src/HOL/Finite_Set.thy
 author nipkow Tue Feb 23 16:25:08 2016 +0100 (2016-02-23) changeset 62390 842917225d56 parent 62093 bd73a2279fcd child 62481 b5d8e57826df permissions -rw-r--r--
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```     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 Nat
```
```    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_surj:
```
```   295   "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
```
```   296   by (erule finite_subset) (rule finite_imageI)
```
```   297
```
```   298 lemma finite_range_imageI:
```
```   299   "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
```
```   300   by (drule finite_imageI) (simp add: range_composition)
```
```   301
```
```   302 lemma finite_subset_image:
```
```   303   assumes "finite B"
```
```   304   shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
```
```   305 using assms
```
```   306 proof induct
```
```   307   case empty then show ?case by simp
```
```   308 next
```
```   309   case insert then show ?case
```
```   310     by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
```
```   311        blast
```
```   312 qed
```
```   313
```
```   314 lemma finite_vimage_IntI:
```
```   315   "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
```
```   316   apply (induct rule: finite_induct)
```
```   317    apply simp_all
```
```   318   apply (subst vimage_insert)
```
```   319   apply (simp add: finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2)
```
```   320   done
```
```   321
```
```   322 lemma finite_finite_vimage_IntI:
```
```   323   assumes "finite F" and "\<And>y. y \<in> F \<Longrightarrow> finite ((h -` {y}) \<inter> A)"
```
```   324   shows "finite (h -` F \<inter> A)"
```
```   325 proof -
```
```   326   have *: "h -` F \<inter> A = (\<Union> y\<in>F. (h -` {y}) \<inter> A)"
```
```   327     by blast
```
```   328   show ?thesis
```
```   329     by (simp only: * assms finite_UN_I)
```
```   330 qed
```
```   331
```
```   332 lemma finite_vimageI:
```
```   333   "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
```
```   334   using finite_vimage_IntI[of F h UNIV] by auto
```
```   335
```
```   336 lemma finite_vimageD': "\<lbrakk> finite (f -` A); A \<subseteq> range f \<rbrakk> \<Longrightarrow> finite A"
```
```   337 by(auto simp add: subset_image_iff intro: finite_subset[rotated])
```
```   338
```
```   339 lemma finite_vimageD: "\<lbrakk> finite (h -` F); surj h \<rbrakk> \<Longrightarrow> finite F"
```
```   340 by(auto dest: finite_vimageD')
```
```   341
```
```   342 lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
```
```   343   unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
```
```   344
```
```   345 lemma finite_Collect_bex [simp]:
```
```   346   assumes "finite A"
```
```   347   shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"
```
```   348 proof -
```
```   349   have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto
```
```   350   with assms show ?thesis by simp
```
```   351 qed
```
```   352
```
```   353 lemma finite_Collect_bounded_ex [simp]:
```
```   354   assumes "finite {y. P y}"
```
```   355   shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
```
```   356 proof -
```
```   357   have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
```
```   358   with assms show ?thesis by simp
```
```   359 qed
```
```   360
```
```   361 lemma finite_Plus:
```
```   362   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
```
```   363   by (simp add: Plus_def)
```
```   364
```
```   365 lemma finite_PlusD:
```
```   366   fixes A :: "'a set" and B :: "'b set"
```
```   367   assumes fin: "finite (A <+> B)"
```
```   368   shows "finite A" "finite B"
```
```   369 proof -
```
```   370   have "Inl ` A \<subseteq> A <+> B" by auto
```
```   371   then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
```
```   372   then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
```
```   373 next
```
```   374   have "Inr ` B \<subseteq> A <+> B" by auto
```
```   375   then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
```
```   376   then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
```
```   377 qed
```
```   378
```
```   379 lemma finite_Plus_iff [simp]:
```
```   380   "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
```
```   381   by (auto intro: finite_PlusD finite_Plus)
```
```   382
```
```   383 lemma finite_Plus_UNIV_iff [simp]:
```
```   384   "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
```
```   385   by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
```
```   386
```
```   387 lemma finite_SigmaI [simp, intro]:
```
```   388   "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
```
```   389   by (unfold Sigma_def) blast
```
```   390
```
```   391 lemma finite_SigmaI2:
```
```   392   assumes "finite {x\<in>A. B x \<noteq> {}}"
```
```   393   and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)"
```
```   394   shows "finite (Sigma A B)"
```
```   395 proof -
```
```   396   from assms have "finite (Sigma {x\<in>A. B x \<noteq> {}} B)" by auto
```
```   397   also have "Sigma {x:A. B x \<noteq> {}} B = Sigma A B" by auto
```
```   398   finally show ?thesis .
```
```   399 qed
```
```   400
```
```   401 lemma finite_cartesian_product:
```
```   402   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
```
```   403   by (rule finite_SigmaI)
```
```   404
```
```   405 lemma finite_Prod_UNIV:
```
```   406   "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
```
```   407   by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
```
```   408
```
```   409 lemma finite_cartesian_productD1:
```
```   410   assumes "finite (A \<times> B)" and "B \<noteq> {}"
```
```   411   shows "finite A"
```
```   412 proof -
```
```   413   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
```
```   414     by (auto simp add: finite_conv_nat_seg_image)
```
```   415   then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
```
```   416   with \<open>B \<noteq> {}\<close> have "A = (fst \<circ> f) ` {i::nat. i < n}"
```
```   417     by (simp add: image_comp)
```
```   418   then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
```
```   419   then show ?thesis
```
```   420     by (auto simp add: finite_conv_nat_seg_image)
```
```   421 qed
```
```   422
```
```   423 lemma finite_cartesian_productD2:
```
```   424   assumes "finite (A \<times> B)" and "A \<noteq> {}"
```
```   425   shows "finite B"
```
```   426 proof -
```
```   427   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
```
```   428     by (auto simp add: finite_conv_nat_seg_image)
```
```   429   then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
```
```   430   with \<open>A \<noteq> {}\<close> have "B = (snd \<circ> f) ` {i::nat. i < n}"
```
```   431     by (simp add: image_comp)
```
```   432   then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
```
```   433   then show ?thesis
```
```   434     by (auto simp add: finite_conv_nat_seg_image)
```
```   435 qed
```
```   436
```
```   437 lemma finite_cartesian_product_iff:
```
```   438   "finite (A \<times> B) \<longleftrightarrow> (A = {} \<or> B = {} \<or> (finite A \<and> finite B))"
```
```   439   by (auto dest: finite_cartesian_productD1 finite_cartesian_productD2 finite_cartesian_product)
```
```   440
```
```   441 lemma finite_prod:
```
```   442   "finite (UNIV :: ('a \<times> 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
```
```   443   using finite_cartesian_product_iff[of UNIV UNIV] by simp
```
```   444
```
```   445 lemma finite_Pow_iff [iff]:
```
```   446   "finite (Pow A) \<longleftrightarrow> finite A"
```
```   447 proof
```
```   448   assume "finite (Pow A)"
```
```   449   then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
```
```   450   then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
```
```   451 next
```
```   452   assume "finite A"
```
```   453   then show "finite (Pow A)"
```
```   454     by induct (simp_all add: Pow_insert)
```
```   455 qed
```
```   456
```
```   457 corollary finite_Collect_subsets [simp, intro]:
```
```   458   "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
```
```   459   by (simp add: Pow_def [symmetric])
```
```   460
```
```   461 lemma finite_set: "finite (UNIV :: 'a set set) \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   462 by(simp only: finite_Pow_iff Pow_UNIV[symmetric])
```
```   463
```
```   464 lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
```
```   465   by (blast intro: finite_subset [OF subset_Pow_Union])
```
```   466
```
```   467 lemma finite_set_of_finite_funs: assumes "finite A" "finite B"
```
```   468 shows "finite{f. \<forall>x. (x \<in> A \<longrightarrow> f x \<in> B) \<and> (x \<notin> A \<longrightarrow> f x = d)}" (is "finite ?S")
```
```   469 proof-
```
```   470   let ?F = "\<lambda>f. {(a,b). a \<in> A \<and> b = f a}"
```
```   471   have "?F ` ?S \<subseteq> Pow(A \<times> B)" by auto
```
```   472   from finite_subset[OF this] assms have 1: "finite (?F ` ?S)" by simp
```
```   473   have 2: "inj_on ?F ?S"
```
```   474     by(fastforce simp add: inj_on_def set_eq_iff fun_eq_iff)
```
```   475   show ?thesis by(rule finite_imageD[OF 1 2])
```
```   476 qed
```
```   477
```
```   478 lemma not_finite_existsD:
```
```   479   assumes "\<not> finite {a. P a}"
```
```   480   shows "\<exists>a. P a"
```
```   481 proof (rule classical)
```
```   482   assume "\<not> (\<exists>a. P a)"
```
```   483   with assms show ?thesis by auto
```
```   484 qed
```
```   485
```
```   486
```
```   487 subsubsection \<open>Further induction rules on finite sets\<close>
```
```   488
```
```   489 lemma finite_ne_induct [case_names singleton insert, consumes 2]:
```
```   490   assumes "finite F" and "F \<noteq> {}"
```
```   491   assumes "\<And>x. P {x}"
```
```   492     and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
```
```   493   shows "P F"
```
```   494 using assms
```
```   495 proof induct
```
```   496   case empty then show ?case by simp
```
```   497 next
```
```   498   case (insert x F) then show ?case by cases auto
```
```   499 qed
```
```   500
```
```   501 lemma finite_subset_induct [consumes 2, case_names empty insert]:
```
```   502   assumes "finite F" and "F \<subseteq> A"
```
```   503   assumes empty: "P {}"
```
```   504     and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
```
```   505   shows "P F"
```
```   506 using \<open>finite F\<close> \<open>F \<subseteq> A\<close>
```
```   507 proof induct
```
```   508   show "P {}" by fact
```
```   509 next
```
```   510   fix x F
```
```   511   assume "finite F" and "x \<notin> F" and
```
```   512     P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
```
```   513   show "P (insert x F)"
```
```   514   proof (rule insert)
```
```   515     from i show "x \<in> A" by blast
```
```   516     from i have "F \<subseteq> A" by blast
```
```   517     with P show "P F" .
```
```   518     show "finite F" by fact
```
```   519     show "x \<notin> F" by fact
```
```   520   qed
```
```   521 qed
```
```   522
```
```   523 lemma finite_empty_induct:
```
```   524   assumes "finite A"
```
```   525   assumes "P A"
```
```   526     and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
```
```   527   shows "P {}"
```
```   528 proof -
```
```   529   have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
```
```   530   proof -
```
```   531     fix B :: "'a set"
```
```   532     assume "B \<subseteq> A"
```
```   533     with \<open>finite A\<close> have "finite B" by (rule rev_finite_subset)
```
```   534     from this \<open>B \<subseteq> A\<close> show "P (A - B)"
```
```   535     proof induct
```
```   536       case empty
```
```   537       from \<open>P A\<close> show ?case by simp
```
```   538     next
```
```   539       case (insert b B)
```
```   540       have "P (A - B - {b})"
```
```   541       proof (rule remove)
```
```   542         from \<open>finite A\<close> show "finite (A - B)" by induct auto
```
```   543         from insert show "b \<in> A - B" by simp
```
```   544         from insert show "P (A - B)" by simp
```
```   545       qed
```
```   546       also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
```
```   547       finally show ?case .
```
```   548     qed
```
```   549   qed
```
```   550   then have "P (A - A)" by blast
```
```   551   then show ?thesis by simp
```
```   552 qed
```
```   553
```
```   554 lemma finite_update_induct [consumes 1, case_names const update]:
```
```   555   assumes finite: "finite {a. f a \<noteq> c}"
```
```   556   assumes const: "P (\<lambda>a. c)"
```
```   557   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))"
```
```   558   shows "P f"
```
```   559 using finite proof (induct "{a. f a \<noteq> c}" arbitrary: f)
```
```   560   case empty with const show ?case by simp
```
```   561 next
```
```   562   case (insert a A)
```
```   563   then have "A = {a'. (f(a := c)) a' \<noteq> c}" and "f a \<noteq> c"
```
```   564     by auto
```
```   565   with \<open>finite A\<close> have "finite {a'. (f(a := c)) a' \<noteq> c}"
```
```   566     by simp
```
```   567   have "(f(a := c)) a = c"
```
```   568     by simp
```
```   569   from insert \<open>A = {a'. (f(a := c)) a' \<noteq> c}\<close> have "P (f(a := c))"
```
```   570     by simp
```
```   571   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))"
```
```   572     by (rule update)
```
```   573   then show ?case by simp
```
```   574 qed
```
```   575
```
```   576
```
```   577 subsection \<open>Class \<open>finite\<close>\<close>
```
```   578
```
```   579 class finite =
```
```   580   assumes finite_UNIV: "finite (UNIV :: 'a set)"
```
```   581 begin
```
```   582
```
```   583 lemma finite [simp]: "finite (A :: 'a set)"
```
```   584   by (rule subset_UNIV finite_UNIV finite_subset)+
```
```   585
```
```   586 lemma finite_code [code]: "finite (A :: 'a set) \<longleftrightarrow> True"
```
```   587   by simp
```
```   588
```
```   589 end
```
```   590
```
```   591 instance prod :: (finite, finite) finite
```
```   592   by standard (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
```
```   593
```
```   594 lemma inj_graph: "inj (%f. {(x, y). y = f x})"
```
```   595   by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
```
```   596
```
```   597 instance "fun" :: (finite, finite) finite
```
```   598 proof
```
```   599   show "finite (UNIV :: ('a => 'b) set)"
```
```   600   proof (rule finite_imageD)
```
```   601     let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
```
```   602     have "range ?graph \<subseteq> Pow UNIV" by simp
```
```   603     moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
```
```   604       by (simp only: finite_Pow_iff finite)
```
```   605     ultimately show "finite (range ?graph)"
```
```   606       by (rule finite_subset)
```
```   607     show "inj ?graph" by (rule inj_graph)
```
```   608   qed
```
```   609 qed
```
```   610
```
```   611 instance bool :: finite
```
```   612   by standard (simp add: UNIV_bool)
```
```   613
```
```   614 instance set :: (finite) finite
```
```   615   by standard (simp only: Pow_UNIV [symmetric] finite_Pow_iff finite)
```
```   616
```
```   617 instance unit :: finite
```
```   618   by standard (simp add: UNIV_unit)
```
```   619
```
```   620 instance sum :: (finite, finite) finite
```
```   621   by standard (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
```
```   622
```
```   623
```
```   624 subsection \<open>A basic fold functional for finite sets\<close>
```
```   625
```
```   626 text \<open>The intended behaviour is
```
```   627 \<open>fold f z {x\<^sub>1, ..., x\<^sub>n} = f x\<^sub>1 (\<dots> (f x\<^sub>n z)\<dots>)\<close>
```
```   628 if \<open>f\<close> is ``left-commutative'':
```
```   629 \<close>
```
```   630
```
```   631 locale comp_fun_commute =
```
```   632   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
```
```   633   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
```
```   634 begin
```
```   635
```
```   636 lemma fun_left_comm: "f y (f x z) = f x (f y z)"
```
```   637   using comp_fun_commute by (simp add: fun_eq_iff)
```
```   638
```
```   639 lemma commute_left_comp:
```
```   640   "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
```
```   641   by (simp add: o_assoc comp_fun_commute)
```
```   642
```
```   643 end
```
```   644
```
```   645 inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
```
```   646 for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
```
```   647   emptyI [intro]: "fold_graph f z {} z" |
```
```   648   insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
```
```   649       \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
```
```   650
```
```   651 inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
```
```   652
```
```   653 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
```
```   654   "fold f z A = (if finite A then (THE y. fold_graph f z A y) else z)"
```
```   655
```
```   656 text\<open>A tempting alternative for the definiens is
```
```   657 @{term "if finite A then THE y. fold_graph f z A y else e"}.
```
```   658 It allows the removal of finiteness assumptions from the theorems
```
```   659 \<open>fold_comm\<close>, \<open>fold_reindex\<close> and \<open>fold_distrib\<close>.
```
```   660 The proofs become ugly. It is not worth the effort. (???)\<close>
```
```   661
```
```   662 lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
```
```   663 by (induct rule: finite_induct) auto
```
```   664
```
```   665
```
```   666 subsubsection\<open>From @{const fold_graph} to @{term fold}\<close>
```
```   667
```
```   668 context comp_fun_commute
```
```   669 begin
```
```   670
```
```   671 lemma fold_graph_finite:
```
```   672   assumes "fold_graph f z A y"
```
```   673   shows "finite A"
```
```   674   using assms by induct simp_all
```
```   675
```
```   676 lemma fold_graph_insertE_aux:
```
```   677   "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'"
```
```   678 proof (induct set: fold_graph)
```
```   679   case (insertI x A y) show ?case
```
```   680   proof (cases "x = a")
```
```   681     assume "x = a" with insertI show ?case by auto
```
```   682   next
```
```   683     assume "x \<noteq> a"
```
```   684     then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
```
```   685       using insertI by auto
```
```   686     have "f x y = f a (f x y')"
```
```   687       unfolding y by (rule fun_left_comm)
```
```   688     moreover have "fold_graph f z (insert x A - {a}) (f x y')"
```
```   689       using y' and \<open>x \<noteq> a\<close> and \<open>x \<notin> A\<close>
```
```   690       by (simp add: insert_Diff_if fold_graph.insertI)
```
```   691     ultimately show ?case by fast
```
```   692   qed
```
```   693 qed simp
```
```   694
```
```   695 lemma fold_graph_insertE:
```
```   696   assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
```
```   697   obtains y where "v = f x y" and "fold_graph f z A y"
```
```   698 using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
```
```   699
```
```   700 lemma fold_graph_determ:
```
```   701   "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
```
```   702 proof (induct arbitrary: y set: fold_graph)
```
```   703   case (insertI x A y v)
```
```   704   from \<open>fold_graph f z (insert x A) v\<close> and \<open>x \<notin> A\<close>
```
```   705   obtain y' where "v = f x y'" and "fold_graph f z A y'"
```
```   706     by (rule fold_graph_insertE)
```
```   707   from \<open>fold_graph f z A y'\<close> have "y' = y" by (rule insertI)
```
```   708   with \<open>v = f x y'\<close> show "v = f x y" by simp
```
```   709 qed fast
```
```   710
```
```   711 lemma fold_equality:
```
```   712   "fold_graph f z A y \<Longrightarrow> fold f z A = y"
```
```   713   by (cases "finite A") (auto simp add: fold_def intro: fold_graph_determ dest: fold_graph_finite)
```
```   714
```
```   715 lemma fold_graph_fold:
```
```   716   assumes "finite A"
```
```   717   shows "fold_graph f z A (fold f z A)"
```
```   718 proof -
```
```   719   from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
```
```   720   moreover note fold_graph_determ
```
```   721   ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
```
```   722   then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
```
```   723   with assms show ?thesis by (simp add: fold_def)
```
```   724 qed
```
```   725
```
```   726 text \<open>The base case for \<open>fold\<close>:\<close>
```
```   727
```
```   728 lemma (in -) fold_infinite [simp]:
```
```   729   assumes "\<not> finite A"
```
```   730   shows "fold f z A = z"
```
```   731   using assms by (auto simp add: fold_def)
```
```   732
```
```   733 lemma (in -) fold_empty [simp]:
```
```   734   "fold f z {} = z"
```
```   735   by (auto simp add: fold_def)
```
```   736
```
```   737 text\<open>The various recursion equations for @{const fold}:\<close>
```
```   738
```
```   739 lemma fold_insert [simp]:
```
```   740   assumes "finite A" and "x \<notin> A"
```
```   741   shows "fold f z (insert x A) = f x (fold f z A)"
```
```   742 proof (rule fold_equality)
```
```   743   fix z
```
```   744   from \<open>finite A\<close> have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
```
```   745   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)
```
```   746   then show "fold_graph f z (insert x A) (f x (fold f z A))" by simp
```
```   747 qed
```
```   748
```
```   749 declare (in -) empty_fold_graphE [rule del] fold_graph.intros [rule del]
```
```   750   \<comment> \<open>No more proofs involve these.\<close>
```
```   751
```
```   752 lemma fold_fun_left_comm:
```
```   753   "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
```
```   754 proof (induct rule: finite_induct)
```
```   755   case empty then show ?case by simp
```
```   756 next
```
```   757   case (insert y A) then show ?case
```
```   758     by (simp add: fun_left_comm [of x])
```
```   759 qed
```
```   760
```
```   761 lemma fold_insert2:
```
```   762   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A)  = fold f (f x z) A"
```
```   763   by (simp add: fold_fun_left_comm)
```
```   764
```
```   765 lemma fold_rec:
```
```   766   assumes "finite A" and "x \<in> A"
```
```   767   shows "fold f z A = f x (fold f z (A - {x}))"
```
```   768 proof -
```
```   769   have A: "A = insert x (A - {x})" using \<open>x \<in> A\<close> by blast
```
```   770   then have "fold f z A = fold f z (insert x (A - {x}))" by simp
```
```   771   also have "\<dots> = f x (fold f z (A - {x}))"
```
```   772     by (rule fold_insert) (simp add: \<open>finite A\<close>)+
```
```   773   finally show ?thesis .
```
```   774 qed
```
```   775
```
```   776 lemma fold_insert_remove:
```
```   777   assumes "finite A"
```
```   778   shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
```
```   779 proof -
```
```   780   from \<open>finite A\<close> have "finite (insert x A)" by auto
```
```   781   moreover have "x \<in> insert x A" by auto
```
```   782   ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
```
```   783     by (rule fold_rec)
```
```   784   then show ?thesis by simp
```
```   785 qed
```
```   786
```
```   787 lemma fold_set_union_disj:
```
```   788   assumes "finite A" "finite B" "A \<inter> B = {}"
```
```   789   shows "Finite_Set.fold f z (A \<union> B) = Finite_Set.fold f (Finite_Set.fold f z A) B"
```
```   790 using assms(2,1,3) by induction simp_all
```
```   791
```
```   792 end
```
```   793
```
```   794 text\<open>Other properties of @{const fold}:\<close>
```
```   795
```
```   796 lemma fold_image:
```
```   797   assumes "inj_on g A"
```
```   798   shows "fold f z (g ` A) = fold (f \<circ> g) z A"
```
```   799 proof (cases "finite A")
```
```   800   case False with assms show ?thesis by (auto dest: finite_imageD simp add: fold_def)
```
```   801 next
```
```   802   case True
```
```   803   have "fold_graph f z (g ` A) = fold_graph (f \<circ> g) z A"
```
```   804   proof
```
```   805     fix w
```
```   806     show "fold_graph f z (g ` A) w \<longleftrightarrow> fold_graph (f \<circ> g) z A w" (is "?P \<longleftrightarrow> ?Q")
```
```   807     proof
```
```   808       assume ?P then show ?Q using assms
```
```   809       proof (induct "g ` A" w arbitrary: A)
```
```   810         case emptyI then show ?case by (auto intro: fold_graph.emptyI)
```
```   811       next
```
```   812         case (insertI x A r B)
```
```   813         from \<open>inj_on g B\<close> \<open>x \<notin> A\<close> \<open>insert x A = image g B\<close> obtain x' A' where
```
```   814           "x' \<notin> A'" and [simp]: "B = insert x' A'" "x = g x'" "A = g ` A'"
```
```   815           by (rule inj_img_insertE)
```
```   816         from insertI.prems have "fold_graph (f o g) z A' r"
```
```   817           by (auto intro: insertI.hyps)
```
```   818         with \<open>x' \<notin> A'\<close> have "fold_graph (f \<circ> g) z (insert x' A') ((f \<circ> g) x' r)"
```
```   819           by (rule fold_graph.insertI)
```
```   820         then show ?case by simp
```
```   821       qed
```
```   822     next
```
```   823       assume ?Q then show ?P using assms
```
```   824       proof induct
```
```   825         case emptyI thus ?case by (auto intro: fold_graph.emptyI)
```
```   826       next
```
```   827         case (insertI x A r)
```
```   828         from \<open>x \<notin> A\<close> insertI.prems have "g x \<notin> g ` A" by auto
```
```   829         moreover from insertI have "fold_graph f z (g ` A) r" by simp
```
```   830         ultimately have "fold_graph f z (insert (g x) (g ` A)) (f (g x) r)"
```
```   831           by (rule fold_graph.insertI)
```
```   832         then show ?case by simp
```
```   833       qed
```
```   834     qed
```
```   835   qed
```
```   836   with True assms show ?thesis by (auto simp add: fold_def)
```
```   837 qed
```
```   838
```
```   839 lemma fold_cong:
```
```   840   assumes "comp_fun_commute f" "comp_fun_commute g"
```
```   841   assumes "finite A" and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
```
```   842     and "s = t" and "A = B"
```
```   843   shows "fold f s A = fold g t B"
```
```   844 proof -
```
```   845   have "fold f s A = fold g s A"
```
```   846   using \<open>finite A\<close> cong proof (induct A)
```
```   847     case empty then show ?case by simp
```
```   848   next
```
```   849     case (insert x A)
```
```   850     interpret f: comp_fun_commute f by (fact \<open>comp_fun_commute f\<close>)
```
```   851     interpret g: comp_fun_commute g by (fact \<open>comp_fun_commute g\<close>)
```
```   852     from insert show ?case by simp
```
```   853   qed
```
```   854   with assms show ?thesis by simp
```
```   855 qed
```
```   856
```
```   857
```
```   858 text \<open>A simplified version for idempotent functions:\<close>
```
```   859
```
```   860 locale comp_fun_idem = comp_fun_commute +
```
```   861   assumes comp_fun_idem: "f x \<circ> f x = f x"
```
```   862 begin
```
```   863
```
```   864 lemma fun_left_idem: "f x (f x z) = f x z"
```
```   865   using comp_fun_idem by (simp add: fun_eq_iff)
```
```   866
```
```   867 lemma fold_insert_idem:
```
```   868   assumes fin: "finite A"
```
```   869   shows "fold f z (insert x A)  = f x (fold f z A)"
```
```   870 proof cases
```
```   871   assume "x \<in> A"
```
```   872   then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
```
```   873   then show ?thesis using assms by (simp add: comp_fun_idem fun_left_idem)
```
```   874 next
```
```   875   assume "x \<notin> A" then show ?thesis using assms by simp
```
```   876 qed
```
```   877
```
```   878 declare fold_insert [simp del] fold_insert_idem [simp]
```
```   879
```
```   880 lemma fold_insert_idem2:
```
```   881   "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
```
```   882   by (simp add: fold_fun_left_comm)
```
```   883
```
```   884 end
```
```   885
```
```   886
```
```   887 subsubsection \<open>Liftings to \<open>comp_fun_commute\<close> etc.\<close>
```
```   888
```
```   889 lemma (in comp_fun_commute) comp_comp_fun_commute:
```
```   890   "comp_fun_commute (f \<circ> g)"
```
```   891 proof
```
```   892 qed (simp_all add: comp_fun_commute)
```
```   893
```
```   894 lemma (in comp_fun_idem) comp_comp_fun_idem:
```
```   895   "comp_fun_idem (f \<circ> g)"
```
```   896   by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
```
```   897     (simp_all add: comp_fun_idem)
```
```   898
```
```   899 lemma (in comp_fun_commute) comp_fun_commute_funpow:
```
```   900   "comp_fun_commute (\<lambda>x. f x ^^ g x)"
```
```   901 proof
```
```   902   fix y x
```
```   903   show "f y ^^ g y \<circ> f x ^^ g x = f x ^^ g x \<circ> f y ^^ g y"
```
```   904   proof (cases "x = y")
```
```   905     case True then show ?thesis by simp
```
```   906   next
```
```   907     case False show ?thesis
```
```   908     proof (induct "g x" arbitrary: g)
```
```   909       case 0 then show ?case by simp
```
```   910     next
```
```   911       case (Suc n g)
```
```   912       have hyp1: "f y ^^ g y \<circ> f x = f x \<circ> f y ^^ g y"
```
```   913       proof (induct "g y" arbitrary: g)
```
```   914         case 0 then show ?case by simp
```
```   915       next
```
```   916         case (Suc n g)
```
```   917         def h \<equiv> "\<lambda>z. g z - 1"
```
```   918         with Suc have "n = h y" by simp
```
```   919         with Suc have hyp: "f y ^^ h y \<circ> f x = f x \<circ> f y ^^ h y"
```
```   920           by auto
```
```   921         from Suc h_def have "g y = Suc (h y)" by simp
```
```   922         then show ?case by (simp add: comp_assoc hyp)
```
```   923           (simp add: o_assoc comp_fun_commute)
```
```   924       qed
```
```   925       def h \<equiv> "\<lambda>z. if z = x then g x - 1 else g z"
```
```   926       with Suc have "n = h x" by simp
```
```   927       with Suc have "f y ^^ h y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ h y"
```
```   928         by auto
```
```   929       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
```
```   930       from Suc h_def have "g x = Suc (h x)" by simp
```
```   931       then show ?case by (simp del: funpow.simps add: funpow_Suc_right o_assoc hyp2)
```
```   932         (simp add: comp_assoc hyp1)
```
```   933     qed
```
```   934   qed
```
```   935 qed
```
```   936
```
```   937
```
```   938 subsubsection \<open>Expressing set operations via @{const fold}\<close>
```
```   939
```
```   940 lemma comp_fun_commute_const:
```
```   941   "comp_fun_commute (\<lambda>_. f)"
```
```   942 proof
```
```   943 qed rule
```
```   944
```
```   945 lemma comp_fun_idem_insert:
```
```   946   "comp_fun_idem insert"
```
```   947 proof
```
```   948 qed auto
```
```   949
```
```   950 lemma comp_fun_idem_remove:
```
```   951   "comp_fun_idem Set.remove"
```
```   952 proof
```
```   953 qed auto
```
```   954
```
```   955 lemma (in semilattice_inf) comp_fun_idem_inf:
```
```   956   "comp_fun_idem inf"
```
```   957 proof
```
```   958 qed (auto simp add: inf_left_commute)
```
```   959
```
```   960 lemma (in semilattice_sup) comp_fun_idem_sup:
```
```   961   "comp_fun_idem sup"
```
```   962 proof
```
```   963 qed (auto simp add: sup_left_commute)
```
```   964
```
```   965 lemma union_fold_insert:
```
```   966   assumes "finite A"
```
```   967   shows "A \<union> B = fold insert B A"
```
```   968 proof -
```
```   969   interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
```
```   970   from \<open>finite A\<close> show ?thesis by (induct A arbitrary: B) simp_all
```
```   971 qed
```
```   972
```
```   973 lemma minus_fold_remove:
```
```   974   assumes "finite A"
```
```   975   shows "B - A = fold Set.remove B A"
```
```   976 proof -
```
```   977   interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove)
```
```   978   from \<open>finite A\<close> have "fold Set.remove B A = B - A" by (induct A arbitrary: B) auto
```
```   979   then show ?thesis ..
```
```   980 qed
```
```   981
```
```   982 lemma comp_fun_commute_filter_fold:
```
```   983   "comp_fun_commute (\<lambda>x A'. if P x then Set.insert x A' else A')"
```
```   984 proof -
```
```   985   interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
```
```   986   show ?thesis by standard (auto simp: fun_eq_iff)
```
```   987 qed
```
```   988
```
```   989 lemma Set_filter_fold:
```
```   990   assumes "finite A"
```
```   991   shows "Set.filter P A = fold (\<lambda>x A'. if P x then Set.insert x A' else A') {} A"
```
```   992 using assms
```
```   993 by (induct A)
```
```   994   (auto simp add: Set.filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold])
```
```   995
```
```   996 lemma inter_Set_filter:
```
```   997   assumes "finite B"
```
```   998   shows "A \<inter> B = Set.filter (\<lambda>x. x \<in> A) B"
```
```   999 using assms
```
```  1000 by (induct B) (auto simp: Set.filter_def)
```
```  1001
```
```  1002 lemma image_fold_insert:
```
```  1003   assumes "finite A"
```
```  1004   shows "image f A = fold (\<lambda>k A. Set.insert (f k) A) {} A"
```
```  1005 using assms
```
```  1006 proof -
```
```  1007   interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A" by standard auto
```
```  1008   show ?thesis using assms by (induct A) auto
```
```  1009 qed
```
```  1010
```
```  1011 lemma Ball_fold:
```
```  1012   assumes "finite A"
```
```  1013   shows "Ball A P = fold (\<lambda>k s. s \<and> P k) True A"
```
```  1014 using assms
```
```  1015 proof -
```
```  1016   interpret comp_fun_commute "\<lambda>k s. s \<and> P k" by standard auto
```
```  1017   show ?thesis using assms by (induct A) auto
```
```  1018 qed
```
```  1019
```
```  1020 lemma Bex_fold:
```
```  1021   assumes "finite A"
```
```  1022   shows "Bex A P = fold (\<lambda>k s. s \<or> P k) False A"
```
```  1023 using assms
```
```  1024 proof -
```
```  1025   interpret comp_fun_commute "\<lambda>k s. s \<or> P k" by standard auto
```
```  1026   show ?thesis using assms by (induct A) auto
```
```  1027 qed
```
```  1028
```
```  1029 lemma comp_fun_commute_Pow_fold:
```
```  1030   "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)"
```
```  1031   by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast
```
```  1032
```
```  1033 lemma Pow_fold:
```
```  1034   assumes "finite A"
```
```  1035   shows "Pow A = fold (\<lambda>x A. A \<union> Set.insert x ` A) {{}} A"
```
```  1036 using assms
```
```  1037 proof -
```
```  1038   interpret comp_fun_commute "\<lambda>x A. A \<union> Set.insert x ` A" by (rule comp_fun_commute_Pow_fold)
```
```  1039   show ?thesis using assms by (induct A) (auto simp: Pow_insert)
```
```  1040 qed
```
```  1041
```
```  1042 lemma fold_union_pair:
```
```  1043   assumes "finite B"
```
```  1044   shows "(\<Union>y\<in>B. {(x, y)}) \<union> A = fold (\<lambda>y. Set.insert (x, y)) A B"
```
```  1045 proof -
```
```  1046   interpret comp_fun_commute "\<lambda>y. Set.insert (x, y)" by standard auto
```
```  1047   show ?thesis using assms  by (induct B arbitrary: A) simp_all
```
```  1048 qed
```
```  1049
```
```  1050 lemma comp_fun_commute_product_fold:
```
```  1051   assumes "finite B"
```
```  1052   shows "comp_fun_commute (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B)"
```
```  1053   by standard (auto simp: fold_union_pair[symmetric] assms)
```
```  1054
```
```  1055 lemma product_fold:
```
```  1056   assumes "finite A"
```
```  1057   assumes "finite B"
```
```  1058   shows "A \<times> B = fold (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B) {} A"
```
```  1059 using assms unfolding Sigma_def
```
```  1060 by (induct A)
```
```  1061   (simp_all add: comp_fun_commute.fold_insert[OF comp_fun_commute_product_fold] fold_union_pair)
```
```  1062
```
```  1063
```
```  1064 context complete_lattice
```
```  1065 begin
```
```  1066
```
```  1067 lemma inf_Inf_fold_inf:
```
```  1068   assumes "finite A"
```
```  1069   shows "inf (Inf A) B = fold inf B A"
```
```  1070 proof -
```
```  1071   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
```
```  1072   from \<open>finite A\<close> fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
```
```  1073     (simp_all add: inf_commute fun_eq_iff)
```
```  1074 qed
```
```  1075
```
```  1076 lemma sup_Sup_fold_sup:
```
```  1077   assumes "finite A"
```
```  1078   shows "sup (Sup A) B = fold sup B A"
```
```  1079 proof -
```
```  1080   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
```
```  1081   from \<open>finite A\<close> fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
```
```  1082     (simp_all add: sup_commute fun_eq_iff)
```
```  1083 qed
```
```  1084
```
```  1085 lemma Inf_fold_inf:
```
```  1086   assumes "finite A"
```
```  1087   shows "Inf A = fold inf top A"
```
```  1088   using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
```
```  1089
```
```  1090 lemma Sup_fold_sup:
```
```  1091   assumes "finite A"
```
```  1092   shows "Sup A = fold sup bot A"
```
```  1093   using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
```
```  1094
```
```  1095 lemma inf_INF_fold_inf:
```
```  1096   assumes "finite A"
```
```  1097   shows "inf B (INFIMUM A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold")
```
```  1098 proof (rule sym)
```
```  1099   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
```
```  1100   interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
```
```  1101   from \<open>finite A\<close> show "?fold = ?inf"
```
```  1102     by (induct A arbitrary: B)
```
```  1103       (simp_all add: inf_left_commute)
```
```  1104 qed
```
```  1105
```
```  1106 lemma sup_SUP_fold_sup:
```
```  1107   assumes "finite A"
```
```  1108   shows "sup B (SUPREMUM A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold")
```
```  1109 proof (rule sym)
```
```  1110   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
```
```  1111   interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
```
```  1112   from \<open>finite A\<close> show "?fold = ?sup"
```
```  1113     by (induct A arbitrary: B)
```
```  1114       (simp_all add: sup_left_commute)
```
```  1115 qed
```
```  1116
```
```  1117 lemma INF_fold_inf:
```
```  1118   assumes "finite A"
```
```  1119   shows "INFIMUM A f = fold (inf \<circ> f) top A"
```
```  1120   using assms inf_INF_fold_inf [of A top] by simp
```
```  1121
```
```  1122 lemma SUP_fold_sup:
```
```  1123   assumes "finite A"
```
```  1124   shows "SUPREMUM A f = fold (sup \<circ> f) bot A"
```
```  1125   using assms sup_SUP_fold_sup [of A bot] by simp
```
```  1126
```
```  1127 end
```
```  1128
```
```  1129
```
```  1130 subsection \<open>Locales as mini-packages for fold operations\<close>
```
```  1131
```
```  1132 subsubsection \<open>The natural case\<close>
```
```  1133
```
```  1134 locale folding =
```
```  1135   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
```
```  1136   fixes z :: "'b"
```
```  1137   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
```
```  1138 begin
```
```  1139
```
```  1140 interpretation fold?: comp_fun_commute f
```
```  1141   by standard (insert comp_fun_commute, simp add: fun_eq_iff)
```
```  1142
```
```  1143 definition F :: "'a set \<Rightarrow> 'b"
```
```  1144 where
```
```  1145   eq_fold: "F A = fold f z A"
```
```  1146
```
```  1147 lemma empty [simp]:"F {} = z"
```
```  1148   by (simp add: eq_fold)
```
```  1149
```
```  1150 lemma infinite [simp]: "\<not> finite A \<Longrightarrow> F A = z"
```
```  1151   by (simp add: eq_fold)
```
```  1152
```
```  1153 lemma insert [simp]:
```
```  1154   assumes "finite A" and "x \<notin> A"
```
```  1155   shows "F (insert x A) = f x (F A)"
```
```  1156 proof -
```
```  1157   from fold_insert assms
```
```  1158   have "fold f z (insert x A) = f x (fold f z A)" by simp
```
```  1159   with \<open>finite A\<close> show ?thesis by (simp add: eq_fold fun_eq_iff)
```
```  1160 qed
```
```  1161
```
```  1162 lemma remove:
```
```  1163   assumes "finite A" and "x \<in> A"
```
```  1164   shows "F A = f x (F (A - {x}))"
```
```  1165 proof -
```
```  1166   from \<open>x \<in> A\<close> obtain B where A: "A = insert x B" and "x \<notin> B"
```
```  1167     by (auto dest: mk_disjoint_insert)
```
```  1168   moreover from \<open>finite A\<close> A have "finite B" by simp
```
```  1169   ultimately show ?thesis by simp
```
```  1170 qed
```
```  1171
```
```  1172 lemma insert_remove:
```
```  1173   assumes "finite A"
```
```  1174   shows "F (insert x A) = f x (F (A - {x}))"
```
```  1175   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
```
```  1176
```
```  1177 end
```
```  1178
```
```  1179
```
```  1180 subsubsection \<open>With idempotency\<close>
```
```  1181
```
```  1182 locale folding_idem = folding +
```
```  1183   assumes comp_fun_idem: "f x \<circ> f x = f x"
```
```  1184 begin
```
```  1185
```
```  1186 declare insert [simp del]
```
```  1187
```
```  1188 interpretation fold?: comp_fun_idem f
```
```  1189   by standard (insert comp_fun_commute comp_fun_idem, simp add: fun_eq_iff)
```
```  1190
```
```  1191 lemma insert_idem [simp]:
```
```  1192   assumes "finite A"
```
```  1193   shows "F (insert x A) = f x (F A)"
```
```  1194 proof -
```
```  1195   from fold_insert_idem assms
```
```  1196   have "fold f z (insert x A) = f x (fold f z A)" by simp
```
```  1197   with \<open>finite A\<close> show ?thesis by (simp add: eq_fold fun_eq_iff)
```
```  1198 qed
```
```  1199
```
```  1200 end
```
```  1201
```
```  1202
```
```  1203 subsection \<open>Finite cardinality\<close>
```
```  1204
```
```  1205 text \<open>
```
```  1206   The traditional definition
```
```  1207   @{prop "card A \<equiv> LEAST n. EX f. A = {f i | i. i < n}"}
```
```  1208   is ugly to work with.
```
```  1209   But now that we have @{const fold} things are easy:
```
```  1210 \<close>
```
```  1211
```
```  1212 global_interpretation card: folding "\<lambda>_. Suc" 0
```
```  1213   defines card = "folding.F (\<lambda>_. Suc) 0"
```
```  1214   by standard rule
```
```  1215
```
```  1216 lemma card_infinite:
```
```  1217   "\<not> finite A \<Longrightarrow> card A = 0"
```
```  1218   by (fact card.infinite)
```
```  1219
```
```  1220 lemma card_empty:
```
```  1221   "card {} = 0"
```
```  1222   by (fact card.empty)
```
```  1223
```
```  1224 lemma card_insert_disjoint:
```
```  1225   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> card (insert x A) = Suc (card A)"
```
```  1226   by (fact card.insert)
```
```  1227
```
```  1228 lemma card_insert_if:
```
```  1229   "finite A \<Longrightarrow> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
```
```  1230   by auto (simp add: card.insert_remove card.remove)
```
```  1231
```
```  1232 lemma card_ge_0_finite:
```
```  1233   "card A > 0 \<Longrightarrow> finite A"
```
```  1234   by (rule ccontr) simp
```
```  1235
```
```  1236 lemma card_0_eq [simp]:
```
```  1237   "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
```
```  1238   by (auto dest: mk_disjoint_insert)
```
```  1239
```
```  1240 lemma finite_UNIV_card_ge_0:
```
```  1241   "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
```
```  1242   by (rule ccontr) simp
```
```  1243
```
```  1244 lemma card_eq_0_iff:
```
```  1245   "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
```
```  1246   by auto
```
```  1247
```
```  1248 lemma card_gt_0_iff:
```
```  1249   "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
```
```  1250   by (simp add: neq0_conv [symmetric] card_eq_0_iff)
```
```  1251
```
```  1252 lemma card_Suc_Diff1:
```
```  1253   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> Suc (card (A - {x})) = card A"
```
```  1254 apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
```
```  1255 apply(simp del:insert_Diff_single)
```
```  1256 done
```
```  1257
```
```  1258 lemma card_insert_le_m1: "n>0 \<Longrightarrow> card y \<le> n-1 \<Longrightarrow> card (insert x y) \<le> n"
```
```  1259   apply (cases "finite y")
```
```  1260   apply (cases "x \<in> y")
```
```  1261   apply (auto simp: insert_absorb)
```
```  1262   done
```
```  1263
```
```  1264 lemma card_Diff_singleton:
```
```  1265   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) = card A - 1"
```
```  1266   by (simp add: card_Suc_Diff1 [symmetric])
```
```  1267
```
```  1268 lemma card_Diff_singleton_if:
```
```  1269   "finite A \<Longrightarrow> card (A - {x}) = (if x \<in> A then card A - 1 else card A)"
```
```  1270   by (simp add: card_Diff_singleton)
```
```  1271
```
```  1272 lemma card_Diff_insert[simp]:
```
```  1273   assumes "finite A" and "a \<in> A" and "a \<notin> B"
```
```  1274   shows "card (A - insert a B) = card (A - B) - 1"
```
```  1275 proof -
```
```  1276   have "A - insert a B = (A - B) - {a}" using assms by blast
```
```  1277   then show ?thesis using assms by(simp add: card_Diff_singleton)
```
```  1278 qed
```
```  1279
```
```  1280 lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
```
```  1281   by (fact card.insert_remove)
```
```  1282
```
```  1283 lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
```
```  1284 by (simp add: card_insert_if)
```
```  1285
```
```  1286 lemma card_Collect_less_nat[simp]: "card{i::nat. i < n} = n"
```
```  1287 by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
```
```  1288
```
```  1289 lemma card_Collect_le_nat[simp]: "card{i::nat. i <= n} = Suc n"
```
```  1290 using card_Collect_less_nat[of "Suc n"] by(simp add: less_Suc_eq_le)
```
```  1291
```
```  1292 lemma card_mono:
```
```  1293   assumes "finite B" and "A \<subseteq> B"
```
```  1294   shows "card A \<le> card B"
```
```  1295 proof -
```
```  1296   from assms have "finite A" by (auto intro: finite_subset)
```
```  1297   then show ?thesis using assms proof (induct A arbitrary: B)
```
```  1298     case empty then show ?case by simp
```
```  1299   next
```
```  1300     case (insert x A)
```
```  1301     then have "x \<in> B" by simp
```
```  1302     from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
```
```  1303     with insert.hyps have "card A \<le> card (B - {x})" by auto
```
```  1304     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)
```
```  1305   qed
```
```  1306 qed
```
```  1307
```
```  1308 lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
```
```  1309 apply (induct rule: finite_induct)
```
```  1310 apply simp
```
```  1311 apply clarify
```
```  1312 apply (subgoal_tac "finite A & A - {x} <= F")
```
```  1313  prefer 2 apply (blast intro: finite_subset, atomize)
```
```  1314 apply (drule_tac x = "A - {x}" in spec)
```
```  1315 apply (simp add: card_Diff_singleton_if split add: if_split_asm)
```
```  1316 apply (case_tac "card A", auto)
```
```  1317 done
```
```  1318
```
```  1319 lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
```
```  1320 apply (simp add: psubset_eq linorder_not_le [symmetric])
```
```  1321 apply (blast dest: card_seteq)
```
```  1322 done
```
```  1323
```
```  1324 lemma card_Un_Int:
```
```  1325   assumes "finite A" and "finite B"
```
```  1326   shows "card A + card B = card (A \<union> B) + card (A \<inter> B)"
```
```  1327 using assms proof (induct A)
```
```  1328   case empty then show ?case by simp
```
```  1329 next
```
```  1330  case (insert x A) then show ?case
```
```  1331     by (auto simp add: insert_absorb Int_insert_left)
```
```  1332 qed
```
```  1333
```
```  1334 lemma card_Un_disjoint:
```
```  1335   assumes "finite A" and "finite B"
```
```  1336   assumes "A \<inter> B = {}"
```
```  1337   shows "card (A \<union> B) = card A + card B"
```
```  1338 using assms card_Un_Int [of A B] by simp
```
```  1339
```
```  1340 lemma card_Un_le: "card (A \<union> B) \<le> card A + card B"
```
```  1341 apply(cases "finite A")
```
```  1342  apply(cases "finite B")
```
```  1343   using le_iff_add card_Un_Int apply blast
```
```  1344  apply simp
```
```  1345 apply simp
```
```  1346 done
```
```  1347
```
```  1348 lemma card_Diff_subset:
```
```  1349   assumes "finite B" and "B \<subseteq> A"
```
```  1350   shows "card (A - B) = card A - card B"
```
```  1351 proof (cases "finite A")
```
```  1352   case False with assms show ?thesis by simp
```
```  1353 next
```
```  1354   case True with assms show ?thesis by (induct B arbitrary: A) simp_all
```
```  1355 qed
```
```  1356
```
```  1357 lemma card_Diff_subset_Int:
```
```  1358   assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
```
```  1359 proof -
```
```  1360   have "A - B = A - A \<inter> B" by auto
```
```  1361   thus ?thesis
```
```  1362     by (simp add: card_Diff_subset AB)
```
```  1363 qed
```
```  1364
```
```  1365 lemma diff_card_le_card_Diff:
```
```  1366 assumes "finite B" shows "card A - card B \<le> card(A - B)"
```
```  1367 proof-
```
```  1368   have "card A - card B \<le> card A - card (A \<inter> B)"
```
```  1369     using card_mono[OF assms Int_lower2, of A] by arith
```
```  1370   also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
```
```  1371   finally show ?thesis .
```
```  1372 qed
```
```  1373
```
```  1374 lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
```
```  1375 apply (rule Suc_less_SucD)
```
```  1376 apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
```
```  1377 done
```
```  1378
```
```  1379 lemma card_Diff2_less:
```
```  1380   "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
```
```  1381 apply (case_tac "x = y")
```
```  1382  apply (simp add: card_Diff1_less del:card_Diff_insert)
```
```  1383 apply (rule less_trans)
```
```  1384  prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
```
```  1385 done
```
```  1386
```
```  1387 lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
```
```  1388 apply (case_tac "x : A")
```
```  1389  apply (simp_all add: card_Diff1_less less_imp_le)
```
```  1390 done
```
```  1391
```
```  1392 lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
```
```  1393 by (erule psubsetI, blast)
```
```  1394
```
```  1395 lemma card_le_inj:
```
```  1396   assumes fA: "finite A"
```
```  1397     and fB: "finite B"
```
```  1398     and c: "card A \<le> card B"
```
```  1399   shows "\<exists>f. f ` A \<subseteq> B \<and> inj_on f A"
```
```  1400   using fA fB c
```
```  1401 proof (induct arbitrary: B rule: finite_induct)
```
```  1402   case empty
```
```  1403   then show ?case by simp
```
```  1404 next
```
```  1405   case (insert x s t)
```
```  1406   then show ?case
```
```  1407   proof (induct rule: finite_induct[OF "insert.prems"(1)])
```
```  1408     case 1
```
```  1409     then show ?case by simp
```
```  1410   next
```
```  1411     case (2 y t)
```
```  1412     from "2.prems"(1,2,5) "2.hyps"(1,2) have cst: "card s \<le> card t"
```
```  1413       by simp
```
```  1414     from "2.prems"(3) [OF "2.hyps"(1) cst]
```
```  1415     obtain f where "f ` s \<subseteq> t" "inj_on f s"
```
```  1416       by blast
```
```  1417     with "2.prems"(2) "2.hyps"(2) show ?case
```
```  1418       apply -
```
```  1419       apply (rule exI[where x = "\<lambda>z. if z = x then y else f z"])
```
```  1420       apply (auto simp add: inj_on_def)
```
```  1421       done
```
```  1422   qed
```
```  1423 qed
```
```  1424
```
```  1425 lemma card_subset_eq:
```
```  1426   assumes fB: "finite B"
```
```  1427     and AB: "A \<subseteq> B"
```
```  1428     and c: "card A = card B"
```
```  1429   shows "A = B"
```
```  1430 proof -
```
```  1431   from fB AB have fA: "finite A"
```
```  1432     by (auto intro: finite_subset)
```
```  1433   from fA fB have fBA: "finite (B - A)"
```
```  1434     by auto
```
```  1435   have e: "A \<inter> (B - A) = {}"
```
```  1436     by blast
```
```  1437   have eq: "A \<union> (B - A) = B"
```
```  1438     using AB by blast
```
```  1439   from card_Un_disjoint[OF fA fBA e, unfolded eq c] have "card (B - A) = 0"
```
```  1440     by arith
```
```  1441   then have "B - A = {}"
```
```  1442     unfolding card_eq_0_iff using fA fB by simp
```
```  1443   with AB show "A = B"
```
```  1444     by blast
```
```  1445 qed
```
```  1446
```
```  1447 lemma insert_partition:
```
```  1448   "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
```
```  1449   \<Longrightarrow> x \<inter> \<Union>F = {}"
```
```  1450 by auto
```
```  1451
```
```  1452 lemma finite_psubset_induct[consumes 1, case_names psubset]:
```
```  1453   assumes fin: "finite A"
```
```  1454   and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A"
```
```  1455   shows "P A"
```
```  1456 using fin
```
```  1457 proof (induct A taking: card rule: measure_induct_rule)
```
```  1458   case (less A)
```
```  1459   have fin: "finite A" by fact
```
```  1460   have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
```
```  1461   { fix B
```
```  1462     assume asm: "B \<subset> A"
```
```  1463     from asm have "card B < card A" using psubset_card_mono fin by blast
```
```  1464     moreover
```
```  1465     from asm have "B \<subseteq> A" by auto
```
```  1466     then have "finite B" using fin finite_subset by blast
```
```  1467     ultimately
```
```  1468     have "P B" using ih by simp
```
```  1469   }
```
```  1470   with fin show "P A" using major by blast
```
```  1471 qed
```
```  1472
```
```  1473 lemma finite_induct_select[consumes 1, case_names empty select]:
```
```  1474   assumes "finite S"
```
```  1475   assumes "P {}"
```
```  1476   assumes select: "\<And>T. T \<subset> S \<Longrightarrow> P T \<Longrightarrow> \<exists>s\<in>S - T. P (insert s T)"
```
```  1477   shows "P S"
```
```  1478 proof -
```
```  1479   have "0 \<le> card S" by simp
```
```  1480   then have "\<exists>T \<subseteq> S. card T = card S \<and> P T"
```
```  1481   proof (induct rule: dec_induct)
```
```  1482     case base with \<open>P {}\<close> show ?case
```
```  1483       by (intro exI[of _ "{}"]) auto
```
```  1484   next
```
```  1485     case (step n)
```
```  1486     then obtain T where T: "T \<subseteq> S" "card T = n" "P T"
```
```  1487       by auto
```
```  1488     with \<open>n < card S\<close> have "T \<subset> S" "P T"
```
```  1489       by auto
```
```  1490     with select[of T] obtain s where "s \<in> S" "s \<notin> T" "P (insert s T)"
```
```  1491       by auto
```
```  1492     with step(2) T \<open>finite S\<close> show ?case
```
```  1493       by (intro exI[of _ "insert s T"]) (auto dest: finite_subset)
```
```  1494   qed
```
```  1495   with \<open>finite S\<close> show "P S"
```
```  1496     by (auto dest: card_subset_eq)
```
```  1497 qed
```
```  1498
```
```  1499 text\<open>main cardinality theorem\<close>
```
```  1500 lemma card_partition [rule_format]:
```
```  1501   "finite C ==>
```
```  1502      finite (\<Union>C) -->
```
```  1503      (\<forall>c\<in>C. card c = k) -->
```
```  1504      (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
```
```  1505      k * card(C) = card (\<Union>C)"
```
```  1506 apply (erule finite_induct, simp)
```
```  1507 apply (simp add: card_Un_disjoint insert_partition
```
```  1508        finite_subset [of _ "\<Union>(insert x F)"])
```
```  1509 done
```
```  1510
```
```  1511 lemma card_eq_UNIV_imp_eq_UNIV:
```
```  1512   assumes fin: "finite (UNIV :: 'a set)"
```
```  1513   and card: "card A = card (UNIV :: 'a set)"
```
```  1514   shows "A = (UNIV :: 'a set)"
```
```  1515 proof
```
```  1516   show "A \<subseteq> UNIV" by simp
```
```  1517   show "UNIV \<subseteq> A"
```
```  1518   proof
```
```  1519     fix x
```
```  1520     show "x \<in> A"
```
```  1521     proof (rule ccontr)
```
```  1522       assume "x \<notin> A"
```
```  1523       then have "A \<subset> UNIV" by auto
```
```  1524       with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
```
```  1525       with card show False by simp
```
```  1526     qed
```
```  1527   qed
```
```  1528 qed
```
```  1529
```
```  1530 text\<open>The form of a finite set of given cardinality\<close>
```
```  1531
```
```  1532 lemma card_eq_SucD:
```
```  1533 assumes "card A = Suc k"
```
```  1534 shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
```
```  1535 proof -
```
```  1536   have fin: "finite A" using assms by (auto intro: ccontr)
```
```  1537   moreover have "card A \<noteq> 0" using assms by auto
```
```  1538   ultimately obtain b where b: "b \<in> A" by auto
```
```  1539   show ?thesis
```
```  1540   proof (intro exI conjI)
```
```  1541     show "A = insert b (A-{b})" using b by blast
```
```  1542     show "b \<notin> A - {b}" by blast
```
```  1543     show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
```
```  1544       using assms b fin by(fastforce dest:mk_disjoint_insert)+
```
```  1545   qed
```
```  1546 qed
```
```  1547
```
```  1548 lemma card_Suc_eq:
```
```  1549   "(card A = Suc k) =
```
```  1550    (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
```
```  1551  apply(auto elim!: card_eq_SucD)
```
```  1552  apply(subst card.insert)
```
```  1553  apply(auto simp add: intro:ccontr)
```
```  1554  done
```
```  1555
```
```  1556 lemma card_1_singletonE:
```
```  1557     assumes "card A = 1" obtains x where "A = {x}"
```
```  1558   using assms by (auto simp: card_Suc_eq)
```
```  1559
```
```  1560 lemma card_le_Suc_iff: "finite A \<Longrightarrow>
```
```  1561   Suc n \<le> card A = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
```
```  1562 by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
```
```  1563   dest: subset_singletonD split: nat.splits if_splits)
```
```  1564
```
```  1565 lemma finite_fun_UNIVD2:
```
```  1566   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
```
```  1567   shows "finite (UNIV :: 'b set)"
```
```  1568 proof -
```
```  1569   from fin have "\<And>arbitrary. finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
```
```  1570     by (rule finite_imageI)
```
```  1571   moreover have "\<And>arbitrary. UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
```
```  1572     by (rule UNIV_eq_I) auto
```
```  1573   ultimately show "finite (UNIV :: 'b set)" by simp
```
```  1574 qed
```
```  1575
```
```  1576 lemma card_UNIV_unit [simp]: "card (UNIV :: unit set) = 1"
```
```  1577   unfolding UNIV_unit by simp
```
```  1578
```
```  1579 lemma infinite_arbitrarily_large:
```
```  1580   assumes "\<not> finite A"
```
```  1581   shows "\<exists>B. finite B \<and> card B = n \<and> B \<subseteq> A"
```
```  1582 proof (induction n)
```
```  1583   case 0 show ?case by (intro exI[of _ "{}"]) auto
```
```  1584 next
```
```  1585   case (Suc n)
```
```  1586   then guess B .. note B = this
```
```  1587   with \<open>\<not> finite A\<close> have "A \<noteq> B" by auto
```
```  1588   with B have "B \<subset> A" by auto
```
```  1589   hence "\<exists>x. x \<in> A - B" by (elim psubset_imp_ex_mem)
```
```  1590   then guess x .. note x = this
```
```  1591   with B have "finite (insert x B) \<and> card (insert x B) = Suc n \<and> insert x B \<subseteq> A"
```
```  1592     by auto
```
```  1593   thus "\<exists>B. finite B \<and> card B = Suc n \<and> B \<subseteq> A" ..
```
```  1594 qed
```
```  1595
```
```  1596 subsubsection \<open>Cardinality of image\<close>
```
```  1597
```
```  1598 lemma card_image_le: "finite A ==> card (f ` A) \<le> card A"
```
```  1599   by (induct rule: finite_induct) (simp_all add: le_SucI card_insert_if)
```
```  1600
```
```  1601 lemma card_image:
```
```  1602   assumes "inj_on f A"
```
```  1603   shows "card (f ` A) = card A"
```
```  1604 proof (cases "finite A")
```
```  1605   case True then show ?thesis using assms by (induct A) simp_all
```
```  1606 next
```
```  1607   case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
```
```  1608   with False show ?thesis by simp
```
```  1609 qed
```
```  1610
```
```  1611 lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
```
```  1612 by(auto simp: card_image bij_betw_def)
```
```  1613
```
```  1614 lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
```
```  1615 by (simp add: card_seteq card_image)
```
```  1616
```
```  1617 lemma eq_card_imp_inj_on:
```
```  1618   assumes "finite A" "card(f ` A) = card A" shows "inj_on f A"
```
```  1619 using assms
```
```  1620 proof (induct rule:finite_induct)
```
```  1621   case empty show ?case by simp
```
```  1622 next
```
```  1623   case (insert x A)
```
```  1624   then show ?case using card_image_le [of A f]
```
```  1625     by (simp add: card_insert_if split: if_splits)
```
```  1626 qed
```
```  1627
```
```  1628 lemma inj_on_iff_eq_card: "finite A \<Longrightarrow> inj_on f A \<longleftrightarrow> card(f ` A) = card A"
```
```  1629   by (blast intro: card_image eq_card_imp_inj_on)
```
```  1630
```
```  1631 lemma card_inj_on_le:
```
```  1632   assumes "inj_on f A" "f ` A \<subseteq> B" "finite B" shows "card A \<le> card B"
```
```  1633 proof -
```
```  1634   have "finite A" using assms
```
```  1635     by (blast intro: finite_imageD dest: finite_subset)
```
```  1636   then show ?thesis using assms
```
```  1637    by (force intro: card_mono simp: card_image [symmetric])
```
```  1638 qed
```
```  1639
```
```  1640 lemma surj_card_le: "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> card B \<le> card A"
```
```  1641   by (blast intro: card_image_le card_mono le_trans)
```
```  1642
```
```  1643 lemma card_bij_eq:
```
```  1644   "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
```
```  1645      finite A; finite B |] ==> card A = card B"
```
```  1646 by (auto intro: le_antisym card_inj_on_le)
```
```  1647
```
```  1648 lemma bij_betw_finite:
```
```  1649   assumes "bij_betw f A B"
```
```  1650   shows "finite A \<longleftrightarrow> finite B"
```
```  1651 using assms unfolding bij_betw_def
```
```  1652 using finite_imageD[of f A] by auto
```
```  1653
```
```  1654 lemma inj_on_finite:
```
```  1655 assumes "inj_on f A" "f ` A \<le> B" "finite B"
```
```  1656 shows "finite A"
```
```  1657 using assms finite_imageD finite_subset by blast
```
```  1658
```
```  1659 lemma card_vimage_inj: "\<lbrakk> inj f; A \<subseteq> range f \<rbrakk> \<Longrightarrow> card (f -` A) = card A"
```
```  1660 by(auto 4 3 simp add: subset_image_iff inj_vimage_image_eq intro: card_image[symmetric, OF subset_inj_on])
```
```  1661
```
```  1662 subsubsection \<open>Pigeonhole Principles\<close>
```
```  1663
```
```  1664 lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
```
```  1665 by (auto dest: card_image less_irrefl_nat)
```
```  1666
```
```  1667 lemma pigeonhole_infinite:
```
```  1668 assumes  "~ finite A" and "finite(f`A)"
```
```  1669 shows "EX a0:A. ~finite{a:A. f a = f a0}"
```
```  1670 proof -
```
```  1671   have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
```
```  1672   proof(induct "f`A" arbitrary: A rule: finite_induct)
```
```  1673     case empty thus ?case by simp
```
```  1674   next
```
```  1675     case (insert b F)
```
```  1676     show ?case
```
```  1677     proof cases
```
```  1678       assume "finite{a:A. f a = b}"
```
```  1679       hence "~ finite(A - {a:A. f a = b})" using \<open>\<not> finite A\<close> by simp
```
```  1680       also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
```
```  1681       finally have "~ finite({a:A. f a \<noteq> b})" .
```
```  1682       from insert(3)[OF _ this]
```
```  1683       show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
```
```  1684     next
```
```  1685       assume 1: "~finite{a:A. f a = b}"
```
```  1686       hence "{a \<in> A. f a = b} \<noteq> {}" by force
```
```  1687       thus ?thesis using 1 by blast
```
```  1688     qed
```
```  1689   qed
```
```  1690   from this[OF assms(2,1)] show ?thesis .
```
```  1691 qed
```
```  1692
```
```  1693 lemma pigeonhole_infinite_rel:
```
```  1694 assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
```
```  1695 shows "EX b:B. ~finite{a:A. R a b}"
```
```  1696 proof -
```
```  1697    let ?F = "%a. {b:B. R a b}"
```
```  1698    from finite_Pow_iff[THEN iffD2, OF \<open>finite B\<close>]
```
```  1699    have "finite(?F ` A)" by(blast intro: rev_finite_subset)
```
```  1700    from pigeonhole_infinite[where f = ?F, OF assms(1) this]
```
```  1701    obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
```
```  1702    obtain b0 where "b0 : B" and "R a0 b0" using \<open>a0:A\<close> assms(3) by blast
```
```  1703    { assume "finite{a:A. R a b0}"
```
```  1704      then have "finite {a\<in>A. ?F a = ?F a0}"
```
```  1705        using \<open>b0 : B\<close> \<open>R a0 b0\<close> by(blast intro: rev_finite_subset)
```
```  1706    }
```
```  1707    with 1 \<open>b0 : B\<close> show ?thesis by blast
```
```  1708 qed
```
```  1709
```
```  1710
```
```  1711 subsubsection \<open>Cardinality of sums\<close>
```
```  1712
```
```  1713 lemma card_Plus:
```
```  1714   assumes "finite A" and "finite B"
```
```  1715   shows "card (A <+> B) = card A + card B"
```
```  1716 proof -
```
```  1717   have "Inl`A \<inter> Inr`B = {}" by fast
```
```  1718   with assms show ?thesis
```
```  1719     unfolding Plus_def
```
```  1720     by (simp add: card_Un_disjoint card_image)
```
```  1721 qed
```
```  1722
```
```  1723 lemma card_Plus_conv_if:
```
```  1724   "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
```
```  1725   by (auto simp add: card_Plus)
```
```  1726
```
```  1727 text \<open>Relates to equivalence classes.  Based on a theorem of F. Kamm\"uller.\<close>
```
```  1728
```
```  1729 lemma dvd_partition:
```
```  1730   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 = {}"
```
```  1731     shows "k dvd card (\<Union>C)"
```
```  1732 proof -
```
```  1733   have "finite C"
```
```  1734     by (rule finite_UnionD [OF f])
```
```  1735   then show ?thesis using assms
```
```  1736   proof (induct rule: finite_induct)
```
```  1737     case empty show ?case by simp
```
```  1738   next
```
```  1739     case (insert c C)
```
```  1740     then show ?case
```
```  1741       apply simp
```
```  1742       apply (subst card_Un_disjoint)
```
```  1743       apply (auto simp add: disjoint_eq_subset_Compl)
```
```  1744       done
```
```  1745   qed
```
```  1746 qed
```
```  1747
```
```  1748 subsubsection \<open>Relating injectivity and surjectivity\<close>
```
```  1749
```
```  1750 lemma finite_surj_inj: assumes "finite A" "A \<subseteq> f ` A" shows "inj_on f A"
```
```  1751 proof -
```
```  1752   have "f ` A = A"
```
```  1753     by (rule card_seteq [THEN sym]) (auto simp add: assms card_image_le)
```
```  1754   then show ?thesis using assms
```
```  1755     by (simp add: eq_card_imp_inj_on)
```
```  1756 qed
```
```  1757
```
```  1758 lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
```
```  1759 shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
```
```  1760 by (blast intro: finite_surj_inj subset_UNIV)
```
```  1761
```
```  1762 lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
```
```  1763 shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
```
```  1764 by(fastforce simp:surj_def dest!: endo_inj_surj)
```
```  1765
```
```  1766 corollary infinite_UNIV_nat [iff]:
```
```  1767   "\<not> finite (UNIV :: nat set)"
```
```  1768 proof
```
```  1769   assume "finite (UNIV :: nat set)"
```
```  1770   with finite_UNIV_inj_surj [of Suc]
```
```  1771   show False by simp (blast dest: Suc_neq_Zero surjD)
```
```  1772 qed
```
```  1773
```
```  1774 lemma infinite_UNIV_char_0:
```
```  1775   "\<not> finite (UNIV :: 'a::semiring_char_0 set)"
```
```  1776 proof
```
```  1777   assume "finite (UNIV :: 'a set)"
```
```  1778   with subset_UNIV have "finite (range of_nat :: 'a set)"
```
```  1779     by (rule finite_subset)
```
```  1780   moreover have "inj (of_nat :: nat \<Rightarrow> 'a)"
```
```  1781     by (simp add: inj_on_def)
```
```  1782   ultimately have "finite (UNIV :: nat set)"
```
```  1783     by (rule finite_imageD)
```
```  1784   then show False
```
```  1785     by simp
```
```  1786 qed
```
```  1787
```
```  1788 hide_const (open) Finite_Set.fold
```
```  1789
```
```  1790
```
```  1791 subsection "Infinite Sets"
```
```  1792
```
```  1793 text \<open>
```
```  1794   Some elementary facts about infinite sets, mostly by Stephan Merz.
```
```  1795   Beware! Because "infinite" merely abbreviates a negation, these
```
```  1796   lemmas may not work well with \<open>blast\<close>.
```
```  1797 \<close>
```
```  1798
```
```  1799 abbreviation infinite :: "'a set \<Rightarrow> bool"
```
```  1800   where "infinite S \<equiv> \<not> finite S"
```
```  1801
```
```  1802 text \<open>
```
```  1803   Infinite sets are non-empty, and if we remove some elements from an
```
```  1804   infinite set, the result is still infinite.
```
```  1805 \<close>
```
```  1806
```
```  1807 lemma infinite_imp_nonempty: "infinite S \<Longrightarrow> S \<noteq> {}"
```
```  1808   by auto
```
```  1809
```
```  1810 lemma infinite_remove: "infinite S \<Longrightarrow> infinite (S - {a})"
```
```  1811   by simp
```
```  1812
```
```  1813 lemma Diff_infinite_finite:
```
```  1814   assumes T: "finite T" and S: "infinite S"
```
```  1815   shows "infinite (S - T)"
```
```  1816   using T
```
```  1817 proof induct
```
```  1818   from S
```
```  1819   show "infinite (S - {})" by auto
```
```  1820 next
```
```  1821   fix T x
```
```  1822   assume ih: "infinite (S - T)"
```
```  1823   have "S - (insert x T) = (S - T) - {x}"
```
```  1824     by (rule Diff_insert)
```
```  1825   with ih
```
```  1826   show "infinite (S - (insert x T))"
```
```  1827     by (simp add: infinite_remove)
```
```  1828 qed
```
```  1829
```
```  1830 lemma Un_infinite: "infinite S \<Longrightarrow> infinite (S \<union> T)"
```
```  1831   by simp
```
```  1832
```
```  1833 lemma infinite_Un: "infinite (S \<union> T) \<longleftrightarrow> infinite S \<or> infinite T"
```
```  1834   by simp
```
```  1835
```
```  1836 lemma infinite_super:
```
```  1837   assumes T: "S \<subseteq> T" and S: "infinite S"
```
```  1838   shows "infinite T"
```
```  1839 proof
```
```  1840   assume "finite T"
```
```  1841   with T have "finite S" by (simp add: finite_subset)
```
```  1842   with S show False by simp
```
```  1843 qed
```
```  1844
```
```  1845 proposition infinite_coinduct [consumes 1, case_names infinite]:
```
```  1846   assumes "X A"
```
```  1847   and step: "\<And>A. X A \<Longrightarrow> \<exists>x\<in>A. X (A - {x}) \<or> infinite (A - {x})"
```
```  1848   shows "infinite A"
```
```  1849 proof
```
```  1850   assume "finite A"
```
```  1851   then show False using \<open>X A\<close>
```
```  1852   proof (induction rule: finite_psubset_induct)
```
```  1853     case (psubset A)
```
```  1854     then obtain x where "x \<in> A" "X (A - {x}) \<or> infinite (A - {x})"
```
```  1855       using local.step psubset.prems by blast
```
```  1856     then have "X (A - {x})"
```
```  1857       using psubset.hyps by blast
```
```  1858     show False
```
```  1859       apply (rule psubset.IH [where B = "A - {x}"])
```
```  1860       using \<open>x \<in> A\<close> apply blast
```
```  1861       by (simp add: \<open>X (A - {x})\<close>)
```
```  1862   qed
```
```  1863 qed
```
```  1864
```
```  1865 text \<open>
```
```  1866   For any function with infinite domain and finite range there is some
```
```  1867   element that is the image of infinitely many domain elements.  In
```
```  1868   particular, any infinite sequence of elements from a finite set
```
```  1869   contains some element that occurs infinitely often.
```
```  1870 \<close>
```
```  1871
```
```  1872 lemma inf_img_fin_dom':
```
```  1873   assumes img: "finite (f ` A)" and dom: "infinite A"
```
```  1874   shows "\<exists>y \<in> f ` A. infinite (f -` {y} \<inter> A)"
```
```  1875 proof (rule ccontr)
```
```  1876   have "A \<subseteq> (\<Union>y\<in>f ` A. f -` {y} \<inter> A)" by auto
```
```  1877   moreover
```
```  1878   assume "\<not> ?thesis"
```
```  1879   with img have "finite (\<Union>y\<in>f ` A. f -` {y} \<inter> A)" by blast
```
```  1880   ultimately have "finite A" by(rule finite_subset)
```
```  1881   with dom show False by contradiction
```
```  1882 qed
```
```  1883
```
```  1884 lemma inf_img_fin_domE':
```
```  1885   assumes "finite (f ` A)" and "infinite A"
```
```  1886   obtains y where "y \<in> f`A" and "infinite (f -` {y} \<inter> A)"
```
```  1887   using assms by (blast dest: inf_img_fin_dom')
```
```  1888
```
```  1889 lemma inf_img_fin_dom:
```
```  1890   assumes img: "finite (f`A)" and dom: "infinite A"
```
```  1891   shows "\<exists>y \<in> f`A. infinite (f -` {y})"
```
```  1892 using inf_img_fin_dom'[OF assms] by auto
```
```  1893
```
```  1894 lemma inf_img_fin_domE:
```
```  1895   assumes "finite (f`A)" and "infinite A"
```
```  1896   obtains y where "y \<in> f`A" and "infinite (f -` {y})"
```
```  1897   using assms by (blast dest: inf_img_fin_dom)
```
```  1898
```
```  1899 proposition finite_image_absD:
```
```  1900     fixes S :: "'a::linordered_ring set"
```
```  1901     shows "finite (abs ` S) \<Longrightarrow> finite S"
```
```  1902   by (rule ccontr) (auto simp: abs_eq_iff vimage_def dest: inf_img_fin_dom)
```
```  1903
```
```  1904 end
```