src/HOL/Finite_Set.thy
author haftmann
Fri Jan 21 09:41:59 2011 +0100 (2011-01-21)
changeset 41656 011fcb70e32f
parent 41550 efa734d9b221
child 41657 89451110ba8e
permissions -rw-r--r--
restructured theory;
tuned proofs
wenzelm@12396
     1
(*  Title:      HOL/Finite_Set.thy
wenzelm@12396
     2
    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
avigad@16775
     3
                with contributions by Jeremy Avigad
wenzelm@12396
     4
*)
wenzelm@12396
     5
wenzelm@12396
     6
header {* Finite sets *}
wenzelm@12396
     7
nipkow@15131
     8
theory Finite_Set
haftmann@38400
     9
imports Option Power
nipkow@15131
    10
begin
wenzelm@12396
    11
haftmann@41656
    12
-- {* FIXME move *}
haftmann@41656
    13
haftmann@41656
    14
lemma inj_vimage_singleton: "inj f \<Longrightarrow> f -` {a} \<subseteq> {THE x. f x = a}"
haftmann@41656
    15
  -- {* The inverse image of a singleton under an injective function
haftmann@41656
    16
         is included in a singleton. *}
haftmann@41656
    17
  apply (auto simp add: inj_on_def)
haftmann@41656
    18
  apply (blast intro: the_equality [symmetric])
haftmann@41656
    19
  done
haftmann@41656
    20
haftmann@35817
    21
subsection {* Predicate for finite sets *}
wenzelm@12396
    22
haftmann@41656
    23
inductive finite :: "'a set \<Rightarrow> bool"
berghofe@22262
    24
  where
berghofe@22262
    25
    emptyI [simp, intro!]: "finite {}"
haftmann@41656
    26
  | insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)"
haftmann@41656
    27
haftmann@41656
    28
lemma finite_induct [case_names empty insert, induct set: finite]:
haftmann@41656
    29
  -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
haftmann@41656
    30
  assumes "finite F"
haftmann@41656
    31
  assumes "P {}"
haftmann@41656
    32
    and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
haftmann@41656
    33
  shows "P F"
haftmann@41656
    34
using `finite F` proof induct
haftmann@41656
    35
  show "P {}" by fact
haftmann@41656
    36
  fix x F assume F: "finite F" and P: "P F"
haftmann@41656
    37
  show "P (insert x F)"
haftmann@41656
    38
  proof cases
haftmann@41656
    39
    assume "x \<in> F"
haftmann@41656
    40
    hence "insert x F = F" by (rule insert_absorb)
haftmann@41656
    41
    with P show ?thesis by (simp only:)
haftmann@41656
    42
  next
haftmann@41656
    43
    assume "x \<notin> F"
haftmann@41656
    44
    from F this P show ?thesis by (rule insert)
haftmann@41656
    45
  qed
haftmann@41656
    46
qed
haftmann@41656
    47
haftmann@41656
    48
haftmann@41656
    49
subsubsection {* Choice principles *}
wenzelm@12396
    50
nipkow@13737
    51
lemma ex_new_if_finite: -- "does not depend on def of finite at all"
wenzelm@14661
    52
  assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
wenzelm@14661
    53
  shows "\<exists>a::'a. a \<notin> A"
wenzelm@14661
    54
proof -
haftmann@28823
    55
  from assms have "A \<noteq> UNIV" by blast
haftmann@41656
    56
  then show ?thesis by blast
wenzelm@12396
    57
qed
wenzelm@12396
    58
haftmann@41656
    59
text {* A finite choice principle. Does not need the SOME choice operator. *}
nipkow@15484
    60
nipkow@29923
    61
lemma finite_set_choice:
haftmann@41656
    62
  "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
haftmann@41656
    63
proof (induct rule: finite_induct)
haftmann@41656
    64
  case empty then show ?case by simp
nipkow@29923
    65
next
nipkow@29923
    66
  case (insert a A)
nipkow@29923
    67
  then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
nipkow@29923
    68
  show ?case (is "EX f. ?P f")
nipkow@29923
    69
  proof
nipkow@29923
    70
    show "?P(%x. if x = a then b else f x)" using f ab by auto
nipkow@29923
    71
  qed
nipkow@29923
    72
qed
nipkow@29923
    73
haftmann@23878
    74
haftmann@41656
    75
subsubsection {* Finite sets are the images of initial segments of natural numbers *}
nipkow@15392
    76
paulson@15510
    77
lemma finite_imp_nat_seg_image_inj_on:
haftmann@41656
    78
  assumes "finite A" 
haftmann@41656
    79
  shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
haftmann@41656
    80
using assms proof induct
nipkow@15392
    81
  case empty
haftmann@41656
    82
  show ?case
haftmann@41656
    83
  proof
haftmann@41656
    84
    show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp 
paulson@15510
    85
  qed
nipkow@15392
    86
next
nipkow@15392
    87
  case (insert a A)
wenzelm@23389
    88
  have notinA: "a \<notin> A" by fact
paulson@15510
    89
  from insert.hyps obtain n f
paulson@15510
    90
    where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
paulson@15510
    91
  hence "insert a A = f(n:=a) ` {i. i < Suc n}"
paulson@15510
    92
        "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
paulson@15510
    93
    by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
nipkow@15392
    94
  thus ?case by blast
nipkow@15392
    95
qed
nipkow@15392
    96
nipkow@15392
    97
lemma nat_seg_image_imp_finite:
haftmann@41656
    98
  "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
haftmann@41656
    99
proof (induct n arbitrary: A)
nipkow@15392
   100
  case 0 thus ?case by simp
nipkow@15392
   101
next
nipkow@15392
   102
  case (Suc n)
nipkow@15392
   103
  let ?B = "f ` {i. i < n}"
nipkow@15392
   104
  have finB: "finite ?B" by(rule Suc.hyps[OF refl])
nipkow@15392
   105
  show ?case
nipkow@15392
   106
  proof cases
nipkow@15392
   107
    assume "\<exists>k<n. f n = f k"
nipkow@15392
   108
    hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
nipkow@15392
   109
    thus ?thesis using finB by simp
nipkow@15392
   110
  next
nipkow@15392
   111
    assume "\<not>(\<exists> k<n. f n = f k)"
nipkow@15392
   112
    hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
nipkow@15392
   113
    thus ?thesis using finB by simp
nipkow@15392
   114
  qed
nipkow@15392
   115
qed
nipkow@15392
   116
nipkow@15392
   117
lemma finite_conv_nat_seg_image:
haftmann@41656
   118
  "finite A \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"
haftmann@41656
   119
  by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
nipkow@15392
   120
nipkow@32988
   121
lemma finite_imp_inj_to_nat_seg:
haftmann@41656
   122
  assumes "finite A"
haftmann@41656
   123
  shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> inj_on f A"
nipkow@32988
   124
proof -
nipkow@32988
   125
  from finite_imp_nat_seg_image_inj_on[OF `finite A`]
nipkow@32988
   126
  obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
nipkow@32988
   127
    by (auto simp:bij_betw_def)
nipkow@33057
   128
  let ?f = "the_inv_into {i. i<n} f"
nipkow@32988
   129
  have "inj_on ?f A & ?f ` A = {i. i<n}"
nipkow@33057
   130
    by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
nipkow@32988
   131
  thus ?thesis by blast
nipkow@32988
   132
qed
nipkow@32988
   133
haftmann@41656
   134
lemma finite_Collect_less_nat [iff]:
haftmann@41656
   135
  "finite {n::nat. n < k}"
haftmann@41656
   136
  by (fastsimp simp: finite_conv_nat_seg_image)
nipkow@29920
   137
haftmann@41656
   138
lemma finite_Collect_le_nat [iff]:
haftmann@41656
   139
  "finite {n::nat. n \<le> k}"
haftmann@41656
   140
  by (simp add: le_eq_less_or_eq Collect_disj_eq)
nipkow@15392
   141
haftmann@41656
   142
haftmann@41656
   143
subsubsection {* Finiteness and common set operations *}
wenzelm@12396
   144
haftmann@41656
   145
lemma rev_finite_subset:
haftmann@41656
   146
  "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
haftmann@41656
   147
proof (induct arbitrary: A rule: finite_induct)
haftmann@41656
   148
  case empty
haftmann@41656
   149
  then show ?case by simp
haftmann@41656
   150
next
haftmann@41656
   151
  case (insert x F A)
haftmann@41656
   152
  have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+
haftmann@41656
   153
  show "finite A"
haftmann@41656
   154
  proof cases
haftmann@41656
   155
    assume x: "x \<in> A"
haftmann@41656
   156
    with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
haftmann@41656
   157
    with r have "finite (A - {x})" .
haftmann@41656
   158
    hence "finite (insert x (A - {x}))" ..
haftmann@41656
   159
    also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
haftmann@41656
   160
    finally show ?thesis .
wenzelm@12396
   161
  next
haftmann@41656
   162
    show "A \<subseteq> F ==> ?thesis" by fact
haftmann@41656
   163
    assume "x \<notin> A"
haftmann@41656
   164
    with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
wenzelm@12396
   165
  qed
wenzelm@12396
   166
qed
wenzelm@12396
   167
haftmann@41656
   168
lemma finite_subset:
haftmann@41656
   169
  "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
haftmann@41656
   170
  by (rule rev_finite_subset)
nipkow@29901
   171
haftmann@41656
   172
lemma finite_UnI:
haftmann@41656
   173
  assumes "finite F" and "finite G"
haftmann@41656
   174
  shows "finite (F \<union> G)"
haftmann@41656
   175
  using assms by induct simp_all
nipkow@31992
   176
haftmann@41656
   177
lemma finite_Un [iff]:
haftmann@41656
   178
  "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
haftmann@41656
   179
  by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])
nipkow@31992
   180
haftmann@41656
   181
lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"
wenzelm@12396
   182
proof -
haftmann@41656
   183
  have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp
haftmann@41656
   184
  then have "finite ({a} \<union> A) \<longleftrightarrow> finite A" by (simp only: finite_Un)
wenzelm@23389
   185
  then show ?thesis by simp
wenzelm@12396
   186
qed
wenzelm@12396
   187
haftmann@41656
   188
lemma finite_Int [simp, intro]:
haftmann@41656
   189
  "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
haftmann@41656
   190
  by (blast intro: finite_subset)
haftmann@41656
   191
haftmann@41656
   192
lemma finite_Collect_conjI [simp, intro]:
haftmann@41656
   193
  "finite {x. P x} \<or> finite {x. Q x} \<Longrightarrow> finite {x. P x \<and> Q x}"
haftmann@41656
   194
  by (simp add: Collect_conj_eq)
haftmann@41656
   195
haftmann@41656
   196
lemma finite_Collect_disjI [simp]:
haftmann@41656
   197
  "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"
haftmann@41656
   198
  by (simp add: Collect_disj_eq)
haftmann@41656
   199
haftmann@41656
   200
lemma finite_Diff [simp, intro]:
haftmann@41656
   201
  "finite A \<Longrightarrow> finite (A - B)"
haftmann@41656
   202
  by (rule finite_subset, rule Diff_subset)
nipkow@29901
   203
nipkow@29901
   204
lemma finite_Diff2 [simp]:
haftmann@41656
   205
  assumes "finite B"
haftmann@41656
   206
  shows "finite (A - B) \<longleftrightarrow> finite A"
nipkow@29901
   207
proof -
haftmann@41656
   208
  have "finite A \<longleftrightarrow> finite((A - B) \<union> (A \<inter> B))" by (simp add: Un_Diff_Int)
haftmann@41656
   209
  also have "\<dots> \<longleftrightarrow> finite (A - B)" using `finite B` by simp
nipkow@29901
   210
  finally show ?thesis ..
nipkow@29901
   211
qed
nipkow@29901
   212
haftmann@41656
   213
lemma finite_Diff_insert [iff]:
haftmann@41656
   214
  "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
haftmann@41656
   215
proof -
haftmann@41656
   216
  have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
haftmann@41656
   217
  moreover have "A - insert a B = A - B - {a}" by auto
haftmann@41656
   218
  ultimately show ?thesis by simp
haftmann@41656
   219
qed
haftmann@41656
   220
nipkow@29901
   221
lemma finite_compl[simp]:
haftmann@41656
   222
  "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
haftmann@41656
   223
  by (simp add: Compl_eq_Diff_UNIV)
wenzelm@12396
   224
nipkow@29916
   225
lemma finite_Collect_not[simp]:
haftmann@41656
   226
  "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
haftmann@41656
   227
  by (simp add: Collect_neg_eq)
haftmann@41656
   228
haftmann@41656
   229
lemma finite_Union [simp, intro]:
haftmann@41656
   230
  "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite(\<Union>A)"
haftmann@41656
   231
  by (induct rule: finite_induct) simp_all
haftmann@41656
   232
haftmann@41656
   233
lemma finite_UN_I [intro]:
haftmann@41656
   234
  "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"
haftmann@41656
   235
  by (induct rule: finite_induct) simp_all
nipkow@29903
   236
haftmann@41656
   237
lemma finite_UN [simp]:
haftmann@41656
   238
  "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
haftmann@41656
   239
  by (blast intro: finite_subset)
haftmann@41656
   240
haftmann@41656
   241
lemma finite_Inter [intro]:
haftmann@41656
   242
  "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
haftmann@41656
   243
  by (blast intro: Inter_lower finite_subset)
wenzelm@12396
   244
haftmann@41656
   245
lemma finite_INT [intro]:
haftmann@41656
   246
  "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
haftmann@41656
   247
  by (blast intro: INT_lower finite_subset)
paulson@13825
   248
haftmann@41656
   249
lemma finite_imageI [simp, intro]:
haftmann@41656
   250
  "finite F \<Longrightarrow> finite (h ` F)"
haftmann@41656
   251
  by (induct rule: finite_induct) simp_all
paulson@13825
   252
haftmann@31768
   253
lemma finite_image_set [simp]:
haftmann@31768
   254
  "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
haftmann@31768
   255
  by (simp add: image_Collect [symmetric])
haftmann@31768
   256
haftmann@41656
   257
lemma finite_imageD:
haftmann@41656
   258
  "finite (f ` A) \<Longrightarrow> inj_on f A \<Longrightarrow> finite A"
wenzelm@12396
   259
proof -
wenzelm@12396
   260
  have aux: "!!A. finite (A - {}) = finite A" by simp
wenzelm@12396
   261
  fix B :: "'a set"
wenzelm@12396
   262
  assume "finite B"
wenzelm@12396
   263
  thus "!!A. f`A = B ==> inj_on f A ==> finite A"
wenzelm@12396
   264
    apply induct
wenzelm@12396
   265
     apply simp
wenzelm@12396
   266
    apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})")
wenzelm@12396
   267
     apply clarify
wenzelm@12396
   268
     apply (simp (no_asm_use) add: inj_on_def)
paulson@14208
   269
     apply (blast dest!: aux [THEN iffD1], atomize)
wenzelm@12396
   270
    apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
paulson@14208
   271
    apply (frule subsetD [OF equalityD2 insertI1], clarify)
wenzelm@12396
   272
    apply (rule_tac x = xa in bexI)
wenzelm@12396
   273
     apply (simp_all add: inj_on_image_set_diff)
wenzelm@12396
   274
    done
wenzelm@12396
   275
qed (rule refl)
wenzelm@12396
   276
haftmann@41656
   277
lemma finite_surj:
haftmann@41656
   278
  "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
haftmann@41656
   279
  by (erule finite_subset) (rule finite_imageI)
wenzelm@12396
   280
haftmann@41656
   281
lemma finite_range_imageI:
haftmann@41656
   282
  "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
haftmann@41656
   283
  by (drule finite_imageI) (simp add: range_composition)
paulson@13825
   284
haftmann@41656
   285
lemma finite_subset_image:
haftmann@41656
   286
  assumes "finite B"
haftmann@41656
   287
  shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
haftmann@41656
   288
using assms proof induct
haftmann@41656
   289
  case empty then show ?case by simp
haftmann@41656
   290
next
haftmann@41656
   291
  case insert then show ?case
haftmann@41656
   292
    by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
haftmann@41656
   293
       blast
haftmann@41656
   294
qed
haftmann@41656
   295
haftmann@41656
   296
lemma finite_vimageI:
haftmann@41656
   297
  "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
haftmann@41656
   298
  apply (induct rule: finite_induct)
wenzelm@21575
   299
   apply simp_all
paulson@14430
   300
  apply (subst vimage_insert)
huffman@35216
   301
  apply (simp add: finite_subset [OF inj_vimage_singleton])
paulson@13825
   302
  done
paulson@13825
   303
huffman@34111
   304
lemma finite_vimageD:
huffman@34111
   305
  assumes fin: "finite (h -` F)" and surj: "surj h"
huffman@34111
   306
  shows "finite F"
huffman@34111
   307
proof -
huffman@34111
   308
  have "finite (h ` (h -` F))" using fin by (rule finite_imageI)
huffman@34111
   309
  also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)
huffman@34111
   310
  finally show "finite F" .
huffman@34111
   311
qed
huffman@34111
   312
huffman@34111
   313
lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
huffman@34111
   314
  unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
huffman@34111
   315
haftmann@41656
   316
lemma finite_Collect_bex [simp]:
haftmann@41656
   317
  assumes "finite A"
haftmann@41656
   318
  shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"
haftmann@41656
   319
proof -
haftmann@41656
   320
  have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto
haftmann@41656
   321
  with assms show ?thesis by simp
haftmann@41656
   322
qed
wenzelm@12396
   323
haftmann@41656
   324
lemma finite_Collect_bounded_ex [simp]:
haftmann@41656
   325
  assumes "finite {y. P y}"
haftmann@41656
   326
  shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
haftmann@41656
   327
proof -
haftmann@41656
   328
  have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
haftmann@41656
   329
  with assms show ?thesis by simp
haftmann@41656
   330
qed
nipkow@29920
   331
haftmann@41656
   332
lemma finite_Plus:
haftmann@41656
   333
  "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
haftmann@41656
   334
  by (simp add: Plus_def)
nipkow@17022
   335
nipkow@31080
   336
lemma finite_PlusD: 
nipkow@31080
   337
  fixes A :: "'a set" and B :: "'b set"
nipkow@31080
   338
  assumes fin: "finite (A <+> B)"
nipkow@31080
   339
  shows "finite A" "finite B"
nipkow@31080
   340
proof -
nipkow@31080
   341
  have "Inl ` A \<subseteq> A <+> B" by auto
haftmann@41656
   342
  then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
haftmann@41656
   343
  then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
nipkow@31080
   344
next
nipkow@31080
   345
  have "Inr ` B \<subseteq> A <+> B" by auto
haftmann@41656
   346
  then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
haftmann@41656
   347
  then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
nipkow@31080
   348
qed
nipkow@31080
   349
haftmann@41656
   350
lemma finite_Plus_iff [simp]:
haftmann@41656
   351
  "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
haftmann@41656
   352
  by (auto intro: finite_PlusD finite_Plus)
nipkow@31080
   353
haftmann@41656
   354
lemma finite_Plus_UNIV_iff [simp]:
haftmann@41656
   355
  "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
haftmann@41656
   356
  by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
wenzelm@12396
   357
nipkow@40786
   358
lemma finite_SigmaI [simp, intro]:
haftmann@41656
   359
  "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
nipkow@40786
   360
  by (unfold Sigma_def) blast
wenzelm@12396
   361
haftmann@41656
   362
lemma finite_cartesian_product:
haftmann@41656
   363
  "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
nipkow@15402
   364
  by (rule finite_SigmaI)
nipkow@15402
   365
wenzelm@12396
   366
lemma finite_Prod_UNIV:
haftmann@41656
   367
  "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
haftmann@41656
   368
  by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
wenzelm@12396
   369
paulson@15409
   370
lemma finite_cartesian_productD1:
haftmann@41656
   371
  "finite (A \<times> B) \<Longrightarrow> B \<noteq> {} \<Longrightarrow> finite A"
paulson@15409
   372
apply (auto simp add: finite_conv_nat_seg_image) 
paulson@15409
   373
apply (drule_tac x=n in spec) 
paulson@15409
   374
apply (drule_tac x="fst o f" in spec) 
paulson@15409
   375
apply (auto simp add: o_def) 
paulson@15409
   376
 prefer 2 apply (force dest!: equalityD2) 
paulson@15409
   377
apply (drule equalityD1) 
paulson@15409
   378
apply (rename_tac y x)
paulson@15409
   379
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
paulson@15409
   380
 prefer 2 apply force
paulson@15409
   381
apply clarify
paulson@15409
   382
apply (rule_tac x=k in image_eqI, auto)
paulson@15409
   383
done
paulson@15409
   384
paulson@15409
   385
lemma finite_cartesian_productD2:
paulson@15409
   386
     "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
paulson@15409
   387
apply (auto simp add: finite_conv_nat_seg_image) 
paulson@15409
   388
apply (drule_tac x=n in spec) 
paulson@15409
   389
apply (drule_tac x="snd o f" in spec) 
paulson@15409
   390
apply (auto simp add: o_def) 
paulson@15409
   391
 prefer 2 apply (force dest!: equalityD2) 
paulson@15409
   392
apply (drule equalityD1)
paulson@15409
   393
apply (rename_tac x y)
paulson@15409
   394
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
paulson@15409
   395
 prefer 2 apply force
paulson@15409
   396
apply clarify
paulson@15409
   397
apply (rule_tac x=k in image_eqI, auto)
paulson@15409
   398
done
paulson@15409
   399
haftmann@41656
   400
lemma finite_Pow_iff [iff]:
haftmann@41656
   401
  "finite (Pow A) \<longleftrightarrow> finite A"
wenzelm@12396
   402
proof
wenzelm@12396
   403
  assume "finite (Pow A)"
haftmann@41656
   404
  then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
haftmann@41656
   405
  then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
wenzelm@12396
   406
next
wenzelm@12396
   407
  assume "finite A"
haftmann@41656
   408
  then show "finite (Pow A)"
huffman@35216
   409
    by induct (simp_all add: Pow_insert)
wenzelm@12396
   410
qed
wenzelm@12396
   411
haftmann@41656
   412
corollary finite_Collect_subsets [simp, intro]:
haftmann@41656
   413
  "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
haftmann@41656
   414
  by (simp add: Pow_def [symmetric])
nipkow@29918
   415
nipkow@15392
   416
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
haftmann@41656
   417
  by (blast intro: finite_subset [OF subset_Pow_Union])
nipkow@15392
   418
nipkow@15392
   419
haftmann@41656
   420
subsubsection {* Further induction rules on finite sets *}
haftmann@41656
   421
haftmann@41656
   422
lemma finite_ne_induct [case_names singleton insert, consumes 2]:
haftmann@41656
   423
  assumes "finite F" and "F \<noteq> {}"
haftmann@41656
   424
  assumes "\<And>x. P {x}"
haftmann@41656
   425
    and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
haftmann@41656
   426
  shows "P F"
haftmann@41656
   427
using assms proof induct
haftmann@41656
   428
  case empty then show ?case by simp
haftmann@41656
   429
next
haftmann@41656
   430
  case (insert x F) then show ?case by cases auto
haftmann@41656
   431
qed
haftmann@41656
   432
haftmann@41656
   433
lemma finite_subset_induct [consumes 2, case_names empty insert]:
haftmann@41656
   434
  assumes "finite F" and "F \<subseteq> A"
haftmann@41656
   435
  assumes empty: "P {}"
haftmann@41656
   436
    and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
haftmann@41656
   437
  shows "P F"
haftmann@41656
   438
using `finite F` `F \<subseteq> A` proof induct
haftmann@41656
   439
  show "P {}" by fact
nipkow@31441
   440
next
haftmann@41656
   441
  fix x F
haftmann@41656
   442
  assume "finite F" and "x \<notin> F" and
haftmann@41656
   443
    P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
haftmann@41656
   444
  show "P (insert x F)"
haftmann@41656
   445
  proof (rule insert)
haftmann@41656
   446
    from i show "x \<in> A" by blast
haftmann@41656
   447
    from i have "F \<subseteq> A" by blast
haftmann@41656
   448
    with P show "P F" .
haftmann@41656
   449
    show "finite F" by fact
haftmann@41656
   450
    show "x \<notin> F" by fact
haftmann@41656
   451
  qed
haftmann@41656
   452
qed
haftmann@41656
   453
haftmann@41656
   454
lemma finite_empty_induct:
haftmann@41656
   455
  assumes "finite A"
haftmann@41656
   456
  assumes "P A"
haftmann@41656
   457
    and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
haftmann@41656
   458
  shows "P {}"
haftmann@41656
   459
proof -
haftmann@41656
   460
  have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
haftmann@41656
   461
  proof -
haftmann@41656
   462
    fix B :: "'a set"
haftmann@41656
   463
    assume "B \<subseteq> A"
haftmann@41656
   464
    with `finite A` have "finite B" by (rule rev_finite_subset)
haftmann@41656
   465
    from this `B \<subseteq> A` show "P (A - B)"
haftmann@41656
   466
    proof induct
haftmann@41656
   467
      case empty
haftmann@41656
   468
      from `P A` show ?case by simp
haftmann@41656
   469
    next
haftmann@41656
   470
      case (insert b B)
haftmann@41656
   471
      have "P (A - B - {b})"
haftmann@41656
   472
      proof (rule remove)
haftmann@41656
   473
        from `finite A` show "finite (A - B)" by induct auto
haftmann@41656
   474
        from insert show "b \<in> A - B" by simp
haftmann@41656
   475
        from insert show "P (A - B)" by simp
haftmann@41656
   476
      qed
haftmann@41656
   477
      also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
haftmann@41656
   478
      finally show ?case .
haftmann@41656
   479
    qed
haftmann@41656
   480
  qed
haftmann@41656
   481
  then have "P (A - A)" by blast
haftmann@41656
   482
  then show ?thesis by simp
nipkow@31441
   483
qed
nipkow@31441
   484
nipkow@31441
   485
haftmann@26441
   486
subsection {* Class @{text finite}  *}
haftmann@26041
   487
haftmann@29797
   488
class finite =
haftmann@26041
   489
  assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
huffman@27430
   490
begin
huffman@27430
   491
huffman@27430
   492
lemma finite [simp]: "finite (A \<Colon> 'a set)"
haftmann@26441
   493
  by (rule subset_UNIV finite_UNIV finite_subset)+
haftmann@26041
   494
bulwahn@40922
   495
lemma finite_code [code]: "finite (A \<Colon> 'a set) = True"
bulwahn@40922
   496
  by simp
bulwahn@40922
   497
huffman@27430
   498
end
huffman@27430
   499
blanchet@35828
   500
lemma UNIV_unit [no_atp]:
haftmann@26041
   501
  "UNIV = {()}" by auto
haftmann@26041
   502
haftmann@35719
   503
instance unit :: finite proof
haftmann@35719
   504
qed (simp add: UNIV_unit)
haftmann@26146
   505
blanchet@35828
   506
lemma UNIV_bool [no_atp]:
haftmann@26041
   507
  "UNIV = {False, True}" by auto
haftmann@26041
   508
haftmann@35719
   509
instance bool :: finite proof
haftmann@35719
   510
qed (simp add: UNIV_bool)
haftmann@35719
   511
haftmann@37678
   512
instance prod :: (finite, finite) finite proof
haftmann@35719
   513
qed (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
haftmann@26146
   514
haftmann@35719
   515
lemma finite_option_UNIV [simp]:
haftmann@35719
   516
  "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
haftmann@35719
   517
  by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
haftmann@35719
   518
haftmann@35719
   519
instance option :: (finite) finite proof
haftmann@35719
   520
qed (simp add: UNIV_option_conv)
haftmann@26146
   521
haftmann@26041
   522
lemma inj_graph: "inj (%f. {(x, y). y = f x})"
nipkow@39302
   523
  by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
haftmann@26041
   524
haftmann@26146
   525
instance "fun" :: (finite, finite) finite
haftmann@26146
   526
proof
haftmann@26041
   527
  show "finite (UNIV :: ('a => 'b) set)"
haftmann@26041
   528
  proof (rule finite_imageD)
haftmann@26041
   529
    let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
berghofe@26792
   530
    have "range ?graph \<subseteq> Pow UNIV" by simp
berghofe@26792
   531
    moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
berghofe@26792
   532
      by (simp only: finite_Pow_iff finite)
berghofe@26792
   533
    ultimately show "finite (range ?graph)"
berghofe@26792
   534
      by (rule finite_subset)
haftmann@26041
   535
    show "inj ?graph" by (rule inj_graph)
haftmann@26041
   536
  qed
haftmann@26041
   537
qed
haftmann@26041
   538
haftmann@37678
   539
instance sum :: (finite, finite) finite proof
haftmann@35719
   540
qed (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
haftmann@27981
   541
haftmann@26041
   542
haftmann@35817
   543
subsection {* A basic fold functional for finite sets *}
nipkow@15392
   544
nipkow@15392
   545
text {* The intended behaviour is
wenzelm@31916
   546
@{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
nipkow@28853
   547
if @{text f} is ``left-commutative'':
nipkow@15392
   548
*}
nipkow@15392
   549
nipkow@28853
   550
locale fun_left_comm =
nipkow@28853
   551
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@28853
   552
  assumes fun_left_comm: "f x (f y z) = f y (f x z)"
nipkow@28853
   553
begin
nipkow@28853
   554
nipkow@28853
   555
text{* On a functional level it looks much nicer: *}
nipkow@28853
   556
nipkow@28853
   557
lemma fun_comp_comm:  "f x \<circ> f y = f y \<circ> f x"
nipkow@39302
   558
by (simp add: fun_left_comm fun_eq_iff)
nipkow@28853
   559
nipkow@28853
   560
end
nipkow@28853
   561
nipkow@28853
   562
inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
nipkow@28853
   563
for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
nipkow@28853
   564
  emptyI [intro]: "fold_graph f z {} z" |
nipkow@28853
   565
  insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
nipkow@28853
   566
      \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
nipkow@28853
   567
nipkow@28853
   568
inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
nipkow@28853
   569
nipkow@28853
   570
definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
haftmann@37767
   571
  "fold f z A = (THE y. fold_graph f z A y)"
nipkow@15392
   572
paulson@15498
   573
text{*A tempting alternative for the definiens is
nipkow@28853
   574
@{term "if finite A then THE y. fold_graph f z A y else e"}.
paulson@15498
   575
It allows the removal of finiteness assumptions from the theorems
nipkow@28853
   576
@{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
nipkow@28853
   577
The proofs become ugly. It is not worth the effort. (???) *}
nipkow@28853
   578
nipkow@28853
   579
nipkow@28853
   580
lemma Diff1_fold_graph:
nipkow@28853
   581
  "fold_graph f z (A - {x}) y \<Longrightarrow> x \<in> A \<Longrightarrow> fold_graph f z A (f x y)"
nipkow@28853
   582
by (erule insert_Diff [THEN subst], rule fold_graph.intros, auto)
nipkow@28853
   583
nipkow@28853
   584
lemma fold_graph_imp_finite: "fold_graph f z A x \<Longrightarrow> finite A"
nipkow@28853
   585
by (induct set: fold_graph) auto
nipkow@28853
   586
nipkow@28853
   587
lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
haftmann@41656
   588
by (induct rule: finite_induct) auto
nipkow@28853
   589
nipkow@28853
   590
nipkow@28853
   591
subsubsection{*From @{const fold_graph} to @{term fold}*}
nipkow@15392
   592
nipkow@28853
   593
context fun_left_comm
haftmann@26041
   594
begin
haftmann@26041
   595
huffman@36045
   596
lemma fold_graph_insertE_aux:
huffman@36045
   597
  "fold_graph f z A y \<Longrightarrow> a \<in> A \<Longrightarrow> \<exists>y'. y = f a y' \<and> fold_graph f z (A - {a}) y'"
huffman@36045
   598
proof (induct set: fold_graph)
huffman@36045
   599
  case (insertI x A y) show ?case
huffman@36045
   600
  proof (cases "x = a")
huffman@36045
   601
    assume "x = a" with insertI show ?case by auto
nipkow@28853
   602
  next
huffman@36045
   603
    assume "x \<noteq> a"
huffman@36045
   604
    then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
huffman@36045
   605
      using insertI by auto
huffman@36045
   606
    have 1: "f x y = f a (f x y')"
huffman@36045
   607
      unfolding y by (rule fun_left_comm)
huffman@36045
   608
    have 2: "fold_graph f z (insert x A - {a}) (f x y')"
huffman@36045
   609
      using y' and `x \<noteq> a` and `x \<notin> A`
huffman@36045
   610
      by (simp add: insert_Diff_if fold_graph.insertI)
huffman@36045
   611
    from 1 2 show ?case by fast
nipkow@15392
   612
  qed
huffman@36045
   613
qed simp
huffman@36045
   614
huffman@36045
   615
lemma fold_graph_insertE:
huffman@36045
   616
  assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
huffman@36045
   617
  obtains y where "v = f x y" and "fold_graph f z A y"
huffman@36045
   618
using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
nipkow@28853
   619
nipkow@28853
   620
lemma fold_graph_determ:
nipkow@28853
   621
  "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
huffman@36045
   622
proof (induct arbitrary: y set: fold_graph)
huffman@36045
   623
  case (insertI x A y v)
huffman@36045
   624
  from `fold_graph f z (insert x A) v` and `x \<notin> A`
huffman@36045
   625
  obtain y' where "v = f x y'" and "fold_graph f z A y'"
huffman@36045
   626
    by (rule fold_graph_insertE)
huffman@36045
   627
  from `fold_graph f z A y'` have "y' = y" by (rule insertI)
huffman@36045
   628
  with `v = f x y'` show "v = f x y" by simp
huffman@36045
   629
qed fast
nipkow@15392
   630
nipkow@28853
   631
lemma fold_equality:
nipkow@28853
   632
  "fold_graph f z A y \<Longrightarrow> fold f z A = y"
nipkow@28853
   633
by (unfold fold_def) (blast intro: fold_graph_determ)
nipkow@15392
   634
huffman@36045
   635
lemma fold_graph_fold: "finite A \<Longrightarrow> fold_graph f z A (fold f z A)"
huffman@36045
   636
unfolding fold_def
huffman@36045
   637
apply (rule theI')
huffman@36045
   638
apply (rule ex_ex1I)
huffman@36045
   639
apply (erule finite_imp_fold_graph)
huffman@36045
   640
apply (erule (1) fold_graph_determ)
huffman@36045
   641
done
huffman@36045
   642
nipkow@15392
   643
text{* The base case for @{text fold}: *}
nipkow@15392
   644
nipkow@28853
   645
lemma (in -) fold_empty [simp]: "fold f z {} = z"
nipkow@28853
   646
by (unfold fold_def) blast
nipkow@28853
   647
nipkow@28853
   648
text{* The various recursion equations for @{const fold}: *}
nipkow@28853
   649
haftmann@26041
   650
lemma fold_insert [simp]:
nipkow@28853
   651
  "finite A ==> x \<notin> A ==> fold f z (insert x A) = f x (fold f z A)"
huffman@36045
   652
apply (rule fold_equality)
huffman@36045
   653
apply (erule fold_graph.insertI)
huffman@36045
   654
apply (erule fold_graph_fold)
nipkow@28853
   655
done
nipkow@28853
   656
nipkow@28853
   657
lemma fold_fun_comm:
nipkow@28853
   658
  "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
nipkow@28853
   659
proof (induct rule: finite_induct)
nipkow@28853
   660
  case empty then show ?case by simp
nipkow@28853
   661
next
nipkow@28853
   662
  case (insert y A) then show ?case
nipkow@28853
   663
    by (simp add: fun_left_comm[of x])
nipkow@28853
   664
qed
nipkow@28853
   665
nipkow@28853
   666
lemma fold_insert2:
nipkow@28853
   667
  "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
huffman@35216
   668
by (simp add: fold_fun_comm)
nipkow@15392
   669
haftmann@26041
   670
lemma fold_rec:
nipkow@28853
   671
assumes "finite A" and "x \<in> A"
nipkow@28853
   672
shows "fold f z A = f x (fold f z (A - {x}))"
nipkow@28853
   673
proof -
nipkow@28853
   674
  have A: "A = insert x (A - {x})" using `x \<in> A` by blast
nipkow@28853
   675
  then have "fold f z A = fold f z (insert x (A - {x}))" by simp
nipkow@28853
   676
  also have "\<dots> = f x (fold f z (A - {x}))"
nipkow@28853
   677
    by (rule fold_insert) (simp add: `finite A`)+
nipkow@15535
   678
  finally show ?thesis .
nipkow@15535
   679
qed
nipkow@15535
   680
nipkow@28853
   681
lemma fold_insert_remove:
nipkow@28853
   682
  assumes "finite A"
nipkow@28853
   683
  shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
nipkow@28853
   684
proof -
nipkow@28853
   685
  from `finite A` have "finite (insert x A)" by auto
nipkow@28853
   686
  moreover have "x \<in> insert x A" by auto
nipkow@28853
   687
  ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
nipkow@28853
   688
    by (rule fold_rec)
nipkow@28853
   689
  then show ?thesis by simp
nipkow@28853
   690
qed
nipkow@28853
   691
haftmann@26041
   692
end
nipkow@15392
   693
nipkow@15480
   694
text{* A simplified version for idempotent functions: *}
nipkow@15480
   695
nipkow@28853
   696
locale fun_left_comm_idem = fun_left_comm +
nipkow@28853
   697
  assumes fun_left_idem: "f x (f x z) = f x z"
haftmann@26041
   698
begin
haftmann@26041
   699
nipkow@28853
   700
text{* The nice version: *}
nipkow@28853
   701
lemma fun_comp_idem : "f x o f x = f x"
nipkow@39302
   702
by (simp add: fun_left_idem fun_eq_iff)
nipkow@28853
   703
haftmann@26041
   704
lemma fold_insert_idem:
nipkow@28853
   705
  assumes fin: "finite A"
nipkow@28853
   706
  shows "fold f z (insert x A) = f x (fold f z A)"
nipkow@15480
   707
proof cases
nipkow@28853
   708
  assume "x \<in> A"
nipkow@28853
   709
  then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
nipkow@28853
   710
  then show ?thesis using assms by (simp add:fun_left_idem)
nipkow@15480
   711
next
nipkow@28853
   712
  assume "x \<notin> A" then show ?thesis using assms by simp
nipkow@15480
   713
qed
nipkow@15480
   714
nipkow@28853
   715
declare fold_insert[simp del] fold_insert_idem[simp]
nipkow@28853
   716
nipkow@28853
   717
lemma fold_insert_idem2:
nipkow@28853
   718
  "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
nipkow@28853
   719
by(simp add:fold_fun_comm)
nipkow@15484
   720
haftmann@26041
   721
end
haftmann@26041
   722
haftmann@35817
   723
haftmann@35817
   724
subsubsection {* Expressing set operations via @{const fold} *}
haftmann@35817
   725
haftmann@35817
   726
lemma (in fun_left_comm) fun_left_comm_apply:
haftmann@35817
   727
  "fun_left_comm (\<lambda>x. f (g x))"
haftmann@35817
   728
proof
haftmann@35817
   729
qed (simp_all add: fun_left_comm)
haftmann@35817
   730
haftmann@35817
   731
lemma (in fun_left_comm_idem) fun_left_comm_idem_apply:
haftmann@35817
   732
  "fun_left_comm_idem (\<lambda>x. f (g x))"
haftmann@35817
   733
  by (rule fun_left_comm_idem.intro, rule fun_left_comm_apply, unfold_locales)
haftmann@35817
   734
    (simp_all add: fun_left_idem)
haftmann@35817
   735
haftmann@35817
   736
lemma fun_left_comm_idem_insert:
haftmann@35817
   737
  "fun_left_comm_idem insert"
haftmann@35817
   738
proof
haftmann@35817
   739
qed auto
haftmann@35817
   740
haftmann@35817
   741
lemma fun_left_comm_idem_remove:
haftmann@35817
   742
  "fun_left_comm_idem (\<lambda>x A. A - {x})"
haftmann@35817
   743
proof
haftmann@35817
   744
qed auto
nipkow@31992
   745
haftmann@35817
   746
lemma (in semilattice_inf) fun_left_comm_idem_inf:
haftmann@35817
   747
  "fun_left_comm_idem inf"
haftmann@35817
   748
proof
haftmann@35817
   749
qed (auto simp add: inf_left_commute)
haftmann@35817
   750
haftmann@35817
   751
lemma (in semilattice_sup) fun_left_comm_idem_sup:
haftmann@35817
   752
  "fun_left_comm_idem sup"
haftmann@35817
   753
proof
haftmann@35817
   754
qed (auto simp add: sup_left_commute)
nipkow@31992
   755
haftmann@35817
   756
lemma union_fold_insert:
haftmann@35817
   757
  assumes "finite A"
haftmann@35817
   758
  shows "A \<union> B = fold insert B A"
haftmann@35817
   759
proof -
haftmann@35817
   760
  interpret fun_left_comm_idem insert by (fact fun_left_comm_idem_insert)
haftmann@35817
   761
  from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
haftmann@35817
   762
qed
nipkow@31992
   763
haftmann@35817
   764
lemma minus_fold_remove:
haftmann@35817
   765
  assumes "finite A"
haftmann@35817
   766
  shows "B - A = fold (\<lambda>x A. A - {x}) B A"
haftmann@35817
   767
proof -
haftmann@35817
   768
  interpret fun_left_comm_idem "\<lambda>x A. A - {x}" by (fact fun_left_comm_idem_remove)
haftmann@35817
   769
  from `finite A` show ?thesis by (induct A arbitrary: B) auto
haftmann@35817
   770
qed
haftmann@35817
   771
haftmann@35817
   772
context complete_lattice
nipkow@31992
   773
begin
nipkow@31992
   774
haftmann@35817
   775
lemma inf_Inf_fold_inf:
haftmann@35817
   776
  assumes "finite A"
haftmann@35817
   777
  shows "inf B (Inf A) = fold inf B A"
haftmann@35817
   778
proof -
haftmann@35817
   779
  interpret fun_left_comm_idem inf by (fact fun_left_comm_idem_inf)
haftmann@35817
   780
  from `finite A` show ?thesis by (induct A arbitrary: B)
wenzelm@41550
   781
    (simp_all add: Inf_insert inf_commute fold_fun_comm)
haftmann@35817
   782
qed
nipkow@31992
   783
haftmann@35817
   784
lemma sup_Sup_fold_sup:
haftmann@35817
   785
  assumes "finite A"
haftmann@35817
   786
  shows "sup B (Sup A) = fold sup B A"
haftmann@35817
   787
proof -
haftmann@35817
   788
  interpret fun_left_comm_idem sup by (fact fun_left_comm_idem_sup)
haftmann@35817
   789
  from `finite A` show ?thesis by (induct A arbitrary: B)
wenzelm@41550
   790
    (simp_all add: Sup_insert sup_commute fold_fun_comm)
nipkow@31992
   791
qed
nipkow@31992
   792
haftmann@35817
   793
lemma Inf_fold_inf:
haftmann@35817
   794
  assumes "finite A"
haftmann@35817
   795
  shows "Inf A = fold inf top A"
haftmann@35817
   796
  using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
haftmann@35817
   797
haftmann@35817
   798
lemma Sup_fold_sup:
haftmann@35817
   799
  assumes "finite A"
haftmann@35817
   800
  shows "Sup A = fold sup bot A"
haftmann@35817
   801
  using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
nipkow@31992
   802
haftmann@35817
   803
lemma inf_INFI_fold_inf:
haftmann@35817
   804
  assumes "finite A"
haftmann@35817
   805
  shows "inf B (INFI A f) = fold (\<lambda>A. inf (f A)) B A" (is "?inf = ?fold") 
haftmann@35817
   806
proof (rule sym)
haftmann@35817
   807
  interpret fun_left_comm_idem inf by (fact fun_left_comm_idem_inf)
haftmann@35817
   808
  interpret fun_left_comm_idem "\<lambda>A. inf (f A)" by (fact fun_left_comm_idem_apply)
haftmann@35817
   809
  from `finite A` show "?fold = ?inf"
haftmann@35817
   810
  by (induct A arbitrary: B)
wenzelm@41550
   811
    (simp_all add: INFI_def Inf_insert inf_left_commute)
haftmann@35817
   812
qed
nipkow@31992
   813
haftmann@35817
   814
lemma sup_SUPR_fold_sup:
haftmann@35817
   815
  assumes "finite A"
haftmann@35817
   816
  shows "sup B (SUPR A f) = fold (\<lambda>A. sup (f A)) B A" (is "?sup = ?fold") 
haftmann@35817
   817
proof (rule sym)
haftmann@35817
   818
  interpret fun_left_comm_idem sup by (fact fun_left_comm_idem_sup)
haftmann@35817
   819
  interpret fun_left_comm_idem "\<lambda>A. sup (f A)" by (fact fun_left_comm_idem_apply)
haftmann@35817
   820
  from `finite A` show "?fold = ?sup"
haftmann@35817
   821
  by (induct A arbitrary: B)
wenzelm@41550
   822
    (simp_all add: SUPR_def Sup_insert sup_left_commute)
haftmann@35817
   823
qed
nipkow@31992
   824
haftmann@35817
   825
lemma INFI_fold_inf:
haftmann@35817
   826
  assumes "finite A"
haftmann@35817
   827
  shows "INFI A f = fold (\<lambda>A. inf (f A)) top A"
haftmann@35817
   828
  using assms inf_INFI_fold_inf [of A top] by simp
nipkow@31992
   829
haftmann@35817
   830
lemma SUPR_fold_sup:
haftmann@35817
   831
  assumes "finite A"
haftmann@35817
   832
  shows "SUPR A f = fold (\<lambda>A. sup (f A)) bot A"
haftmann@35817
   833
  using assms sup_SUPR_fold_sup [of A bot] by simp
nipkow@31992
   834
nipkow@31992
   835
end
nipkow@31992
   836
nipkow@31992
   837
haftmann@35817
   838
subsection {* The derived combinator @{text fold_image} *}
nipkow@28853
   839
nipkow@28853
   840
definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
nipkow@28853
   841
where "fold_image f g = fold (%x y. f (g x) y)"
nipkow@28853
   842
nipkow@28853
   843
lemma fold_image_empty[simp]: "fold_image f g z {} = z"
nipkow@28853
   844
by(simp add:fold_image_def)
nipkow@15392
   845
haftmann@26041
   846
context ab_semigroup_mult
haftmann@26041
   847
begin
haftmann@26041
   848
nipkow@28853
   849
lemma fold_image_insert[simp]:
nipkow@28853
   850
assumes "finite A" and "a \<notin> A"
nipkow@28853
   851
shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
nipkow@28853
   852
proof -
ballarin@29223
   853
  interpret I: fun_left_comm "%x y. (g x) * y"
nipkow@28853
   854
    by unfold_locales (simp add: mult_ac)
nipkow@31992
   855
  show ?thesis using assms by(simp add:fold_image_def)
nipkow@28853
   856
qed
nipkow@28853
   857
nipkow@28853
   858
(*
haftmann@26041
   859
lemma fold_commute:
haftmann@26041
   860
  "finite A ==> (!!z. x * (fold times g z A) = fold times g (x * z) A)"
berghofe@22262
   861
  apply (induct set: finite)
wenzelm@21575
   862
   apply simp
haftmann@26041
   863
  apply (simp add: mult_left_commute [of x])
nipkow@15392
   864
  done
nipkow@15392
   865
haftmann@26041
   866
lemma fold_nest_Un_Int:
nipkow@15392
   867
  "finite A ==> finite B
haftmann@26041
   868
    ==> fold times g (fold times g z B) A = fold times g (fold times g z (A Int B)) (A Un B)"
berghofe@22262
   869
  apply (induct set: finite)
wenzelm@21575
   870
   apply simp
nipkow@15392
   871
  apply (simp add: fold_commute Int_insert_left insert_absorb)
nipkow@15392
   872
  done
nipkow@15392
   873
haftmann@26041
   874
lemma fold_nest_Un_disjoint:
nipkow@15392
   875
  "finite A ==> finite B ==> A Int B = {}
haftmann@26041
   876
    ==> fold times g z (A Un B) = fold times g (fold times g z B) A"
nipkow@15392
   877
  by (simp add: fold_nest_Un_Int)
nipkow@28853
   878
*)
nipkow@28853
   879
nipkow@28853
   880
lemma fold_image_reindex:
paulson@15487
   881
assumes fin: "finite A"
nipkow@28853
   882
shows "inj_on h A \<Longrightarrow> fold_image times g z (h`A) = fold_image times (g\<circ>h) z A"
nipkow@31992
   883
using fin by induct auto
nipkow@15392
   884
nipkow@28853
   885
(*
haftmann@26041
   886
text{*
haftmann@26041
   887
  Fusion theorem, as described in Graham Hutton's paper,
haftmann@26041
   888
  A Tutorial on the Universality and Expressiveness of Fold,
haftmann@26041
   889
  JFP 9:4 (355-372), 1999.
haftmann@26041
   890
*}
haftmann@26041
   891
haftmann@26041
   892
lemma fold_fusion:
ballarin@27611
   893
  assumes "ab_semigroup_mult g"
haftmann@26041
   894
  assumes fin: "finite A"
haftmann@26041
   895
    and hyp: "\<And>x y. h (g x y) = times x (h y)"
haftmann@26041
   896
  shows "h (fold g j w A) = fold times j (h w) A"
ballarin@27611
   897
proof -
ballarin@29223
   898
  class_interpret ab_semigroup_mult [g] by fact
ballarin@27611
   899
  show ?thesis using fin hyp by (induct set: finite) simp_all
ballarin@27611
   900
qed
nipkow@28853
   901
*)
nipkow@28853
   902
nipkow@28853
   903
lemma fold_image_cong:
nipkow@28853
   904
  "finite A \<Longrightarrow>
nipkow@28853
   905
  (!!x. x:A ==> g x = h x) ==> fold_image times g z A = fold_image times h z A"
nipkow@28853
   906
apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C")
nipkow@28853
   907
 apply simp
nipkow@28853
   908
apply (erule finite_induct, simp)
nipkow@28853
   909
apply (simp add: subset_insert_iff, clarify)
nipkow@28853
   910
apply (subgoal_tac "finite C")
nipkow@28853
   911
 prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
nipkow@28853
   912
apply (subgoal_tac "C = insert x (C - {x})")
nipkow@28853
   913
 prefer 2 apply blast
nipkow@28853
   914
apply (erule ssubst)
nipkow@28853
   915
apply (drule spec)
nipkow@28853
   916
apply (erule (1) notE impE)
nipkow@28853
   917
apply (simp add: Ball_def del: insert_Diff_single)
nipkow@28853
   918
done
nipkow@15392
   919
haftmann@26041
   920
end
haftmann@26041
   921
haftmann@26041
   922
context comm_monoid_mult
haftmann@26041
   923
begin
haftmann@26041
   924
haftmann@35817
   925
lemma fold_image_1:
haftmann@35817
   926
  "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
haftmann@41656
   927
  apply (induct rule: finite_induct)
haftmann@35817
   928
  apply simp by auto
haftmann@35817
   929
nipkow@28853
   930
lemma fold_image_Un_Int:
haftmann@26041
   931
  "finite A ==> finite B ==>
nipkow@28853
   932
    fold_image times g 1 A * fold_image times g 1 B =
nipkow@28853
   933
    fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
haftmann@41656
   934
  apply (induct rule: finite_induct)
nipkow@28853
   935
by (induct set: finite) 
nipkow@28853
   936
   (auto simp add: mult_ac insert_absorb Int_insert_left)
haftmann@26041
   937
haftmann@35817
   938
lemma fold_image_Un_one:
haftmann@35817
   939
  assumes fS: "finite S" and fT: "finite T"
haftmann@35817
   940
  and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
haftmann@35817
   941
  shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
haftmann@35817
   942
proof-
haftmann@35817
   943
  have "fold_image op * f 1 (S \<inter> T) = 1" 
haftmann@35817
   944
    apply (rule fold_image_1)
haftmann@35817
   945
    using fS fT I0 by auto 
haftmann@35817
   946
  with fold_image_Un_Int[OF fS fT] show ?thesis by simp
haftmann@35817
   947
qed
haftmann@35817
   948
haftmann@26041
   949
corollary fold_Un_disjoint:
haftmann@26041
   950
  "finite A ==> finite B ==> A Int B = {} ==>
nipkow@28853
   951
   fold_image times g 1 (A Un B) =
nipkow@28853
   952
   fold_image times g 1 A * fold_image times g 1 B"
nipkow@28853
   953
by (simp add: fold_image_Un_Int)
nipkow@28853
   954
nipkow@28853
   955
lemma fold_image_UN_disjoint:
haftmann@26041
   956
  "\<lbrakk> finite I; ALL i:I. finite (A i);
haftmann@26041
   957
     ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
nipkow@28853
   958
   \<Longrightarrow> fold_image times g 1 (UNION I A) =
nipkow@28853
   959
       fold_image times (%i. fold_image times g 1 (A i)) 1 I"
haftmann@41656
   960
apply (induct rule: finite_induct)
haftmann@41656
   961
apply simp
haftmann@41656
   962
apply atomize
nipkow@28853
   963
apply (subgoal_tac "ALL i:F. x \<noteq> i")
nipkow@28853
   964
 prefer 2 apply blast
nipkow@28853
   965
apply (subgoal_tac "A x Int UNION F A = {}")
nipkow@28853
   966
 prefer 2 apply blast
nipkow@28853
   967
apply (simp add: fold_Un_disjoint)
nipkow@28853
   968
done
nipkow@28853
   969
nipkow@28853
   970
lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@28853
   971
  fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A =
nipkow@28853
   972
  fold_image times (split g) 1 (SIGMA x:A. B x)"
nipkow@15392
   973
apply (subst Sigma_def)
nipkow@28853
   974
apply (subst fold_image_UN_disjoint, assumption, simp)
nipkow@15392
   975
 apply blast
nipkow@28853
   976
apply (erule fold_image_cong)
nipkow@28853
   977
apply (subst fold_image_UN_disjoint, simp, simp)
nipkow@15392
   978
 apply blast
paulson@15506
   979
apply simp
nipkow@15392
   980
done
nipkow@15392
   981
nipkow@28853
   982
lemma fold_image_distrib: "finite A \<Longrightarrow>
nipkow@28853
   983
   fold_image times (%x. g x * h x) 1 A =
nipkow@28853
   984
   fold_image times g 1 A *  fold_image times h 1 A"
nipkow@28853
   985
by (erule finite_induct) (simp_all add: mult_ac)
haftmann@26041
   986
chaieb@30260
   987
lemma fold_image_related: 
chaieb@30260
   988
  assumes Re: "R e e" 
chaieb@30260
   989
  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
chaieb@30260
   990
  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
chaieb@30260
   991
  shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
chaieb@30260
   992
  using fS by (rule finite_subset_induct) (insert assms, auto)
chaieb@30260
   993
chaieb@30260
   994
lemma  fold_image_eq_general:
chaieb@30260
   995
  assumes fS: "finite S"
chaieb@30260
   996
  and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y" 
chaieb@30260
   997
  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
chaieb@30260
   998
  shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
chaieb@30260
   999
proof-
chaieb@30260
  1000
  from h f12 have hS: "h ` S = S'" by auto
chaieb@30260
  1001
  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
chaieb@30260
  1002
    from f12 h H  have "x = y" by auto }
chaieb@30260
  1003
  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
chaieb@30260
  1004
  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
chaieb@30260
  1005
  from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
chaieb@30260
  1006
  also have "\<dots> = fold_image (op *) (f2 o h) e S" 
chaieb@30260
  1007
    using fold_image_reindex[OF fS hinj, of f2 e] .
chaieb@30260
  1008
  also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
chaieb@30260
  1009
    by blast
chaieb@30260
  1010
  finally show ?thesis ..
chaieb@30260
  1011
qed
chaieb@30260
  1012
chaieb@30260
  1013
lemma fold_image_eq_general_inverses:
chaieb@30260
  1014
  assumes fS: "finite S" 
chaieb@30260
  1015
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
chaieb@30260
  1016
  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
chaieb@30260
  1017
  shows "fold_image (op *) f e S = fold_image (op *) g e T"
chaieb@30260
  1018
  (* metis solves it, but not yet available here *)
chaieb@30260
  1019
  apply (rule fold_image_eq_general[OF fS, of T h g f e])
chaieb@30260
  1020
  apply (rule ballI)
chaieb@30260
  1021
  apply (frule kh)
chaieb@30260
  1022
  apply (rule ex1I[])
chaieb@30260
  1023
  apply blast
chaieb@30260
  1024
  apply clarsimp
chaieb@30260
  1025
  apply (drule hk) apply simp
chaieb@30260
  1026
  apply (rule sym)
chaieb@30260
  1027
  apply (erule conjunct1[OF conjunct2[OF hk]])
chaieb@30260
  1028
  apply (rule ballI)
chaieb@30260
  1029
  apply (drule  hk)
chaieb@30260
  1030
  apply blast
chaieb@30260
  1031
  done
chaieb@30260
  1032
haftmann@26041
  1033
end
haftmann@22917
  1034
nipkow@25162
  1035
haftmann@35817
  1036
subsection {* A fold functional for non-empty sets *}
nipkow@15392
  1037
nipkow@15392
  1038
text{* Does not require start value. *}
wenzelm@12396
  1039
berghofe@23736
  1040
inductive
berghofe@22262
  1041
  fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
berghofe@22262
  1042
  for f :: "'a => 'a => 'a"
berghofe@22262
  1043
where
paulson@15506
  1044
  fold1Set_insertI [intro]:
nipkow@28853
  1045
   "\<lbrakk> fold_graph f a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
wenzelm@12396
  1046
haftmann@35416
  1047
definition fold1 :: "('a => 'a => 'a) => 'a set => 'a" where
berghofe@22262
  1048
  "fold1 f A == THE x. fold1Set f A x"
paulson@15506
  1049
paulson@15506
  1050
lemma fold1Set_nonempty:
haftmann@22917
  1051
  "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
nipkow@28853
  1052
by(erule fold1Set.cases, simp_all)
nipkow@15392
  1053
berghofe@23736
  1054
inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
berghofe@23736
  1055
berghofe@23736
  1056
inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
berghofe@22262
  1057
berghofe@22262
  1058
berghofe@22262
  1059
lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
huffman@35216
  1060
by (blast elim: fold_graph.cases)
nipkow@15392
  1061
haftmann@22917
  1062
lemma fold1_singleton [simp]: "fold1 f {a} = a"
nipkow@28853
  1063
by (unfold fold1_def) blast
wenzelm@12396
  1064
paulson@15508
  1065
lemma finite_nonempty_imp_fold1Set:
berghofe@22262
  1066
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
paulson@15508
  1067
apply (induct A rule: finite_induct)
nipkow@28853
  1068
apply (auto dest: finite_imp_fold_graph [of _ f])
paulson@15508
  1069
done
paulson@15506
  1070
nipkow@28853
  1071
text{*First, some lemmas about @{const fold_graph}.*}
nipkow@15392
  1072
haftmann@26041
  1073
context ab_semigroup_mult
haftmann@26041
  1074
begin
haftmann@26041
  1075
nipkow@28853
  1076
lemma fun_left_comm: "fun_left_comm(op *)"
nipkow@28853
  1077
by unfold_locales (simp add: mult_ac)
nipkow@28853
  1078
nipkow@28853
  1079
lemma fold_graph_insert_swap:
nipkow@28853
  1080
assumes fold: "fold_graph times (b::'a) A y" and "b \<notin> A"
nipkow@28853
  1081
shows "fold_graph times z (insert b A) (z * y)"
nipkow@28853
  1082
proof -
ballarin@29223
  1083
  interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
nipkow@28853
  1084
from assms show ?thesis
nipkow@28853
  1085
proof (induct rule: fold_graph.induct)
huffman@36045
  1086
  case emptyI show ?case by (subst mult_commute [of z b], fast)
paulson@15508
  1087
next
berghofe@22262
  1088
  case (insertI x A y)
nipkow@28853
  1089
    have "fold_graph times z (insert x (insert b A)) (x * (z * y))"
paulson@15521
  1090
      using insertI by force  --{*how does @{term id} get unfolded?*}
haftmann@26041
  1091
    thus ?case by (simp add: insert_commute mult_ac)
paulson@15508
  1092
qed
nipkow@28853
  1093
qed
nipkow@28853
  1094
nipkow@28853
  1095
lemma fold_graph_permute_diff:
nipkow@28853
  1096
assumes fold: "fold_graph times b A x"
nipkow@28853
  1097
shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> fold_graph times a (insert b (A-{a})) x"
paulson@15508
  1098
using fold
nipkow@28853
  1099
proof (induct rule: fold_graph.induct)
paulson@15508
  1100
  case emptyI thus ?case by simp
paulson@15508
  1101
next
berghofe@22262
  1102
  case (insertI x A y)
paulson@15521
  1103
  have "a = x \<or> a \<in> A" using insertI by simp
paulson@15521
  1104
  thus ?case
paulson@15521
  1105
  proof
paulson@15521
  1106
    assume "a = x"
paulson@15521
  1107
    with insertI show ?thesis
nipkow@28853
  1108
      by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap)
paulson@15521
  1109
  next
paulson@15521
  1110
    assume ainA: "a \<in> A"
nipkow@28853
  1111
    hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)"
nipkow@28853
  1112
      using insertI by force
paulson@15521
  1113
    moreover
paulson@15521
  1114
    have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
paulson@15521
  1115
      using ainA insertI by blast
nipkow@28853
  1116
    ultimately show ?thesis by simp
paulson@15508
  1117
  qed
paulson@15508
  1118
qed
paulson@15508
  1119
haftmann@26041
  1120
lemma fold1_eq_fold:
nipkow@28853
  1121
assumes "finite A" "a \<notin> A" shows "fold1 times (insert a A) = fold times a A"
nipkow@28853
  1122
proof -
ballarin@29223
  1123
  interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
nipkow@28853
  1124
  from assms show ?thesis
nipkow@28853
  1125
apply (simp add: fold1_def fold_def)
paulson@15508
  1126
apply (rule the_equality)
nipkow@28853
  1127
apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times])
paulson@15508
  1128
apply (rule sym, clarify)
paulson@15508
  1129
apply (case_tac "Aa=A")
huffman@35216
  1130
 apply (best intro: fold_graph_determ)
nipkow@28853
  1131
apply (subgoal_tac "fold_graph times a A x")
huffman@35216
  1132
 apply (best intro: fold_graph_determ)
nipkow@28853
  1133
apply (subgoal_tac "insert aa (Aa - {a}) = A")
nipkow@28853
  1134
 prefer 2 apply (blast elim: equalityE)
nipkow@28853
  1135
apply (auto dest: fold_graph_permute_diff [where a=a])
paulson@15508
  1136
done
nipkow@28853
  1137
qed
paulson@15508
  1138
paulson@15521
  1139
lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
paulson@15521
  1140
apply safe
nipkow@28853
  1141
 apply simp
nipkow@28853
  1142
 apply (drule_tac x=x in spec)
nipkow@28853
  1143
 apply (drule_tac x="A-{x}" in spec, auto)
paulson@15508
  1144
done
paulson@15508
  1145
haftmann@26041
  1146
lemma fold1_insert:
paulson@15521
  1147
  assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
haftmann@26041
  1148
  shows "fold1 times (insert x A) = x * fold1 times A"
paulson@15521
  1149
proof -
ballarin@29223
  1150
  interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
nipkow@28853
  1151
  from nonempty obtain a A' where "A = insert a A' & a ~: A'"
paulson@15521
  1152
    by (auto simp add: nonempty_iff)
paulson@15521
  1153
  with A show ?thesis
nipkow@28853
  1154
    by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
paulson@15521
  1155
qed
paulson@15521
  1156
haftmann@26041
  1157
end
haftmann@26041
  1158
haftmann@26041
  1159
context ab_semigroup_idem_mult
haftmann@26041
  1160
begin
haftmann@26041
  1161
haftmann@35817
  1162
lemma fun_left_comm_idem: "fun_left_comm_idem(op *)"
haftmann@35817
  1163
apply unfold_locales
haftmann@35817
  1164
 apply (rule mult_left_commute)
haftmann@35817
  1165
apply (rule mult_left_idem)
haftmann@35817
  1166
done
haftmann@35817
  1167
haftmann@26041
  1168
lemma fold1_insert_idem [simp]:
paulson@15521
  1169
  assumes nonempty: "A \<noteq> {}" and A: "finite A" 
haftmann@26041
  1170
  shows "fold1 times (insert x A) = x * fold1 times A"
paulson@15521
  1171
proof -
ballarin@29223
  1172
  interpret fun_left_comm_idem "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a"
nipkow@28853
  1173
    by (rule fun_left_comm_idem)
nipkow@28853
  1174
  from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
paulson@15521
  1175
    by (auto simp add: nonempty_iff)
paulson@15521
  1176
  show ?thesis
paulson@15521
  1177
  proof cases
wenzelm@41550
  1178
    assume a: "a = x"
wenzelm@41550
  1179
    show ?thesis
paulson@15521
  1180
    proof cases
paulson@15521
  1181
      assume "A' = {}"
wenzelm@41550
  1182
      with A' a show ?thesis by simp
paulson@15521
  1183
    next
paulson@15521
  1184
      assume "A' \<noteq> {}"
wenzelm@41550
  1185
      with A A' a show ?thesis
huffman@35216
  1186
        by (simp add: fold1_insert mult_assoc [symmetric])
paulson@15521
  1187
    qed
paulson@15521
  1188
  next
paulson@15521
  1189
    assume "a \<noteq> x"
wenzelm@41550
  1190
    with A A' show ?thesis
huffman@35216
  1191
      by (simp add: insert_commute fold1_eq_fold)
paulson@15521
  1192
  qed
paulson@15521
  1193
qed
paulson@15506
  1194
haftmann@26041
  1195
lemma hom_fold1_commute:
haftmann@26041
  1196
assumes hom: "!!x y. h (x * y) = h x * h y"
haftmann@26041
  1197
and N: "finite N" "N \<noteq> {}" shows "h (fold1 times N) = fold1 times (h ` N)"
haftmann@22917
  1198
using N proof (induct rule: finite_ne_induct)
haftmann@22917
  1199
  case singleton thus ?case by simp
haftmann@22917
  1200
next
haftmann@22917
  1201
  case (insert n N)
haftmann@26041
  1202
  then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp
haftmann@26041
  1203
  also have "\<dots> = h n * h (fold1 times N)" by(rule hom)
haftmann@26041
  1204
  also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert)
haftmann@26041
  1205
  also have "times (h n) \<dots> = fold1 times (insert (h n) (h ` N))"
haftmann@22917
  1206
    using insert by(simp)
haftmann@22917
  1207
  also have "insert (h n) (h ` N) = h ` insert n N" by simp
haftmann@22917
  1208
  finally show ?case .
haftmann@22917
  1209
qed
haftmann@22917
  1210
haftmann@32679
  1211
lemma fold1_eq_fold_idem:
haftmann@32679
  1212
  assumes "finite A"
haftmann@32679
  1213
  shows "fold1 times (insert a A) = fold times a A"
haftmann@32679
  1214
proof (cases "a \<in> A")
haftmann@32679
  1215
  case False
haftmann@32679
  1216
  with assms show ?thesis by (simp add: fold1_eq_fold)
haftmann@32679
  1217
next
haftmann@32679
  1218
  interpret fun_left_comm_idem times by (fact fun_left_comm_idem)
haftmann@32679
  1219
  case True then obtain b B
haftmann@32679
  1220
    where A: "A = insert a B" and "a \<notin> B" by (rule set_insert)
haftmann@32679
  1221
  with assms have "finite B" by auto
haftmann@32679
  1222
  then have "fold times a (insert a B) = fold times (a * a) B"
haftmann@32679
  1223
    using `a \<notin> B` by (rule fold_insert2)
haftmann@32679
  1224
  then show ?thesis
haftmann@32679
  1225
    using `a \<notin> B` `finite B` by (simp add: fold1_eq_fold A)
haftmann@32679
  1226
qed
haftmann@32679
  1227
haftmann@26041
  1228
end
haftmann@26041
  1229
paulson@15506
  1230
paulson@15508
  1231
text{* Now the recursion rules for definitions: *}
paulson@15508
  1232
haftmann@22917
  1233
lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
huffman@35216
  1234
by simp
paulson@15508
  1235
haftmann@26041
  1236
lemma (in ab_semigroup_mult) fold1_insert_def:
haftmann@26041
  1237
  "\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
haftmann@26041
  1238
by (simp add:fold1_insert)
haftmann@26041
  1239
haftmann@26041
  1240
lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:
haftmann@26041
  1241
  "\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
haftmann@26041
  1242
by simp
paulson@15508
  1243
paulson@15508
  1244
subsubsection{* Determinacy for @{term fold1Set} *}
paulson@15508
  1245
nipkow@28853
  1246
(*Not actually used!!*)
nipkow@28853
  1247
(*
haftmann@26041
  1248
context ab_semigroup_mult
haftmann@26041
  1249
begin
haftmann@26041
  1250
nipkow@28853
  1251
lemma fold_graph_permute:
nipkow@28853
  1252
  "[|fold_graph times id b (insert a A) x; a \<notin> A; b \<notin> A|]
nipkow@28853
  1253
   ==> fold_graph times id a (insert b A) x"
haftmann@26041
  1254
apply (cases "a=b") 
nipkow@28853
  1255
apply (auto dest: fold_graph_permute_diff) 
paulson@15506
  1256
done
nipkow@15376
  1257
haftmann@26041
  1258
lemma fold1Set_determ:
haftmann@26041
  1259
  "fold1Set times A x ==> fold1Set times A y ==> y = x"
paulson@15506
  1260
proof (clarify elim!: fold1Set.cases)
paulson@15506
  1261
  fix A x B y a b
nipkow@28853
  1262
  assume Ax: "fold_graph times id a A x"
nipkow@28853
  1263
  assume By: "fold_graph times id b B y"
paulson@15506
  1264
  assume anotA:  "a \<notin> A"
paulson@15506
  1265
  assume bnotB:  "b \<notin> B"
paulson@15506
  1266
  assume eq: "insert a A = insert b B"
paulson@15506
  1267
  show "y=x"
paulson@15506
  1268
  proof cases
paulson@15506
  1269
    assume same: "a=b"
paulson@15506
  1270
    hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
nipkow@28853
  1271
    thus ?thesis using Ax By same by (blast intro: fold_graph_determ)
nipkow@15392
  1272
  next
paulson@15506
  1273
    assume diff: "a\<noteq>b"
paulson@15506
  1274
    let ?D = "B - {a}"
paulson@15506
  1275
    have B: "B = insert a ?D" and A: "A = insert b ?D"
paulson@15506
  1276
     and aB: "a \<in> B" and bA: "b \<in> A"
paulson@15506
  1277
      using eq anotA bnotB diff by (blast elim!:equalityE)+
paulson@15506
  1278
    with aB bnotB By
nipkow@28853
  1279
    have "fold_graph times id a (insert b ?D) y" 
nipkow@28853
  1280
      by (auto intro: fold_graph_permute simp add: insert_absorb)
paulson@15506
  1281
    moreover
nipkow@28853
  1282
    have "fold_graph times id a (insert b ?D) x"
paulson@15506
  1283
      by (simp add: A [symmetric] Ax) 
nipkow@28853
  1284
    ultimately show ?thesis by (blast intro: fold_graph_determ) 
nipkow@15392
  1285
  qed
wenzelm@12396
  1286
qed
wenzelm@12396
  1287
haftmann@26041
  1288
lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y"
paulson@15506
  1289
  by (unfold fold1_def) (blast intro: fold1Set_determ)
paulson@15506
  1290
haftmann@26041
  1291
end
nipkow@28853
  1292
*)
haftmann@26041
  1293
paulson@15506
  1294
declare
nipkow@28853
  1295
  empty_fold_graphE [rule del]  fold_graph.intros [rule del]
paulson@15506
  1296
  empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
ballarin@19931
  1297
  -- {* No more proofs involve these relations. *}
nipkow@15376
  1298
haftmann@26041
  1299
subsubsection {* Lemmas about @{text fold1} *}
haftmann@26041
  1300
haftmann@26041
  1301
context ab_semigroup_mult
haftmann@22917
  1302
begin
haftmann@22917
  1303
haftmann@26041
  1304
lemma fold1_Un:
nipkow@15484
  1305
assumes A: "finite A" "A \<noteq> {}"
nipkow@15484
  1306
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
haftmann@26041
  1307
       fold1 times (A Un B) = fold1 times A * fold1 times B"
haftmann@26041
  1308
using A by (induct rule: finite_ne_induct)
haftmann@26041
  1309
  (simp_all add: fold1_insert mult_assoc)
haftmann@26041
  1310
haftmann@26041
  1311
lemma fold1_in:
haftmann@26041
  1312
  assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x,y}"
haftmann@26041
  1313
  shows "fold1 times A \<in> A"
nipkow@15484
  1314
using A
nipkow@15484
  1315
proof (induct rule:finite_ne_induct)
paulson@15506
  1316
  case singleton thus ?case by simp
nipkow@15484
  1317
next
nipkow@15484
  1318
  case insert thus ?case using elem by (force simp add:fold1_insert)
nipkow@15484
  1319
qed
nipkow@15484
  1320
haftmann@26041
  1321
end
haftmann@26041
  1322
haftmann@26041
  1323
lemma (in ab_semigroup_idem_mult) fold1_Un2:
nipkow@15497
  1324
assumes A: "finite A" "A \<noteq> {}"
haftmann@26041
  1325
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
haftmann@26041
  1326
       fold1 times (A Un B) = fold1 times A * fold1 times B"
nipkow@15497
  1327
using A
haftmann@26041
  1328
proof(induct rule:finite_ne_induct)
nipkow@15497
  1329
  case singleton thus ?case by simp
nipkow@15484
  1330
next
haftmann@26041
  1331
  case insert thus ?case by (simp add: mult_assoc)
nipkow@18423
  1332
qed
nipkow@18423
  1333
nipkow@18423
  1334
haftmann@35817
  1335
subsection {* Locales as mini-packages for fold operations *}
haftmann@34007
  1336
haftmann@35817
  1337
subsubsection {* The natural case *}
haftmann@35719
  1338
haftmann@35719
  1339
locale folding =
haftmann@35719
  1340
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@35719
  1341
  fixes F :: "'a set \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@35817
  1342
  assumes commute_comp: "f y \<circ> f x = f x \<circ> f y"
haftmann@35722
  1343
  assumes eq_fold: "finite A \<Longrightarrow> F A s = fold f s A"
haftmann@35719
  1344
begin
haftmann@35719
  1345
haftmann@35719
  1346
lemma empty [simp]:
haftmann@35719
  1347
  "F {} = id"
nipkow@39302
  1348
  by (simp add: eq_fold fun_eq_iff)
haftmann@35719
  1349
haftmann@35719
  1350
lemma insert [simp]:
haftmann@35719
  1351
  assumes "finite A" and "x \<notin> A"
haftmann@35719
  1352
  shows "F (insert x A) = F A \<circ> f x"
haftmann@35719
  1353
proof -
haftmann@35817
  1354
  interpret fun_left_comm f proof
nipkow@39302
  1355
  qed (insert commute_comp, simp add: fun_eq_iff)
haftmann@35719
  1356
  from fold_insert2 assms
haftmann@35722
  1357
  have "\<And>s. fold f s (insert x A) = fold f (f x s) A" .
nipkow@39302
  1358
  with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
haftmann@35719
  1359
qed
haftmann@35719
  1360
haftmann@35719
  1361
lemma remove:
haftmann@35719
  1362
  assumes "finite A" and "x \<in> A"
haftmann@35719
  1363
  shows "F A = F (A - {x}) \<circ> f x"
haftmann@35719
  1364
proof -
haftmann@35719
  1365
  from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
haftmann@35719
  1366
    by (auto dest: mk_disjoint_insert)
haftmann@35719
  1367
  moreover from `finite A` this have "finite B" by simp
haftmann@35719
  1368
  ultimately show ?thesis by simp
haftmann@35719
  1369
qed
haftmann@35719
  1370
haftmann@35719
  1371
lemma insert_remove:
haftmann@35719
  1372
  assumes "finite A"
haftmann@35719
  1373
  shows "F (insert x A) = F (A - {x}) \<circ> f x"
haftmann@35722
  1374
  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
haftmann@35719
  1375
haftmann@35817
  1376
lemma commute_left_comp:
haftmann@35817
  1377
  "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
haftmann@35817
  1378
  by (simp add: o_assoc commute_comp)
haftmann@35817
  1379
haftmann@35719
  1380
lemma commute_comp':
haftmann@35719
  1381
  assumes "finite A"
haftmann@35719
  1382
  shows "f x \<circ> F A = F A \<circ> f x"
haftmann@35817
  1383
  using assms by (induct A)
haftmann@35817
  1384
    (simp, simp del: o_apply add: o_assoc, simp del: o_apply add: o_assoc [symmetric] commute_comp)
haftmann@35817
  1385
haftmann@35817
  1386
lemma commute_left_comp':
haftmann@35817
  1387
  assumes "finite A"
haftmann@35817
  1388
  shows "f x \<circ> (F A \<circ> g) = F A \<circ> (f x \<circ> g)"
haftmann@35817
  1389
  using assms by (simp add: o_assoc commute_comp')
haftmann@35817
  1390
haftmann@35817
  1391
lemma commute_comp'':
haftmann@35817
  1392
  assumes "finite A" and "finite B"
haftmann@35817
  1393
  shows "F B \<circ> F A = F A \<circ> F B"
haftmann@35817
  1394
  using assms by (induct A)
haftmann@35817
  1395
    (simp_all add: o_assoc, simp add: o_assoc [symmetric] commute_comp')
haftmann@35719
  1396
haftmann@35817
  1397
lemma commute_left_comp'':
haftmann@35817
  1398
  assumes "finite A" and "finite B"
haftmann@35817
  1399
  shows "F B \<circ> (F A \<circ> g) = F A \<circ> (F B \<circ> g)"
haftmann@35817
  1400
  using assms by (simp add: o_assoc commute_comp'')
haftmann@35817
  1401
haftmann@35817
  1402
lemmas commute_comps = o_assoc [symmetric] commute_comp commute_left_comp
haftmann@35817
  1403
  commute_comp' commute_left_comp' commute_comp'' commute_left_comp''
haftmann@35817
  1404
haftmann@35817
  1405
lemma union_inter:
haftmann@35817
  1406
  assumes "finite A" and "finite B"
haftmann@35817
  1407
  shows "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B"
haftmann@35817
  1408
  using assms by (induct A)
haftmann@35817
  1409
    (simp_all del: o_apply add: insert_absorb Int_insert_left commute_comps,
haftmann@35817
  1410
      simp add: o_assoc)
haftmann@35719
  1411
haftmann@35719
  1412
lemma union:
haftmann@35719
  1413
  assumes "finite A" and "finite B"
haftmann@35719
  1414
  and "A \<inter> B = {}"
haftmann@35719
  1415
  shows "F (A \<union> B) = F A \<circ> F B"
haftmann@35817
  1416
proof -
haftmann@35817
  1417
  from union_inter `finite A` `finite B` have "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B" .
haftmann@35817
  1418
  with `A \<inter> B = {}` show ?thesis by simp
haftmann@35719
  1419
qed
haftmann@35719
  1420
haftmann@34007
  1421
end
haftmann@35719
  1422
haftmann@35817
  1423
haftmann@35817
  1424
subsubsection {* The natural case with idempotency *}
haftmann@35817
  1425
haftmann@35719
  1426
locale folding_idem = folding +
haftmann@35719
  1427
  assumes idem_comp: "f x \<circ> f x = f x"
haftmann@35719
  1428
begin
haftmann@35719
  1429
haftmann@35817
  1430
lemma idem_left_comp:
haftmann@35817
  1431
  "f x \<circ> (f x \<circ> g) = f x \<circ> g"
haftmann@35817
  1432
  by (simp add: o_assoc idem_comp)
haftmann@35817
  1433
haftmann@35817
  1434
lemma in_comp_idem:
haftmann@35817
  1435
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1436
  shows "F A \<circ> f x = F A"
haftmann@35817
  1437
using assms by (induct A)
haftmann@35817
  1438
  (auto simp add: commute_comps idem_comp, simp add: commute_left_comp' [symmetric] commute_comp')
haftmann@35719
  1439
haftmann@35817
  1440
lemma subset_comp_idem:
haftmann@35817
  1441
  assumes "finite A" and "B \<subseteq> A"
haftmann@35817
  1442
  shows "F A \<circ> F B = F A"
haftmann@35817
  1443
proof -
haftmann@35817
  1444
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1445
  then show ?thesis using `B \<subseteq> A` by (induct B)
haftmann@35817
  1446
    (simp_all add: o_assoc in_comp_idem `finite A`)
haftmann@35817
  1447
qed
haftmann@35719
  1448
haftmann@35817
  1449
declare insert [simp del]
haftmann@35719
  1450
haftmann@35719
  1451
lemma insert_idem [simp]:
haftmann@35719
  1452
  assumes "finite A"
haftmann@35719
  1453
  shows "F (insert x A) = F A \<circ> f x"
haftmann@35817
  1454
  using assms by (cases "x \<in> A") (simp_all add: insert in_comp_idem insert_absorb)
haftmann@35719
  1455
haftmann@35719
  1456
lemma union_idem:
haftmann@35719
  1457
  assumes "finite A" and "finite B"
haftmann@35719
  1458
  shows "F (A \<union> B) = F A \<circ> F B"
haftmann@35817
  1459
proof -
haftmann@35817
  1460
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1461
  then have "F (A \<union> B) \<circ> F (A \<inter> B) = F (A \<union> B)" by (rule subset_comp_idem)
haftmann@35817
  1462
  with assms show ?thesis by (simp add: union_inter)
haftmann@35719
  1463
qed
haftmann@35719
  1464
haftmann@35719
  1465
end
haftmann@35719
  1466
haftmann@35817
  1467
haftmann@35817
  1468
subsubsection {* The image case with fixed function *}
haftmann@35817
  1469
haftmann@35796
  1470
no_notation times (infixl "*" 70)
haftmann@35796
  1471
no_notation Groups.one ("1")
haftmann@35722
  1472
haftmann@35722
  1473
locale folding_image_simple = comm_monoid +
haftmann@35722
  1474
  fixes g :: "('b \<Rightarrow> 'a)"
haftmann@35722
  1475
  fixes F :: "'b set \<Rightarrow> 'a"
haftmann@35817
  1476
  assumes eq_fold_g: "finite A \<Longrightarrow> F A = fold_image f g 1 A"
haftmann@35722
  1477
begin
haftmann@35722
  1478
haftmann@35722
  1479
lemma empty [simp]:
haftmann@35722
  1480
  "F {} = 1"
haftmann@35817
  1481
  by (simp add: eq_fold_g)
haftmann@35722
  1482
haftmann@35722
  1483
lemma insert [simp]:
haftmann@35722
  1484
  assumes "finite A" and "x \<notin> A"
haftmann@35722
  1485
  shows "F (insert x A) = g x * F A"
haftmann@35722
  1486
proof -
haftmann@35722
  1487
  interpret fun_left_comm "%x y. (g x) * y" proof
haftmann@35722
  1488
  qed (simp add: ac_simps)
haftmann@35722
  1489
  with assms have "fold_image (op *) g 1 (insert x A) = g x * fold_image (op *) g 1 A"
haftmann@35722
  1490
    by (simp add: fold_image_def)
haftmann@35817
  1491
  with `finite A` show ?thesis by (simp add: eq_fold_g)
haftmann@35722
  1492
qed
haftmann@35722
  1493
haftmann@35722
  1494
lemma remove:
haftmann@35722
  1495
  assumes "finite A" and "x \<in> A"
haftmann@35722
  1496
  shows "F A = g x * F (A - {x})"
haftmann@35722
  1497
proof -
haftmann@35722
  1498
  from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
haftmann@35722
  1499
    by (auto dest: mk_disjoint_insert)
haftmann@35722
  1500
  moreover from `finite A` this have "finite B" by simp
haftmann@35722
  1501
  ultimately show ?thesis by simp
haftmann@35722
  1502
qed
haftmann@35722
  1503
haftmann@35722
  1504
lemma insert_remove:
haftmann@35722
  1505
  assumes "finite A"
haftmann@35722
  1506
  shows "F (insert x A) = g x * F (A - {x})"
haftmann@35722
  1507
  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
haftmann@35722
  1508
haftmann@35817
  1509
lemma neutral:
haftmann@35817
  1510
  assumes "finite A" and "\<forall>x\<in>A. g x = 1"
haftmann@35817
  1511
  shows "F A = 1"
haftmann@35817
  1512
  using assms by (induct A) simp_all
haftmann@35817
  1513
haftmann@35722
  1514
lemma union_inter:
haftmann@35722
  1515
  assumes "finite A" and "finite B"
haftmann@35817
  1516
  shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
haftmann@35722
  1517
using assms proof (induct A)
haftmann@35722
  1518
  case empty then show ?case by simp
haftmann@35722
  1519
next
haftmann@35722
  1520
  case (insert x A) then show ?case
haftmann@35722
  1521
    by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
haftmann@35722
  1522
qed
haftmann@35722
  1523
haftmann@35817
  1524
corollary union_inter_neutral:
haftmann@35817
  1525
  assumes "finite A" and "finite B"
haftmann@35817
  1526
  and I0: "\<forall>x \<in> A\<inter>B. g x = 1"
haftmann@35817
  1527
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1528
  using assms by (simp add: union_inter [symmetric] neutral)
haftmann@35817
  1529
haftmann@35722
  1530
corollary union_disjoint:
haftmann@35722
  1531
  assumes "finite A" and "finite B"
haftmann@35722
  1532
  assumes "A \<inter> B = {}"
haftmann@35722
  1533
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1534
  using assms by (simp add: union_inter_neutral)
haftmann@35722
  1535
haftmann@35719
  1536
end
haftmann@35722
  1537
haftmann@35817
  1538
haftmann@35817
  1539
subsubsection {* The image case with flexible function *}
haftmann@35817
  1540
haftmann@35722
  1541
locale folding_image = comm_monoid +
haftmann@35722
  1542
  fixes F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
haftmann@35722
  1543
  assumes eq_fold: "\<And>g. finite A \<Longrightarrow> F g A = fold_image f g 1 A"
haftmann@35722
  1544
haftmann@35722
  1545
sublocale folding_image < folding_image_simple "op *" 1 g "F g" proof
haftmann@35722
  1546
qed (fact eq_fold)
haftmann@35722
  1547
haftmann@35722
  1548
context folding_image
haftmann@35722
  1549
begin
haftmann@35722
  1550
haftmann@35817
  1551
lemma reindex: (* FIXME polymorhism *)
haftmann@35722
  1552
  assumes "finite A" and "inj_on h A"
haftmann@35722
  1553
  shows "F g (h ` A) = F (g \<circ> h) A"
haftmann@35722
  1554
  using assms by (induct A) auto
haftmann@35722
  1555
haftmann@35722
  1556
lemma cong:
haftmann@35722
  1557
  assumes "finite A" and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"
haftmann@35722
  1558
  shows "F g A = F h A"
haftmann@35722
  1559
proof -
haftmann@35722
  1560
  from assms have "ALL C. C <= A --> (ALL x:C. g x = h x) --> F g C = F h C"
haftmann@35722
  1561
  apply - apply (erule finite_induct) apply simp
haftmann@35722
  1562
  apply (simp add: subset_insert_iff, clarify)
haftmann@35722
  1563
  apply (subgoal_tac "finite C")
haftmann@35722
  1564
  prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
haftmann@35722
  1565
  apply (subgoal_tac "C = insert x (C - {x})")
haftmann@35722
  1566
  prefer 2 apply blast
haftmann@35722
  1567
  apply (erule ssubst)
haftmann@35722
  1568
  apply (drule spec)
haftmann@35722
  1569
  apply (erule (1) notE impE)
haftmann@35722
  1570
  apply (simp add: Ball_def del: insert_Diff_single)
haftmann@35722
  1571
  done
haftmann@35722
  1572
  with assms show ?thesis by simp
haftmann@35722
  1573
qed
haftmann@35722
  1574
haftmann@35722
  1575
lemma UNION_disjoint:
haftmann@35722
  1576
  assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
haftmann@35722
  1577
  and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
haftmann@35722
  1578
  shows "F g (UNION I A) = F (F g \<circ> A) I"
haftmann@35722
  1579
apply (insert assms)
haftmann@41656
  1580
apply (induct rule: finite_induct)
haftmann@41656
  1581
apply simp
haftmann@41656
  1582
apply atomize
haftmann@35722
  1583
apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
haftmann@35722
  1584
 prefer 2 apply blast
haftmann@35722
  1585
apply (subgoal_tac "A x Int UNION Fa A = {}")
haftmann@35722
  1586
 prefer 2 apply blast
haftmann@35722
  1587
apply (simp add: union_disjoint)
haftmann@35722
  1588
done
haftmann@35722
  1589
haftmann@35722
  1590
lemma distrib:
haftmann@35722
  1591
  assumes "finite A"
haftmann@35722
  1592
  shows "F (\<lambda>x. g x * h x) A = F g A * F h A"
haftmann@35722
  1593
  using assms by (rule finite_induct) (simp_all add: assoc commute left_commute)
haftmann@35722
  1594
haftmann@35722
  1595
lemma related: 
haftmann@35722
  1596
  assumes Re: "R 1 1" 
haftmann@35722
  1597
  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
haftmann@35722
  1598
  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
haftmann@35722
  1599
  shows "R (F h S) (F g S)"
haftmann@35722
  1600
  using fS by (rule finite_subset_induct) (insert assms, auto)
haftmann@35722
  1601
haftmann@35722
  1602
lemma eq_general:
haftmann@35722
  1603
  assumes fS: "finite S"
haftmann@35722
  1604
  and h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y" 
haftmann@35722
  1605
  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
haftmann@35722
  1606
  shows "F f1 S = F f2 S'"
haftmann@35722
  1607
proof-
haftmann@35722
  1608
  from h f12 have hS: "h ` S = S'" by blast
haftmann@35722
  1609
  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
haftmann@35722
  1610
    from f12 h H  have "x = y" by auto }
haftmann@35722
  1611
  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
haftmann@35722
  1612
  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
haftmann@35722
  1613
  from hS have "F f2 S' = F f2 (h ` S)" by simp
haftmann@35722
  1614
  also have "\<dots> = F (f2 o h) S" using reindex [OF fS hinj, of f2] .
haftmann@35722
  1615
  also have "\<dots> = F f1 S " using th cong [OF fS, of "f2 o h" f1]
haftmann@35722
  1616
    by blast
haftmann@35722
  1617
  finally show ?thesis ..
haftmann@35722
  1618
qed
haftmann@35722
  1619
haftmann@35722
  1620
lemma eq_general_inverses:
haftmann@35722
  1621
  assumes fS: "finite S" 
haftmann@35722
  1622
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
haftmann@35722
  1623
  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"
haftmann@35722
  1624
  shows "F j S = F g T"
haftmann@35722
  1625
  (* metis solves it, but not yet available here *)
haftmann@35722
  1626
  apply (rule eq_general [OF fS, of T h g j])
haftmann@35722
  1627
  apply (rule ballI)
haftmann@35722
  1628
  apply (frule kh)
haftmann@35722
  1629
  apply (rule ex1I[])
haftmann@35722
  1630
  apply blast
haftmann@35722
  1631
  apply clarsimp
haftmann@35722
  1632
  apply (drule hk) apply simp
haftmann@35722
  1633
  apply (rule sym)
haftmann@35722
  1634
  apply (erule conjunct1[OF conjunct2[OF hk]])
haftmann@35722
  1635
  apply (rule ballI)
haftmann@35722
  1636
  apply (drule hk)
haftmann@35722
  1637
  apply blast
haftmann@35722
  1638
  done
haftmann@35722
  1639
haftmann@35722
  1640
end
haftmann@35722
  1641
haftmann@35817
  1642
haftmann@35817
  1643
subsubsection {* The image case with fixed function and idempotency *}
haftmann@35817
  1644
haftmann@35817
  1645
locale folding_image_simple_idem = folding_image_simple +
haftmann@35817
  1646
  assumes idem: "x * x = x"
haftmann@35817
  1647
haftmann@35817
  1648
sublocale folding_image_simple_idem < semilattice proof
haftmann@35817
  1649
qed (fact idem)
haftmann@35817
  1650
haftmann@35817
  1651
context folding_image_simple_idem
haftmann@35817
  1652
begin
haftmann@35817
  1653
haftmann@35817
  1654
lemma in_idem:
haftmann@35817
  1655
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1656
  shows "g x * F A = F A"
haftmann@35817
  1657
  using assms by (induct A) (auto simp add: left_commute)
haftmann@35817
  1658
haftmann@35817
  1659
lemma subset_idem:
haftmann@35817
  1660
  assumes "finite A" and "B \<subseteq> A"
haftmann@35817
  1661
  shows "F B * F A = F A"
haftmann@35817
  1662
proof -
haftmann@35817
  1663
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1664
  then show ?thesis using `B \<subseteq> A` by (induct B)
haftmann@35817
  1665
    (auto simp add: assoc in_idem `finite A`)
haftmann@35817
  1666
qed
haftmann@35817
  1667
haftmann@35817
  1668
declare insert [simp del]
haftmann@35817
  1669
haftmann@35817
  1670
lemma insert_idem [simp]:
haftmann@35817
  1671
  assumes "finite A"
haftmann@35817
  1672
  shows "F (insert x A) = g x * F A"
haftmann@35817
  1673
  using assms by (cases "x \<in> A") (simp_all add: insert in_idem insert_absorb)
haftmann@35817
  1674
haftmann@35817
  1675
lemma union_idem:
haftmann@35817
  1676
  assumes "finite A" and "finite B"
haftmann@35817
  1677
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1678
proof -
haftmann@35817
  1679
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1680
  then have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (rule subset_idem)
haftmann@35817
  1681
  with assms show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
haftmann@35817
  1682
qed
haftmann@35817
  1683
haftmann@35817
  1684
end
haftmann@35817
  1685
haftmann@35817
  1686
haftmann@35817
  1687
subsubsection {* The image case with flexible function and idempotency *}
haftmann@35817
  1688
haftmann@35817
  1689
locale folding_image_idem = folding_image +
haftmann@35817
  1690
  assumes idem: "x * x = x"
haftmann@35817
  1691
haftmann@35817
  1692
sublocale folding_image_idem < folding_image_simple_idem "op *" 1 g "F g" proof
haftmann@35817
  1693
qed (fact idem)
haftmann@35817
  1694
haftmann@35817
  1695
haftmann@35817
  1696
subsubsection {* The neutral-less case *}
haftmann@35817
  1697
haftmann@35817
  1698
locale folding_one = abel_semigroup +
haftmann@35817
  1699
  fixes F :: "'a set \<Rightarrow> 'a"
haftmann@35817
  1700
  assumes eq_fold: "finite A \<Longrightarrow> F A = fold1 f A"
haftmann@35817
  1701
begin
haftmann@35817
  1702
haftmann@35817
  1703
lemma singleton [simp]:
haftmann@35817
  1704
  "F {x} = x"
haftmann@35817
  1705
  by (simp add: eq_fold)
haftmann@35817
  1706
haftmann@35817
  1707
lemma eq_fold':
haftmann@35817
  1708
  assumes "finite A" and "x \<notin> A"
haftmann@35817
  1709
  shows "F (insert x A) = fold (op *) x A"
haftmann@35817
  1710
proof -
haftmann@35817
  1711
  interpret ab_semigroup_mult "op *" proof qed (simp_all add: ac_simps)
haftmann@35817
  1712
  with assms show ?thesis by (simp add: eq_fold fold1_eq_fold)
haftmann@35817
  1713
qed
haftmann@35817
  1714
haftmann@35817
  1715
lemma insert [simp]:
haftmann@36637
  1716
  assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
haftmann@36637
  1717
  shows "F (insert x A) = x * F A"
haftmann@36637
  1718
proof -
haftmann@36637
  1719
  from `A \<noteq> {}` obtain b where "b \<in> A" by blast
haftmann@35817
  1720
  then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@35817
  1721
  with `finite A` have "finite B" by simp
haftmann@35817
  1722
  interpret fold: folding "op *" "\<lambda>a b. fold (op *) b a" proof
nipkow@39302
  1723
  qed (simp_all add: fun_eq_iff ac_simps)
nipkow@39302
  1724
  thm fold.commute_comp' [of B b, simplified fun_eq_iff, simplified]
haftmann@35817
  1725
  from `finite B` fold.commute_comp' [of B x]
haftmann@35817
  1726
    have "op * x \<circ> (\<lambda>b. fold op * b B) = (\<lambda>b. fold op * b B) \<circ> op * x" by simp
nipkow@39302
  1727
  then have A: "x * fold op * b B = fold op * (b * x) B" by (simp add: fun_eq_iff commute)
haftmann@35817
  1728
  from `finite B` * fold.insert [of B b]
haftmann@35817
  1729
    have "(\<lambda>x. fold op * x (insert b B)) = (\<lambda>x. fold op * x B) \<circ> op * b" by simp
nipkow@39302
  1730
  then have B: "fold op * x (insert b B) = fold op * (b * x) B" by (simp add: fun_eq_iff)
haftmann@35817
  1731
  from A B assms * show ?thesis by (simp add: eq_fold' del: fold.insert)
haftmann@35817
  1732
qed
haftmann@35817
  1733
haftmann@35817
  1734
lemma remove:
haftmann@35817
  1735
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1736
  shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
haftmann@35817
  1737
proof -
haftmann@35817
  1738
  from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@35817
  1739
  with assms show ?thesis by simp
haftmann@35817
  1740
qed
haftmann@35817
  1741
haftmann@35817
  1742
lemma insert_remove:
haftmann@35817
  1743
  assumes "finite A"
haftmann@35817
  1744
  shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
haftmann@35817
  1745
  using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
haftmann@35817
  1746
haftmann@35817
  1747
lemma union_disjoint:
haftmann@35817
  1748
  assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}" and "A \<inter> B = {}"
haftmann@35817
  1749
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1750
  using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
haftmann@35817
  1751
haftmann@35817
  1752
lemma union_inter:
haftmann@35817
  1753
  assumes "finite A" and "finite B" and "A \<inter> B \<noteq> {}"
haftmann@35817
  1754
  shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
haftmann@35817
  1755
proof -
haftmann@35817
  1756
  from assms have "A \<noteq> {}" and "B \<noteq> {}" by auto
haftmann@35817
  1757
  from `finite A` `A \<noteq> {}` `A \<inter> B \<noteq> {}` show ?thesis proof (induct A rule: finite_ne_induct)
haftmann@35817
  1758
    case (singleton x) then show ?case by (simp add: insert_absorb ac_simps)
haftmann@35817
  1759
  next
haftmann@35817
  1760
    case (insert x A) show ?case proof (cases "x \<in> B")
haftmann@35817
  1761
      case True then have "B \<noteq> {}" by auto
haftmann@35817
  1762
      with insert True `finite B` show ?thesis by (cases "A \<inter> B = {}")
haftmann@35817
  1763
        (simp_all add: insert_absorb ac_simps union_disjoint)
haftmann@35817
  1764
    next
haftmann@35817
  1765
      case False with insert have "F (A \<union> B) * F (A \<inter> B) = F A * F B" by simp
haftmann@35817
  1766
      moreover from False `finite B` insert have "finite (A \<union> B)" "x \<notin> A \<union> B" "A \<union> B \<noteq> {}"
haftmann@35817
  1767
        by auto
haftmann@35817
  1768
      ultimately show ?thesis using False `finite A` `x \<notin> A` `A \<noteq> {}` by (simp add: assoc)
haftmann@35817
  1769
    qed
haftmann@35817
  1770
  qed
haftmann@35817
  1771
qed
haftmann@35817
  1772
haftmann@35817
  1773
lemma closed:
haftmann@35817
  1774
  assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
haftmann@35817
  1775
  shows "F A \<in> A"
haftmann@35817
  1776
using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
haftmann@35817
  1777
  case singleton then show ?case by simp
haftmann@35817
  1778
next
haftmann@35817
  1779
  case insert with elem show ?case by force
haftmann@35817
  1780
qed
haftmann@35817
  1781
haftmann@35817
  1782
end
haftmann@35817
  1783
haftmann@35817
  1784
haftmann@35817
  1785
subsubsection {* The neutral-less case with idempotency *}
haftmann@35817
  1786
haftmann@35817
  1787
locale folding_one_idem = folding_one +
haftmann@35817
  1788
  assumes idem: "x * x = x"
haftmann@35817
  1789
haftmann@35817
  1790
sublocale folding_one_idem < semilattice proof
haftmann@35817
  1791
qed (fact idem)
haftmann@35817
  1792
haftmann@35817
  1793
context folding_one_idem
haftmann@35817
  1794
begin
haftmann@35817
  1795
haftmann@35817
  1796
lemma in_idem:
haftmann@35817
  1797
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1798
  shows "x * F A = F A"
haftmann@35817
  1799
proof -
haftmann@35817
  1800
  from assms have "A \<noteq> {}" by auto
haftmann@35817
  1801
  with `finite A` show ?thesis using `x \<in> A` by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
haftmann@35817
  1802
qed
haftmann@35817
  1803
haftmann@35817
  1804
lemma subset_idem:
haftmann@35817
  1805
  assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
haftmann@35817
  1806
  shows "F B * F A = F A"
haftmann@35817
  1807
proof -
haftmann@35817
  1808
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1809
  then show ?thesis using `B \<noteq> {}` `B \<subseteq> A` by (induct B rule: finite_ne_induct)
haftmann@35817
  1810
    (simp_all add: assoc in_idem `finite A`)
haftmann@35817
  1811
qed
haftmann@35817
  1812
haftmann@35817
  1813
lemma eq_fold_idem':
haftmann@35817
  1814
  assumes "finite A"
haftmann@35817
  1815
  shows "F (insert a A) = fold (op *) a A"
haftmann@35817
  1816
proof -
haftmann@35817
  1817
  interpret ab_semigroup_idem_mult "op *" proof qed (simp_all add: ac_simps)
haftmann@35817
  1818
  with assms show ?thesis by (simp add: eq_fold fold1_eq_fold_idem)
haftmann@35817
  1819
qed
haftmann@35817
  1820
haftmann@35817
  1821
lemma insert_idem [simp]:
haftmann@36637
  1822
  assumes "finite A" and "A \<noteq> {}"
haftmann@36637
  1823
  shows "F (insert x A) = x * F A"
haftmann@35817
  1824
proof (cases "x \<in> A")
haftmann@36637
  1825
  case False from `finite A` `x \<notin> A` `A \<noteq> {}` show ?thesis by (rule insert)
haftmann@35817
  1826
next
haftmann@36637
  1827
  case True
haftmann@36637
  1828
  from `finite A` `A \<noteq> {}` show ?thesis by (simp add: in_idem insert_absorb True)
haftmann@35817
  1829
qed
haftmann@35817
  1830
  
haftmann@35817
  1831
lemma union_idem:
haftmann@35817
  1832
  assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
haftmann@35817
  1833
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1834
proof (cases "A \<inter> B = {}")
haftmann@35817
  1835
  case True with assms show ?thesis by (simp add: union_disjoint)
haftmann@35817
  1836
next
haftmann@35817
  1837
  case False
haftmann@35817
  1838
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1839
  with False have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (auto intro: subset_idem)
haftmann@35817
  1840
  with assms False show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
haftmann@35817
  1841
qed
haftmann@35817
  1842
haftmann@35817
  1843
lemma hom_commute:
haftmann@35817
  1844
  assumes hom: "\<And>x y. h (x * y) = h x * h y"
haftmann@35817
  1845
  and N: "finite N" "N \<noteq> {}" shows "h (F N) = F (h ` N)"
haftmann@35817
  1846
using N proof (induct rule: finite_ne_induct)
haftmann@35817
  1847
  case singleton thus ?case by simp
haftmann@35817
  1848
next
haftmann@35817
  1849
  case (insert n N)
haftmann@35817
  1850
  then have "h (F (insert n N)) = h (n * F N)" by simp
haftmann@35817
  1851
  also have "\<dots> = h n * h (F N)" by (rule hom)
haftmann@35817
  1852
  also have "h (F N) = F (h ` N)" by(rule insert)
haftmann@35817
  1853
  also have "h n * \<dots> = F (insert (h n) (h ` N))"
haftmann@35817
  1854
    using insert by(simp)
haftmann@35817
  1855
  also have "insert (h n) (h ` N) = h ` insert n N" by simp
haftmann@35817
  1856
  finally show ?case .
haftmann@35817
  1857
qed
haftmann@35817
  1858
haftmann@35817
  1859
end
haftmann@35817
  1860
haftmann@35796
  1861
notation times (infixl "*" 70)
haftmann@35796
  1862
notation Groups.one ("1")
haftmann@35722
  1863
haftmann@35722
  1864
haftmann@35722
  1865
subsection {* Finite cardinality *}
haftmann@35722
  1866
haftmann@35722
  1867
text {* This definition, although traditional, is ugly to work with:
haftmann@35722
  1868
@{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
haftmann@35722
  1869
But now that we have @{text fold_image} things are easy:
haftmann@35722
  1870
*}
haftmann@35722
  1871
haftmann@35722
  1872
definition card :: "'a set \<Rightarrow> nat" where
haftmann@35722
  1873
  "card A = (if finite A then fold_image (op +) (\<lambda>x. 1) 0 A else 0)"
haftmann@35722
  1874
haftmann@37770
  1875
interpretation card: folding_image_simple "op +" 0 "\<lambda>x. 1" card proof
haftmann@35722
  1876
qed (simp add: card_def)
haftmann@35722
  1877
haftmann@35722
  1878
lemma card_infinite [simp]:
haftmann@35722
  1879
  "\<not> finite A \<Longrightarrow> card A = 0"
haftmann@35722
  1880
  by (simp add: card_def)
haftmann@35722
  1881
haftmann@35722
  1882
lemma card_empty:
haftmann@35722
  1883
  "card {} = 0"
haftmann@35722
  1884
  by (fact card.empty)
haftmann@35722
  1885
haftmann@35722
  1886
lemma card_insert_disjoint:
haftmann@35722
  1887
  "finite A ==> x \<notin> A ==> card (insert x A) = Suc (card A)"
haftmann@35722
  1888
  by simp
haftmann@35722
  1889
haftmann@35722
  1890
lemma card_insert_if:
haftmann@35722
  1891
  "finite A ==> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
haftmann@35722
  1892
  by auto (simp add: card.insert_remove card.remove)
haftmann@35722
  1893
haftmann@35722
  1894
lemma card_ge_0_finite:
haftmann@35722
  1895
  "card A > 0 \<Longrightarrow> finite A"
haftmann@35722
  1896
  by (rule ccontr) simp
haftmann@35722
  1897
blanchet@35828
  1898
lemma card_0_eq [simp, no_atp]:
haftmann@35722
  1899
  "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
haftmann@35722
  1900
  by (auto dest: mk_disjoint_insert)
haftmann@35722
  1901
haftmann@35722
  1902
lemma finite_UNIV_card_ge_0:
haftmann@35722
  1903
  "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
haftmann@35722
  1904
  by (rule ccontr) simp
haftmann@35722
  1905
haftmann@35722
  1906
lemma card_eq_0_iff:
haftmann@35722
  1907
  "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
haftmann@35722
  1908
  by auto
haftmann@35722
  1909
haftmann@35722
  1910
lemma card_gt_0_iff:
haftmann@35722
  1911
  "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
haftmann@35722
  1912
  by (simp add: neq0_conv [symmetric] card_eq_0_iff) 
haftmann@35722
  1913
haftmann@35722
  1914
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
haftmann@35722
  1915
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
haftmann@35722
  1916
apply(simp del:insert_Diff_single)
haftmann@35722
  1917
done
haftmann@35722
  1918
haftmann@35722
  1919
lemma card_Diff_singleton:
haftmann@35722
  1920
  "finite A ==> x: A ==> card (A - {x}) = card A - 1"
haftmann@35722
  1921
by (simp add: card_Suc_Diff1 [symmetric])
haftmann@35722
  1922
haftmann@35722
  1923
lemma card_Diff_singleton_if:
haftmann@35722
  1924
  "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
haftmann@35722
  1925
by (simp add: card_Diff_singleton)
haftmann@35722
  1926
haftmann@35722
  1927
lemma card_Diff_insert[simp]:
haftmann@35722
  1928
assumes "finite A" and "a:A" and "a ~: B"
haftmann@35722
  1929
shows "card(A - insert a B) = card(A - B) - 1"
haftmann@35722
  1930
proof -
haftmann@35722
  1931
  have "A - insert a B = (A - B) - {a}" using assms by blast
haftmann@35722
  1932
  then show ?thesis using assms by(simp add:card_Diff_singleton)
haftmann@35722
  1933
qed
haftmann@35722
  1934
haftmann@35722
  1935
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
haftmann@35722
  1936
by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert)
haftmann@35722
  1937
haftmann@35722
  1938
lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
haftmann@35722
  1939
by (simp add: card_insert_if)
haftmann@35722
  1940
haftmann@35722
  1941
lemma card_mono:
haftmann@35722
  1942
  assumes "finite B" and "A \<subseteq> B"
haftmann@35722
  1943
  shows "card A \<le> card B"
haftmann@35722
  1944
proof -
haftmann@35722
  1945
  from assms have "finite A" by (auto intro: finite_subset)
haftmann@35722
  1946
  then show ?thesis using assms proof (induct A arbitrary: B)
haftmann@35722
  1947
    case empty then show ?case by simp
haftmann@35722
  1948
  next
haftmann@35722
  1949
    case (insert x A)
haftmann@35722
  1950
    then have "x \<in> B" by simp
haftmann@35722
  1951
    from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
haftmann@35722
  1952
    with insert.hyps have "card A \<le> card (B - {x})" by auto
haftmann@35722
  1953
    with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
haftmann@35722
  1954
  qed
haftmann@35722
  1955
qed
haftmann@35722
  1956
haftmann@35722
  1957
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
haftmann@41656
  1958
apply (induct rule: finite_induct)
haftmann@41656
  1959
apply simp
haftmann@41656
  1960
apply clarify
haftmann@35722
  1961
apply (subgoal_tac "finite A & A - {x} <= F")
haftmann@35722
  1962
 prefer 2 apply (blast intro: finite_subset, atomize)
haftmann@35722
  1963
apply (drule_tac x = "A - {x}" in spec)
haftmann@35722
  1964
apply (simp add: card_Diff_singleton_if split add: split_if_asm)
haftmann@35722
  1965
apply (case_tac "card A", auto)
haftmann@35722
  1966
done
haftmann@35722
  1967
haftmann@35722
  1968
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
haftmann@35722
  1969
apply (simp add: psubset_eq linorder_not_le [symmetric])
haftmann@35722
  1970
apply (blast dest: card_seteq)
haftmann@35722
  1971
done
haftmann@35722
  1972
haftmann@35722
  1973
lemma card_Un_Int: "finite A ==> finite B
haftmann@35722
  1974
    ==> card A + card B = card (A Un B) + card (A Int B)"
haftmann@35817
  1975
  by (fact card.union_inter [symmetric])
haftmann@35722
  1976
haftmann@35722
  1977
lemma card_Un_disjoint: "finite A ==> finite B
haftmann@35722
  1978
    ==> A Int B = {} ==> card (A Un B) = card A + card B"
haftmann@35722
  1979
  by (fact card.union_disjoint)
haftmann@35722
  1980
haftmann@35722
  1981
lemma card_Diff_subset:
haftmann@35722
  1982
  assumes "finite B" and "B \<subseteq> A"
haftmann@35722
  1983
  shows "card (A - B) = card A - card B"
haftmann@35722
  1984
proof (cases "finite A")
haftmann@35722
  1985
  case False with assms show ?thesis by simp
haftmann@35722
  1986
next
haftmann@35722
  1987
  case True with assms show ?thesis by (induct B arbitrary: A) simp_all
haftmann@35722
  1988
qed
haftmann@35722
  1989
haftmann@35722
  1990
lemma card_Diff_subset_Int:
haftmann@35722
  1991
  assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
haftmann@35722
  1992
proof -
haftmann@35722
  1993
  have "A - B = A - A \<inter> B" by auto
haftmann@35722
  1994
  thus ?thesis
haftmann@35722
  1995
    by (simp add: card_Diff_subset AB) 
haftmann@35722
  1996
qed
haftmann@35722
  1997
nipkow@40716
  1998
lemma diff_card_le_card_Diff:
nipkow@40716
  1999
assumes "finite B" shows "card A - card B \<le> card(A - B)"
nipkow@40716
  2000
proof-
nipkow@40716
  2001
  have "card A - card B \<le> card A - card (A \<inter> B)"
nipkow@40716
  2002
    using card_mono[OF assms Int_lower2, of A] by arith
nipkow@40716
  2003
  also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
nipkow@40716
  2004
  finally show ?thesis .
nipkow@40716
  2005
qed
nipkow@40716
  2006
haftmann@35722
  2007
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
haftmann@35722
  2008
apply (rule Suc_less_SucD)
haftmann@35722
  2009
apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
haftmann@35722
  2010
done
haftmann@35722
  2011
haftmann@35722
  2012
lemma card_Diff2_less:
haftmann@35722
  2013
  "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
haftmann@35722
  2014
apply (case_tac "x = y")
haftmann@35722
  2015
 apply (simp add: card_Diff1_less del:card_Diff_insert)
haftmann@35722
  2016
apply (rule less_trans)
haftmann@35722
  2017
 prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
haftmann@35722
  2018
done
haftmann@35722
  2019
haftmann@35722
  2020
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
haftmann@35722
  2021
apply (case_tac "x : A")
haftmann@35722
  2022
 apply (simp_all add: card_Diff1_less less_imp_le)
haftmann@35722
  2023
done
haftmann@35722
  2024
haftmann@35722
  2025
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
haftmann@35722
  2026
by (erule psubsetI, blast)
haftmann@35722
  2027
haftmann@35722
  2028
lemma insert_partition:
haftmann@35722
  2029
  "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
haftmann@35722
  2030
  \<Longrightarrow> x \<inter> \<Union> F = {}"
haftmann@35722
  2031
by auto
haftmann@35722
  2032
haftmann@35722
  2033
lemma finite_psubset_induct[consumes 1, case_names psubset]:
urbanc@36079
  2034
  assumes fin: "finite A" 
urbanc@36079
  2035
  and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A" 
urbanc@36079
  2036
  shows "P A"
urbanc@36079
  2037
using fin
urbanc@36079
  2038
proof (induct A taking: card rule: measure_induct_rule)
haftmann@35722
  2039
  case (less A)
urbanc@36079
  2040
  have fin: "finite A" by fact
urbanc@36079
  2041
  have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
urbanc@36079
  2042
  { fix B 
urbanc@36079
  2043
    assume asm: "B \<subset> A"
urbanc@36079
  2044
    from asm have "card B < card A" using psubset_card_mono fin by blast
urbanc@36079
  2045
    moreover
urbanc@36079
  2046
    from asm have "B \<subseteq> A" by auto
urbanc@36079
  2047
    then have "finite B" using fin finite_subset by blast
urbanc@36079
  2048
    ultimately 
urbanc@36079
  2049
    have "P B" using ih by simp
urbanc@36079
  2050
  }
urbanc@36079
  2051
  with fin show "P A" using major by blast
haftmann@35722
  2052
qed
haftmann@35722
  2053
haftmann@35722
  2054
text{* main cardinality theorem *}
haftmann@35722
  2055
lemma card_partition [rule_format]:
haftmann@35722
  2056
  "finite C ==>
haftmann@35722
  2057
     finite (\<Union> C) -->
haftmann@35722
  2058
     (\<forall>c\<in>C. card c = k) -->
haftmann@35722
  2059
     (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
haftmann@35722
  2060
     k * card(C) = card (\<Union> C)"
haftmann@35722
  2061
apply (erule finite_induct, simp)
haftmann@35722
  2062
apply (simp add: card_Un_disjoint insert_partition 
haftmann@35722
  2063
       finite_subset [of _ "\<Union> (insert x F)"])
haftmann@35722
  2064
done
haftmann@35722
  2065
haftmann@35722
  2066
lemma card_eq_UNIV_imp_eq_UNIV:
haftmann@35722
  2067
  assumes fin: "finite (UNIV :: 'a set)"
haftmann@35722
  2068
  and card: "card A = card (UNIV :: 'a set)"
haftmann@35722
  2069
  shows "A = (UNIV :: 'a set)"
haftmann@35722
  2070
proof
haftmann@35722
  2071
  show "A \<subseteq> UNIV" by simp
haftmann@35722
  2072
  show "UNIV \<subseteq> A"
haftmann@35722
  2073
  proof
haftmann@35722
  2074
    fix x
haftmann@35722
  2075
    show "x \<in> A"
haftmann@35722
  2076
    proof (rule ccontr)
haftmann@35722
  2077
      assume "x \<notin> A"
haftmann@35722
  2078
      then have "A \<subset> UNIV" by auto
haftmann@35722
  2079
      with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
haftmann@35722
  2080
      with card show False by simp
haftmann@35722
  2081
    qed
haftmann@35722
  2082
  qed
haftmann@35722
  2083
qed
haftmann@35722
  2084
haftmann@35722
  2085
text{*The form of a finite set of given cardinality*}
haftmann@35722
  2086
haftmann@35722
  2087
lemma card_eq_SucD:
haftmann@35722
  2088
assumes "card A = Suc k"
haftmann@35722
  2089
shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
haftmann@35722
  2090
proof -
haftmann@35722
  2091
  have fin: "finite A" using assms by (auto intro: ccontr)
haftmann@35722
  2092
  moreover have "card A \<noteq> 0" using assms by auto
haftmann@35722
  2093
  ultimately obtain b where b: "b \<in> A" by auto
haftmann@35722
  2094
  show ?thesis
haftmann@35722
  2095
  proof (intro exI conjI)
haftmann@35722
  2096
    show "A = insert b (A-{b})" using b by blast
haftmann@35722
  2097
    show "b \<notin> A - {b}" by blast
haftmann@35722
  2098
    show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
haftmann@35722
  2099
      using assms b fin by(fastsimp dest:mk_disjoint_insert)+
haftmann@35722
  2100
  qed
haftmann@35722
  2101
qed
haftmann@35722
  2102
haftmann@35722
  2103
lemma card_Suc_eq:
haftmann@35722
  2104
  "(card A = Suc k) =
haftmann@35722
  2105
   (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
haftmann@35722
  2106
apply(rule iffI)
haftmann@35722
  2107
 apply(erule card_eq_SucD)
haftmann@35722
  2108
apply(auto)
haftmann@35722
  2109
apply(subst card_insert)
haftmann@35722
  2110
 apply(auto intro:ccontr)
haftmann@35722
  2111
done
haftmann@35722
  2112
haftmann@35722
  2113
lemma finite_fun_UNIVD2:
haftmann@35722
  2114
  assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
haftmann@35722
  2115
  shows "finite (UNIV :: 'b set)"
haftmann@35722
  2116
proof -
haftmann@35722
  2117
  from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
haftmann@35722
  2118
    by(rule finite_imageI)
haftmann@35722
  2119
  moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
haftmann@35722
  2120
    by(rule UNIV_eq_I) auto
haftmann@35722
  2121
  ultimately show "finite (UNIV :: 'b set)" by simp
haftmann@35722
  2122
qed
haftmann@35722
  2123
haftmann@35722
  2124
lemma card_UNIV_unit: "card (UNIV :: unit set) = 1"
haftmann@35722
  2125
  unfolding UNIV_unit by simp
haftmann@35722
  2126
haftmann@35722
  2127
haftmann@35722
  2128
subsubsection {* Cardinality of image *}
haftmann@35722
  2129
haftmann@35722
  2130
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
haftmann@41656
  2131
apply (induct rule: finite_induct)
haftmann@35722
  2132
 apply simp
haftmann@35722
  2133
apply (simp add: le_SucI card_insert_if)
haftmann@35722
  2134
done
haftmann@35722
  2135
haftmann@35722
  2136
lemma card_image:
haftmann@35722
  2137
  assumes "inj_on f A"
haftmann@35722
  2138
  shows "card (f ` A) = card A"
haftmann@35722
  2139
proof (cases "finite A")
haftmann@35722
  2140
  case True then show ?thesis using assms by (induct A) simp_all
haftmann@35722
  2141
next
haftmann@35722
  2142
  case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
haftmann@35722
  2143
  with False show ?thesis by simp
haftmann@35722
  2144
qed
haftmann@35722
  2145
haftmann@35722
  2146
lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
haftmann@35722
  2147
by(auto simp: card_image bij_betw_def)
haftmann@35722
  2148
haftmann@35722
  2149
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
haftmann@35722
  2150
by (simp add: card_seteq card_image)
haftmann@35722
  2151
haftmann@35722
  2152
lemma eq_card_imp_inj_on:
haftmann@35722
  2153
  "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
haftmann@35722
  2154
apply (induct rule:finite_induct)
haftmann@35722
  2155
apply simp
haftmann@35722
  2156
apply(frule card_image_le[where f = f])
haftmann@35722
  2157
apply(simp add:card_insert_if split:if_splits)
haftmann@35722
  2158
done
haftmann@35722
  2159
haftmann@35722
  2160
lemma inj_on_iff_eq_card:
haftmann@35722
  2161
  "finite A ==> inj_on f A = (card(f ` A) = card A)"
haftmann@35722
  2162
by(blast intro: card_image eq_card_imp_inj_on)
haftmann@35722
  2163
haftmann@35722
  2164
haftmann@35722
  2165
lemma card_inj_on_le:
haftmann@35722
  2166
  "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
haftmann@35722
  2167
apply (subgoal_tac "finite A") 
haftmann@35722
  2168
 apply (force intro: card_mono simp add: card_image [symmetric])
haftmann@35722
  2169
apply (blast intro: finite_imageD dest: finite_subset) 
haftmann@35722
  2170
done
haftmann@35722
  2171
haftmann@35722
  2172
lemma card_bij_eq:
haftmann@35722
  2173
  "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
haftmann@35722
  2174
     finite A; finite B |] ==> card A = card B"
haftmann@35722
  2175
by (auto intro: le_antisym card_inj_on_le)
haftmann@35722
  2176
hoelzl@40703
  2177
lemma bij_betw_finite:
hoelzl@40703
  2178
  assumes "bij_betw f A B"
hoelzl@40703
  2179
  shows "finite A \<longleftrightarrow> finite B"
hoelzl@40703
  2180
using assms unfolding bij_betw_def
hoelzl@40703
  2181
using finite_imageD[of f A] by auto
haftmann@35722
  2182
haftmann@41656
  2183
nipkow@37466
  2184
subsubsection {* Pigeonhole Principles *}
nipkow@37466
  2185
nipkow@40311
  2186
lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
nipkow@37466
  2187
by (auto dest: card_image less_irrefl_nat)
nipkow@37466
  2188
nipkow@37466
  2189
lemma pigeonhole_infinite:
nipkow@37466
  2190
assumes  "~ finite A" and "finite(f`A)"
nipkow@37466
  2191
shows "EX a0:A. ~finite{a:A. f a = f a0}"
nipkow@37466
  2192
proof -
nipkow@37466
  2193
  have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
nipkow@37466
  2194
  proof(induct "f`A" arbitrary: A rule: finite_induct)
nipkow@37466
  2195
    case empty thus ?case by simp
nipkow@37466
  2196
  next
nipkow@37466
  2197
    case (insert b F)
nipkow@37466
  2198
    show ?case
nipkow@37466
  2199
    proof cases
nipkow@37466
  2200
      assume "finite{a:A. f a = b}"
nipkow@37466
  2201
      hence "~ finite(A - {a:A. f a = b})" using `\<not> finite A` by simp
nipkow@37466
  2202
      also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
nipkow@37466
  2203
      finally have "~ finite({a:A. f a \<noteq> b})" .
nipkow@37466
  2204
      from insert(3)[OF _ this]
nipkow@37466
  2205
      show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
nipkow@37466
  2206
    next
nipkow@37466
  2207
      assume 1: "~finite{a:A. f a = b}"
nipkow@37466
  2208
      hence "{a \<in> A. f a = b} \<noteq> {}" by force
nipkow@37466
  2209
      thus ?thesis using 1 by blast
nipkow@37466
  2210
    qed
nipkow@37466
  2211
  qed
nipkow@37466
  2212
  from this[OF assms(2,1)] show ?thesis .
nipkow@37466
  2213
qed
nipkow@37466
  2214
nipkow@37466
  2215
lemma pigeonhole_infinite_rel:
nipkow@37466
  2216
assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
nipkow@37466
  2217
shows "EX b:B. ~finite{a:A. R a b}"
nipkow@37466
  2218
proof -
nipkow@37466
  2219
   let ?F = "%a. {b:B. R a b}"
nipkow@37466
  2220
   from finite_Pow_iff[THEN iffD2, OF `finite B`]
nipkow@37466
  2221
   have "finite(?F ` A)" by(blast intro: rev_finite_subset)
nipkow@37466
  2222
   from pigeonhole_infinite[where f = ?F, OF assms(1) this]
nipkow@37466
  2223
   obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
nipkow@37466
  2224
   obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast
nipkow@37466
  2225
   { assume "finite{a:A. R a b0}"
nipkow@37466
  2226
     then have "finite {a\<in>A. ?F a = ?F a0}"
nipkow@37466
  2227
       using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)
nipkow@37466
  2228
   }
nipkow@37466
  2229
   with 1 `b0 : B` show ?thesis by blast
nipkow@37466
  2230
qed
nipkow@37466
  2231
nipkow@37466
  2232
haftmann@35722
  2233
subsubsection {* Cardinality of sums *}
haftmann@35722
  2234
haftmann@35722
  2235
lemma card_Plus:
haftmann@35722
  2236
  assumes "finite A" and "finite B"
haftmann@35722
  2237
  shows "card (A <+> B) = card A + card B"
haftmann@35722
  2238
proof -
haftmann@35722
  2239
  have "Inl`A \<inter> Inr`B = {}" by fast
haftmann@35722
  2240
  with assms show ?thesis
haftmann@35722
  2241
    unfolding Plus_def
haftmann@35722
  2242
    by (simp add: card_Un_disjoint card_image)
haftmann@35722
  2243
qed
haftmann@35722
  2244
haftmann@35722
  2245
lemma card_Plus_conv_if:
haftmann@35722
  2246
  "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
haftmann@35722
  2247
  by (auto simp add: card_Plus)
haftmann@35722
  2248
haftmann@35722
  2249
haftmann@35722
  2250
subsubsection {* Cardinality of the Powerset *}
haftmann@35722
  2251
haftmann@35722
  2252
lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
haftmann@41656
  2253
apply (induct rule: finite_induct)
haftmann@35722
  2254
 apply (simp_all add: Pow_insert)
haftmann@35722
  2255
apply (subst card_Un_disjoint, blast)
nipkow@40786
  2256
  apply (blast, blast)
haftmann@35722
  2257
apply (subgoal_tac "inj_on (insert x) (Pow F)")
haftmann@35722
  2258
 apply (simp add: card_image Pow_insert)
haftmann@35722
  2259
apply (unfold inj_on_def)
haftmann@35722
  2260
apply (blast elim!: equalityE)
haftmann@35722
  2261
done
haftmann@35722
  2262
wenzelm@40945
  2263
text {* Relates to equivalence classes.  Based on a theorem of F. Kammüller.  *}
haftmann@35722
  2264
haftmann@35722
  2265
lemma dvd_partition:
haftmann@35722
  2266
  "finite (Union C) ==>
haftmann@35722
  2267
    ALL c : C. k dvd card c ==>
haftmann@35722
  2268
    (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
haftmann@35722
  2269
  k dvd card (Union C)"
haftmann@41656
  2270
apply (frule finite_UnionD)
haftmann@41656
  2271
apply (rotate_tac -1)
haftmann@41656
  2272
apply (induct rule: finite_induct)
haftmann@41656
  2273
apply simp_all
haftmann@41656
  2274
apply clarify
haftmann@35722
  2275
apply (subst card_Un_disjoint)
haftmann@35722
  2276
   apply (auto simp add: disjoint_eq_subset_Compl)
haftmann@35722
  2277
done
haftmann@35722
  2278
haftmann@35722
  2279
haftmann@35722
  2280
subsubsection {* Relating injectivity and surjectivity *}
haftmann@35722
  2281
haftmann@41656
  2282
lemma finite_surj_inj: "finite A \<Longrightarrow> A \<subseteq> f ` A \<Longrightarrow> inj_on f A"
haftmann@35722
  2283
apply(rule eq_card_imp_inj_on, assumption)
haftmann@35722
  2284
apply(frule finite_imageI)
haftmann@35722
  2285
apply(drule (1) card_seteq)
haftmann@35722
  2286
 apply(erule card_image_le)
haftmann@35722
  2287
apply simp
haftmann@35722
  2288
done
haftmann@35722
  2289
haftmann@35722
  2290
lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
haftmann@35722
  2291
shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
hoelzl@40702
  2292
by (blast intro: finite_surj_inj subset_UNIV)
haftmann@35722
  2293
haftmann@35722
  2294
lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
haftmann@35722
  2295
shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
haftmann@35722
  2296
by(fastsimp simp:surj_def dest!: endo_inj_surj)
haftmann@35722
  2297
haftmann@35722
  2298
corollary infinite_UNIV_nat[iff]: "~finite(UNIV::nat set)"
haftmann@35722
  2299
proof
haftmann@35722
  2300
  assume "finite(UNIV::nat set)"
haftmann@35722
  2301
  with finite_UNIV_inj_surj[of Suc]
haftmann@35722
  2302
  show False by simp (blast dest: Suc_neq_Zero surjD)
haftmann@35722
  2303
qed
haftmann@35722
  2304
blanchet@35828
  2305
(* Often leads to bogus ATP proofs because of reduced type information, hence no_atp *)
blanchet@35828
  2306
lemma infinite_UNIV_char_0[no_atp]:
haftmann@35722
  2307
  "\<not> finite (UNIV::'a::semiring_char_0 set)"
haftmann@35722
  2308
proof
haftmann@35722
  2309
  assume "finite (UNIV::'a set)"
haftmann@35722
  2310
  with subset_UNIV have "finite (range of_nat::'a set)"
haftmann@35722
  2311
    by (rule finite_subset)
haftmann@35722
  2312
  moreover have "inj (of_nat::nat \<Rightarrow> 'a)"
haftmann@35722
  2313
    by (simp add: inj_on_def)
haftmann@35722
  2314
  ultimately have "finite (UNIV::nat set)"
haftmann@35722
  2315
    by (rule finite_imageD)
haftmann@35722
  2316
  then show "False"
haftmann@35722
  2317
    by simp
haftmann@35722
  2318
qed
haftmann@35722
  2319
haftmann@35722
  2320
end