src/HOL/Finite_Set.thy
 author paulson Mon May 23 15:33:24 2016 +0100 (2016-05-23) changeset 63114 27afe7af7379 parent 63099 af0e964aad7b child 63365 5340fb6633d0 permissions -rw-r--r--
Lots of new material for multivariate analysis
```     1 (*  Title:      HOL/Finite_Set.thy
```
```     2     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
```
```     3                 with contributions by Jeremy Avigad and Andrei Popescu
```
```     4 *)
```
```     5
```
```     6 section \<open>Finite sets\<close>
```
```     7
```
```     8 theory Finite_Set
```
```     9 imports Product_Type Sum_Type Fields
```
```    10 begin
```
```    11
```
```    12 subsection \<open>Predicate for finite sets\<close>
```
```    13
```
```    14 context
```
```    15   notes [[inductive_internals]]
```
```    16 begin
```
```    17
```
```    18 inductive finite :: "'a set \<Rightarrow> bool"
```
```    19   where
```
```    20     emptyI [simp, intro!]: "finite {}"
```
```    21   | insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)"
```
```    22
```
```    23 end
```
```    24
```
```    25 simproc_setup finite_Collect ("finite (Collect P)") = \<open>K Set_Comprehension_Pointfree.simproc\<close>
```
```    26
```
```    27 declare [[simproc del: finite_Collect]]
```
```    28
```
```    29 lemma finite_induct [case_names empty insert, induct set: finite]:
```
```    30   \<comment> \<open>Discharging \<open>x \<notin> F\<close> entails extra work.\<close>
```
```    31   assumes "finite F"
```
```    32   assumes "P {}"
```
```    33     and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
```
```    34   shows "P F"
```
```    35 using \<open>finite F\<close>
```
```    36 proof induct
```
```    37   show "P {}" by fact
```
```    38   fix x F assume F: "finite F" and P: "P F"
```
```    39   show "P (insert x F)"
```
```    40   proof cases
```
```    41     assume "x \<in> F"
```
```    42     hence "insert x F = F" by (rule insert_absorb)
```
```    43     with P show ?thesis by (simp only:)
```
```    44   next
```
```    45     assume "x \<notin> F"
```
```    46     from F this P show ?thesis by (rule insert)
```
```    47   qed
```
```    48 qed
```
```    49
```
```    50 lemma infinite_finite_induct [case_names infinite empty insert]:
```
```    51   assumes infinite: "\<And>A. \<not> finite A \<Longrightarrow> P A"
```
```    52   assumes empty: "P {}"
```
```    53   assumes insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
```
```    54   shows "P A"
```
```    55 proof (cases "finite A")
```
```    56   case False with infinite show ?thesis .
```
```    57 next
```
```    58   case True then show ?thesis by (induct A) (fact empty insert)+
```
```    59 qed
```
```    60
```
```    61
```
```    62 subsubsection \<open>Choice principles\<close>
```
```    63
```
```    64 lemma ex_new_if_finite: \<comment> "does not depend on def of finite at all"
```
```    65   assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
```
```    66   shows "\<exists>a::'a. a \<notin> A"
```
```    67 proof -
```
```    68   from assms have "A \<noteq> UNIV" by blast
```
```    69   then show ?thesis by blast
```
```    70 qed
```
```    71
```
```    72 text \<open>A finite choice principle. Does not need the SOME choice operator.\<close>
```
```    73
```
```    74 lemma finite_set_choice:
```
```    75   "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
```
```    76 proof (induct rule: finite_induct)
```
```    77   case empty then show ?case by simp
```
```    78 next
```
```    79   case (insert a A)
```
```    80   then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
```
```    81   show ?case (is "EX f. ?P f")
```
```    82   proof
```
```    83     show "?P(%x. if x = a then b else f x)" using f ab by auto
```
```    84   qed
```
```    85 qed
```
```    86
```
```    87
```
```    88 subsubsection \<open>Finite sets are the images of initial segments of natural numbers\<close>
```
```    89
```
```    90 lemma finite_imp_nat_seg_image_inj_on:
```
```    91   assumes "finite A"
```
```    92   shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
```
```    93 using assms
```
```    94 proof induct
```
```    95   case empty
```
```    96   show ?case
```
```    97   proof
```
```    98     show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp
```
```    99   qed
```
```   100 next
```
```   101   case (insert a A)
```
```   102   have notinA: "a \<notin> A" by fact
```
```   103   from insert.hyps obtain n f
```
```   104     where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
```
```   105   hence "insert a A = f(n:=a) ` {i. i < Suc n}"
```
```   106         "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
```
```   107     by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
```
```   108   thus ?case by blast
```
```   109 qed
```
```   110
```
```   111 lemma nat_seg_image_imp_finite:
```
```   112   "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
```
```   113 proof (induct n arbitrary: A)
```
```   114   case 0 thus ?case by simp
```
```   115 next
```
```   116   case (Suc n)
```
```   117   let ?B = "f ` {i. i < n}"
```
```   118   have finB: "finite ?B" by(rule Suc.hyps[OF refl])
```
```   119   show ?case
```
```   120   proof cases
```
```   121     assume "\<exists>k<n. f n = f k"
```
```   122     hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   123     thus ?thesis using finB by simp
```
```   124   next
```
```   125     assume "\<not>(\<exists> k<n. f n = f k)"
```
```   126     hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   127     thus ?thesis using finB by simp
```
```   128   qed
```
```   129 qed
```
```   130
```
```   131 lemma finite_conv_nat_seg_image:
```
```   132   "finite A \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"
```
```   133   by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
```
```   134
```
```   135 lemma finite_imp_inj_to_nat_seg:
```
```   136   assumes "finite A"
```
```   137   shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> inj_on f A"
```
```   138 proof -
```
```   139   from finite_imp_nat_seg_image_inj_on[OF \<open>finite A\<close>]
```
```   140   obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
```
```   141     by (auto simp:bij_betw_def)
```
```   142   let ?f = "the_inv_into {i. i<n} f"
```
```   143   have "inj_on ?f A & ?f ` A = {i. i<n}"
```
```   144     by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
```
```   145   thus ?thesis by blast
```
```   146 qed
```
```   147
```
```   148 lemma finite_Collect_less_nat [iff]:
```
```   149   "finite {n::nat. n < k}"
```
```   150   by (fastforce simp: finite_conv_nat_seg_image)
```
```   151
```
```   152 lemma finite_Collect_le_nat [iff]:
```
```   153   "finite {n::nat. n \<le> k}"
```
```   154   by (simp add: le_eq_less_or_eq Collect_disj_eq)
```
```   155
```
```   156
```
```   157 subsubsection \<open>Finiteness and common set operations\<close>
```
```   158
```
```   159 lemma rev_finite_subset:
```
```   160   "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
```
```   161 proof (induct arbitrary: A rule: finite_induct)
```
```   162   case empty
```
```   163   then show ?case by simp
```
```   164 next
```
```   165   case (insert x F A)
```
```   166   have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+
```
```   167   show "finite A"
```
```   168   proof cases
```
```   169     assume x: "x \<in> A"
```
```   170     with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
```
```   171     with r have "finite (A - {x})" .
```
```   172     hence "finite (insert x (A - {x}))" ..
```
```   173     also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
```
```   174     finally show ?thesis .
```
```   175   next
```
```   176     show ?thesis when "A \<subseteq> F"
```
```   177       using that by fact
```
```   178     assume "x \<notin> A"
```
```   179     with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
```
```   180   qed
```
```   181 qed
```
```   182
```
```   183 lemma finite_subset:
```
```   184   "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
```
```   185   by (rule rev_finite_subset)
```
```   186
```
```   187 lemma finite_UnI:
```
```   188   assumes "finite F" and "finite G"
```
```   189   shows "finite (F \<union> G)"
```
```   190   using assms by induct simp_all
```
```   191
```
```   192 lemma finite_Un [iff]:
```
```   193   "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
```
```   194   by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])
```
```   195
```
```   196 lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"
```
```   197 proof -
```
```   198   have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp
```
```   199   then have "finite ({a} \<union> A) \<longleftrightarrow> finite A" by (simp only: finite_Un)
```
```   200   then show ?thesis by simp
```
```   201 qed
```
```   202
```
```   203 lemma finite_Int [simp, intro]:
```
```   204   "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
```
```   205   by (blast intro: finite_subset)
```
```   206
```
```   207 lemma finite_Collect_conjI [simp, intro]:
```
```   208   "finite {x. P x} \<or> finite {x. Q x} \<Longrightarrow> finite {x. P x \<and> Q x}"
```
```   209   by (simp add: Collect_conj_eq)
```
```   210
```
```   211 lemma finite_Collect_disjI [simp]:
```
```   212   "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"
```
```   213   by (simp add: Collect_disj_eq)
```
```   214
```
```   215 lemma finite_Diff [simp, intro]:
```
```   216   "finite A \<Longrightarrow> finite (A - B)"
```
```   217   by (rule finite_subset, rule Diff_subset)
```
```   218
```
```   219 lemma finite_Diff2 [simp]:
```
```   220   assumes "finite B"
```
```   221   shows "finite (A - B) \<longleftrightarrow> finite A"
```
```   222 proof -
```
```   223   have "finite A \<longleftrightarrow> finite((A - B) \<union> (A \<inter> B))" by (simp add: Un_Diff_Int)
```
```   224   also have "\<dots> \<longleftrightarrow> finite (A - B)" using \<open>finite B\<close> by simp
```
```   225   finally show ?thesis ..
```
```   226 qed
```
```   227
```
```   228 lemma finite_Diff_insert [iff]:
```
```   229   "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
```
```   230 proof -
```
```   231   have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
```
```   232   moreover have "A - insert a B = A - B - {a}" by auto
```
```   233   ultimately show ?thesis by simp
```
```   234 qed
```
```   235
```
```   236 lemma finite_compl[simp]:
```
```   237   "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   238   by (simp add: Compl_eq_Diff_UNIV)
```
```   239
```
```   240 lemma finite_Collect_not[simp]:
```
```   241   "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   242   by (simp add: Collect_neg_eq)
```
```   243
```
```   244 lemma finite_Union [simp, intro]:
```
```   245   "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite(\<Union>A)"
```
```   246   by (induct rule: finite_induct) simp_all
```
```   247
```
```   248 lemma finite_UN_I [intro]:
```
```   249   "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"
```
```   250   by (induct rule: finite_induct) simp_all
```
```   251
```
```   252 lemma finite_UN [simp]:
```
```   253   "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
```
```   254   by (blast intro: finite_subset)
```
```   255
```
```   256 lemma finite_Inter [intro]:
```
```   257   "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
```
```   258   by (blast intro: Inter_lower finite_subset)
```
```   259
```
```   260 lemma finite_INT [intro]:
```
```   261   "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
```
```   262   by (blast intro: INT_lower finite_subset)
```
```   263
```
```   264 lemma finite_imageI [simp, intro]:
```
```   265   "finite F \<Longrightarrow> finite (h ` F)"
```
```   266   by (induct rule: finite_induct) simp_all
```
```   267
```
```   268 lemma finite_image_set [simp]:
```
```   269   "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
```
```   270   by (simp add: image_Collect [symmetric])
```
```   271
```
```   272 lemma finite_image_set2:
```
```   273   "finite {x. P x} \<Longrightarrow> finite {y. Q y} \<Longrightarrow> finite {f x y | x y. P x \<and> Q y}"
```
```   274   by (rule finite_subset [where B = "\<Union>x \<in> {x. P x}. \<Union>y \<in> {y. Q y}. {f x y}"]) auto
```
```   275
```
```   276 lemma finite_imageD:
```
```   277   assumes "finite (f ` A)" and "inj_on f A"
```
```   278   shows "finite A"
```
```   279 using assms
```
```   280 proof (induct "f ` A" arbitrary: A)
```
```   281   case empty then show ?case by simp
```
```   282 next
```
```   283   case (insert x B)
```
```   284   then have B_A: "insert x B = f ` A" by simp
```
```   285   then obtain y where "x = f y" and "y \<in> A" by blast
```
```   286   from B_A \<open>x \<notin> B\<close> have "B = f ` A - {x}" by blast
```
```   287   with B_A \<open>x \<notin> B\<close> \<open>x = f y\<close> \<open>inj_on f A\<close> \<open>y \<in> A\<close> have "B = f ` (A - {y})"
```
```   288     by (simp add: inj_on_image_set_diff Set.Diff_subset)
```
```   289   moreover from \<open>inj_on f A\<close> have "inj_on f (A - {y})" by (rule inj_on_diff)
```
```   290   ultimately have "finite (A - {y})" by (rule insert.hyps)
```
```   291   then show "finite A" by simp
```
```   292 qed
```
```   293
```
```   294 lemma finite_image_iff:
```
```   295   assumes "inj_on f A"
```
```   296   shows "finite (f ` A) \<longleftrightarrow> finite A"
```
```   297 using assms finite_imageD by blast
```
```   298
```
```   299 lemma finite_surj:
```
```   300   "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
```
```   301   by (erule finite_subset) (rule finite_imageI)
```
```   302
```
```   303 lemma finite_range_imageI:
```
```   304   "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
```
```   305   by (drule finite_imageI) (simp add: range_composition)
```
```   306
```
```   307 lemma finite_subset_image:
```
```   308   assumes "finite B"
```
```   309   shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
```
```   310 using assms
```
```   311 proof induct
```
```   312   case empty then show ?case by simp
```
```   313 next
```
```   314   case insert then show ?case
```
```   315     by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
```
```   316        blast
```
```   317 qed
```
```   318
```
```   319 lemma finite_vimage_IntI:
```
```   320   "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
```
```   321   apply (induct rule: finite_induct)
```
```   322    apply simp_all
```
```   323   apply (subst vimage_insert)
```
```   324   apply (simp add: finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2)
```
```   325   done
```
```   326
```
```   327 lemma finite_finite_vimage_IntI:
```
```   328   assumes "finite F" and "\<And>y. y \<in> F \<Longrightarrow> finite ((h -` {y}) \<inter> A)"
```
```   329   shows "finite (h -` F \<inter> A)"
```
```   330 proof -
```
```   331   have *: "h -` F \<inter> A = (\<Union> y\<in>F. (h -` {y}) \<inter> A)"
```
```   332     by blast
```
```   333   show ?thesis
```
```   334     by (simp only: * assms finite_UN_I)
```
```   335 qed
```
```   336
```
```   337 lemma finite_vimageI:
```
```   338   "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
```
```   339   using finite_vimage_IntI[of F h UNIV] by auto
```
```   340
```
```   341 lemma finite_vimageD': "\<lbrakk> finite (f -` A); A \<subseteq> range f \<rbrakk> \<Longrightarrow> finite A"
```
```   342 by(auto simp add: subset_image_iff intro: finite_subset[rotated])
```
```   343
```
```   344 lemma finite_vimageD: "\<lbrakk> finite (h -` F); surj h \<rbrakk> \<Longrightarrow> finite F"
```
```   345 by(auto dest: finite_vimageD')
```
```   346
```
```   347 lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
```
```   348   unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
```
```   349
```
```   350 lemma finite_Collect_bex [simp]:
```
```   351   assumes "finite A"
```
```   352   shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"
```
```   353 proof -
```
```   354   have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto
```
```   355   with assms show ?thesis by simp
```
```   356 qed
```
```   357
```
```   358 lemma finite_Collect_bounded_ex [simp]:
```
```   359   assumes "finite {y. P y}"
```
```   360   shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
```
```   361 proof -
```
```   362   have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
```
```   363   with assms show ?thesis by simp
```
```   364 qed
```
```   365
```
```   366 lemma finite_Plus:
```
```   367   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
```
```   368   by (simp add: Plus_def)
```
```   369
```
```   370 lemma finite_PlusD:
```
```   371   fixes A :: "'a set" and B :: "'b set"
```
```   372   assumes fin: "finite (A <+> B)"
```
```   373   shows "finite A" "finite B"
```
```   374 proof -
```
```   375   have "Inl ` A \<subseteq> A <+> B" by auto
```
```   376   then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
```
```   377   then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
```
```   378 next
```
```   379   have "Inr ` B \<subseteq> A <+> B" by auto
```
```   380   then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
```
```   381   then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
```
```   382 qed
```
```   383
```
```   384 lemma finite_Plus_iff [simp]:
```
```   385   "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
```
```   386   by (auto intro: finite_PlusD finite_Plus)
```
```   387
```
```   388 lemma finite_Plus_UNIV_iff [simp]:
```
```   389   "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
```
```   390   by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
```
```   391
```
```   392 lemma finite_SigmaI [simp, intro]:
```
```   393   "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
```
```   394   by (unfold Sigma_def) blast
```
```   395
```
```   396 lemma finite_SigmaI2:
```
```   397   assumes "finite {x\<in>A. B x \<noteq> {}}"
```
```   398   and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)"
```
```   399   shows "finite (Sigma A B)"
```
```   400 proof -
```
```   401   from assms have "finite (Sigma {x\<in>A. B x \<noteq> {}} B)" by auto
```
```   402   also have "Sigma {x:A. B x \<noteq> {}} B = Sigma A B" by auto
```
```   403   finally show ?thesis .
```
```   404 qed
```
```   405
```
```   406 lemma finite_cartesian_product:
```
```   407   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
```
```   408   by (rule finite_SigmaI)
```
```   409
```
```   410 lemma finite_Prod_UNIV:
```
```   411   "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
```
```   412   by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
```
```   413
```
```   414 lemma finite_cartesian_productD1:
```
```   415   assumes "finite (A \<times> B)" and "B \<noteq> {}"
```
```   416   shows "finite A"
```
```   417 proof -
```
```   418   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
```
```   419     by (auto simp add: finite_conv_nat_seg_image)
```
```   420   then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
```
```   421   with \<open>B \<noteq> {}\<close> have "A = (fst \<circ> f) ` {i::nat. i < n}"
```
```   422     by (simp add: image_comp)
```
```   423   then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
```
```   424   then show ?thesis
```
```   425     by (auto simp add: finite_conv_nat_seg_image)
```
```   426 qed
```
```   427
```
```   428 lemma finite_cartesian_productD2:
```
```   429   assumes "finite (A \<times> B)" and "A \<noteq> {}"
```
```   430   shows "finite B"
```
```   431 proof -
```
```   432   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
```
```   433     by (auto simp add: finite_conv_nat_seg_image)
```
```   434   then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
```
```   435   with \<open>A \<noteq> {}\<close> have "B = (snd \<circ> f) ` {i::nat. i < n}"
```
```   436     by (simp add: image_comp)
```
```   437   then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
```
```   438   then show ?thesis
```
```   439     by (auto simp add: finite_conv_nat_seg_image)
```
```   440 qed
```
```   441
```
```   442 lemma finite_cartesian_product_iff:
```
```   443   "finite (A \<times> B) \<longleftrightarrow> (A = {} \<or> B = {} \<or> (finite A \<and> finite B))"
```
```   444   by (auto dest: finite_cartesian_productD1 finite_cartesian_productD2 finite_cartesian_product)
```
```   445
```
```   446 lemma finite_prod:
```
```   447   "finite (UNIV :: ('a \<times> 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
```
```   448   using finite_cartesian_product_iff[of UNIV UNIV] by simp
```
```   449
```
```   450 lemma finite_Pow_iff [iff]:
```
```   451   "finite (Pow A) \<longleftrightarrow> finite A"
```
```   452 proof
```
```   453   assume "finite (Pow A)"
```
```   454   then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
```
```   455   then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
```
```   456 next
```
```   457   assume "finite A"
```
```   458   then show "finite (Pow A)"
```
```   459     by induct (simp_all add: Pow_insert)
```
```   460 qed
```
```   461
```
```   462 corollary finite_Collect_subsets [simp, intro]:
```
```   463   "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
```
```   464   by (simp add: Pow_def [symmetric])
```
```   465
```
```   466 lemma finite_set: "finite (UNIV :: 'a set set) \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   467 by(simp only: finite_Pow_iff Pow_UNIV[symmetric])
```
```   468
```
```   469 lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
```
```   470   by (blast intro: finite_subset [OF subset_Pow_Union])
```
```   471
```
```   472 lemma finite_set_of_finite_funs: assumes "finite A" "finite B"
```
```   473 shows "finite{f. \<forall>x. (x \<in> A \<longrightarrow> f x \<in> B) \<and> (x \<notin> A \<longrightarrow> f x = d)}" (is "finite ?S")
```
```   474 proof-
```
```   475   let ?F = "\<lambda>f. {(a,b). a \<in> A \<and> b = f a}"
```
```   476   have "?F ` ?S \<subseteq> Pow(A \<times> B)" by auto
```
```   477   from finite_subset[OF this] assms have 1: "finite (?F ` ?S)" by simp
```
```   478   have 2: "inj_on ?F ?S"
```
```   479     by(fastforce simp add: inj_on_def set_eq_iff fun_eq_iff)
```
```   480   show ?thesis by(rule finite_imageD[OF 1 2])
```
```   481 qed
```
```   482
```
```   483 lemma not_finite_existsD:
```
```   484   assumes "\<not> finite {a. P a}"
```
```   485   shows "\<exists>a. P a"
```
```   486 proof (rule classical)
```
```   487   assume "\<not> (\<exists>a. P a)"
```
```   488   with assms show ?thesis by auto
```
```   489 qed
```
```   490
```
```   491
```
```   492 subsubsection \<open>Further induction rules on finite sets\<close>
```
```   493
```
```   494 lemma finite_ne_induct [case_names singleton insert, consumes 2]:
```
```   495   assumes "finite F" and "F \<noteq> {}"
```
```   496   assumes "\<And>x. P {x}"
```
```   497     and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
```
```   498   shows "P F"
```
```   499 using assms
```
```   500 proof induct
```
```   501   case empty then show ?case by simp
```
```   502 next
```
```   503   case (insert x F) then show ?case by cases auto
```
```   504 qed
```
```   505
```
```   506 lemma finite_subset_induct [consumes 2, case_names empty insert]:
```
```   507   assumes "finite F" and "F \<subseteq> A"
```
```   508   assumes empty: "P {}"
```
```   509     and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
```
```   510   shows "P F"
```
```   511 using \<open>finite F\<close> \<open>F \<subseteq> A\<close>
```
```   512 proof induct
```
```   513   show "P {}" by fact
```
```   514 next
```
```   515   fix x F
```
```   516   assume "finite F" and "x \<notin> F" and
```
```   517     P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
```
```   518   show "P (insert x F)"
```
```   519   proof (rule insert)
```
```   520     from i show "x \<in> A" by blast
```
```   521     from i have "F \<subseteq> A" by blast
```
```   522     with P show "P F" .
```
```   523     show "finite F" by fact
```
```   524     show "x \<notin> F" by fact
```
```   525   qed
```
```   526 qed
```
```   527
```
```   528 lemma finite_empty_induct:
```
```   529   assumes "finite A"
```
```   530   assumes "P A"
```
```   531     and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
```
```   532   shows "P {}"
```
```   533 proof -
```
```   534   have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
```
```   535   proof -
```
```   536     fix B :: "'a set"
```
```   537     assume "B \<subseteq> A"
```
```   538     with \<open>finite A\<close> have "finite B" by (rule rev_finite_subset)
```
```   539     from this \<open>B \<subseteq> A\<close> show "P (A - B)"
```
```   540     proof induct
```
```   541       case empty
```
```   542       from \<open>P A\<close> show ?case by simp
```
```   543     next
```
```   544       case (insert b B)
```
```   545       have "P (A - B - {b})"
```
```   546       proof (rule remove)
```
```   547         from \<open>finite A\<close> show "finite (A - B)" by induct auto
```
```   548         from insert show "b \<in> A - B" by simp
```
```   549         from insert show "P (A - B)" by simp
```
```   550       qed
```
```   551       also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
```
```   552       finally show ?case .
```
```   553     qed
```
```   554   qed
```
```   555   then have "P (A - A)" by blast
```
```   556   then show ?thesis by simp
```
```   557 qed
```
```   558
```
```   559 lemma finite_update_induct [consumes 1, case_names const update]:
```
```   560   assumes finite: "finite {a. f a \<noteq> c}"
```
```   561   assumes const: "P (\<lambda>a. c)"
```
```   562   assumes update: "\<And>a b f. finite {a. f a \<noteq> c} \<Longrightarrow> f a = c \<Longrightarrow> b \<noteq> c \<Longrightarrow> P f \<Longrightarrow> P (f(a := b))"
```
```   563   shows "P f"
```
```   564 using finite proof (induct "{a. f a \<noteq> c}" arbitrary: f)
```
```   565   case empty with const show ?case by simp
```
```   566 next
```
```   567   case (insert a A)
```
```   568   then have "A = {a'. (f(a := c)) a' \<noteq> c}" and "f a \<noteq> c"
```
```   569     by auto
```
```   570   with \<open>finite A\<close> have "finite {a'. (f(a := c)) a' \<noteq> c}"
```
```   571     by simp
```
```   572   have "(f(a := c)) a = c"
```
```   573     by simp
```
```   574   from insert \<open>A = {a'. (f(a := c)) a' \<noteq> c}\<close> have "P (f(a := c))"
```
```   575     by simp
```
```   576   with \<open>finite {a'. (f(a := c)) a' \<noteq> c}\<close> \<open>(f(a := c)) a = c\<close> \<open>f a \<noteq> c\<close> have "P ((f(a := c))(a := f a))"
```
```   577     by (rule update)
```
```   578   then show ?case by simp
```
```   579 qed
```
```   580
```
```   581
```
```   582 subsection \<open>Class \<open>finite\<close>\<close>
```
```   583
```
```   584 class finite =
```
```   585   assumes finite_UNIV: "finite (UNIV :: 'a set)"
```
```   586 begin
```
```   587
```
```   588 lemma finite [simp]: "finite (A :: 'a set)"
```
```   589   by (rule subset_UNIV finite_UNIV finite_subset)+
```
```   590
```
```   591 lemma finite_code [code]: "finite (A :: 'a set) \<longleftrightarrow> True"
```
```   592   by simp
```
```   593
```
```   594 end
```
```   595
```
```   596 instance prod :: (finite, finite) finite
```
```   597   by standard (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
```
```   598
```
```   599 lemma inj_graph: "inj (%f. {(x, y). y = f x})"
```
```   600   by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
```
```   601
```
```   602 instance "fun" :: (finite, finite) finite
```
```   603 proof
```
```   604   show "finite (UNIV :: ('a => 'b) set)"
```
```   605   proof (rule finite_imageD)
```
```   606     let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
```
```   607     have "range ?graph \<subseteq> Pow UNIV" by simp
```
```   608     moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
```
```   609       by (simp only: finite_Pow_iff finite)
```
```   610     ultimately show "finite (range ?graph)"
```
```   611       by (rule finite_subset)
```
```   612     show "inj ?graph" by (rule inj_graph)
```
```   613   qed
```
```   614 qed
```
```   615
```
```   616 instance bool :: finite
```
```   617   by standard (simp add: UNIV_bool)
```
```   618
```
```   619 instance set :: (finite) finite
```
```   620   by standard (simp only: Pow_UNIV [symmetric] finite_Pow_iff finite)
```
```   621
```
```   622 instance unit :: finite
```
```   623   by standard (simp add: UNIV_unit)
```
```   624
```
```   625 instance sum :: (finite, finite) finite
```
```   626   by standard (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
```
```   627
```
```   628
```
```   629 subsection \<open>A basic fold functional for finite sets\<close>
```
```   630
```
```   631 text \<open>The intended behaviour is
```
```   632 \<open>fold f z {x\<^sub>1, ..., x\<^sub>n} = f x\<^sub>1 (\<dots> (f x\<^sub>n z)\<dots>)\<close>
```
```   633 if \<open>f\<close> is ``left-commutative'':
```
```   634 \<close>
```
```   635
```
```   636 locale comp_fun_commute =
```
```   637   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
```
```   638   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
```
```   639 begin
```
```   640
```
```   641 lemma fun_left_comm: "f y (f x z) = f x (f y z)"
```
```   642   using comp_fun_commute by (simp add: fun_eq_iff)
```
```   643
```
```   644 lemma commute_left_comp:
```
```   645   "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
```
```   646   by (simp add: o_assoc comp_fun_commute)
```
```   647
```
```   648 end
```
```   649
```
```   650 inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
```
```   651 for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
```
```   652   emptyI [intro]: "fold_graph f z {} z" |
```
```   653   insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
```
```   654       \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
```
```   655
```
```   656 inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
```
```   657
```
```   658 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
```
```   659   "fold f z A = (if finite A then (THE y. fold_graph f z A y) else z)"
```
```   660
```
```   661 text\<open>A tempting alternative for the definiens is
```
```   662 @{term "if finite A then THE y. fold_graph f z A y else e"}.
```
```   663 It allows the removal of finiteness assumptions from the theorems
```
```   664 \<open>fold_comm\<close>, \<open>fold_reindex\<close> and \<open>fold_distrib\<close>.
```
```   665 The proofs become ugly. It is not worth the effort. (???)\<close>
```
```   666
```
```   667 lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
```
```   668 by (induct rule: finite_induct) auto
```
```   669
```
```   670
```
```   671 subsubsection\<open>From @{const fold_graph} to @{term fold}\<close>
```
```   672
```
```   673 context comp_fun_commute
```
```   674 begin
```
```   675
```
```   676 lemma fold_graph_finite:
```
```   677   assumes "fold_graph f z A y"
```
```   678   shows "finite A"
```
```   679   using assms by induct simp_all
```
```   680
```
```   681 lemma fold_graph_insertE_aux:
```
```   682   "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'"
```
```   683 proof (induct set: fold_graph)
```
```   684   case (insertI x A y) show ?case
```
```   685   proof (cases "x = a")
```
```   686     assume "x = a" with insertI show ?case by auto
```
```   687   next
```
```   688     assume "x \<noteq> a"
```
```   689     then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
```
```   690       using insertI by auto
```
```   691     have "f x y = f a (f x y')"
```
```   692       unfolding y by (rule fun_left_comm)
```
```   693     moreover have "fold_graph f z (insert x A - {a}) (f x y')"
```
```   694       using y' and \<open>x \<noteq> a\<close> and \<open>x \<notin> A\<close>
```
```   695       by (simp add: insert_Diff_if fold_graph.insertI)
```
```   696     ultimately show ?case by fast
```
```   697   qed
```
```   698 qed simp
```
```   699
```
```   700 lemma fold_graph_insertE:
```
```   701   assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
```
```   702   obtains y where "v = f x y" and "fold_graph f z A y"
```
```   703 using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
```
```   704
```
```   705 lemma fold_graph_determ:
```
```   706   "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
```
```   707 proof (induct arbitrary: y set: fold_graph)
```
```   708   case (insertI x A y v)
```
```   709   from \<open>fold_graph f z (insert x A) v\<close> and \<open>x \<notin> A\<close>
```
```   710   obtain y' where "v = f x y'" and "fold_graph f z A y'"
```
```   711     by (rule fold_graph_insertE)
```
```   712   from \<open>fold_graph f z A y'\<close> have "y' = y" by (rule insertI)
```
```   713   with \<open>v = f x y'\<close> show "v = f x y" by simp
```
```   714 qed fast
```
```   715
```
```   716 lemma fold_equality:
```
```   717   "fold_graph f z A y \<Longrightarrow> fold f z A = y"
```
```   718   by (cases "finite A") (auto simp add: fold_def intro: fold_graph_determ dest: fold_graph_finite)
```
```   719
```
```   720 lemma fold_graph_fold:
```
```   721   assumes "finite A"
```
```   722   shows "fold_graph f z A (fold f z A)"
```
```   723 proof -
```
```   724   from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
```
```   725   moreover note fold_graph_determ
```
```   726   ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
```
```   727   then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
```
```   728   with assms show ?thesis by (simp add: fold_def)
```
```   729 qed
```
```   730
```
```   731 text \<open>The base case for \<open>fold\<close>:\<close>
```
```   732
```
```   733 lemma (in -) fold_infinite [simp]:
```
```   734   assumes "\<not> finite A"
```
```   735   shows "fold f z A = z"
```
```   736   using assms by (auto simp add: fold_def)
```
```   737
```
```   738 lemma (in -) fold_empty [simp]:
```
```   739   "fold f z {} = z"
```
```   740   by (auto simp add: fold_def)
```
```   741
```
```   742 text\<open>The various recursion equations for @{const fold}:\<close>
```
```   743
```
```   744 lemma fold_insert [simp]:
```
```   745   assumes "finite A" and "x \<notin> A"
```
```   746   shows "fold f z (insert x A) = f x (fold f z A)"
```
```   747 proof (rule fold_equality)
```
```   748   fix z
```
```   749   from \<open>finite A\<close> have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
```
```   750   with \<open>x \<notin> A\<close> have "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)
```
```   751   then show "fold_graph f z (insert x A) (f x (fold f z A))" by simp
```
```   752 qed
```
```   753
```
```   754 declare (in -) empty_fold_graphE [rule del] fold_graph.intros [rule del]
```
```   755   \<comment> \<open>No more proofs involve these.\<close>
```
```   756
```
```   757 lemma fold_fun_left_comm:
```
```   758   "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
```
```   759 proof (induct rule: finite_induct)
```
```   760   case empty then show ?case by simp
```
```   761 next
```
```   762   case (insert y A) then show ?case
```
```   763     by (simp add: fun_left_comm [of x])
```
```   764 qed
```
```   765
```
```   766 lemma fold_insert2:
```
```   767   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A)  = fold f (f x z) A"
```
```   768   by (simp add: fold_fun_left_comm)
```
```   769
```
```   770 lemma fold_rec:
```
```   771   assumes "finite A" and "x \<in> A"
```
```   772   shows "fold f z A = f x (fold f z (A - {x}))"
```
```   773 proof -
```
```   774   have A: "A = insert x (A - {x})" using \<open>x \<in> A\<close> by blast
```
```   775   then have "fold f z A = fold f z (insert x (A - {x}))" by simp
```
```   776   also have "\<dots> = f x (fold f z (A - {x}))"
```
```   777     by (rule fold_insert) (simp add: \<open>finite A\<close>)+
```
```   778   finally show ?thesis .
```
```   779 qed
```
```   780
```
```   781 lemma fold_insert_remove:
```
```   782   assumes "finite A"
```
```   783   shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
```
```   784 proof -
```
```   785   from \<open>finite A\<close> have "finite (insert x A)" by auto
```
```   786   moreover have "x \<in> insert x A" by auto
```
```   787   ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
```
```   788     by (rule fold_rec)
```
```   789   then show ?thesis by simp
```
```   790 qed
```
```   791
```
```   792 lemma fold_set_union_disj:
```
```   793   assumes "finite A" "finite B" "A \<inter> B = {}"
```
```   794   shows "Finite_Set.fold f z (A \<union> B) = Finite_Set.fold f (Finite_Set.fold f z A) B"
```
```   795 using assms(2,1,3) by induction simp_all
```
```   796
```
```   797 end
```
```   798
```
```   799 text\<open>Other properties of @{const fold}:\<close>
```
```   800
```
```   801 lemma fold_image:
```
```   802   assumes "inj_on g A"
```
```   803   shows "fold f z (g ` A) = fold (f \<circ> g) z A"
```
```   804 proof (cases "finite A")
```
```   805   case False with assms show ?thesis by (auto dest: finite_imageD simp add: fold_def)
```
```   806 next
```
```   807   case True
```
```   808   have "fold_graph f z (g ` A) = fold_graph (f \<circ> g) z A"
```
```   809   proof
```
```   810     fix w
```
```   811     show "fold_graph f z (g ` A) w \<longleftrightarrow> fold_graph (f \<circ> g) z A w" (is "?P \<longleftrightarrow> ?Q")
```
```   812     proof
```
```   813       assume ?P then show ?Q using assms
```
```   814       proof (induct "g ` A" w arbitrary: A)
```
```   815         case emptyI then show ?case by (auto intro: fold_graph.emptyI)
```
```   816       next
```
```   817         case (insertI x A r B)
```
```   818         from \<open>inj_on g B\<close> \<open>x \<notin> A\<close> \<open>insert x A = image g B\<close> obtain x' A' where
```
```   819           "x' \<notin> A'" and [simp]: "B = insert x' A'" "x = g x'" "A = g ` A'"
```
```   820           by (rule inj_img_insertE)
```
```   821         from insertI.prems have "fold_graph (f o g) z A' r"
```
```   822           by (auto intro: insertI.hyps)
```
```   823         with \<open>x' \<notin> A'\<close> have "fold_graph (f \<circ> g) z (insert x' A') ((f \<circ> g) x' r)"
```
```   824           by (rule fold_graph.insertI)
```
```   825         then show ?case by simp
```
```   826       qed
```
```   827     next
```
```   828       assume ?Q then show ?P using assms
```
```   829       proof induct
```
```   830         case emptyI thus ?case by (auto intro: fold_graph.emptyI)
```
```   831       next
```
```   832         case (insertI x A r)
```
```   833         from \<open>x \<notin> A\<close> insertI.prems have "g x \<notin> g ` A" by auto
```
```   834         moreover from insertI have "fold_graph f z (g ` A) r" by simp
```
```   835         ultimately have "fold_graph f z (insert (g x) (g ` A)) (f (g x) r)"
```
```   836           by (rule fold_graph.insertI)
```
```   837         then show ?case by simp
```
```   838       qed
```
```   839     qed
```
```   840   qed
```
```   841   with True assms show ?thesis by (auto simp add: fold_def)
```
```   842 qed
```
```   843
```
```   844 lemma fold_cong:
```
```   845   assumes "comp_fun_commute f" "comp_fun_commute g"
```
```   846   assumes "finite A" and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
```
```   847     and "s = t" and "A = B"
```
```   848   shows "fold f s A = fold g t B"
```
```   849 proof -
```
```   850   have "fold f s A = fold g s A"
```
```   851   using \<open>finite A\<close> cong proof (induct A)
```
```   852     case empty then show ?case by simp
```
```   853   next
```
```   854     case (insert x A)
```
```   855     interpret f: comp_fun_commute f by (fact \<open>comp_fun_commute f\<close>)
```
```   856     interpret g: comp_fun_commute g by (fact \<open>comp_fun_commute g\<close>)
```
```   857     from insert show ?case by simp
```
```   858   qed
```
```   859   with assms show ?thesis by simp
```
```   860 qed
```
```   861
```
```   862
```
```   863 text \<open>A simplified version for idempotent functions:\<close>
```
```   864
```
```   865 locale comp_fun_idem = comp_fun_commute +
```
```   866   assumes comp_fun_idem: "f x \<circ> f x = f x"
```
```   867 begin
```
```   868
```
```   869 lemma fun_left_idem: "f x (f x z) = f x z"
```
```   870   using comp_fun_idem by (simp add: fun_eq_iff)
```
```   871
```
```   872 lemma fold_insert_idem:
```
```   873   assumes fin: "finite A"
```
```   874   shows "fold f z (insert x A)  = f x (fold f z A)"
```
```   875 proof cases
```
```   876   assume "x \<in> A"
```
```   877   then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
```
```   878   then show ?thesis using assms by (simp add: comp_fun_idem fun_left_idem)
```
```   879 next
```
```   880   assume "x \<notin> A" then show ?thesis using assms by simp
```
```   881 qed
```
```   882
```
```   883 declare fold_insert [simp del] fold_insert_idem [simp]
```
```   884
```
```   885 lemma fold_insert_idem2:
```
```   886   "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
```
```   887   by (simp add: fold_fun_left_comm)
```
```   888
```
```   889 end
```
```   890
```
```   891
```
```   892 subsubsection \<open>Liftings to \<open>comp_fun_commute\<close> etc.\<close>
```
```   893
```
```   894 lemma (in comp_fun_commute) comp_comp_fun_commute:
```
```   895   "comp_fun_commute (f \<circ> g)"
```
```   896 proof
```
```   897 qed (simp_all add: comp_fun_commute)
```
```   898
```
```   899 lemma (in comp_fun_idem) comp_comp_fun_idem:
```
```   900   "comp_fun_idem (f \<circ> g)"
```
```   901   by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
```
```   902     (simp_all add: comp_fun_idem)
```
```   903
```
```   904 lemma (in comp_fun_commute) comp_fun_commute_funpow:
```
```   905   "comp_fun_commute (\<lambda>x. f x ^^ g x)"
```
```   906 proof
```
```   907   fix y x
```
```   908   show "f y ^^ g y \<circ> f x ^^ g x = f x ^^ g x \<circ> f y ^^ g y"
```
```   909   proof (cases "x = y")
```
```   910     case True then show ?thesis by simp
```
```   911   next
```
```   912     case False show ?thesis
```
```   913     proof (induct "g x" arbitrary: g)
```
```   914       case 0 then show ?case by simp
```
```   915     next
```
```   916       case (Suc n g)
```
```   917       have hyp1: "f y ^^ g y \<circ> f x = f x \<circ> f y ^^ g y"
```
```   918       proof (induct "g y" arbitrary: g)
```
```   919         case 0 then show ?case by simp
```
```   920       next
```
```   921         case (Suc n g)
```
```   922         define h where "h z = g z - 1" for z
```
```   923         with Suc have "n = h y" by simp
```
```   924         with Suc have hyp: "f y ^^ h y \<circ> f x = f x \<circ> f y ^^ h y"
```
```   925           by auto
```
```   926         from Suc h_def have "g y = Suc (h y)" by simp
```
```   927         then show ?case by (simp add: comp_assoc hyp)
```
```   928           (simp add: o_assoc comp_fun_commute)
```
```   929       qed
```
```   930       define h where "h z = (if z = x then g x - 1 else g z)" for z
```
```   931       with Suc have "n = h x" by simp
```
```   932       with Suc have "f y ^^ h y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ h y"
```
```   933         by auto
```
```   934       with False h_def have hyp2: "f y ^^ g y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ g y" by simp
```
```   935       from Suc h_def have "g x = Suc (h x)" by simp
```
```   936       then show ?case by (simp del: funpow.simps add: funpow_Suc_right o_assoc hyp2)
```
```   937         (simp add: comp_assoc hyp1)
```
```   938     qed
```
```   939   qed
```
```   940 qed
```
```   941
```
```   942
```
```   943 subsubsection \<open>Expressing set operations via @{const fold}\<close>
```
```   944
```
```   945 lemma comp_fun_commute_const:
```
```   946   "comp_fun_commute (\<lambda>_. f)"
```
```   947 proof
```
```   948 qed rule
```
```   949
```
```   950 lemma comp_fun_idem_insert:
```
```   951   "comp_fun_idem insert"
```
```   952 proof
```
```   953 qed auto
```
```   954
```
```   955 lemma comp_fun_idem_remove:
```
```   956   "comp_fun_idem Set.remove"
```
```   957 proof
```
```   958 qed auto
```
```   959
```
```   960 lemma (in semilattice_inf) comp_fun_idem_inf:
```
```   961   "comp_fun_idem inf"
```
```   962 proof
```
```   963 qed (auto simp add: inf_left_commute)
```
```   964
```
```   965 lemma (in semilattice_sup) comp_fun_idem_sup:
```
```   966   "comp_fun_idem sup"
```
```   967 proof
```
```   968 qed (auto simp add: sup_left_commute)
```
```   969
```
```   970 lemma union_fold_insert:
```
```   971   assumes "finite A"
```
```   972   shows "A \<union> B = fold insert B A"
```
```   973 proof -
```
```   974   interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
```
```   975   from \<open>finite A\<close> show ?thesis by (induct A arbitrary: B) simp_all
```
```   976 qed
```
```   977
```
```   978 lemma minus_fold_remove:
```
```   979   assumes "finite A"
```
```   980   shows "B - A = fold Set.remove B A"
```
```   981 proof -
```
```   982   interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove)
```
```   983   from \<open>finite A\<close> have "fold Set.remove B A = B - A" by (induct A arbitrary: B) auto
```
```   984   then show ?thesis ..
```
```   985 qed
```
```   986
```
```   987 lemma comp_fun_commute_filter_fold:
```
```   988   "comp_fun_commute (\<lambda>x A'. if P x then Set.insert x A' else A')"
```
```   989 proof -
```
```   990   interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
```
```   991   show ?thesis by standard (auto simp: fun_eq_iff)
```
```   992 qed
```
```   993
```
```   994 lemma Set_filter_fold:
```
```   995   assumes "finite A"
```
```   996   shows "Set.filter P A = fold (\<lambda>x A'. if P x then Set.insert x A' else A') {} A"
```
```   997 using assms
```
```   998 by (induct A)
```
```   999   (auto simp add: Set.filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold])
```
```  1000
```
```  1001 lemma inter_Set_filter:
```
```  1002   assumes "finite B"
```
```  1003   shows "A \<inter> B = Set.filter (\<lambda>x. x \<in> A) B"
```
```  1004 using assms
```
```  1005 by (induct B) (auto simp: Set.filter_def)
```
```  1006
```
```  1007 lemma image_fold_insert:
```
```  1008   assumes "finite A"
```
```  1009   shows "image f A = fold (\<lambda>k A. Set.insert (f k) A) {} A"
```
```  1010 using assms
```
```  1011 proof -
```
```  1012   interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A" by standard auto
```
```  1013   show ?thesis using assms by (induct A) auto
```
```  1014 qed
```
```  1015
```
```  1016 lemma Ball_fold:
```
```  1017   assumes "finite A"
```
```  1018   shows "Ball A P = fold (\<lambda>k s. s \<and> P k) True A"
```
```  1019 using assms
```
```  1020 proof -
```
```  1021   interpret comp_fun_commute "\<lambda>k s. s \<and> P k" by standard auto
```
```  1022   show ?thesis using assms by (induct A) auto
```
```  1023 qed
```
```  1024
```
```  1025 lemma Bex_fold:
```
```  1026   assumes "finite A"
```
```  1027   shows "Bex A P = fold (\<lambda>k s. s \<or> P k) False A"
```
```  1028 using assms
```
```  1029 proof -
```
```  1030   interpret comp_fun_commute "\<lambda>k s. s \<or> P k" by standard auto
```
```  1031   show ?thesis using assms by (induct A) auto
```
```  1032 qed
```
```  1033
```
```  1034 lemma comp_fun_commute_Pow_fold:
```
```  1035   "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)"
```
```  1036   by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast
```
```  1037
```
```  1038 lemma Pow_fold:
```
```  1039   assumes "finite A"
```
```  1040   shows "Pow A = fold (\<lambda>x A. A \<union> Set.insert x ` A) {{}} A"
```
```  1041 using assms
```
```  1042 proof -
```
```  1043   interpret comp_fun_commute "\<lambda>x A. A \<union> Set.insert x ` A" by (rule comp_fun_commute_Pow_fold)
```
```  1044   show ?thesis using assms by (induct A) (auto simp: Pow_insert)
```
```  1045 qed
```
```  1046
```
```  1047 lemma fold_union_pair:
```
```  1048   assumes "finite B"
```
```  1049   shows "(\<Union>y\<in>B. {(x, y)}) \<union> A = fold (\<lambda>y. Set.insert (x, y)) A B"
```
```  1050 proof -
```
```  1051   interpret comp_fun_commute "\<lambda>y. Set.insert (x, y)" by standard auto
```
```  1052   show ?thesis using assms  by (induct B arbitrary: A) simp_all
```
```  1053 qed
```
```  1054
```
```  1055 lemma comp_fun_commute_product_fold:
```
```  1056   assumes "finite B"
```
```  1057   shows "comp_fun_commute (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B)"
```
```  1058   by standard (auto simp: fold_union_pair[symmetric] assms)
```
```  1059
```
```  1060 lemma product_fold:
```
```  1061   assumes "finite A"
```
```  1062   assumes "finite B"
```
```  1063   shows "A \<times> B = fold (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B) {} A"
```
```  1064 using assms unfolding Sigma_def
```
```  1065 by (induct A)
```
```  1066   (simp_all add: comp_fun_commute.fold_insert[OF comp_fun_commute_product_fold] fold_union_pair)
```
```  1067
```
```  1068
```
```  1069 context complete_lattice
```
```  1070 begin
```
```  1071
```
```  1072 lemma inf_Inf_fold_inf:
```
```  1073   assumes "finite A"
```
```  1074   shows "inf (Inf A) B = fold inf B A"
```
```  1075 proof -
```
```  1076   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
```
```  1077   from \<open>finite A\<close> fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
```
```  1078     (simp_all add: inf_commute fun_eq_iff)
```
```  1079 qed
```
```  1080
```
```  1081 lemma sup_Sup_fold_sup:
```
```  1082   assumes "finite A"
```
```  1083   shows "sup (Sup A) B = fold sup B A"
```
```  1084 proof -
```
```  1085   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
```
```  1086   from \<open>finite A\<close> fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
```
```  1087     (simp_all add: sup_commute fun_eq_iff)
```
```  1088 qed
```
```  1089
```
```  1090 lemma Inf_fold_inf:
```
```  1091   assumes "finite A"
```
```  1092   shows "Inf A = fold inf top A"
```
```  1093   using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
```
```  1094
```
```  1095 lemma Sup_fold_sup:
```
```  1096   assumes "finite A"
```
```  1097   shows "Sup A = fold sup bot A"
```
```  1098   using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
```
```  1099
```
```  1100 lemma inf_INF_fold_inf:
```
```  1101   assumes "finite A"
```
```  1102   shows "inf B (INFIMUM A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold")
```
```  1103 proof (rule sym)
```
```  1104   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
```
```  1105   interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
```
```  1106   from \<open>finite A\<close> show "?fold = ?inf"
```
```  1107     by (induct A arbitrary: B)
```
```  1108       (simp_all add: inf_left_commute)
```
```  1109 qed
```
```  1110
```
```  1111 lemma sup_SUP_fold_sup:
```
```  1112   assumes "finite A"
```
```  1113   shows "sup B (SUPREMUM A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold")
```
```  1114 proof (rule sym)
```
```  1115   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
```
```  1116   interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
```
```  1117   from \<open>finite A\<close> show "?fold = ?sup"
```
```  1118     by (induct A arbitrary: B)
```
```  1119       (simp_all add: sup_left_commute)
```
```  1120 qed
```
```  1121
```
```  1122 lemma INF_fold_inf:
```
```  1123   assumes "finite A"
```
```  1124   shows "INFIMUM A f = fold (inf \<circ> f) top A"
```
```  1125   using assms inf_INF_fold_inf [of A top] by simp
```
```  1126
```
```  1127 lemma SUP_fold_sup:
```
```  1128   assumes "finite A"
```
```  1129   shows "SUPREMUM A f = fold (sup \<circ> f) bot A"
```
```  1130   using assms sup_SUP_fold_sup [of A bot] by simp
```
```  1131
```
```  1132 end
```
```  1133
```
```  1134
```
```  1135 subsection \<open>Locales as mini-packages for fold operations\<close>
```
```  1136
```
```  1137 subsubsection \<open>The natural case\<close>
```
```  1138
```
```  1139 locale folding =
```
```  1140   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
```
```  1141   fixes z :: "'b"
```
```  1142   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
```
```  1143 begin
```
```  1144
```
```  1145 interpretation fold?: comp_fun_commute f
```
```  1146   by standard (insert comp_fun_commute, simp add: fun_eq_iff)
```
```  1147
```
```  1148 definition F :: "'a set \<Rightarrow> 'b"
```
```  1149 where
```
```  1150   eq_fold: "F A = fold f z A"
```
```  1151
```
```  1152 lemma empty [simp]:"F {} = z"
```
```  1153   by (simp add: eq_fold)
```
```  1154
```
```  1155 lemma infinite [simp]: "\<not> finite A \<Longrightarrow> F A = z"
```
```  1156   by (simp add: eq_fold)
```
```  1157
```
```  1158 lemma insert [simp]:
```
```  1159   assumes "finite A" and "x \<notin> A"
```
```  1160   shows "F (insert x A) = f x (F A)"
```
```  1161 proof -
```
```  1162   from fold_insert assms
```
```  1163   have "fold f z (insert x A) = f x (fold f z A)" by simp
```
```  1164   with \<open>finite A\<close> show ?thesis by (simp add: eq_fold fun_eq_iff)
```
```  1165 qed
```
```  1166
```
```  1167 lemma remove:
```
```  1168   assumes "finite A" and "x \<in> A"
```
```  1169   shows "F A = f x (F (A - {x}))"
```
```  1170 proof -
```
```  1171   from \<open>x \<in> A\<close> obtain B where A: "A = insert x B" and "x \<notin> B"
```
```  1172     by (auto dest: mk_disjoint_insert)
```
```  1173   moreover from \<open>finite A\<close> A have "finite B" by simp
```
```  1174   ultimately show ?thesis by simp
```
```  1175 qed
```
```  1176
```
```  1177 lemma insert_remove:
```
```  1178   assumes "finite A"
```
```  1179   shows "F (insert x A) = f x (F (A - {x}))"
```
```  1180   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
```
```  1181
```
```  1182 end
```
```  1183
```
```  1184
```
```  1185 subsubsection \<open>With idempotency\<close>
```
```  1186
```
```  1187 locale folding_idem = folding +
```
```  1188   assumes comp_fun_idem: "f x \<circ> f x = f x"
```
```  1189 begin
```
```  1190
```
```  1191 declare insert [simp del]
```
```  1192
```
```  1193 interpretation fold?: comp_fun_idem f
```
```  1194   by standard (insert comp_fun_commute comp_fun_idem, simp add: fun_eq_iff)
```
```  1195
```
```  1196 lemma insert_idem [simp]:
```
```  1197   assumes "finite A"
```
```  1198   shows "F (insert x A) = f x (F A)"
```
```  1199 proof -
```
```  1200   from fold_insert_idem assms
```
```  1201   have "fold f z (insert x A) = f x (fold f z A)" by simp
```
```  1202   with \<open>finite A\<close> show ?thesis by (simp add: eq_fold fun_eq_iff)
```
```  1203 qed
```
```  1204
```
```  1205 end
```
```  1206
```
```  1207
```
```  1208 subsection \<open>Finite cardinality\<close>
```
```  1209
```
```  1210 text \<open>
```
```  1211   The traditional definition
```
```  1212   @{prop "card A \<equiv> LEAST n. EX f. A = {f i | i. i < n}"}
```
```  1213   is ugly to work with.
```
```  1214   But now that we have @{const fold} things are easy:
```
```  1215 \<close>
```
```  1216
```
```  1217 global_interpretation card: folding "\<lambda>_. Suc" 0
```
```  1218   defines card = "folding.F (\<lambda>_. Suc) 0"
```
```  1219   by standard rule
```
```  1220
```
```  1221 lemma card_infinite:
```
```  1222   "\<not> finite A \<Longrightarrow> card A = 0"
```
```  1223   by (fact card.infinite)
```
```  1224
```
```  1225 lemma card_empty:
```
```  1226   "card {} = 0"
```
```  1227   by (fact card.empty)
```
```  1228
```
```  1229 lemma card_insert_disjoint:
```
```  1230   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> card (insert x A) = Suc (card A)"
```
```  1231   by (fact card.insert)
```
```  1232
```
```  1233 lemma card_insert_if:
```
```  1234   "finite A \<Longrightarrow> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
```
```  1235   by auto (simp add: card.insert_remove card.remove)
```
```  1236
```
```  1237 lemma card_ge_0_finite:
```
```  1238   "card A > 0 \<Longrightarrow> finite A"
```
```  1239   by (rule ccontr) simp
```
```  1240
```
```  1241 lemma card_0_eq [simp]:
```
```  1242   "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
```
```  1243   by (auto dest: mk_disjoint_insert)
```
```  1244
```
```  1245 lemma finite_UNIV_card_ge_0:
```
```  1246   "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
```
```  1247   by (rule ccontr) simp
```
```  1248
```
```  1249 lemma card_eq_0_iff:
```
```  1250   "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
```
```  1251   by auto
```
```  1252
```
```  1253 lemma card_gt_0_iff:
```
```  1254   "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
```
```  1255   by (simp add: neq0_conv [symmetric] card_eq_0_iff)
```
```  1256
```
```  1257 lemma card_Suc_Diff1:
```
```  1258   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> Suc (card (A - {x})) = card A"
```
```  1259 apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
```
```  1260 apply(simp del:insert_Diff_single)
```
```  1261 done
```
```  1262
```
```  1263 lemma card_insert_le_m1: "n>0 \<Longrightarrow> card y \<le> n-1 \<Longrightarrow> card (insert x y) \<le> n"
```
```  1264   apply (cases "finite y")
```
```  1265   apply (cases "x \<in> y")
```
```  1266   apply (auto simp: insert_absorb)
```
```  1267   done
```
```  1268
```
```  1269 lemma card_Diff_singleton:
```
```  1270   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) = card A - 1"
```
```  1271   by (simp add: card_Suc_Diff1 [symmetric])
```
```  1272
```
```  1273 lemma card_Diff_singleton_if:
```
```  1274   "finite A \<Longrightarrow> card (A - {x}) = (if x \<in> A then card A - 1 else card A)"
```
```  1275   by (simp add: card_Diff_singleton)
```
```  1276
```
```  1277 lemma card_Diff_insert[simp]:
```
```  1278   assumes "finite A" and "a \<in> A" and "a \<notin> B"
```
```  1279   shows "card (A - insert a B) = card (A - B) - 1"
```
```  1280 proof -
```
```  1281   have "A - insert a B = (A - B) - {a}" using assms by blast
```
```  1282   then show ?thesis using assms by(simp add: card_Diff_singleton)
```
```  1283 qed
```
```  1284
```
```  1285 lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
```
```  1286   by (fact card.insert_remove)
```
```  1287
```
```  1288 lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
```
```  1289 by (simp add: card_insert_if)
```
```  1290
```
```  1291 lemma card_Collect_less_nat[simp]: "card{i::nat. i < n} = n"
```
```  1292 by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
```
```  1293
```
```  1294 lemma card_Collect_le_nat[simp]: "card{i::nat. i <= n} = Suc n"
```
```  1295 using card_Collect_less_nat[of "Suc n"] by(simp add: less_Suc_eq_le)
```
```  1296
```
```  1297 lemma card_mono:
```
```  1298   assumes "finite B" and "A \<subseteq> B"
```
```  1299   shows "card A \<le> card B"
```
```  1300 proof -
```
```  1301   from assms have "finite A" by (auto intro: finite_subset)
```
```  1302   then show ?thesis using assms proof (induct A arbitrary: B)
```
```  1303     case empty then show ?case by simp
```
```  1304   next
```
```  1305     case (insert x A)
```
```  1306     then have "x \<in> B" by simp
```
```  1307     from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
```
```  1308     with insert.hyps have "card A \<le> card (B - {x})" by auto
```
```  1309     with \<open>finite A\<close> \<open>x \<notin> A\<close> \<open>finite B\<close> \<open>x \<in> B\<close> show ?case by simp (simp only: card.remove)
```
```  1310   qed
```
```  1311 qed
```
```  1312
```
```  1313 lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
```
```  1314 apply (induct rule: finite_induct)
```
```  1315 apply simp
```
```  1316 apply clarify
```
```  1317 apply (subgoal_tac "finite A & A - {x} <= F")
```
```  1318  prefer 2 apply (blast intro: finite_subset, atomize)
```
```  1319 apply (drule_tac x = "A - {x}" in spec)
```
```  1320 apply (simp add: card_Diff_singleton_if split add: if_split_asm)
```
```  1321 apply (case_tac "card A", auto)
```
```  1322 done
```
```  1323
```
```  1324 lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
```
```  1325 apply (simp add: psubset_eq linorder_not_le [symmetric])
```
```  1326 apply (blast dest: card_seteq)
```
```  1327 done
```
```  1328
```
```  1329 lemma card_Un_Int:
```
```  1330   assumes "finite A" and "finite B"
```
```  1331   shows "card A + card B = card (A \<union> B) + card (A \<inter> B)"
```
```  1332 using assms proof (induct A)
```
```  1333   case empty then show ?case by simp
```
```  1334 next
```
```  1335  case (insert x A) then show ?case
```
```  1336     by (auto simp add: insert_absorb Int_insert_left)
```
```  1337 qed
```
```  1338
```
```  1339 lemma card_Un_disjoint:
```
```  1340   assumes "finite A" and "finite B"
```
```  1341   assumes "A \<inter> B = {}"
```
```  1342   shows "card (A \<union> B) = card A + card B"
```
```  1343 using assms card_Un_Int [of A B] by simp
```
```  1344
```
```  1345 lemma card_Un_le: "card (A \<union> B) \<le> card A + card B"
```
```  1346 apply(cases "finite A")
```
```  1347  apply(cases "finite B")
```
```  1348   using le_iff_add card_Un_Int apply blast
```
```  1349  apply simp
```
```  1350 apply simp
```
```  1351 done
```
```  1352
```
```  1353 lemma card_Diff_subset:
```
```  1354   assumes "finite B" and "B \<subseteq> A"
```
```  1355   shows "card (A - B) = card A - card B"
```
```  1356 proof (cases "finite A")
```
```  1357   case False with assms show ?thesis by simp
```
```  1358 next
```
```  1359   case True with assms show ?thesis by (induct B arbitrary: A) simp_all
```
```  1360 qed
```
```  1361
```
```  1362 lemma card_Diff_subset_Int:
```
```  1363   assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
```
```  1364 proof -
```
```  1365   have "A - B = A - A \<inter> B" by auto
```
```  1366   thus ?thesis
```
```  1367     by (simp add: card_Diff_subset AB)
```
```  1368 qed
```
```  1369
```
```  1370 lemma diff_card_le_card_Diff:
```
```  1371 assumes "finite B" shows "card A - card B \<le> card(A - B)"
```
```  1372 proof-
```
```  1373   have "card A - card B \<le> card A - card (A \<inter> B)"
```
```  1374     using card_mono[OF assms Int_lower2, of A] by arith
```
```  1375   also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
```
```  1376   finally show ?thesis .
```
```  1377 qed
```
```  1378
```
```  1379 lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
```
```  1380 apply (rule Suc_less_SucD)
```
```  1381 apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
```
```  1382 done
```
```  1383
```
```  1384 lemma card_Diff2_less:
```
```  1385   "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
```
```  1386 apply (case_tac "x = y")
```
```  1387  apply (simp add: card_Diff1_less del:card_Diff_insert)
```
```  1388 apply (rule less_trans)
```
```  1389  prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
```
```  1390 done
```
```  1391
```
```  1392 lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
```
```  1393 apply (case_tac "x : A")
```
```  1394  apply (simp_all add: card_Diff1_less less_imp_le)
```
```  1395 done
```
```  1396
```
```  1397 lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
```
```  1398 by (erule psubsetI, blast)
```
```  1399
```
```  1400 lemma card_le_inj:
```
```  1401   assumes fA: "finite A"
```
```  1402     and fB: "finite B"
```
```  1403     and c: "card A \<le> card B"
```
```  1404   shows "\<exists>f. f ` A \<subseteq> B \<and> inj_on f A"
```
```  1405   using fA fB c
```
```  1406 proof (induct arbitrary: B rule: finite_induct)
```
```  1407   case empty
```
```  1408   then show ?case by simp
```
```  1409 next
```
```  1410   case (insert x s t)
```
```  1411   then show ?case
```
```  1412   proof (induct rule: finite_induct[OF "insert.prems"(1)])
```
```  1413     case 1
```
```  1414     then show ?case by simp
```
```  1415   next
```
```  1416     case (2 y t)
```
```  1417     from "2.prems"(1,2,5) "2.hyps"(1,2) have cst: "card s \<le> card t"
```
```  1418       by simp
```
```  1419     from "2.prems"(3) [OF "2.hyps"(1) cst]
```
```  1420     obtain f where "f ` s \<subseteq> t" "inj_on f s"
```
```  1421       by blast
```
```  1422     with "2.prems"(2) "2.hyps"(2) show ?case
```
```  1423       apply -
```
```  1424       apply (rule exI[where x = "\<lambda>z. if z = x then y else f z"])
```
```  1425       apply (auto simp add: inj_on_def)
```
```  1426       done
```
```  1427   qed
```
```  1428 qed
```
```  1429
```
```  1430 lemma card_subset_eq:
```
```  1431   assumes fB: "finite B"
```
```  1432     and AB: "A \<subseteq> B"
```
```  1433     and c: "card A = card B"
```
```  1434   shows "A = B"
```
```  1435 proof -
```
```  1436   from fB AB have fA: "finite A"
```
```  1437     by (auto intro: finite_subset)
```
```  1438   from fA fB have fBA: "finite (B - A)"
```
```  1439     by auto
```
```  1440   have e: "A \<inter> (B - A) = {}"
```
```  1441     by blast
```
```  1442   have eq: "A \<union> (B - A) = B"
```
```  1443     using AB by blast
```
```  1444   from card_Un_disjoint[OF fA fBA e, unfolded eq c] have "card (B - A) = 0"
```
```  1445     by arith
```
```  1446   then have "B - A = {}"
```
```  1447     unfolding card_eq_0_iff using fA fB by simp
```
```  1448   with AB show "A = B"
```
```  1449     by blast
```
```  1450 qed
```
```  1451
```
```  1452 lemma insert_partition:
```
```  1453   "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
```
```  1454   \<Longrightarrow> x \<inter> \<Union>F = {}"
```
```  1455 by auto
```
```  1456
```
```  1457 lemma finite_psubset_induct[consumes 1, case_names psubset]:
```
```  1458   assumes fin: "finite A"
```
```  1459   and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A"
```
```  1460   shows "P A"
```
```  1461 using fin
```
```  1462 proof (induct A taking: card rule: measure_induct_rule)
```
```  1463   case (less A)
```
```  1464   have fin: "finite A" by fact
```
```  1465   have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
```
```  1466   { fix B
```
```  1467     assume asm: "B \<subset> A"
```
```  1468     from asm have "card B < card A" using psubset_card_mono fin by blast
```
```  1469     moreover
```
```  1470     from asm have "B \<subseteq> A" by auto
```
```  1471     then have "finite B" using fin finite_subset by blast
```
```  1472     ultimately
```
```  1473     have "P B" using ih by simp
```
```  1474   }
```
```  1475   with fin show "P A" using major by blast
```
```  1476 qed
```
```  1477
```
```  1478 lemma finite_induct_select[consumes 1, case_names empty select]:
```
```  1479   assumes "finite S"
```
```  1480   assumes "P {}"
```
```  1481   assumes select: "\<And>T. T \<subset> S \<Longrightarrow> P T \<Longrightarrow> \<exists>s\<in>S - T. P (insert s T)"
```
```  1482   shows "P S"
```
```  1483 proof -
```
```  1484   have "0 \<le> card S" by simp
```
```  1485   then have "\<exists>T \<subseteq> S. card T = card S \<and> P T"
```
```  1486   proof (induct rule: dec_induct)
```
```  1487     case base with \<open>P {}\<close> show ?case
```
```  1488       by (intro exI[of _ "{}"]) auto
```
```  1489   next
```
```  1490     case (step n)
```
```  1491     then obtain T where T: "T \<subseteq> S" "card T = n" "P T"
```
```  1492       by auto
```
```  1493     with \<open>n < card S\<close> have "T \<subset> S" "P T"
```
```  1494       by auto
```
```  1495     with select[of T] obtain s where "s \<in> S" "s \<notin> T" "P (insert s T)"
```
```  1496       by auto
```
```  1497     with step(2) T \<open>finite S\<close> show ?case
```
```  1498       by (intro exI[of _ "insert s T"]) (auto dest: finite_subset)
```
```  1499   qed
```
```  1500   with \<open>finite S\<close> show "P S"
```
```  1501     by (auto dest: card_subset_eq)
```
```  1502 qed
```
```  1503
```
```  1504 lemma remove_induct [case_names empty infinite remove]:
```
```  1505   assumes empty: "P ({} :: 'a set)" and infinite: "\<not>finite B \<Longrightarrow> P B"
```
```  1506       and remove: "\<And>A. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P (A - {x})) \<Longrightarrow> P A"
```
```  1507   shows "P B"
```
```  1508 proof (cases "finite B")
```
```  1509   assume "\<not>finite B"
```
```  1510   thus ?thesis by (rule infinite)
```
```  1511 next
```
```  1512   define A where "A = B"
```
```  1513   assume "finite B"
```
```  1514   hence "finite A" "A \<subseteq> B" by (simp_all add: A_def)
```
```  1515   thus "P A"
```
```  1516   proof (induction "card A" arbitrary: A)
```
```  1517     case 0
```
```  1518     hence "A = {}" by auto
```
```  1519     with empty show ?case by simp
```
```  1520   next
```
```  1521     case (Suc n A)
```
```  1522     from \<open>A \<subseteq> B\<close> and \<open>finite B\<close> have "finite A" by (rule finite_subset)
```
```  1523     moreover from Suc.hyps have "A \<noteq> {}" by auto
```
```  1524     moreover note \<open>A \<subseteq> B\<close>
```
```  1525     moreover have "P (A - {x})" if x: "x \<in> A" for x
```
```  1526       using x Suc.prems \<open>Suc n = card A\<close> by (intro Suc) auto
```
```  1527     ultimately show ?case by (rule remove)
```
```  1528   qed
```
```  1529 qed
```
```  1530
```
```  1531 lemma finite_remove_induct [consumes 1, case_names empty remove]:
```
```  1532   assumes finite: "finite B" and empty: "P ({} :: 'a set)"
```
```  1533       and rm: "\<And>A. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P (A - {x})) \<Longrightarrow> P A"
```
```  1534   defines "B' \<equiv> B"
```
```  1535   shows   "P B'"
```
```  1536   by (induction B' rule: remove_induct) (simp_all add: assms)
```
```  1537
```
```  1538
```
```  1539 text\<open>main cardinality theorem\<close>
```
```  1540 lemma card_partition [rule_format]:
```
```  1541   "finite C ==>
```
```  1542      finite (\<Union>C) -->
```
```  1543      (\<forall>c\<in>C. card c = k) -->
```
```  1544      (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
```
```  1545      k * card(C) = card (\<Union>C)"
```
```  1546 apply (erule finite_induct, simp)
```
```  1547 apply (simp add: card_Un_disjoint insert_partition
```
```  1548        finite_subset [of _ "\<Union>(insert x F)"])
```
```  1549 done
```
```  1550
```
```  1551 lemma card_eq_UNIV_imp_eq_UNIV:
```
```  1552   assumes fin: "finite (UNIV :: 'a set)"
```
```  1553   and card: "card A = card (UNIV :: 'a set)"
```
```  1554   shows "A = (UNIV :: 'a set)"
```
```  1555 proof
```
```  1556   show "A \<subseteq> UNIV" by simp
```
```  1557   show "UNIV \<subseteq> A"
```
```  1558   proof
```
```  1559     fix x
```
```  1560     show "x \<in> A"
```
```  1561     proof (rule ccontr)
```
```  1562       assume "x \<notin> A"
```
```  1563       then have "A \<subset> UNIV" by auto
```
```  1564       with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
```
```  1565       with card show False by simp
```
```  1566     qed
```
```  1567   qed
```
```  1568 qed
```
```  1569
```
```  1570 text\<open>The form of a finite set of given cardinality\<close>
```
```  1571
```
```  1572 lemma card_eq_SucD:
```
```  1573 assumes "card A = Suc k"
```
```  1574 shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
```
```  1575 proof -
```
```  1576   have fin: "finite A" using assms by (auto intro: ccontr)
```
```  1577   moreover have "card A \<noteq> 0" using assms by auto
```
```  1578   ultimately obtain b where b: "b \<in> A" by auto
```
```  1579   show ?thesis
```
```  1580   proof (intro exI conjI)
```
```  1581     show "A = insert b (A-{b})" using b by blast
```
```  1582     show "b \<notin> A - {b}" by blast
```
```  1583     show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
```
```  1584       using assms b fin by(fastforce dest:mk_disjoint_insert)+
```
```  1585   qed
```
```  1586 qed
```
```  1587
```
```  1588 lemma card_Suc_eq:
```
```  1589   "(card A = Suc k) =
```
```  1590    (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
```
```  1591  apply(auto elim!: card_eq_SucD)
```
```  1592  apply(subst card.insert)
```
```  1593  apply(auto simp add: intro:ccontr)
```
```  1594  done
```
```  1595
```
```  1596 lemma card_1_singletonE:
```
```  1597     assumes "card A = 1" obtains x where "A = {x}"
```
```  1598   using assms by (auto simp: card_Suc_eq)
```
```  1599
```
```  1600 lemma is_singleton_altdef: "is_singleton A \<longleftrightarrow> card A = 1"
```
```  1601   unfolding is_singleton_def
```
```  1602   by (auto elim!: card_1_singletonE is_singletonE simp del: One_nat_def)
```
```  1603
```
```  1604 lemma card_le_Suc_iff: "finite A \<Longrightarrow>
```
```  1605   Suc n \<le> card A = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
```
```  1606 by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
```
```  1607   dest: subset_singletonD split: nat.splits if_splits)
```
```  1608
```
```  1609 lemma finite_fun_UNIVD2:
```
```  1610   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
```
```  1611   shows "finite (UNIV :: 'b set)"
```
```  1612 proof -
```
```  1613   from fin have "\<And>arbitrary. finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
```
```  1614     by (rule finite_imageI)
```
```  1615   moreover have "\<And>arbitrary. UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
```
```  1616     by (rule UNIV_eq_I) auto
```
```  1617   ultimately show "finite (UNIV :: 'b set)" by simp
```
```  1618 qed
```
```  1619
```
```  1620 lemma card_UNIV_unit [simp]: "card (UNIV :: unit set) = 1"
```
```  1621   unfolding UNIV_unit by simp
```
```  1622
```
```  1623 lemma infinite_arbitrarily_large:
```
```  1624   assumes "\<not> finite A"
```
```  1625   shows "\<exists>B. finite B \<and> card B = n \<and> B \<subseteq> A"
```
```  1626 proof (induction n)
```
```  1627   case 0 show ?case by (intro exI[of _ "{}"]) auto
```
```  1628 next
```
```  1629   case (Suc n)
```
```  1630   then guess B .. note B = this
```
```  1631   with \<open>\<not> finite A\<close> have "A \<noteq> B" by auto
```
```  1632   with B have "B \<subset> A" by auto
```
```  1633   hence "\<exists>x. x \<in> A - B" by (elim psubset_imp_ex_mem)
```
```  1634   then guess x .. note x = this
```
```  1635   with B have "finite (insert x B) \<and> card (insert x B) = Suc n \<and> insert x B \<subseteq> A"
```
```  1636     by auto
```
```  1637   thus "\<exists>B. finite B \<and> card B = Suc n \<and> B \<subseteq> A" ..
```
```  1638 qed
```
```  1639
```
```  1640 subsubsection \<open>Cardinality of image\<close>
```
```  1641
```
```  1642 lemma card_image_le: "finite A ==> card (f ` A) \<le> card A"
```
```  1643   by (induct rule: finite_induct) (simp_all add: le_SucI card_insert_if)
```
```  1644
```
```  1645 lemma card_image:
```
```  1646   assumes "inj_on f A"
```
```  1647   shows "card (f ` A) = card A"
```
```  1648 proof (cases "finite A")
```
```  1649   case True then show ?thesis using assms by (induct A) simp_all
```
```  1650 next
```
```  1651   case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
```
```  1652   with False show ?thesis by simp
```
```  1653 qed
```
```  1654
```
```  1655 lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
```
```  1656 by(auto simp: card_image bij_betw_def)
```
```  1657
```
```  1658 lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
```
```  1659 by (simp add: card_seteq card_image)
```
```  1660
```
```  1661 lemma eq_card_imp_inj_on:
```
```  1662   assumes "finite A" "card(f ` A) = card A" shows "inj_on f A"
```
```  1663 using assms
```
```  1664 proof (induct rule:finite_induct)
```
```  1665   case empty show ?case by simp
```
```  1666 next
```
```  1667   case (insert x A)
```
```  1668   then show ?case using card_image_le [of A f]
```
```  1669     by (simp add: card_insert_if split: if_splits)
```
```  1670 qed
```
```  1671
```
```  1672 lemma inj_on_iff_eq_card: "finite A \<Longrightarrow> inj_on f A \<longleftrightarrow> card(f ` A) = card A"
```
```  1673   by (blast intro: card_image eq_card_imp_inj_on)
```
```  1674
```
```  1675 lemma card_inj_on_le:
```
```  1676   assumes "inj_on f A" "f ` A \<subseteq> B" "finite B" shows "card A \<le> card B"
```
```  1677 proof -
```
```  1678   have "finite A" using assms
```
```  1679     by (blast intro: finite_imageD dest: finite_subset)
```
```  1680   then show ?thesis using assms
```
```  1681    by (force intro: card_mono simp: card_image [symmetric])
```
```  1682 qed
```
```  1683
```
```  1684 lemma surj_card_le: "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> card B \<le> card A"
```
```  1685   by (blast intro: card_image_le card_mono le_trans)
```
```  1686
```
```  1687 lemma card_bij_eq:
```
```  1688   "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
```
```  1689      finite A; finite B |] ==> card A = card B"
```
```  1690 by (auto intro: le_antisym card_inj_on_le)
```
```  1691
```
```  1692 lemma bij_betw_finite:
```
```  1693   assumes "bij_betw f A B"
```
```  1694   shows "finite A \<longleftrightarrow> finite B"
```
```  1695 using assms unfolding bij_betw_def
```
```  1696 using finite_imageD[of f A] by auto
```
```  1697
```
```  1698 lemma inj_on_finite:
```
```  1699 assumes "inj_on f A" "f ` A \<le> B" "finite B"
```
```  1700 shows "finite A"
```
```  1701 using assms finite_imageD finite_subset by blast
```
```  1702
```
```  1703 lemma card_vimage_inj: "\<lbrakk> inj f; A \<subseteq> range f \<rbrakk> \<Longrightarrow> card (f -` A) = card A"
```
```  1704 by(auto 4 3 simp add: subset_image_iff inj_vimage_image_eq intro: card_image[symmetric, OF subset_inj_on])
```
```  1705
```
```  1706 subsubsection \<open>Pigeonhole Principles\<close>
```
```  1707
```
```  1708 lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
```
```  1709 by (auto dest: card_image less_irrefl_nat)
```
```  1710
```
```  1711 lemma pigeonhole_infinite:
```
```  1712 assumes  "~ finite A" and "finite(f`A)"
```
```  1713 shows "EX a0:A. ~finite{a:A. f a = f a0}"
```
```  1714 proof -
```
```  1715   have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
```
```  1716   proof(induct "f`A" arbitrary: A rule: finite_induct)
```
```  1717     case empty thus ?case by simp
```
```  1718   next
```
```  1719     case (insert b F)
```
```  1720     show ?case
```
```  1721     proof cases
```
```  1722       assume "finite{a:A. f a = b}"
```
```  1723       hence "~ finite(A - {a:A. f a = b})" using \<open>\<not> finite A\<close> by simp
```
```  1724       also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
```
```  1725       finally have "~ finite({a:A. f a \<noteq> b})" .
```
```  1726       from insert(3)[OF _ this]
```
```  1727       show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
```
```  1728     next
```
```  1729       assume 1: "~finite{a:A. f a = b}"
```
```  1730       hence "{a \<in> A. f a = b} \<noteq> {}" by force
```
```  1731       thus ?thesis using 1 by blast
```
```  1732     qed
```
```  1733   qed
```
```  1734   from this[OF assms(2,1)] show ?thesis .
```
```  1735 qed
```
```  1736
```
```  1737 lemma pigeonhole_infinite_rel:
```
```  1738 assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
```
```  1739 shows "EX b:B. ~finite{a:A. R a b}"
```
```  1740 proof -
```
```  1741    let ?F = "%a. {b:B. R a b}"
```
```  1742    from finite_Pow_iff[THEN iffD2, OF \<open>finite B\<close>]
```
```  1743    have "finite(?F ` A)" by(blast intro: rev_finite_subset)
```
```  1744    from pigeonhole_infinite[where f = ?F, OF assms(1) this]
```
```  1745    obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
```
```  1746    obtain b0 where "b0 : B" and "R a0 b0" using \<open>a0:A\<close> assms(3) by blast
```
```  1747    { assume "finite{a:A. R a b0}"
```
```  1748      then have "finite {a\<in>A. ?F a = ?F a0}"
```
```  1749        using \<open>b0 : B\<close> \<open>R a0 b0\<close> by(blast intro: rev_finite_subset)
```
```  1750    }
```
```  1751    with 1 \<open>b0 : B\<close> show ?thesis by blast
```
```  1752 qed
```
```  1753
```
```  1754
```
```  1755 subsubsection \<open>Cardinality of sums\<close>
```
```  1756
```
```  1757 lemma card_Plus:
```
```  1758   assumes "finite A" and "finite B"
```
```  1759   shows "card (A <+> B) = card A + card B"
```
```  1760 proof -
```
```  1761   have "Inl`A \<inter> Inr`B = {}" by fast
```
```  1762   with assms show ?thesis
```
```  1763     unfolding Plus_def
```
```  1764     by (simp add: card_Un_disjoint card_image)
```
```  1765 qed
```
```  1766
```
```  1767 lemma card_Plus_conv_if:
```
```  1768   "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
```
```  1769   by (auto simp add: card_Plus)
```
```  1770
```
```  1771 text \<open>Relates to equivalence classes.  Based on a theorem of F. Kamm\"uller.\<close>
```
```  1772
```
```  1773 lemma dvd_partition:
```
```  1774   assumes f: "finite (\<Union>C)" and "\<forall>c\<in>C. k dvd card c" "\<forall>c1\<in>C. \<forall>c2\<in>C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {}"
```
```  1775     shows "k dvd card (\<Union>C)"
```
```  1776 proof -
```
```  1777   have "finite C"
```
```  1778     by (rule finite_UnionD [OF f])
```
```  1779   then show ?thesis using assms
```
```  1780   proof (induct rule: finite_induct)
```
```  1781     case empty show ?case by simp
```
```  1782   next
```
```  1783     case (insert c C)
```
```  1784     then show ?case
```
```  1785       apply simp
```
```  1786       apply (subst card_Un_disjoint)
```
```  1787       apply (auto simp add: disjoint_eq_subset_Compl)
```
```  1788       done
```
```  1789   qed
```
```  1790 qed
```
```  1791
```
```  1792 subsubsection \<open>Relating injectivity and surjectivity\<close>
```
```  1793
```
```  1794 lemma finite_surj_inj: assumes "finite A" "A \<subseteq> f ` A" shows "inj_on f A"
```
```  1795 proof -
```
```  1796   have "f ` A = A"
```
```  1797     by (rule card_seteq [THEN sym]) (auto simp add: assms card_image_le)
```
```  1798   then show ?thesis using assms
```
```  1799     by (simp add: eq_card_imp_inj_on)
```
```  1800 qed
```
```  1801
```
```  1802 lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
```
```  1803 shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
```
```  1804 by (blast intro: finite_surj_inj subset_UNIV)
```
```  1805
```
```  1806 lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
```
```  1807 shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
```
```  1808 by(fastforce simp:surj_def dest!: endo_inj_surj)
```
```  1809
```
```  1810 corollary infinite_UNIV_nat [iff]:
```
```  1811   "\<not> finite (UNIV :: nat set)"
```
```  1812 proof
```
```  1813   assume "finite (UNIV :: nat set)"
```
```  1814   with finite_UNIV_inj_surj [of Suc]
```
```  1815   show False by simp (blast dest: Suc_neq_Zero surjD)
```
```  1816 qed
```
```  1817
```
```  1818 lemma infinite_UNIV_char_0:
```
```  1819   "\<not> finite (UNIV :: 'a::semiring_char_0 set)"
```
```  1820 proof
```
```  1821   assume "finite (UNIV :: 'a set)"
```
```  1822   with subset_UNIV have "finite (range of_nat :: 'a set)"
```
```  1823     by (rule finite_subset)
```
```  1824   moreover have "inj (of_nat :: nat \<Rightarrow> 'a)"
```
```  1825     by (simp add: inj_on_def)
```
```  1826   ultimately have "finite (UNIV :: nat set)"
```
```  1827     by (rule finite_imageD)
```
```  1828   then show False
```
```  1829     by simp
```
```  1830 qed
```
```  1831
```
```  1832 hide_const (open) Finite_Set.fold
```
```  1833
```
```  1834
```
```  1835 subsection "Infinite Sets"
```
```  1836
```
```  1837 text \<open>
```
```  1838   Some elementary facts about infinite sets, mostly by Stephan Merz.
```
```  1839   Beware! Because "infinite" merely abbreviates a negation, these
```
```  1840   lemmas may not work well with \<open>blast\<close>.
```
```  1841 \<close>
```
```  1842
```
```  1843 abbreviation infinite :: "'a set \<Rightarrow> bool"
```
```  1844   where "infinite S \<equiv> \<not> finite S"
```
```  1845
```
```  1846 text \<open>
```
```  1847   Infinite sets are non-empty, and if we remove some elements from an
```
```  1848   infinite set, the result is still infinite.
```
```  1849 \<close>
```
```  1850
```
```  1851 lemma infinite_imp_nonempty: "infinite S \<Longrightarrow> S \<noteq> {}"
```
```  1852   by auto
```
```  1853
```
```  1854 lemma infinite_remove: "infinite S \<Longrightarrow> infinite (S - {a})"
```
```  1855   by simp
```
```  1856
```
```  1857 lemma Diff_infinite_finite:
```
```  1858   assumes T: "finite T" and S: "infinite S"
```
```  1859   shows "infinite (S - T)"
```
```  1860   using T
```
```  1861 proof induct
```
```  1862   from S
```
```  1863   show "infinite (S - {})" by auto
```
```  1864 next
```
```  1865   fix T x
```
```  1866   assume ih: "infinite (S - T)"
```
```  1867   have "S - (insert x T) = (S - T) - {x}"
```
```  1868     by (rule Diff_insert)
```
```  1869   with ih
```
```  1870   show "infinite (S - (insert x T))"
```
```  1871     by (simp add: infinite_remove)
```
```  1872 qed
```
```  1873
```
```  1874 lemma Un_infinite: "infinite S \<Longrightarrow> infinite (S \<union> T)"
```
```  1875   by simp
```
```  1876
```
```  1877 lemma infinite_Un: "infinite (S \<union> T) \<longleftrightarrow> infinite S \<or> infinite T"
```
```  1878   by simp
```
```  1879
```
```  1880 lemma infinite_super:
```
```  1881   assumes T: "S \<subseteq> T" and S: "infinite S"
```
```  1882   shows "infinite T"
```
```  1883 proof
```
```  1884   assume "finite T"
```
```  1885   with T have "finite S" by (simp add: finite_subset)
```
```  1886   with S show False by simp
```
```  1887 qed
```
```  1888
```
```  1889 proposition infinite_coinduct [consumes 1, case_names infinite]:
```
```  1890   assumes "X A"
```
```  1891   and step: "\<And>A. X A \<Longrightarrow> \<exists>x\<in>A. X (A - {x}) \<or> infinite (A - {x})"
```
```  1892   shows "infinite A"
```
```  1893 proof
```
```  1894   assume "finite A"
```
```  1895   then show False using \<open>X A\<close>
```
```  1896   proof (induction rule: finite_psubset_induct)
```
```  1897     case (psubset A)
```
```  1898     then obtain x where "x \<in> A" "X (A - {x}) \<or> infinite (A - {x})"
```
```  1899       using local.step psubset.prems by blast
```
```  1900     then have "X (A - {x})"
```
```  1901       using psubset.hyps by blast
```
```  1902     show False
```
```  1903       apply (rule psubset.IH [where B = "A - {x}"])
```
```  1904       using \<open>x \<in> A\<close> apply blast
```
```  1905       by (simp add: \<open>X (A - {x})\<close>)
```
```  1906   qed
```
```  1907 qed
```
```  1908
```
```  1909 text \<open>
```
```  1910   For any function with infinite domain and finite range there is some
```
```  1911   element that is the image of infinitely many domain elements.  In
```
```  1912   particular, any infinite sequence of elements from a finite set
```
```  1913   contains some element that occurs infinitely often.
```
```  1914 \<close>
```
```  1915
```
```  1916 lemma inf_img_fin_dom':
```
```  1917   assumes img: "finite (f ` A)" and dom: "infinite A"
```
```  1918   shows "\<exists>y \<in> f ` A. infinite (f -` {y} \<inter> A)"
```
```  1919 proof (rule ccontr)
```
```  1920   have "A \<subseteq> (\<Union>y\<in>f ` A. f -` {y} \<inter> A)" by auto
```
```  1921   moreover
```
```  1922   assume "\<not> ?thesis"
```
```  1923   with img have "finite (\<Union>y\<in>f ` A. f -` {y} \<inter> A)" by blast
```
```  1924   ultimately have "finite A" by(rule finite_subset)
```
```  1925   with dom show False by contradiction
```
```  1926 qed
```
```  1927
```
```  1928 lemma inf_img_fin_domE':
```
```  1929   assumes "finite (f ` A)" and "infinite A"
```
```  1930   obtains y where "y \<in> f`A" and "infinite (f -` {y} \<inter> A)"
```
```  1931   using assms by (blast dest: inf_img_fin_dom')
```
```  1932
```
```  1933 lemma inf_img_fin_dom:
```
```  1934   assumes img: "finite (f`A)" and dom: "infinite A"
```
```  1935   shows "\<exists>y \<in> f`A. infinite (f -` {y})"
```
```  1936 using inf_img_fin_dom'[OF assms] by auto
```
```  1937
```
```  1938 lemma inf_img_fin_domE:
```
```  1939   assumes "finite (f`A)" and "infinite A"
```
```  1940   obtains y where "y \<in> f`A" and "infinite (f -` {y})"
```
```  1941   using assms by (blast dest: inf_img_fin_dom)
```
```  1942
```
```  1943 proposition finite_image_absD:
```
```  1944     fixes S :: "'a::linordered_ring set"
```
```  1945     shows "finite (abs ` S) \<Longrightarrow> finite S"
```
```  1946   by (rule ccontr) (auto simp: abs_eq_iff vimage_def dest: inf_img_fin_dom)
```
```  1947
```
```  1948 end
```