"
blanchet@48975
   256
  using assms by(unfold bij_betw_def inj_on_def, auto)
blanchet@48975
   257
  thus ?thesis by blast
blanchet@48975
   258
qed
blanchet@48975
   259
blanchet@48975
   260
lemma card_of_Un_Plus_ordIso:
blanchet@48975
   261
assumes "A Int B = {}"
blanchet@48975
   262
shows "|A \<union> B| =o |A <+> B|"
blanchet@48975
   263
using assms card_of_ordIso[of "A \<union> B"] Un_Plus_bij_betw[of A B] by auto
blanchet@48975
   264
blanchet@48975
   265
lemma card_of_Un_Plus_ordIso1:
blanchet@48975
   266
"|A \<union> B| =o |A <+> (B - A)|"
blanchet@48975
   267
using card_of_Un_Plus_ordIso[of A "B - A"] by auto
blanchet@48975
   268
blanchet@48975
   269
lemma card_of_Un_Plus_ordIso2:
blanchet@48975
   270
"|A \<union> B| =o |(A - B) <+> B|"
blanchet@48975
   271
using card_of_Un_Plus_ordIso[of "A - B" B] by auto
blanchet@48975
   272
blanchet@48975
   273
lemma card_of_Times_singl1: "|A| =o |A \<times> {b}|"
blanchet@48975
   274
proof-
blanchet@48975
   275
  have "bij_betw fst (A \<times> {b}) A" unfolding bij_betw_def inj_on_def by force
blanchet@48975
   276
  thus ?thesis using card_of_ordIso ordIso_symmetric by blast
blanchet@48975
   277
qed
blanchet@48975
   278
blanchet@48975
   279
corollary Card_order_Times_singl1:
blanchet@48975
   280
"Card_order r \<Longrightarrow> r =o |(Field r) \<times> {b}|"
blanchet@48975
   281
using card_of_Times_singl1[of _ b] card_of_Field_ordIso ordIso_equivalence by blast
blanchet@48975
   282
blanchet@48975
   283
lemma card_of_Times_singl2: "|A| =o |{b} \<times> A|"
blanchet@48975
   284
proof-
blanchet@48975
   285
  have "bij_betw snd ({b} \<times> A) A" unfolding bij_betw_def inj_on_def by force
blanchet@48975
   286
  thus ?thesis using card_of_ordIso ordIso_symmetric by blast
blanchet@48975
   287
qed
blanchet@48975
   288
blanchet@48975
   289
corollary Card_order_Times_singl2:
blanchet@48975
   290
"Card_order r \<Longrightarrow> r =o |{a} \<times> (Field r)|"
blanchet@48975
   291
using card_of_Times_singl2[of _ a] card_of_Field_ordIso ordIso_equivalence by blast
blanchet@48975
   292
blanchet@48975
   293
lemma card_of_Times_assoc: "|(A \<times> B) \<times> C| =o |A \<times> B \<times> C|"
blanchet@48975
   294
proof -
blanchet@48975
   295
  let ?f = "\<lambda>((a,b),c). (a,(b,c))"
blanchet@48975
   296
  have "A \<times> B \<times> C \<subseteq> ?f ` ((A \<times> B) \<times> C)"
blanchet@48975
   297
  proof
blanchet@48975
   298
    fix x assume "x \<in> A \<times> B \<times> C"
blanchet@48975
   299
    then obtain a b c where *: "a \<in> A" "b \<in> B" "c \<in> C" "x = (a, b, c)" by blast
blanchet@48975
   300
    let ?x = "((a, b), c)"
blanchet@48975
   301
    from * have "?x \<in> (A \<times> B) \<times> C" "x = ?f ?x" by auto
blanchet@48975
   302
    thus "x \<in> ?f ` ((A \<times> B) \<times> C)" by blast
blanchet@48975
   303
  qed
blanchet@48975
   304
  hence "bij_betw ?f ((A \<times> B) \<times> C) (A \<times> B \<times> C)"
blanchet@48975
   305
  unfolding bij_betw_def inj_on_def by auto
blanchet@48975
   306
  thus ?thesis using card_of_ordIso by blast
blanchet@48975
   307
qed
blanchet@48975
   308
blanchet@48975
   309
corollary Card_order_Times3:
blanchet@48975
   310
"Card_order r \<Longrightarrow> |Field r| \<le>o |(Field r) \<times> (Field r)|"
blanchet@48975
   311
using card_of_Times3 card_of_Field_ordIso
blanchet@48975
   312
      ordIso_ordLeq_trans ordIso_symmetric by blast
blanchet@48975
   313
blanchet@54475
   314
lemma card_of_Times_cong1[simp]:
blanchet@54475
   315
assumes "|A| =o |B|"
blanchet@54475
   316
shows "|A \<times> C| =o |B \<times> C|"
blanchet@54475
   317
using assms by (simp add: ordIso_iff_ordLeq card_of_Times_mono1)
blanchet@54475
   318
blanchet@54475
   319
lemma card_of_Times_cong2[simp]:
blanchet@54475
   320
assumes "|A| =o |B|"
blanchet@54475
   321
shows "|C \<times> A| =o |C \<times> B|"
blanchet@54475
   322
using assms by (simp add: ordIso_iff_ordLeq card_of_Times_mono2)
blanchet@54475
   323
blanchet@48975
   324
lemma card_of_Times_mono[simp]:
blanchet@48975
   325
assumes "|A| \<le>o |B|" and "|C| \<le>o |D|"
blanchet@48975
   326
shows "|A \<times> C| \<le>o |B \<times> D|"
blanchet@48975
   327
using assms card_of_Times_mono1[of A B C] card_of_Times_mono2[of C D B]
blanchet@48975
   328
      ordLeq_transitive[of "|A \<times> C|"] by blast
blanchet@48975
   329
blanchet@48975
   330
corollary ordLeq_Times_mono:
blanchet@48975
   331
assumes "r \<le>o r'" and "p \<le>o p'"
blanchet@48975
   332
shows "|(Field r) \<times> (Field p)| \<le>o |(Field r') \<times> (Field p')|"
blanchet@48975
   333
using assms card_of_mono2[of r r'] card_of_mono2[of p p'] card_of_Times_mono by blast
blanchet@48975
   334
blanchet@48975
   335
corollary ordIso_Times_cong1[simp]:
blanchet@48975
   336
assumes "r =o r'"
blanchet@48975
   337
shows "|(Field r) \<times> C| =o |(Field r') \<times> C|"
blanchet@48975
   338
using assms card_of_cong card_of_Times_cong1 by blast
blanchet@48975
   339
blanchet@54475
   340
corollary ordIso_Times_cong2:
blanchet@54475
   341
assumes "r =o r'"
blanchet@54475
   342
shows "|A \<times> (Field r)| =o |A \<times> (Field r')|"
blanchet@54475
   343
using assms card_of_cong card_of_Times_cong2 by blast
blanchet@54475
   344
blanchet@48975
   345
lemma card_of_Times_cong[simp]:
blanchet@48975
   346
assumes "|A| =o |B|" and "|C| =o |D|"
blanchet@48975
   347
shows "|A \<times> C| =o |B \<times> D|"
blanchet@48975
   348
using assms
blanchet@48975
   349
by (auto simp add: ordIso_iff_ordLeq)
blanchet@48975
   350
blanchet@48975
   351
corollary ordIso_Times_cong:
blanchet@48975
   352
assumes "r =o r'" and "p =o p'"
blanchet@48975
   353
shows "|(Field r) \<times> (Field p)| =o |(Field r') \<times> (Field p')|"
blanchet@48975
   354
using assms card_of_cong[of r r'] card_of_cong[of p p'] card_of_Times_cong by blast
blanchet@48975
   355
blanchet@48975
   356
lemma card_of_Sigma_mono2:
blanchet@48975
   357
assumes "inj_on f (I::'i set)" and "f ` I \<le> (J::'j set)"
blanchet@48975
   358
shows "|SIGMA i : I. (A::'j \<Rightarrow> 'a set) (f i)| \<le>o |SIGMA j : J. A j|"
blanchet@48975
   359
proof-
blanchet@48975
   360
  let ?LEFT = "SIGMA i : I. A (f i)"
blanchet@48975
   361
  let ?RIGHT = "SIGMA j : J. A j"
blanchet@48975
   362
  obtain u where u_def: "u = (\<lambda>(i::'i,a::'a). (f i,a))" by blast
blanchet@48975
   363
  have "inj_on u ?LEFT \<and> u `?LEFT \<le> ?RIGHT"
blanchet@48975
   364
  using assms unfolding u_def inj_on_def by auto
blanchet@48975
   365
  thus ?thesis using card_of_ordLeq by blast
blanchet@48975
   366
qed
blanchet@48975
   367
blanchet@48975
   368
lemma card_of_Sigma_mono:
blanchet@48975
   369
assumes INJ: "inj_on f I" and IM: "f ` I \<le> J" and
blanchet@48975
   370
        LEQ: "\<forall>j \<in> J. |A j| \<le>o |B j|"
blanchet@48975
   371
shows "|SIGMA i : I. A (f i)| \<le>o |SIGMA j : J. B j|"
blanchet@48975
   372
proof-
blanchet@48975
   373
  have "\<forall>i \<in> I. |A(f i)| \<le>o |B(f i)|"
blanchet@48975
   374
  using IM LEQ by blast
blanchet@48975
   375
  hence "|SIGMA i : I. A (f i)| \<le>o |SIGMA i : I. B (f i)|"
blanchet@48975
   376
  using card_of_Sigma_mono1[of I] by metis
blanchet@48975
   377
  moreover have "|SIGMA i : I. B (f i)| \<le>o |SIGMA j : J. B j|"
blanchet@48975
   378
  using INJ IM card_of_Sigma_mono2 by blast
blanchet@48975
   379
  ultimately show ?thesis using ordLeq_transitive by blast
blanchet@48975
   380
qed
blanchet@48975
   381
blanchet@48975
   382
blanchet@48975
   383
lemma ordLeq_Sigma_mono1:
blanchet@48975
   384
assumes "\<forall>i \<in> I. p i \<le>o r i"
blanchet@48975
   385
shows "|SIGMA i : I. Field(p i)| \<le>o |SIGMA i : I. Field(r i)|"
blanchet@48975
   386
using assms by (auto simp add: card_of_Sigma_mono1)
blanchet@48975
   387
blanchet@48975
   388
blanchet@48975
   389
lemma ordLeq_Sigma_mono:
blanchet@48975
   390
assumes "inj_on f I" and "f ` I \<le> J" and
blanchet@48975
   391
        "\<forall>j \<in> J. p j \<le>o r j"
blanchet@48975
   392
shows "|SIGMA i : I. Field(p(f i))| \<le>o |SIGMA j : J. Field(r j)|"
blanchet@48975
   393
using assms card_of_mono2 card_of_Sigma_mono
blanchet@48975
   394
      [of f I J "\<lambda> i. Field(p i)" "\<lambda> j. Field(r j)"] by metis
blanchet@48975
   395
blanchet@48975
   396
blanchet@48975
   397
lemma card_of_Sigma_cong1:
blanchet@48975
   398
assumes "\<forall>i \<in> I. |A i| =o |B i|"
blanchet@48975
   399
shows "|SIGMA i : I. A i| =o |SIGMA i : I. B i|"
blanchet@48975
   400
using assms by (auto simp add: card_of_Sigma_mono1 ordIso_iff_ordLeq)
blanchet@48975
   401
blanchet@48975
   402
blanchet@48975
   403
lemma card_of_Sigma_cong2:
blanchet@48975
   404
assumes "bij_betw f (I::'i set) (J::'j set)"
blanchet@48975
   405
shows "|SIGMA i : I. (A::'j \<Rightarrow> 'a set) (f i)| =o |SIGMA j : J. A j|"
blanchet@48975
   406
proof-
blanchet@48975
   407
  let ?LEFT = "SIGMA i : I. A (f i)"
blanchet@48975
   408
  let ?RIGHT = "SIGMA j : J. A j"
blanchet@48975
   409
  obtain u where u_def: "u = (\<lambda>(i::'i,a::'a). (f i,a))" by blast
blanchet@48975
   410
  have "bij_betw u ?LEFT ?RIGHT"
blanchet@48975
   411
  using assms unfolding u_def bij_betw_def inj_on_def by auto
blanchet@48975
   412
  thus ?thesis using card_of_ordIso by blast
blanchet@48975
   413
qed
blanchet@48975
   414
blanchet@48975
   415
lemma card_of_Sigma_cong:
blanchet@48975
   416
assumes BIJ: "bij_betw f I J" and
blanchet@48975
   417
        ISO: "\<forall>j \<in> J. |A j| =o |B j|"
blanchet@48975
   418
shows "|SIGMA i : I. A (f i)| =o |SIGMA j : J. B j|"
blanchet@48975
   419
proof-
blanchet@48975
   420
  have "\<forall>i \<in> I. |A(f i)| =o |B(f i)|"
blanchet@48975
   421
  using ISO BIJ unfolding bij_betw_def by blast
blanchet@48975
   422
  hence "|SIGMA i : I. A (f i)| =o |SIGMA i : I. B (f i)|"
blanchet@48975
   423
  using card_of_Sigma_cong1 by metis
blanchet@48975
   424
  moreover have "|SIGMA i : I. B (f i)| =o |SIGMA j : J. B j|"
blanchet@48975
   425
  using BIJ card_of_Sigma_cong2 by blast
blanchet@48975
   426
  ultimately show ?thesis using ordIso_transitive by blast
blanchet@48975
   427
qed
blanchet@48975
   428
blanchet@48975
   429
lemma ordIso_Sigma_cong1:
blanchet@48975
   430
assumes "\<forall>i \<in> I. p i =o r i"
blanchet@48975
   431
shows "|SIGMA i : I. Field(p i)| =o |SIGMA i : I. Field(r i)|"
blanchet@48975
   432
using assms by (auto simp add: card_of_Sigma_cong1)
blanchet@48975
   433
blanchet@48975
   434
lemma ordLeq_Sigma_cong:
blanchet@48975
   435
assumes "bij_betw f I J" and
blanchet@48975
   436
        "\<forall>j \<in> J. p j =o r j"
blanchet@48975
   437
shows "|SIGMA i : I. Field(p(f i))| =o |SIGMA j : J. Field(r j)|"
blanchet@48975
   438
using assms card_of_cong card_of_Sigma_cong
blanchet@48975
   439
      [of f I J "\<lambda> j. Field(p j)" "\<lambda> j. Field(r j)"] by blast
blanchet@48975
   440
blanchet@48975
   441
corollary ordLeq_Sigma_Times:
blanchet@48975
   442
"\<forall>i \<in> I. p i \<le>o r \<Longrightarrow> |SIGMA i : I. Field (p i)| \<le>o |I \<times> (Field r)|"
blanchet@48975
   443
by (auto simp add: card_of_Sigma_Times)
blanchet@48975
   444
blanchet@48975
   445
lemma card_of_UNION_Sigma2:
blanchet@48975
   446
assumes
blanchet@48975
   447
"!! i j. \<lbrakk>{i,j} <= I; i ~= j\<rbrakk> \<Longrightarrow> A i Int A j = {}"
blanchet@48975
   448
shows
blanchet@48975
   449
"|\<Union>i\<in>I. A i| =o |Sigma I A|"
blanchet@48975
   450
proof-
blanchet@48975
   451
  let ?L = "\<Union>i\<in>I. A i"  let ?R = "Sigma I A"
blanchet@48975
   452
  have "|?L| <=o |?R|" using card_of_UNION_Sigma .
blanchet@48975
   453
  moreover have "|?R| <=o |?L|"
blanchet@48975
   454
  proof-
blanchet@48975
   455
    have "inj_on snd ?R"
blanchet@48975
   456
    unfolding inj_on_def using assms by auto
blanchet@48975
   457
    moreover have "snd ` ?R <= ?L" by auto
blanchet@48975
   458
    ultimately show ?thesis using card_of_ordLeq by blast
blanchet@48975
   459
  qed
blanchet@48975
   460
  ultimately show ?thesis by(simp add: ordIso_iff_ordLeq)
blanchet@48975
   461
qed
blanchet@48975
   462
blanchet@48975
   463
corollary Plus_into_Times:
blanchet@48975
   464
assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
blanchet@48975
   465
        B2: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B"
blanchet@48975
   466
shows "\<exists>f. inj_on f (A <+> B) \<and> f ` (A <+> B) \<le> A \<times> B"
blanchet@48975
   467
using assms by (auto simp add: card_of_Plus_Times card_of_ordLeq)
blanchet@48975
   468
blanchet@48975
   469
corollary Plus_into_Times_types:
blanchet@48975
   470
assumes A2: "(a1::'a) \<noteq> a2" and  B2: "(b1::'b) \<noteq> b2"
blanchet@48975
   471
shows "\<exists>(f::'a + 'b \<Rightarrow> 'a * 'b). inj f"
blanchet@48975
   472
using assms Plus_into_Times[of a1 a2 UNIV b1 b2 UNIV]
blanchet@48975
   473
by auto
blanchet@48975
   474
blanchet@48975
   475
corollary Times_same_infinite_bij_betw:
blanchet@48975
   476
assumes "infinite A"
blanchet@48975
   477
shows "\<exists>f. bij_betw f (A \<times> A) A"
blanchet@48975
   478
using assms by (auto simp add: card_of_ordIso)
blanchet@48975
   479
blanchet@48975
   480
corollary Times_same_infinite_bij_betw_types:
blanchet@48975
   481
assumes INF: "infinite(UNIV::'a set)"
blanchet@48975
   482
shows "\<exists>(f::('a * 'a) => 'a). bij f"
blanchet@48975
   483
using assms Times_same_infinite_bij_betw[of "UNIV::'a set"]
blanchet@48975
   484
by auto
blanchet@48975
   485
blanchet@48975
   486
corollary Times_infinite_bij_betw:
blanchet@48975
   487
assumes INF: "infinite A" and NE: "B \<noteq> {}" and INJ: "inj_on g B \<and> g ` B \<le> A"
blanchet@48975
   488
shows "(\<exists>f. bij_betw f (A \<times> B) A) \<and> (\<exists>h. bij_betw h (B \<times> A) A)"
blanchet@48975
   489
proof-
blanchet@48975
   490
  have "|B| \<le>o |A|" using INJ card_of_ordLeq by blast
blanchet@48975
   491
  thus ?thesis using INF NE
blanchet@48975
   492
  by (auto simp add: card_of_ordIso card_of_Times_infinite)
blanchet@48975
   493
qed
blanchet@48975
   494
blanchet@48975
   495
corollary Times_infinite_bij_betw_types:
blanchet@48975
   496
assumes INF: "infinite(UNIV::'a set)" and
blanchet@48975
   497
        BIJ: "inj(g::'b \<Rightarrow> 'a)"
blanchet@48975
   498
shows "(\<exists>(f::('b * 'a) => 'a). bij f) \<and> (\<exists>(h::('a * 'b) => 'a). bij h)"
blanchet@48975
   499
using assms Times_infinite_bij_betw[of "UNIV::'a set" "UNIV::'b set" g]
blanchet@48975
   500
by auto
blanchet@48975
   501
blanchet@48975
   502
lemma card_of_Times_ordLeq_infinite:
blanchet@48975
   503
"\<lbrakk>infinite C; |A| \<le>o |C|; |B| \<le>o |C|\<rbrakk>
blanchet@48975
   504
 \<Longrightarrow> |A <*> B| \<le>o |C|"
blanchet@48975
   505
by(simp add: card_of_Sigma_ordLeq_infinite)
blanchet@48975
   506
blanchet@48975
   507
corollary Plus_infinite_bij_betw:
blanchet@48975
   508
assumes INF: "infinite A" and INJ: "inj_on g B \<and> g ` B \<le> A"
blanchet@48975
   509
shows "(\<exists>f. bij_betw f (A <+> B) A) \<and> (\<exists>h. bij_betw h (B <+> A) A)"
blanchet@48975
   510
proof-
blanchet@48975
   511
  have "|B| \<le>o |A|" using INJ card_of_ordLeq by blast
blanchet@48975
   512
  thus ?thesis using INF
blanchet@48975
   513
  by (auto simp add: card_of_ordIso)
blanchet@48975
   514
qed
blanchet@48975
   515
blanchet@48975
   516
corollary Plus_infinite_bij_betw_types:
blanchet@48975
   517
assumes INF: "infinite(UNIV::'a set)" and
blanchet@48975
   518
        BIJ: "inj(g::'b \<Rightarrow> 'a)"
blanchet@48975
   519
shows "(\<exists>(f::('b + 'a) => 'a). bij f) \<and> (\<exists>(h::('a + 'b) => 'a). bij h)"
blanchet@48975
   520
using assms Plus_infinite_bij_betw[of "UNIV::'a set" g "UNIV::'b set"]
blanchet@48975
   521
by auto
blanchet@48975
   522
blanchet@54475
   523
lemma card_of_Un_infinite:
blanchet@54475
   524
assumes INF: "infinite A" and LEQ: "|B| \<le>o |A|"
blanchet@54475
   525
shows "|A \<union> B| =o |A| \<and> |B \<union> A| =o |A|"
blanchet@54475
   526
proof-
blanchet@54475
   527
  have "|A \<union> B| \<le>o |A <+> B|" by (rule card_of_Un_Plus_ordLeq)
blanchet@54475
   528
  moreover have "|A <+> B| =o |A|"
blanchet@54475
   529
  using assms by (metis card_of_Plus_infinite)
blanchet@54475
   530
  ultimately have "|A \<union> B| \<le>o |A|" using ordLeq_ordIso_trans by blast
blanchet@54475
   531
  hence "|A \<union> B| =o |A|" using card_of_Un1 ordIso_iff_ordLeq by blast
blanchet@54475
   532
  thus ?thesis using Un_commute[of B A] by auto
blanchet@54475
   533
qed
blanchet@54475
   534
blanchet@48975
   535
lemma card_of_Un_infinite_simps[simp]:
blanchet@48975
   536
"\<lbrakk>infinite A; |B| \<le>o |A| \<rbrakk> \<Longrightarrow> |A \<union> B| =o |A|"
blanchet@48975
   537
"\<lbrakk>infinite A; |B| \<le>o |A| \<rbrakk> \<Longrightarrow> |B \<union> A| =o |A|"
blanchet@48975
   538
using card_of_Un_infinite by auto
blanchet@48975
   539
blanchet@54475
   540
lemma card_of_Un_diff_infinite:
blanchet@54475
   541
assumes INF: "infinite A" and LESS: "|B| <o |A|"
blanchet@54475
   542
shows "|A - B| =o |A|"
blanchet@54475
   543
proof-
blanchet@54475
   544
  obtain C where C_def: "C = A - B" by blast
blanchet@54475
   545
  have "|A \<union> B| =o |A|"
blanchet@54475
   546
  using assms ordLeq_iff_ordLess_or_ordIso card_of_Un_infinite by blast
blanchet@54475
   547
  moreover have "C \<union> B = A \<union> B" unfolding C_def by auto
blanchet@54475
   548
  ultimately have 1: "|C \<union> B| =o |A|" by auto
blanchet@54475
   549
  (*  *)
blanchet@54475
   550
  {assume *: "|C| \<le>o |B|"
blanchet@54475
   551
   moreover
blanchet@54475
   552
   {assume **: "finite B"
blanchet@54475
   553
    hence "finite C"
blanchet@54475
   554
    using card_of_ordLeq_finite * by blast
blanchet@54475
   555
    hence False using ** INF card_of_ordIso_finite 1 by blast
blanchet@54475
   556
   }
blanchet@54475
   557
   hence "infinite B" by auto
blanchet@54475
   558
   ultimately have False
blanchet@54475
   559
   using card_of_Un_infinite 1 ordIso_equivalence(1,3) LESS not_ordLess_ordIso by metis
blanchet@54475
   560
  }
blanchet@54475
   561
  hence 2: "|B| \<le>o |C|" using card_of_Well_order ordLeq_total by blast
blanchet@54475
   562
  {assume *: "finite C"
blanchet@54475
   563
    hence "finite B" using card_of_ordLeq_finite 2 by blast
blanchet@54475
   564
    hence False using * INF card_of_ordIso_finite 1 by blast
blanchet@54475
   565
  }
blanchet@54475
   566
  hence "infinite C" by auto
blanchet@54475
   567
  hence "|C| =o |A|"
blanchet@54475
   568
  using  card_of_Un_infinite 1 2 ordIso_equivalence(1,3) by metis
blanchet@54475
   569
  thus ?thesis unfolding C_def .
blanchet@54475
   570
qed
blanchet@54475
   571
blanchet@48975
   572
corollary Card_order_Un_infinite:
blanchet@48975
   573
assumes INF: "infinite(Field r)" and CARD: "Card_order r" and
blanchet@48975
   574
        LEQ: "p \<le>o r"
blanchet@48975
   575
shows "| (Field r) \<union> (Field p) | =o r \<and> | (Field p) \<union> (Field r) | =o r"
blanchet@48975
   576
proof-
blanchet@48975
   577
  have "| Field r \<union> Field p | =o | Field r | \<and>
blanchet@48975
   578
        | Field p \<union> Field r | =o | Field r |"
blanchet@48975
   579
  using assms by (auto simp add: card_of_Un_infinite)
blanchet@48975
   580
  thus ?thesis
blanchet@48975
   581
  using assms card_of_Field_ordIso[of r]
blanchet@48975
   582
        ordIso_transitive[of "|Field r \<union> Field p|"]
blanchet@48975
   583
        ordIso_transitive[of _ "|Field r|"] by blast
blanchet@48975
   584
qed
blanchet@48975
   585
blanchet@48975
   586
corollary subset_ordLeq_diff_infinite:
blanchet@48975
   587
assumes INF: "infinite B" and SUB: "A \<le> B" and LESS: "|A| <o |B|"
blanchet@48975
   588
shows "infinite (B - A)"
blanchet@48975
   589
using assms card_of_Un_diff_infinite card_of_ordIso_finite by blast
blanchet@48975
   590
blanchet@48975
   591
lemma card_of_Times_ordLess_infinite[simp]:
blanchet@48975
   592
assumes INF: "infinite C" and
blanchet@48975
   593
        LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|"
blanchet@48975
   594
shows "|A \<times> B| <o |C|"
blanchet@48975
   595
proof(cases "A = {} \<or> B = {}")
blanchet@48975
   596
  assume Case1: "A = {} \<or> B = {}"
blanchet@48975
   597
  hence "A \<times> B = {}" by blast
blanchet@48975
   598
  moreover have "C \<noteq> {}" using
blanchet@48975
   599
  LESS1 card_of_empty5 by blast
blanchet@48975
   600
  ultimately show ?thesis by(auto simp add:  card_of_empty4)
blanchet@48975
   601
next
blanchet@48975
   602
  assume Case2: "\<not>(A = {} \<or> B = {})"
blanchet@48975
   603
  {assume *: "|C| \<le>o |A \<times> B|"
blanchet@48975
   604
   hence "infinite (A \<times> B)" using INF card_of_ordLeq_finite by blast
blanchet@48975
   605
   hence 1: "infinite A \<or> infinite B" using finite_cartesian_product by blast
blanchet@48975
   606
   {assume Case21: "|A| \<le>o |B|"
blanchet@48975
   607
    hence "infinite B" using 1 card_of_ordLeq_finite by blast
blanchet@48975
   608
    hence "|A \<times> B| =o |B|" using Case2 Case21
blanchet@48975
   609
    by (auto simp add: card_of_Times_infinite)
blanchet@48975
   610
    hence False using LESS2 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast
blanchet@48975
   611
   }
blanchet@48975
   612
   moreover
blanchet@48975
   613
   {assume Case22: "|B| \<le>o |A|"
blanchet@48975
   614
    hence "infinite A" using 1 card_of_ordLeq_finite by blast
blanchet@48975
   615
    hence "|A \<times> B| =o |A|" using Case2 Case22
blanchet@48975
   616
    by (auto simp add: card_of_Times_infinite)
blanchet@48975
   617
    hence False using LESS1 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast
blanchet@48975
   618
   }
blanchet@48975
   619
   ultimately have False using ordLeq_total card_of_Well_order[of A]
blanchet@48975
   620
   card_of_Well_order[of B] by blast
blanchet@48975
   621
  }
blanchet@48975
   622
  thus ?thesis using ordLess_or_ordLeq[of "|A \<times> B|" "|C|"]
blanchet@48975
   623
  card_of_Well_order[of "A \<times> B"] card_of_Well_order[of "C"] by auto
blanchet@48975
   624
qed
blanchet@48975
   625
blanchet@48975
   626
lemma card_of_Times_ordLess_infinite_Field[simp]:
blanchet@48975
   627
assumes INF: "infinite (Field r)" and r: "Card_order r" and
blanchet@48975
   628
        LESS1: "|A| <o r" and LESS2: "|B| <o r"
blanchet@48975
   629
shows "|A \<times> B| <o r"
blanchet@48975
   630
proof-
blanchet@48975
   631
  let ?C  = "Field r"
blanchet@48975
   632
  have 1: "r =o |?C| \<and> |?C| =o r" using r card_of_Field_ordIso
blanchet@48975
   633
  ordIso_symmetric by blast
blanchet@48975
   634
  hence "|A| <o |?C|"  "|B| <o |?C|"
blanchet@48975
   635
  using LESS1 LESS2 ordLess_ordIso_trans by blast+
blanchet@48975
   636
  hence  "|A <*> B| <o |?C|" using INF
blanchet@48975
   637
  card_of_Times_ordLess_infinite by blast
blanchet@48975
   638
  thus ?thesis using 1 ordLess_ordIso_trans by blast
blanchet@48975
   639
qed
blanchet@48975
   640
blanchet@48975
   641
lemma card_of_Un_ordLess_infinite[simp]:
blanchet@48975
   642
assumes INF: "infinite C" and
blanchet@48975
   643
        LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|"
blanchet@48975
   644
shows "|A \<union> B| <o |C|"
blanchet@48975
   645
using assms card_of_Plus_ordLess_infinite card_of_Un_Plus_ordLeq
blanchet@48975
   646
      ordLeq_ordLess_trans by blast
blanchet@48975
   647
blanchet@48975
   648
lemma card_of_Un_ordLess_infinite_Field[simp]:
blanchet@48975
   649
assumes INF: "infinite (Field r)" and r: "Card_order r" and
blanchet@48975
   650
        LESS1: "|A| <o r" and LESS2: "|B| <o r"
blanchet@48975
   651
shows "|A Un B| <o r"
blanchet@48975
   652
proof-
blanchet@48975
   653
  let ?C  = "Field r"
blanchet@48975
   654
  have 1: "r =o |?C| \<and> |?C| =o r" using r card_of_Field_ordIso
blanchet@48975
   655
  ordIso_symmetric by blast
blanchet@48975
   656
  hence "|A| <o |?C|"  "|B| <o |?C|"
blanchet@48975
   657
  using LESS1 LESS2 ordLess_ordIso_trans by blast+
blanchet@48975
   658
  hence  "|A Un B| <o |?C|" using INF
blanchet@48975
   659
  card_of_Un_ordLess_infinite by blast
blanchet@48975
   660
  thus ?thesis using 1 ordLess_ordIso_trans by blast
blanchet@48975
   661
qed
blanchet@48975
   662
blanchet@54475
   663
blanchet@54475
   664
subsection {* Cardinals versus set operations involving infinite sets *}
blanchet@54475
   665
blanchet@54475
   666
lemma finite_iff_cardOf_nat:
blanchet@54475
   667
"finite A = ( |A| <o |UNIV :: nat set| )"
blanchet@54475
   668
using infinite_iff_card_of_nat[of A]
blanchet@54475
   669
not_ordLeq_iff_ordLess[of "|A|" "|UNIV :: nat set|"]
blanchet@54475
   670
by (fastforce simp: card_of_Well_order)
blanchet@54475
   671
blanchet@54475
   672
lemma finite_ordLess_infinite2[simp]:
blanchet@54475
   673
assumes "finite A" and "infinite B"
blanchet@54475
   674
shows "|A| <o |B|"
blanchet@54475
   675
using assms
blanchet@54475
   676
finite_ordLess_infinite[of "|A|" "|B|"]
blanchet@54475
   677
card_of_Well_order[of A] card_of_Well_order[of B]
blanchet@54475
   678
Field_card_of[of A] Field_card_of[of B] by auto
blanchet@54475
   679
blanchet@54475
   680
lemma infinite_card_of_insert:
blanchet@54475
   681
assumes "infinite A"
blanchet@54475
   682
shows "|insert a A| =o |A|"
blanchet@54475
   683
proof-
blanchet@54475
   684
  have iA: "insert a A = A \<union> {a}" by simp
blanchet@54475
   685
  show ?thesis
blanchet@54475
   686
  using infinite_imp_bij_betw2[OF assms] unfolding iA
blanchet@54475
   687
  by (metis bij_betw_inv card_of_ordIso)
blanchet@54475
   688
qed
blanchet@54475
   689
blanchet@48975
   690
lemma card_of_Un_singl_ordLess_infinite1:
blanchet@48975
   691
assumes "infinite B" and "|A| <o |B|"
blanchet@48975
   692
shows "|{a} Un A| <o |B|"
blanchet@48975
   693
proof-
blanchet@48975
   694
  have "|{a}| <o |B|" using assms by auto
traytel@51764
   695
  thus ?thesis using assms card_of_Un_ordLess_infinite[of B] by blast
blanchet@48975
   696
qed
blanchet@48975
   697
blanchet@48975
   698
lemma card_of_Un_singl_ordLess_infinite:
blanchet@48975
   699
assumes "infinite B"
blanchet@48975
   700
shows "( |A| <o |B| ) = ( |{a} Un A| <o |B| )"
blanchet@48975
   701
using assms card_of_Un_singl_ordLess_infinite1[of B A]
blanchet@48975
   702
proof(auto)
blanchet@48975
   703
  assume "|insert a A| <o |B|"
traytel@51764
   704
  moreover have "|A| <=o |insert a A|" using card_of_mono1[of A "insert a A"] by blast
blanchet@48975
   705
  ultimately show "|A| <o |B|" using ordLeq_ordLess_trans by blast
blanchet@48975
   706
qed
blanchet@48975
   707
blanchet@48975
   708
blanchet@54475
   709
subsection {* Cardinals versus lists *}
blanchet@54475
   710
blanchet@54475
   711
text{* The next is an auxiliary operator, which shall be used for inductive
blanchet@54475
   712
proofs of facts concerning the cardinality of @{text "List"} : *}
blanchet@54475
   713
blanchet@54475
   714
definition nlists :: "'a set \<Rightarrow> nat \<Rightarrow> 'a list set"
blanchet@54475
   715
where "nlists A n \<equiv> {l. set l \<le> A \<and> length l = n}"
blanchet@54475
   716
blanchet@54475
   717
lemma lists_def2: "lists A = {l. set l \<le> A}"
blanchet@54475
   718
using in_listsI by blast
blanchet@54475
   719
blanchet@54475
   720
lemma lists_UNION_nlists: "lists A = (\<Union> n. nlists A n)"
blanchet@54475
   721
unfolding lists_def2 nlists_def by blast
blanchet@54475
   722
blanchet@54475
   723
lemma card_of_lists: "|A| \<le>o |lists A|"
blanchet@54475
   724
proof-
blanchet@54475
   725
  let ?h = "\<lambda> a. [a]"
blanchet@54475
   726
  have "inj_on ?h A \<and> ?h ` A \<le> lists A"
blanchet@54475
   727
  unfolding inj_on_def lists_def2 by auto
blanchet@54475
   728
  thus ?thesis by (metis card_of_ordLeq)
blanchet@54475
   729
qed
blanchet@54475
   730
blanchet@54475
   731
lemma nlists_0: "nlists A 0 = {[]}"
blanchet@54475
   732
unfolding nlists_def by auto
blanchet@54475
   733
blanchet@54475
   734
lemma nlists_not_empty:
blanchet@54475
   735
assumes "A \<noteq> {}"
blanchet@54475
   736
shows "nlists A n \<noteq> {}"
blanchet@54475
   737
proof(induct n, simp add: nlists_0)
blanchet@54475
   738
  fix n assume "nlists A n \<noteq> {}"
blanchet@54475
   739
  then obtain a and l where "a \<in> A \<and> l \<in> nlists A n" using assms by auto
blanchet@54475
   740
  hence "a # l \<in> nlists A (Suc n)" unfolding nlists_def by auto
blanchet@54475
   741
  thus "nlists A (Suc n) \<noteq> {}" by auto
blanchet@54475
   742
qed
blanchet@54475
   743
blanchet@54475
   744
lemma Nil_in_lists: "[] \<in> lists A"
blanchet@54475
   745
unfolding lists_def2 by auto
blanchet@54475
   746
blanchet@54475
   747
lemma lists_not_empty: "lists A \<noteq> {}"
blanchet@54475
   748
using Nil_in_lists by blast
blanchet@54475
   749
blanchet@54475
   750
lemma card_of_nlists_Succ: "|nlists A (Suc n)| =o |A \<times> (nlists A n)|"
blanchet@54475
   751
proof-
blanchet@54475
   752
  let ?B = "A \<times> (nlists A n)"   let ?h = "\<lambda>(a,l). a # l"
blanchet@54475
   753
  have "inj_on ?h ?B \<and> ?h ` ?B \<le> nlists A (Suc n)"
blanchet@54475
   754
  unfolding inj_on_def nlists_def by auto
blanchet@54475
   755
  moreover have "nlists A (Suc n) \<le> ?h ` ?B"
blanchet@54475
   756
  proof(auto)
blanchet@54475
   757
    fix l assume "l \<in> nlists A (Suc n)"
blanchet@54475
   758
    hence 1: "length l = Suc n \<and> set l \<le> A" unfolding nlists_def by auto
blanchet@54475
   759
    then obtain a and l' where 2: "l = a # l'" by (auto simp: length_Suc_conv)
blanchet@54475
   760
    hence "a \<in> A \<and> set l' \<le> A \<and> length l' = n" using 1 by auto
blanchet@54475
   761
    thus "l \<in> ?h ` ?B"  using 2 unfolding nlists_def by auto
blanchet@54475
   762
  qed
blanchet@54475
   763
  ultimately have "bij_betw ?h ?B (nlists A (Suc n))"
blanchet@54475
   764
  unfolding bij_betw_def by auto
blanchet@54475
   765
  thus ?thesis using card_of_ordIso ordIso_symmetric by blast
blanchet@54475
   766
qed
blanchet@54475
   767
blanchet@54475
   768
lemma card_of_nlists_infinite:
blanchet@54475
   769
assumes "infinite A"
blanchet@54475
   770
shows "|nlists A n| \<le>o |A|"
blanchet@54475
   771
proof(induct n)
blanchet@54475
   772
  have "A \<noteq> {}" using assms by auto
blanchet@54475
   773
  thus "|nlists A 0| \<le>o |A|" by (simp add: nlists_0 card_of_singl_ordLeq)
blanchet@54475
   774
next
blanchet@54475
   775
  fix n assume IH: "|nlists A n| \<le>o |A|"
blanchet@54475
   776
  have "|nlists A (Suc n)| =o |A \<times> (nlists A n)|"
blanchet@54475
   777
  using card_of_nlists_Succ by blast
blanchet@54475
   778
  moreover
blanchet@54475
   779
  {have "nlists A n \<noteq> {}" using assms nlists_not_empty[of A] by blast
blanchet@54475
   780
   hence "|A \<times> (nlists A n)| =o |A|"
blanchet@54475
   781
   using assms IH by (auto simp add: card_of_Times_infinite)
blanchet@54475
   782
  }
blanchet@54475
   783
  ultimately show "|nlists A (Suc n)| \<le>o |A|"
blanchet@54475
   784
  using ordIso_transitive ordIso_iff_ordLeq by blast
blanchet@54475
   785
qed
blanchet@48975
   786
blanchet@48975
   787
lemma Card_order_lists: "Card_order r \<Longrightarrow> r \<le>o |lists(Field r) |"
blanchet@48975
   788
using card_of_lists card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
blanchet@48975
   789
blanchet@48975
   790
lemma Union_set_lists:
blanchet@48975
   791
"Union(set ` (lists A)) = A"
blanchet@48975
   792
unfolding lists_def2 proof(auto)
blanchet@48975
   793
  fix a assume "a \<in> A"
blanchet@48975
   794
  hence "set [a] \<le> A \<and> a \<in> set [a]" by auto
blanchet@48975
   795
  thus "\<exists>l. set l \<le> A \<and> a \<in> set l" by blast
blanchet@48975
   796
qed
blanchet@48975
   797
blanchet@48975
   798
lemma inj_on_map_lists:
blanchet@48975
   799
assumes "inj_on f A"
blanchet@48975
   800
shows "inj_on (map f) (lists A)"
blanchet@48975
   801
using assms Union_set_lists[of A] inj_on_mapI[of f "lists A"] by auto
blanchet@48975
   802
blanchet@48975
   803
lemma map_lists_mono:
blanchet@48975
   804
assumes "f ` A \<le> B"
blanchet@48975
   805
shows "(map f) ` (lists A) \<le> lists B"
blanchet@48975
   806
using assms unfolding lists_def2 by (auto, blast) (* lethal combination of methods :)  *)
blanchet@48975
   807
blanchet@48975
   808
lemma map_lists_surjective:
blanchet@48975
   809
assumes "f ` A = B"
blanchet@48975
   810
shows "(map f) ` (lists A) = lists B"
blanchet@48975
   811
using assms unfolding lists_def2
blanchet@48975
   812
proof (auto, blast)
blanchet@48975
   813
  fix l' assume *: "set l' \<le> f ` A"
blanchet@48975
   814
  have "set l' \<le> f ` A \<longrightarrow> l' \<in> map f ` {l. set l \<le> A}"
blanchet@48975
   815
  proof(induct l', auto)
blanchet@48975
   816
    fix l a
blanchet@48975
   817
    assume "a \<in> A" and "set l \<le> A" and
blanchet@48975
   818
           IH: "f ` (set l) \<le> f ` A"
blanchet@48975
   819
    hence "set (a # l) \<le> A" by auto
blanchet@48975
   820
    hence "map f (a # l) \<in> map f ` {l. set l \<le> A}" by blast
blanchet@48975
   821
    thus "f a # map f l \<in> map f ` {l. set l \<le> A}" by auto
blanchet@48975
   822
  qed
blanchet@48975
   823
  thus "l' \<in> map f ` {l. set l \<le> A}" using * by auto
blanchet@48975
   824
qed
blanchet@48975
   825
blanchet@48975
   826
lemma bij_betw_map_lists:
blanchet@48975
   827
assumes "bij_betw f A B"
blanchet@48975
   828
shows "bij_betw (map f) (lists A) (lists B)"
blanchet@48975
   829
using assms unfolding bij_betw_def
blanchet@48975
   830
by(auto simp add: inj_on_map_lists map_lists_surjective)
blanchet@48975
   831
blanchet@48975
   832
lemma card_of_lists_mono[simp]:
blanchet@48975
   833
assumes "|A| \<le>o |B|"
blanchet@48975
   834
shows "|lists A| \<le>o |lists B|"
blanchet@48975
   835
proof-
blanchet@48975
   836
  obtain f where "inj_on f A \<and> f ` A \<le> B"
blanchet@48975
   837
  using assms card_of_ordLeq[of A B] by auto
blanchet@48975
   838
  hence "inj_on (map f) (lists A) \<and> (map f) ` (lists A) \<le> (lists B)"
blanchet@48975
   839
  by (auto simp add: inj_on_map_lists map_lists_mono)
blanchet@48975
   840
  thus ?thesis using card_of_ordLeq[of "lists A"] by metis
blanchet@48975
   841
qed
blanchet@48975
   842
blanchet@48975
   843
lemma ordIso_lists_mono:
blanchet@48975
   844
assumes "r \<le>o r'"
blanchet@48975
   845
shows "|lists(Field r)| \<le>o |lists(Field r')|"
blanchet@48975
   846
using assms card_of_mono2 card_of_lists_mono by blast
blanchet@48975
   847
blanchet@48975
   848
lemma card_of_lists_cong[simp]:
blanchet@48975
   849
assumes "|A| =o |B|"
blanchet@48975
   850
shows "|lists A| =o |lists B|"
blanchet@48975
   851
proof-
blanchet@48975
   852
  obtain f where "bij_betw f A B"
blanchet@48975
   853
  using assms card_of_ordIso[of A B] by auto
blanchet@48975
   854
  hence "bij_betw (map f) (lists A) (lists B)"
blanchet@48975
   855
  by (auto simp add: bij_betw_map_lists)
blanchet@48975
   856
  thus ?thesis using card_of_ordIso[of "lists A"] by auto
blanchet@48975
   857
qed
blanchet@48975
   858
blanchet@54475
   859
lemma card_of_lists_infinite[simp]:
blanchet@54475
   860
assumes "infinite A"
blanchet@54475
   861
shows "|lists A| =o |A|"
blanchet@54475
   862
proof-
blanchet@54475
   863
  have "|lists A| \<le>o |A|"
blanchet@54475
   864
  using assms
blanchet@54475
   865
  by (auto simp add: lists_UNION_nlists card_of_UNION_ordLeq_infinite
blanchet@54475
   866
                     infinite_iff_card_of_nat card_of_nlists_infinite)
blanchet@54475
   867
  thus ?thesis using card_of_lists ordIso_iff_ordLeq by blast
blanchet@54475
   868
qed
blanchet@54475
   869
blanchet@54475
   870
lemma Card_order_lists_infinite:
blanchet@54475
   871
assumes "Card_order r" and "infinite(Field r)"
blanchet@54475
   872
shows "|lists(Field r)| =o r"
blanchet@54475
   873
using assms card_of_lists_infinite card_of_Field_ordIso ordIso_transitive by blast
blanchet@54475
   874
blanchet@48975
   875
lemma ordIso_lists_cong:
blanchet@48975
   876
assumes "r =o r'"
blanchet@48975
   877
shows "|lists(Field r)| =o |lists(Field r')|"
blanchet@48975
   878
using assms card_of_cong card_of_lists_cong by blast
blanchet@48975
   879
blanchet@48975
   880
corollary lists_infinite_bij_betw:
blanchet@48975
   881
assumes "infinite A"
blanchet@48975
   882
shows "\<exists>f. bij_betw f (lists A) A"
blanchet@48975
   883
using assms card_of_lists_infinite card_of_ordIso by blast
blanchet@48975
   884
blanchet@48975
   885
corollary lists_infinite_bij_betw_types:
blanchet@48975
   886
assumes "infinite(UNIV :: 'a set)"
blanchet@48975
   887
shows "\<exists>(f::'a list \<Rightarrow> 'a). bij f"
blanchet@48975
   888
using assms assms lists_infinite_bij_betw[of "UNIV::'a set"]
blanchet@48975
   889
using lists_UNIV by auto
blanchet@48975
   890
blanchet@48975
   891
blanchet@48975
   892
subsection {* Cardinals versus the set-of-finite-sets operator  *}
blanchet@48975
   893
blanchet@48975
   894
definition Fpow :: "'a set \<Rightarrow> 'a set set"
blanchet@48975
   895
where "Fpow A \<equiv> {X. X \<le> A \<and> finite X}"
blanchet@48975
   896
blanchet@48975
   897
lemma Fpow_mono: "A \<le> B \<Longrightarrow> Fpow A \<le> Fpow B"
blanchet@48975
   898
unfolding Fpow_def by auto
blanchet@48975
   899
blanchet@48975
   900
lemma empty_in_Fpow: "{} \<in> Fpow A"
blanchet@48975
   901
unfolding Fpow_def by auto
blanchet@48975
   902
blanchet@48975
   903
lemma Fpow_not_empty: "Fpow A \<noteq> {}"
blanchet@48975
   904
using empty_in_Fpow by blast
blanchet@48975
   905
blanchet@48975
   906
lemma Fpow_subset_Pow: "Fpow A \<le> Pow A"
blanchet@48975
   907
unfolding Fpow_def by auto
blanchet@48975
   908
blanchet@48975
   909
lemma card_of_Fpow[simp]: "|A| \<le>o |Fpow A|"
blanchet@48975
   910
proof-
blanchet@48975
   911
  let ?h = "\<lambda> a. {a}"
blanchet@48975
   912
  have "inj_on ?h A \<and> ?h ` A \<le> Fpow A"
blanchet@48975
   913
  unfolding inj_on_def Fpow_def by auto
blanchet@48975
   914
  thus ?thesis using card_of_ordLeq by metis
blanchet@48975
   915
qed
blanchet@48975
   916
blanchet@48975
   917
lemma Card_order_Fpow: "Card_order r \<Longrightarrow> r \<le>o |Fpow(Field r) |"
blanchet@48975
   918
using card_of_Fpow card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
blanchet@48975
   919
blanchet@48975
   920
lemma Fpow_Pow_finite: "Fpow A = Pow A Int {A. finite A}"
blanchet@48975
   921
unfolding Fpow_def Pow_def by blast
blanchet@48975
   922
blanchet@48975
   923
lemma inj_on_image_Fpow:
blanchet@48975
   924
assumes "inj_on f A"
blanchet@48975
   925
shows "inj_on (image f) (Fpow A)"
blanchet@48975
   926
using assms Fpow_subset_Pow[of A] subset_inj_on[of "image f" "Pow A"]
blanchet@48975
   927
      inj_on_image_Pow by blast
blanchet@48975
   928
blanchet@48975
   929
lemma image_Fpow_mono:
blanchet@48975
   930
assumes "f ` A \<le> B"
blanchet@48975
   931
shows "(image f) ` (Fpow A) \<le> Fpow B"
blanchet@48975
   932
using assms by(unfold Fpow_def, auto)
blanchet@48975
   933
blanchet@48975
   934
lemma image_Fpow_surjective:
blanchet@48975
   935
assumes "f ` A = B"
blanchet@48975
   936
shows "(image f) ` (Fpow A) = Fpow B"
blanchet@48975
   937
using assms proof(unfold Fpow_def, auto)
blanchet@48975
   938
  fix Y assume *: "Y \<le> f ` A" and **: "finite Y"
blanchet@48975
   939
  hence "\<forall>b \<in> Y. \<exists>a. a \<in> A \<and> f a = b" by auto
blanchet@48975
   940
  with bchoice[of Y "\<lambda>b a. a \<in> A \<and> f a = b"]
blanchet@48975
   941
  obtain g where 1: "\<forall>b \<in> Y. g b \<in> A \<and> f(g b) = b" by blast
blanchet@48975
   942
  obtain X where X_def: "X = g ` Y" by blast
blanchet@48975
   943
  have "f ` X = Y \<and> X \<le> A \<and> finite X"
blanchet@48975
   944
  by(unfold X_def, force simp add: ** 1)
blanchet@48975
   945
  thus "Y \<in> (image f) ` {X. X \<le> A \<and> finite X}" by auto
blanchet@48975
   946
qed
blanchet@48975
   947
blanchet@48975
   948
lemma bij_betw_image_Fpow:
blanchet@48975
   949
assumes "bij_betw f A B"
blanchet@48975
   950
shows "bij_betw (image f) (Fpow A) (Fpow B)"
blanchet@48975
   951
using assms unfolding bij_betw_def
blanchet@48975
   952
by (auto simp add: inj_on_image_Fpow image_Fpow_surjective)
blanchet@48975
   953
blanchet@48975
   954
lemma card_of_Fpow_mono[simp]:
blanchet@48975
   955
assumes "|A| \<le>o |B|"
blanchet@48975
   956
shows "|Fpow A| \<le>o |Fpow B|"
blanchet@48975
   957
proof-
blanchet@48975
   958
  obtain f where "inj_on f A \<and> f ` A \<le> B"
blanchet@48975
   959
  using assms card_of_ordLeq[of A B] by auto
blanchet@48975
   960
  hence "inj_on (image f) (Fpow A) \<and> (image f) ` (Fpow A) \<le> (Fpow B)"
blanchet@48975
   961
  by (auto simp add: inj_on_image_Fpow image_Fpow_mono)
blanchet@48975
   962
  thus ?thesis using card_of_ordLeq[of "Fpow A"] by auto
blanchet@48975
   963
qed
blanchet@48975
   964
blanchet@48975
   965
lemma ordIso_Fpow_mono:
blanchet@48975
   966
assumes "r \<le>o r'"
blanchet@48975
   967
shows "|Fpow(Field r)| \<le>o |Fpow(Field r')|"
blanchet@48975
   968
using assms card_of_mono2 card_of_Fpow_mono by blast
blanchet@48975
   969
blanchet@48975
   970
lemma card_of_Fpow_cong[simp]:
blanchet@48975
   971
assumes "|A| =o |B|"
blanchet@48975
   972
shows "|Fpow A| =o |Fpow B|"
blanchet@48975
   973
proof-
blanchet@48975
   974
  obtain f where "bij_betw f A B"
blanchet@48975
   975
  using assms card_of_ordIso[of A B] by auto
blanchet@48975
   976
  hence "bij_betw (image f) (Fpow A) (Fpow B)"
blanchet@48975
   977
  by (auto simp add: bij_betw_image_Fpow)
blanchet@48975
   978
  thus ?thesis using card_of_ordIso[of "Fpow A"] by auto
blanchet@48975
   979
qed
blanchet@48975
   980
blanchet@48975
   981
lemma ordIso_Fpow_cong:
blanchet@48975
   982
assumes "r =o r'"
blanchet@48975
   983
shows "|Fpow(Field r)| =o |Fpow(Field r')|"
blanchet@48975
   984
using assms card_of_cong card_of_Fpow_cong by blast
blanchet@48975
   985
blanchet@48975
   986
lemma card_of_Fpow_lists: "|Fpow A| \<le>o |lists A|"
blanchet@48975
   987
proof-
blanchet@48975
   988
  have "set ` (lists A) = Fpow A"
blanchet@48975
   989
  unfolding lists_def2 Fpow_def using finite_list finite_set by blast
blanchet@48975
   990
  thus ?thesis using card_of_ordLeq2[of "Fpow A"] Fpow_not_empty[of A] by blast
blanchet@48975
   991
qed
blanchet@48975
   992
blanchet@48975
   993
lemma card_of_Fpow_infinite[simp]:
blanchet@48975
   994
assumes "infinite A"
blanchet@48975
   995
shows "|Fpow A| =o |A|"
blanchet@48975
   996
using assms card_of_Fpow_lists card_of_lists_infinite card_of_Fpow
blanchet@48975
   997
      ordLeq_ordIso_trans ordIso_iff_ordLeq by blast
blanchet@48975
   998
blanchet@48975
   999
corollary Fpow_infinite_bij_betw:
blanchet@48975
  1000
assumes "infinite A"
blanchet@48975
  1001
shows "\<exists>f. bij_betw f (Fpow A) A"
blanchet@48975
  1002
using assms card_of_Fpow_infinite card_of_ordIso by blast
blanchet@48975
  1003
blanchet@48975
  1004
blanchet@48975
  1005
subsection {* The cardinal $\omega$ and the finite cardinals  *}
blanchet@48975
  1006
blanchet@48975
  1007
subsubsection {* First as well-orders *}
blanchet@48975
  1008
blanchet@48975
  1009
lemma Field_natLess: "Field natLess = (UNIV::nat set)"
blanchet@48975
  1010
by(unfold Field_def, auto)
blanchet@48975
  1011
blanchet@54475
  1012
lemma natLeq_well_order_on: "well_order_on UNIV natLeq"
blanchet@54475
  1013
using natLeq_Well_order Field_natLeq by auto
blanchet@54475
  1014
blanchet@54475
  1015
lemma natLeq_wo_rel: "wo_rel natLeq"
blanchet@54475
  1016
unfolding wo_rel_def using natLeq_Well_order .
blanchet@54475
  1017
blanchet@48975
  1018
lemma natLeq_ofilter_less: "ofilter natLeq {0 ..< n}"
blanchet@48975
  1019
by(auto simp add: natLeq_wo_rel wo_rel.ofilter_def,
blanchet@54475
  1020
   simp add: Field_natLeq, unfold rel.under_def, auto)
blanchet@48975
  1021
blanchet@48975
  1022
lemma natLeq_ofilter_leq: "ofilter natLeq {0 .. n}"
blanchet@48975
  1023
by(auto simp add: natLeq_wo_rel wo_rel.ofilter_def,
blanchet@54475
  1024
   simp add: Field_natLeq, unfold rel.under_def, auto)
blanchet@54475
  1025
blanchet@54475
  1026
lemma natLeq_UNIV_ofilter: "wo_rel.ofilter natLeq UNIV"
blanchet@54475
  1027
using natLeq_wo_rel Field_natLeq wo_rel.Field_ofilter[of natLeq] by auto
blanchet@48975
  1028
blanchet@48975
  1029
lemma natLeq_ofilter_iff:
blanchet@48975
  1030
"ofilter natLeq A = (A = UNIV \<or> (\<exists>n. A = {0 ..< n}))"
blanchet@48975
  1031
proof(rule iffI)
blanchet@48975
  1032
  assume "ofilter natLeq A"
blanchet@48975
  1033
  hence "\<forall>m n. n \<in> A \<and> m \<le> n \<longrightarrow> m \<in> A"
blanchet@48975
  1034
  by(auto simp add: natLeq_wo_rel wo_rel.ofilter_def rel.under_def)
blanchet@48975
  1035
  thus "A = UNIV \<or> (\<exists>n. A = {0 ..< n})" using closed_nat_set_iff by blast
blanchet@48975
  1036
next
blanchet@48975
  1037
  assume "A = UNIV \<or> (\<exists>n. A = {0 ..< n})"
blanchet@48975
  1038
  thus "ofilter natLeq A"
blanchet@48975
  1039
  by(auto simp add: natLeq_ofilter_less natLeq_UNIV_ofilter)
blanchet@48975
  1040
qed
blanchet@48975
  1041
blanchet@48975
  1042
lemma natLeq_under_leq: "under natLeq n = {0 .. n}"
blanchet@48975
  1043
unfolding rel.under_def by auto
blanchet@48975
  1044
blanchet@48975
  1045
corollary natLeq_on_ofilter:
blanchet@48975
  1046
"ofilter(natLeq_on n) {0 ..< n}"
blanchet@48975
  1047
by (auto simp add: natLeq_on_ofilter_less_eq)
blanchet@48975
  1048
blanchet@48975
  1049
lemma natLeq_on_ofilter_less:
blanchet@48975
  1050
"n < m \<Longrightarrow> ofilter (natLeq_on m) {0 .. n}"
blanchet@48975
  1051
by(auto simp add: natLeq_on_wo_rel wo_rel.ofilter_def,
blanchet@48975
  1052
   simp add: Field_natLeq_on, unfold rel.under_def, auto)
blanchet@48975
  1053
blanchet@48975
  1054
lemma natLeq_on_ordLess_natLeq: "natLeq_on n <o natLeq"
blanchet@48975
  1055
using Field_natLeq Field_natLeq_on[of n] nat_infinite
blanchet@48975
  1056
      finite_ordLess_infinite[of "natLeq_on n" natLeq]
blanchet@48975
  1057
      natLeq_Well_order natLeq_on_Well_order[of n] by auto
blanchet@48975
  1058
blanchet@48975
  1059
lemma natLeq_on_injective:
blanchet@48975
  1060
"natLeq_on m = natLeq_on n \<Longrightarrow> m = n"
blanchet@48975
  1061
using Field_natLeq_on[of m] Field_natLeq_on[of n]
blanchet@48975
  1062
      atLeastLessThan_injective[of m n] by auto
blanchet@48975
  1063
blanchet@48975
  1064
lemma natLeq_on_injective_ordIso:
blanchet@48975
  1065
"(natLeq_on m =o natLeq_on n) = (m = n)"
blanchet@48975
  1066
proof(auto simp add: natLeq_on_Well_order ordIso_reflexive)
blanchet@48975
  1067
  assume "natLeq_on m =o natLeq_on n"
blanchet@48975
  1068
  then obtain f where "bij_betw f {0..<m} {0..<n}"
blanchet@48975
  1069
  using Field_natLeq_on assms unfolding ordIso_def iso_def[abs_def] by auto
blanchet@48975
  1070
  thus "m = n" using atLeastLessThan_injective2 by blast
blanchet@48975
  1071
qed
blanchet@48975
  1072
blanchet@48975
  1073
blanchet@48975
  1074
subsubsection {* Then as cardinals *}
blanchet@48975
  1075
blanchet@48975
  1076
lemma ordIso_natLeq_infinite1:
blanchet@48975
  1077
"|A| =o natLeq \<Longrightarrow> infinite A"
blanchet@48975
  1078
using ordIso_symmetric ordIso_imp_ordLeq infinite_iff_natLeq_ordLeq by blast
blanchet@48975
  1079
blanchet@48975
  1080
lemma ordIso_natLeq_infinite2:
blanchet@48975
  1081
"natLeq =o |A| \<Longrightarrow> infinite A"
blanchet@48975
  1082
using ordIso_imp_ordLeq infinite_iff_natLeq_ordLeq by blast
blanchet@48975
  1083
blanchet@48975
  1084
lemma ordLeq_natLeq_on_imp_finite:
blanchet@48975
  1085
assumes "|A| \<le>o natLeq_on n"
blanchet@48975
  1086
shows "finite A"
blanchet@48975
  1087
proof-
blanchet@48975
  1088
  have "|A| \<le>o |{0 ..< n}|"
blanchet@48975
  1089
  using assms card_of_less ordIso_symmetric ordLeq_ordIso_trans by blast
blanchet@48975
  1090
  thus ?thesis by (auto simp add: card_of_ordLeq_finite)
blanchet@48975
  1091
qed
blanchet@48975
  1092
blanchet@48975
  1093
blanchet@54475
  1094
subsubsection {* "Backward compatibility" with the numeric cardinal operator for finite sets *}
blanchet@48975
  1095
blanchet@48975
  1096
lemma finite_card_of_iff_card:
blanchet@48975
  1097
assumes FIN: "finite A" and FIN': "finite B"
blanchet@48975
  1098
shows "( |A| =o |B| ) = (card A = card B)"
blanchet@48975
  1099
using assms card_of_ordIso[of A B] bij_betw_iff_card[of A B] by blast
blanchet@48975
  1100
blanchet@48975
  1101
lemma finite_card_of_iff_card3:
blanchet@48975
  1102
assumes FIN: "finite A" and FIN': "finite B"
blanchet@48975
  1103
shows "( |A| <o |B| ) = (card A < card B)"
blanchet@48975
  1104
proof-
blanchet@48975
  1105
  have "( |A| <o |B| ) = (~ ( |B| \<le>o |A| ))" by simp
blanchet@48975
  1106
  also have "... = (~ (card B \<le> card A))"
blanchet@48975
  1107
  using assms by(simp add: finite_card_of_iff_card2)
blanchet@48975
  1108
  also have "... = (card A < card B)" by auto
blanchet@48975
  1109
  finally show ?thesis .
blanchet@48975
  1110
qed
blanchet@48975
  1111
blanchet@48975
  1112
lemma card_Field_natLeq_on:
blanchet@48975
  1113
"card(Field(natLeq_on n)) = n"
blanchet@48975
  1114
using Field_natLeq_on card_atLeastLessThan by auto
blanchet@48975
  1115
blanchet@48975
  1116
blanchet@48975
  1117
subsection {* The successor of a cardinal *}
blanchet@48975
  1118
blanchet@48975
  1119
lemma embed_implies_ordIso_Restr:
blanchet@48975
  1120
assumes WELL: "Well_order r" and WELL': "Well_order r'" and EMB: "embed r' r f"
blanchet@48975
  1121
shows "r' =o Restr r (f ` (Field r'))"
blanchet@48975
  1122
using assms embed_implies_iso_Restr Well_order_Restr unfolding ordIso_def by blast
blanchet@48975
  1123
blanchet@48975
  1124
lemma cardSuc_Well_order[simp]:
blanchet@48975
  1125
"Card_order r \<Longrightarrow> Well_order(cardSuc r)"
blanchet@48975
  1126
using cardSuc_Card_order unfolding card_order_on_def by blast
blanchet@48975
  1127
blanchet@48975
  1128
lemma Field_cardSuc_not_empty:
blanchet@48975
  1129
assumes "Card_order r"
blanchet@48975
  1130
shows "Field (cardSuc r) \<noteq> {}"
blanchet@48975
  1131
proof
blanchet@48975
  1132
  assume "Field(cardSuc r) = {}"
blanchet@48975
  1133
  hence "|Field(cardSuc r)| \<le>o r" using assms Card_order_empty[of r] by auto
blanchet@48975
  1134
  hence "cardSuc r \<le>o r" using assms card_of_Field_ordIso
blanchet@48975
  1135
  cardSuc_Card_order ordIso_symmetric ordIso_ordLeq_trans by blast
blanchet@48975
  1136
  thus False using cardSuc_greater not_ordLess_ordLeq assms by blast
blanchet@48975
  1137
qed
blanchet@48975
  1138
blanchet@48975
  1139
lemma cardSuc_mono_ordLess[simp]:
blanchet@48975
  1140
assumes CARD: "Card_order r" and CARD': "Card_order r'"
blanchet@48975
  1141
shows "(cardSuc r <o cardSuc r') = (r <o r')"
blanchet@48975
  1142
proof-
blanchet@48975
  1143
  have 0: "Well_order r \<and> Well_order r' \<and> Well_order(cardSuc r) \<and> Well_order(cardSuc r')"
blanchet@48975
  1144
  using assms by auto
blanchet@48975
  1145
  thus ?thesis
blanchet@48975
  1146
  using not_ordLeq_iff_ordLess not_ordLeq_iff_ordLess[of r r']
blanchet@48975
  1147
  using cardSuc_mono_ordLeq[of r' r] assms by blast
blanchet@48975
  1148
qed
blanchet@48975
  1149
blanchet@48975
  1150
lemma card_of_Plus_ordLeq_infinite[simp]:
blanchet@48975
  1151
assumes C: "infinite C" and A: "|A| \<le>o |C|" and B: "|B| \<le>o |C|"
blanchet@48975
  1152
shows "|A <+> B| \<le>o |C|"
blanchet@48975
  1153
proof-
blanchet@48975
  1154
  let ?r = "cardSuc |C|"
blanchet@48975
  1155
  have "Card_order ?r \<and> infinite (Field ?r)" using assms by simp
blanchet@48975
  1156
  moreover have "|A| <o ?r" and "|B| <o ?r" using A B by auto
blanchet@48975
  1157
  ultimately have "|A <+> B| <o ?r"
blanchet@48975
  1158
  using card_of_Plus_ordLess_infinite_Field by blast
blanchet@48975
  1159
  thus ?thesis using C by simp
blanchet@48975
  1160
qed
blanchet@48975
  1161
blanchet@48975
  1162
lemma card_of_Un_ordLeq_infinite[simp]:
blanchet@48975
  1163
assumes C: "infinite C" and A: "|A| \<le>o |C|" and B: "|B| \<le>o |C|"
blanchet@48975
  1164
shows "|A Un B| \<le>o |C|"
blanchet@48975
  1165
using assms card_of_Plus_ordLeq_infinite card_of_Un_Plus_ordLeq
blanchet@48975
  1166
ordLeq_transitive by metis
blanchet@48975
  1167
blanchet@48975
  1168
blanchet@48975
  1169
subsection {* Others *}
blanchet@48975
  1170
blanchet@48975
  1171
lemma under_mono[simp]:
blanchet@48975
  1172
assumes "Well_order r" and "(i,j) \<in> r"
blanchet@48975
  1173
shows "under r i \<subseteq> under r j"
blanchet@48975
  1174
using assms unfolding rel.under_def order_on_defs
blanchet@48975
  1175
trans_def by blast
blanchet@48975
  1176
blanchet@48975
  1177
lemma underS_under:
blanchet@48975
  1178
assumes "i \<in> Field r"
blanchet@48975
  1179
shows "underS r i = under r i - {i}"
blanchet@48975
  1180
using assms unfolding rel.underS_def rel.under_def by auto
blanchet@48975
  1181
blanchet@48975
  1182
lemma relChain_under:
blanchet@48975
  1183
assumes "Well_order r"
blanchet@48975
  1184
shows "relChain r (\<lambda> i. under r i)"
blanchet@48975
  1185
using assms unfolding relChain_def by auto
blanchet@48975
  1186
blanchet@54475
  1187
lemma card_of_infinite_diff_finite:
blanchet@54475
  1188
assumes "infinite A" and "finite B"
blanchet@54475
  1189
shows "|A - B| =o |A|"
blanchet@54475
  1190
by (metis assms card_of_Un_diff_infinite finite_ordLess_infinite2)
blanchet@54475
  1191
blanchet@48975
  1192
lemma infinite_card_of_diff_singl:
blanchet@48975
  1193
assumes "infinite A"
blanchet@48975
  1194
shows "|A - {a}| =o |A|"
traytel@52544
  1195
by (metis assms card_of_infinite_diff_finite finite.emptyI finite_insert)
blanchet@48975
  1196
blanchet@48975
  1197
lemma card_of_vimage:
blanchet@48975
  1198
assumes "B \<subseteq> range f"
blanchet@48975
  1199
shows "|B| \<le>o |f -` B|"
blanchet@48975
  1200
apply(rule surj_imp_ordLeq[of _ f])
blanchet@48975
  1201
using assms by (metis Int_absorb2 image_vimage_eq order_refl)
blanchet@48975
  1202
blanchet@48975
  1203
lemma surj_card_of_vimage:
blanchet@48975
  1204
assumes "surj f"
blanchet@48975
  1205
shows "|B| \<le>o |f -` B|"
blanchet@48975
  1206
by (metis assms card_of_vimage subset_UNIV)
blanchet@48975
  1207
blanchet@48975
  1208
(* bounded powerset *)
blanchet@48975
  1209
definition Bpow where
blanchet@48975
  1210
"Bpow r A \<equiv> {X . X \<subseteq> A \<and> |X| \<le>o r}"
blanchet@48975
  1211
blanchet@48975
  1212
lemma Bpow_empty[simp]:
blanchet@48975
  1213
assumes "Card_order r"
blanchet@48975
  1214
shows "Bpow r {} = {{}}"
blanchet@48975
  1215
using assms unfolding Bpow_def by auto
blanchet@48975
  1216
blanchet@48975
  1217
lemma singl_in_Bpow:
blanchet@48975
  1218
assumes rc: "Card_order r"
blanchet@48975
  1219
and r: "Field r \<noteq> {}" and a: "a \<in> A"
blanchet@48975
  1220
shows "{a} \<in> Bpow r A"
blanchet@48975
  1221
proof-
blanchet@48975
  1222
  have "|{a}| \<le>o r" using r rc by auto
blanchet@48975
  1223
  thus ?thesis unfolding Bpow_def using a by auto
blanchet@48975
  1224
qed
blanchet@48975
  1225
blanchet@48975
  1226
lemma ordLeq_card_Bpow:
blanchet@48975
  1227
assumes rc: "Card_order r" and r: "Field r \<noteq> {}"
blanchet@48975
  1228
shows "|A| \<le>o |Bpow r A|"
blanchet@48975
  1229
proof-
blanchet@48975
  1230
  have "inj_on (\<lambda> a. {a}) A" unfolding inj_on_def by auto
blanchet@48975
  1231
  moreover have "(\<lambda> a. {a}) ` A \<subseteq> Bpow r A"
blanchet@48975
  1232
  using singl_in_Bpow[OF assms] by auto
blanchet@48975
  1233
  ultimately show ?thesis unfolding card_of_ordLeq[symmetric] by blast
blanchet@48975
  1234
qed
blanchet@48975
  1235
blanchet@48975
  1236
lemma infinite_Bpow:
blanchet@48975
  1237
assumes rc: "Card_order r" and r: "Field r \<noteq> {}"
blanchet@48975
  1238
and A: "infinite A"
blanchet@48975
  1239
shows "infinite (Bpow r A)"
blanchet@48975
  1240
using ordLeq_card_Bpow[OF rc r]
blanchet@48975
  1241
by (metis A card_of_ordLeq_infinite)
blanchet@48975
  1242
traytel@52545
  1243
definition Func_option where
traytel@52545
  1244
 "Func_option A B \<equiv>
traytel@52545
  1245
  {f. (\<forall> a. f a \<noteq> None \<longleftrightarrow> a \<in> A) \<and> (\<forall> a \<in> A. case f a of Some b \<Rightarrow> b \<in> B |None \<Rightarrow> True)}"
traytel@52545
  1246
traytel@52545
  1247
lemma card_of_Func_option_Func:
traytel@52545
  1248
"|Func_option A B| =o |Func A B|"
traytel@52545
  1249
proof (rule ordIso_symmetric, unfold card_of_ordIso[symmetric], intro exI)
traytel@52545
  1250
  let ?F = "\<lambda> f a. if a \<in> A then Some (f a) else None"
traytel@52545
  1251
  show "bij_betw ?F (Func A B) (Func_option A B)"
traytel@52545
  1252
  unfolding bij_betw_def unfolding inj_on_def proof(intro conjI ballI impI)
traytel@52545
  1253
    fix f g assume f: "f \<in> Func A B" and g: "g \<in> Func A B" and eq: "?F f = ?F g"
traytel@52545
  1254
    show "f = g"
traytel@52545
  1255
    proof(rule ext)
traytel@52545
  1256
      fix a
traytel@52545
  1257
      show "f a = g a"
traytel@52545
  1258
      proof(cases "a \<in> A")
traytel@52545
  1259
        case True
traytel@52545
  1260
        have "Some (f a) = ?F f a" using True by auto
traytel@52545
  1261
        also have "... = ?F g a" using eq unfolding fun_eq_iff by(rule allE)
traytel@52545
  1262
        also have "... = Some (g a)" using True by auto
traytel@52545
  1263
        finally have "Some (f a) = Some (g a)" .
traytel@52545
  1264
        thus ?thesis by simp
traytel@52545
  1265
      qed(insert f g, unfold Func_def, auto)
traytel@52545
  1266
    qed
traytel@52545
  1267
  next
traytel@52545
  1268
    show "?F ` Func A B = Func_option A B"
traytel@52545
  1269
    proof safe
traytel@52545
  1270
      fix f assume f: "f \<in> Func_option A B"
traytel@52545
  1271
      def g \<equiv> "\<lambda> a. case f a of Some b \<Rightarrow> b | None \<Rightarrow> undefined"
traytel@52545
  1272
      have "g \<in> Func A B"
traytel@52545
  1273
      using f unfolding g_def Func_def Func_option_def by force+
traytel@52545
  1274
      moreover have "f = ?F g"
traytel@52545
  1275
      proof(rule ext)
traytel@52545
  1276
        fix a show "f a = ?F g a"
traytel@52545
  1277
        using f unfolding Func_option_def g_def by (cases "a \<in> A") force+
traytel@52545
  1278
      qed
traytel@52545
  1279
      ultimately show "f \<in> ?F ` (Func A B)" by blast
traytel@52545
  1280
    qed(unfold Func_def Func_option_def, auto)
traytel@52545
  1281
  qed
traytel@52545
  1282
qed
traytel@52545
  1283
traytel@52545
  1284
(* partial-function space: *)
traytel@52545
  1285
definition Pfunc where
traytel@52545
  1286
"Pfunc A B \<equiv>
traytel@52545
  1287
 {f. (\<forall>a. f a \<noteq> None \<longrightarrow> a \<in> A) \<and>
traytel@52545
  1288
     (\<forall>a. case f a of None \<Rightarrow> True | Some b \<Rightarrow> b \<in> B)}"
traytel@52545
  1289
traytel@52545
  1290
lemma Func_Pfunc:
traytel@52545
  1291
"Func_option A B \<subseteq> Pfunc A B"
traytel@52545
  1292
unfolding Func_option_def Pfunc_def by auto
traytel@52545
  1293
traytel@52545
  1294
lemma Pfunc_Func_option:
traytel@52545
  1295
"Pfunc A B = (\<Union> A' \<in> Pow A. Func_option A' B)"
traytel@52545
  1296
proof safe
traytel@52545
  1297
  fix f assume f: "f \<in> Pfunc A B"
traytel@52545
  1298
  show "f \<in> (\<Union>A'\<in>Pow A. Func_option A' B)"
traytel@52545
  1299
  proof (intro UN_I)
traytel@52545
  1300
    let ?A' = "{a. f a \<noteq> None}"
traytel@52545
  1301
    show "?A' \<in> Pow A" using f unfolding Pow_def Pfunc_def by auto
traytel@52545
  1302
    show "f \<in> Func_option ?A' B" using f unfolding Func_option_def Pfunc_def by auto
traytel@52545
  1303
  qed
traytel@52545
  1304
next
traytel@52545
  1305
  fix f A' assume "f \<in> Func_option A' B" and "A' \<subseteq> A"
traytel@52545
  1306
  thus "f \<in> Pfunc A B" unfolding Func_option_def Pfunc_def by auto
traytel@52545
  1307
qed
traytel@52545
  1308
blanchet@54475
  1309
lemma card_of_Func_mono:
blanchet@54475
  1310
fixes A1 A2 :: "'a set" and B :: "'b set"
blanchet@54475
  1311
assumes A12: "A1 \<subseteq> A2" and B: "B \<noteq> {}"
blanchet@54475
  1312
shows "|Func A1 B| \<le>o |Func A2 B|"
blanchet@54475
  1313
proof-
blanchet@54475
  1314
  obtain bb where bb: "bb \<in> B" using B by auto
blanchet@54475
  1315
  def F \<equiv> "\<lambda> (f1::'a \<Rightarrow> 'b) a. if a \<in> A2 then (if a \<in> A1 then f1 a else bb)
blanchet@54475
  1316
                                                else undefined"
blanchet@54475
  1317
  show ?thesis unfolding card_of_ordLeq[symmetric] proof(intro exI[of _ F] conjI)
blanchet@54475
  1318
    show "inj_on F (Func A1 B)" unfolding inj_on_def proof safe
blanchet@54475
  1319
      fix f g assume f: "f \<in> Func A1 B" and g: "g \<in> Func A1 B" and eq: "F f = F g"
blanchet@54475
  1320
      show "f = g"
blanchet@54475
  1321
      proof(rule ext)
blanchet@54475
  1322
        fix a show "f a = g a"
blanchet@54475
  1323
        proof(cases "a \<in> A1")
blanchet@54475
  1324
          case True
blanchet@54475
  1325
          thus ?thesis using eq A12 unfolding F_def fun_eq_iff
blanchet@54475
  1326
          by (elim allE[of _ a]) auto
blanchet@54475
  1327
        qed(insert f g, unfold Func_def, fastforce)
blanchet@54475
  1328
      qed
blanchet@54475
  1329
    qed
blanchet@54475
  1330
  qed(insert bb, unfold Func_def F_def, force)
blanchet@54475
  1331
qed
blanchet@54475
  1332
traytel@52545
  1333
lemma card_of_Func_option_mono:
traytel@52545
  1334
fixes A1 A2 :: "'a set" and B :: "'b set"
traytel@52545
  1335
assumes A12: "A1 \<subseteq> A2" and B: "B \<noteq> {}"
traytel@52545
  1336
shows "|Func_option A1 B| \<le>o |Func_option A2 B|"
traytel@52545
  1337
by (metis card_of_Func_mono[OF A12 B] card_of_Func_option_Func
traytel@52545
  1338
  ordIso_ordLeq_trans ordLeq_ordIso_trans ordIso_symmetric)
traytel@52545
  1339
traytel@52545
  1340
lemma card_of_Pfunc_Pow_Func_option:
traytel@52545
  1341
assumes "B \<noteq> {}"
traytel@52545
  1342
shows "|Pfunc A B| \<le>o |Pow A <*> Func_option A B|"
traytel@52545
  1343
proof-
traytel@52545
  1344
  have "|Pfunc A B| =o |\<Union> A' \<in> Pow A. Func_option A' B|" (is "_ =o ?K")
traytel@52545
  1345
    unfolding Pfunc_Func_option by(rule card_of_refl)
traytel@52545
  1346
  also have "?K \<le>o |Sigma (Pow A) (\<lambda> A'. Func_option A' B)|" using card_of_UNION_Sigma .
traytel@52545
  1347
  also have "|Sigma (Pow A) (\<lambda> A'. Func_option A' B)| \<le>o |Pow A <*> Func_option A B|"
traytel@52545
  1348
    apply(rule card_of_Sigma_mono1) using card_of_Func_option_mono[OF _ assms] by auto
traytel@52545
  1349
  finally show ?thesis .
traytel@52545
  1350
qed
traytel@52545
  1351
blanchet@48975
  1352
lemma Bpow_ordLeq_Func_Field:
blanchet@48975
  1353
assumes rc: "Card_order r" and r: "Field r \<noteq> {}" and A: "infinite A"
blanchet@48975
  1354
shows "|Bpow r A| \<le>o |Func (Field r) A|"
blanchet@48975
  1355
proof-
traytel@52545
  1356
  let ?F = "\<lambda> f. {x | x a. f a = x \<and> a \<in> Field r}"
blanchet@48975
  1357
  {fix X assume "X \<in> Bpow r A - {{}}"
blanchet@48975
  1358
   hence XA: "X \<subseteq> A" and "|X| \<le>o r"
blanchet@48975
  1359
   and X: "X \<noteq> {}" unfolding Bpow_def by auto
blanchet@48975
  1360
   hence "|X| \<le>o |Field r|" by (metis Field_card_of card_of_mono2)
blanchet@48975
  1361
   then obtain F where 1: "X = F ` (Field r)"
blanchet@48975
  1362
   using card_of_ordLeq2[OF X] by metis
traytel@52545
  1363
   def f \<equiv> "\<lambda> i. if i \<in> Field r then F i else undefined"
blanchet@48975
  1364
   have "\<exists> f \<in> Func (Field r) A. X = ?F f"
blanchet@48975
  1365
   apply (intro bexI[of _ f]) using 1 XA unfolding Func_def f_def by auto
blanchet@48975
  1366
  }
blanchet@48975
  1367
  hence "Bpow r A - {{}} \<subseteq> ?F ` (Func (Field r) A)" by auto
blanchet@48975
  1368
  hence "|Bpow r A - {{}}| \<le>o |Func (Field r) A|"
blanchet@48975
  1369
  by (rule surj_imp_ordLeq)
blanchet@48975
  1370
  moreover
blanchet@48975
  1371
  {have 2: "infinite (Bpow r A)" using infinite_Bpow[OF rc r A] .
blanchet@48975
  1372
   have "|Bpow r A| =o |Bpow r A - {{}}|"
traytel@52544
  1373
   using card_of_infinite_diff_finite
blanchet@48975
  1374
   by (metis Pow_empty 2 finite_Pow_iff infinite_imp_nonempty ordIso_symmetric)
blanchet@48975
  1375
  }
blanchet@48975
  1376
  ultimately show ?thesis by (metis ordIso_ordLeq_trans)
blanchet@48975
  1377
qed
blanchet@48975
  1378
blanchet@48975
  1379
lemma Func_emp2[simp]: "A \<noteq> {} \<Longrightarrow> Func A {} = {}" by auto
blanchet@48975
  1380
blanchet@48975
  1381
lemma empty_in_Func[simp]:
traytel@52545
  1382
"B \<noteq> {} \<Longrightarrow> (\<lambda>x. undefined) \<in> Func {} B"
blanchet@48975
  1383
unfolding Func_def by auto
blanchet@48975
  1384
blanchet@48975
  1385
lemma Func_mono[simp]:
blanchet@48975
  1386
assumes "B1 \<subseteq> B2"
blanchet@48975
  1387
shows "Func A B1 \<subseteq> Func A B2"
blanchet@48975
  1388
using assms unfolding Func_def by force
blanchet@48975
  1389
blanchet@48975
  1390
lemma Pfunc_mono[simp]:
blanchet@48975
  1391
assumes "A1 \<subseteq> A2" and "B1 \<subseteq> B2"
blanchet@48975
  1392
shows "Pfunc A B1 \<subseteq> Pfunc A B2"
blanchet@48975
  1393
using assms in_mono unfolding Pfunc_def apply safe
blanchet@48975
  1394
apply(case_tac "x a", auto)
blanchet@48975
  1395
by (metis in_mono option.simps(5))
blanchet@48975
  1396
blanchet@48975
  1397
lemma card_of_Func_UNIV_UNIV:
blanchet@48975
  1398
"|Func (UNIV::'a set) (UNIV::'b set)| =o |UNIV::('a \<Rightarrow> 'b) set|"
blanchet@48975
  1399
using card_of_Func_UNIV[of "UNIV::'b set"] by auto
blanchet@48975
  1400
blanchet@54475
  1401
lemma ordLeq_Func:
blanchet@54475
  1402
assumes "{b1,b2} \<subseteq> B" "b1 \<noteq> b2"
blanchet@54475
  1403
shows "|A| \<le>o |Func A B|"
blanchet@54475
  1404
unfolding card_of_ordLeq[symmetric] proof(intro exI conjI)
blanchet@54475
  1405
  let ?F = "\<lambda> aa a. if a \<in> A then (if a = aa then b1 else b2) else undefined"
blanchet@54475
  1406
  show "inj_on ?F A" using assms unfolding inj_on_def fun_eq_iff by auto
blanchet@54475
  1407
  show "?F ` A \<subseteq> Func A B" using assms unfolding Func_def by auto
blanchet@54475
  1408
qed
blanchet@54475
  1409
blanchet@54475
  1410
lemma infinite_Func:
blanchet@54475
  1411
assumes A: "infinite A" and B: "{b1,b2} \<subseteq> B" "b1 \<noteq> b2"
blanchet@54475
  1412
shows "infinite (Func A B)"
blanchet@54475
  1413
using ordLeq_Func[OF B] by (metis A card_of_ordLeq_finite)
blanchet@54475
  1414
blanchet@48975
  1415
end