src/HOL/Finite_Set.thy
 author wenzelm Sat Oct 01 19:30:21 2016 +0200 (2016-10-01) changeset 63982 4c4049e3bad8 parent 63915 bab633745c7f child 67443 3abf6a722518 permissions -rw-r--r--
tuned;
```     1 (*  Title:      HOL/Finite_Set.thy
```
```     2     Author:     Tobias Nipkow
```
```     3     Author:     Lawrence C Paulson
```
```     4     Author:     Markus Wenzel
```
```     5     Author:     Jeremy Avigad
```
```     6     Author:     Andrei Popescu
```
```     7 *)
```
```     8
```
```     9 section \<open>Finite sets\<close>
```
```    10
```
```    11 theory Finite_Set
```
```    12   imports Product_Type Sum_Type Fields
```
```    13 begin
```
```    14
```
```    15 subsection \<open>Predicate for finite sets\<close>
```
```    16
```
```    17 context notes [[inductive_internals]]
```
```    18 begin
```
```    19
```
```    20 inductive finite :: "'a set \<Rightarrow> bool"
```
```    21   where
```
```    22     emptyI [simp, intro!]: "finite {}"
```
```    23   | insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)"
```
```    24
```
```    25 end
```
```    26
```
```    27 simproc_setup finite_Collect ("finite (Collect P)") = \<open>K Set_Comprehension_Pointfree.simproc\<close>
```
```    28
```
```    29 declare [[simproc del: finite_Collect]]
```
```    30
```
```    31 lemma finite_induct [case_names empty insert, induct set: finite]:
```
```    32   \<comment> \<open>Discharging \<open>x \<notin> F\<close> entails extra work.\<close>
```
```    33   assumes "finite F"
```
```    34   assumes "P {}"
```
```    35     and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
```
```    36   shows "P F"
```
```    37   using \<open>finite F\<close>
```
```    38 proof induct
```
```    39   show "P {}" by fact
```
```    40 next
```
```    41   fix x F
```
```    42   assume F: "finite F" and P: "P F"
```
```    43   show "P (insert x F)"
```
```    44   proof cases
```
```    45     assume "x \<in> F"
```
```    46     then have "insert x F = F" by (rule insert_absorb)
```
```    47     with P show ?thesis by (simp only:)
```
```    48   next
```
```    49     assume "x \<notin> F"
```
```    50     from F this P show ?thesis by (rule insert)
```
```    51   qed
```
```    52 qed
```
```    53
```
```    54 lemma infinite_finite_induct [case_names infinite empty insert]:
```
```    55   assumes infinite: "\<And>A. \<not> finite A \<Longrightarrow> P A"
```
```    56     and empty: "P {}"
```
```    57     and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
```
```    58   shows "P A"
```
```    59 proof (cases "finite A")
```
```    60   case False
```
```    61   with infinite show ?thesis .
```
```    62 next
```
```    63   case True
```
```    64   then show ?thesis by (induct A) (fact empty insert)+
```
```    65 qed
```
```    66
```
```    67
```
```    68 subsubsection \<open>Choice principles\<close>
```
```    69
```
```    70 lemma ex_new_if_finite: \<comment> "does not depend on def of finite at all"
```
```    71   assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
```
```    72   shows "\<exists>a::'a. a \<notin> A"
```
```    73 proof -
```
```    74   from assms have "A \<noteq> UNIV" by blast
```
```    75   then show ?thesis by blast
```
```    76 qed
```
```    77
```
```    78 text \<open>A finite choice principle. Does not need the SOME choice operator.\<close>
```
```    79
```
```    80 lemma finite_set_choice: "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
```
```    81 proof (induct rule: finite_induct)
```
```    82   case empty
```
```    83   then show ?case by simp
```
```    84 next
```
```    85   case (insert a A)
```
```    86   then obtain f b where f: "\<forall>x\<in>A. P x (f x)" and ab: "P a b"
```
```    87     by auto
```
```    88   show ?case (is "\<exists>f. ?P f")
```
```    89   proof
```
```    90     show "?P (\<lambda>x. if x = a then b else f x)"
```
```    91       using f ab by auto
```
```    92   qed
```
```    93 qed
```
```    94
```
```    95
```
```    96 subsubsection \<open>Finite sets are the images of initial segments of natural numbers\<close>
```
```    97
```
```    98 lemma finite_imp_nat_seg_image_inj_on:
```
```    99   assumes "finite A"
```
```   100   shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
```
```   101   using assms
```
```   102 proof induct
```
```   103   case empty
```
```   104   show ?case
```
```   105   proof
```
```   106     show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}"
```
```   107       by simp
```
```   108   qed
```
```   109 next
```
```   110   case (insert a A)
```
```   111   have notinA: "a \<notin> A" by fact
```
```   112   from insert.hyps obtain n f where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}"
```
```   113     by blast
```
```   114   then have "insert a A = f(n:=a) ` {i. i < Suc n}" and "inj_on (f(n:=a)) {i. i < Suc n}"
```
```   115     using notinA by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
```
```   116   then show ?case by blast
```
```   117 qed
```
```   118
```
```   119 lemma nat_seg_image_imp_finite: "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
```
```   120 proof (induct n arbitrary: A)
```
```   121   case 0
```
```   122   then show ?case by simp
```
```   123 next
```
```   124   case (Suc n)
```
```   125   let ?B = "f ` {i. i < n}"
```
```   126   have finB: "finite ?B" by (rule Suc.hyps[OF refl])
```
```   127   show ?case
```
```   128   proof (cases "\<exists>k<n. f n = f k")
```
```   129     case True
```
```   130     then have "A = ?B"
```
```   131       using Suc.prems by (auto simp:less_Suc_eq)
```
```   132     then show ?thesis
```
```   133       using finB by simp
```
```   134   next
```
```   135     case False
```
```   136     then have "A = insert (f n) ?B"
```
```   137       using Suc.prems by (auto simp:less_Suc_eq)
```
```   138     then show ?thesis using finB by simp
```
```   139   qed
```
```   140 qed
```
```   141
```
```   142 lemma finite_conv_nat_seg_image: "finite A \<longleftrightarrow> (\<exists>n f. A = f ` {i::nat. i < n})"
```
```   143   by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
```
```   144
```
```   145 lemma finite_imp_inj_to_nat_seg:
```
```   146   assumes "finite A"
```
```   147   shows "\<exists>f n. f ` A = {i::nat. i < n} \<and> inj_on f A"
```
```   148 proof -
```
```   149   from finite_imp_nat_seg_image_inj_on [OF \<open>finite A\<close>]
```
```   150   obtain f and n :: nat where bij: "bij_betw f {i. i<n} A"
```
```   151     by (auto simp: bij_betw_def)
```
```   152   let ?f = "the_inv_into {i. i<n} f"
```
```   153   have "inj_on ?f A \<and> ?f ` A = {i. i<n}"
```
```   154     by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
```
```   155   then show ?thesis by blast
```
```   156 qed
```
```   157
```
```   158 lemma finite_Collect_less_nat [iff]: "finite {n::nat. n < k}"
```
```   159   by (fastforce simp: finite_conv_nat_seg_image)
```
```   160
```
```   161 lemma finite_Collect_le_nat [iff]: "finite {n::nat. n \<le> k}"
```
```   162   by (simp add: le_eq_less_or_eq Collect_disj_eq)
```
```   163
```
```   164
```
```   165 subsubsection \<open>Finiteness and common set operations\<close>
```
```   166
```
```   167 lemma rev_finite_subset: "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
```
```   168 proof (induct arbitrary: A rule: finite_induct)
```
```   169   case empty
```
```   170   then show ?case by simp
```
```   171 next
```
```   172   case (insert x F A)
```
```   173   have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})"
```
```   174     by fact+
```
```   175   show "finite A"
```
```   176   proof cases
```
```   177     assume x: "x \<in> A"
```
```   178     with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
```
```   179     with r have "finite (A - {x})" .
```
```   180     then have "finite (insert x (A - {x}))" ..
```
```   181     also have "insert x (A - {x}) = A"
```
```   182       using x by (rule insert_Diff)
```
```   183     finally show ?thesis .
```
```   184   next
```
```   185     show ?thesis when "A \<subseteq> F"
```
```   186       using that by fact
```
```   187     assume "x \<notin> A"
```
```   188     with A show "A \<subseteq> F"
```
```   189       by (simp add: subset_insert_iff)
```
```   190   qed
```
```   191 qed
```
```   192
```
```   193 lemma finite_subset: "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
```
```   194   by (rule rev_finite_subset)
```
```   195
```
```   196 lemma finite_UnI:
```
```   197   assumes "finite F" and "finite G"
```
```   198   shows "finite (F \<union> G)"
```
```   199   using assms by induct simp_all
```
```   200
```
```   201 lemma finite_Un [iff]: "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
```
```   202   by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])
```
```   203
```
```   204 lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"
```
```   205 proof -
```
```   206   have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp
```
```   207   then have "finite ({a} \<union> A) \<longleftrightarrow> finite A" by (simp only: finite_Un)
```
```   208   then show ?thesis by simp
```
```   209 qed
```
```   210
```
```   211 lemma finite_Int [simp, intro]: "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
```
```   212   by (blast intro: finite_subset)
```
```   213
```
```   214 lemma finite_Collect_conjI [simp, intro]:
```
```   215   "finite {x. P x} \<or> finite {x. Q x} \<Longrightarrow> finite {x. P x \<and> Q x}"
```
```   216   by (simp add: Collect_conj_eq)
```
```   217
```
```   218 lemma finite_Collect_disjI [simp]:
```
```   219   "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"
```
```   220   by (simp add: Collect_disj_eq)
```
```   221
```
```   222 lemma finite_Diff [simp, intro]: "finite A \<Longrightarrow> finite (A - B)"
```
```   223   by (rule finite_subset, rule Diff_subset)
```
```   224
```
```   225 lemma finite_Diff2 [simp]:
```
```   226   assumes "finite B"
```
```   227   shows "finite (A - B) \<longleftrightarrow> finite A"
```
```   228 proof -
```
```   229   have "finite A \<longleftrightarrow> finite ((A - B) \<union> (A \<inter> B))"
```
```   230     by (simp add: Un_Diff_Int)
```
```   231   also have "\<dots> \<longleftrightarrow> finite (A - B)"
```
```   232     using \<open>finite B\<close> by simp
```
```   233   finally show ?thesis ..
```
```   234 qed
```
```   235
```
```   236 lemma finite_Diff_insert [iff]: "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
```
```   237 proof -
```
```   238   have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
```
```   239   moreover have "A - insert a B = A - B - {a}" by auto
```
```   240   ultimately show ?thesis by simp
```
```   241 qed
```
```   242
```
```   243 lemma finite_compl [simp]:
```
```   244   "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   245   by (simp add: Compl_eq_Diff_UNIV)
```
```   246
```
```   247 lemma finite_Collect_not [simp]:
```
```   248   "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   249   by (simp add: Collect_neg_eq)
```
```   250
```
```   251 lemma finite_Union [simp, intro]:
```
```   252   "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite (\<Union>A)"
```
```   253   by (induct rule: finite_induct) simp_all
```
```   254
```
```   255 lemma finite_UN_I [intro]:
```
```   256   "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"
```
```   257   by (induct rule: finite_induct) simp_all
```
```   258
```
```   259 lemma finite_UN [simp]: "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
```
```   260   by (blast intro: finite_subset)
```
```   261
```
```   262 lemma finite_Inter [intro]: "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
```
```   263   by (blast intro: Inter_lower finite_subset)
```
```   264
```
```   265 lemma finite_INT [intro]: "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
```
```   266   by (blast intro: INT_lower finite_subset)
```
```   267
```
```   268 lemma finite_imageI [simp, intro]: "finite F \<Longrightarrow> finite (h ` F)"
```
```   269   by (induct rule: finite_induct) simp_all
```
```   270
```
```   271 lemma finite_image_set [simp]: "finite {x. P x} \<Longrightarrow> finite {f x |x. P x}"
```
```   272   by (simp add: image_Collect [symmetric])
```
```   273
```
```   274 lemma finite_image_set2:
```
```   275   "finite {x. P x} \<Longrightarrow> finite {y. Q y} \<Longrightarrow> finite {f x y |x y. P x \<and> Q y}"
```
```   276   by (rule finite_subset [where B = "\<Union>x \<in> {x. P x}. \<Union>y \<in> {y. Q y}. {f x y}"]) auto
```
```   277
```
```   278 lemma finite_imageD:
```
```   279   assumes "finite (f ` A)" and "inj_on f A"
```
```   280   shows "finite A"
```
```   281   using assms
```
```   282 proof (induct "f ` A" arbitrary: A)
```
```   283   case empty
```
```   284   then show ?case by simp
```
```   285 next
```
```   286   case (insert x B)
```
```   287   then have B_A: "insert x B = f ` A"
```
```   288     by simp
```
```   289   then obtain y where "x = f y" and "y \<in> A"
```
```   290     by blast
```
```   291   from B_A \<open>x \<notin> B\<close> have "B = f ` A - {x}"
```
```   292     by blast
```
```   293   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})"
```
```   294     by (simp add: inj_on_image_set_diff Set.Diff_subset)
```
```   295   moreover from \<open>inj_on f A\<close> have "inj_on f (A - {y})"
```
```   296     by (rule inj_on_diff)
```
```   297   ultimately have "finite (A - {y})"
```
```   298     by (rule insert.hyps)
```
```   299   then show "finite A"
```
```   300     by simp
```
```   301 qed
```
```   302
```
```   303 lemma finite_image_iff: "inj_on f A \<Longrightarrow> finite (f ` A) \<longleftrightarrow> finite A"
```
```   304   using finite_imageD by blast
```
```   305
```
```   306 lemma finite_surj: "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
```
```   307   by (erule finite_subset) (rule finite_imageI)
```
```   308
```
```   309 lemma finite_range_imageI: "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
```
```   310   by (drule finite_imageI) (simp add: range_composition)
```
```   311
```
```   312 lemma finite_subset_image:
```
```   313   assumes "finite B"
```
```   314   shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
```
```   315   using assms
```
```   316 proof induct
```
```   317   case empty
```
```   318   then show ?case by simp
```
```   319 next
```
```   320   case insert
```
```   321   then show ?case
```
```   322     by (clarsimp simp del: image_insert simp add: image_insert [symmetric]) blast  (* slow *)
```
```   323 qed
```
```   324
```
```   325 lemma finite_vimage_IntI: "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
```
```   326   apply (induct rule: finite_induct)
```
```   327    apply simp_all
```
```   328   apply (subst vimage_insert)
```
```   329   apply (simp add: finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2)
```
```   330   done
```
```   331
```
```   332 lemma finite_finite_vimage_IntI:
```
```   333   assumes "finite F"
```
```   334     and "\<And>y. y \<in> F \<Longrightarrow> finite ((h -` {y}) \<inter> A)"
```
```   335   shows "finite (h -` F \<inter> A)"
```
```   336 proof -
```
```   337   have *: "h -` F \<inter> A = (\<Union> y\<in>F. (h -` {y}) \<inter> A)"
```
```   338     by blast
```
```   339   show ?thesis
```
```   340     by (simp only: * assms finite_UN_I)
```
```   341 qed
```
```   342
```
```   343 lemma finite_vimageI: "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
```
```   344   using finite_vimage_IntI[of F h UNIV] by auto
```
```   345
```
```   346 lemma finite_vimageD': "finite (f -` A) \<Longrightarrow> A \<subseteq> range f \<Longrightarrow> finite A"
```
```   347   by (auto simp add: subset_image_iff intro: finite_subset[rotated])
```
```   348
```
```   349 lemma finite_vimageD: "finite (h -` F) \<Longrightarrow> surj h \<Longrightarrow> finite F"
```
```   350   by (auto dest: finite_vimageD')
```
```   351
```
```   352 lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
```
```   353   unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
```
```   354
```
```   355 lemma finite_Collect_bex [simp]:
```
```   356   assumes "finite A"
```
```   357   shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"
```
```   358 proof -
```
```   359   have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto
```
```   360   with assms show ?thesis by simp
```
```   361 qed
```
```   362
```
```   363 lemma finite_Collect_bounded_ex [simp]:
```
```   364   assumes "finite {y. P y}"
```
```   365   shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
```
```   366 proof -
```
```   367   have "{x. \<exists>y. P y \<and> Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})"
```
```   368     by auto
```
```   369   with assms show ?thesis
```
```   370     by simp
```
```   371 qed
```
```   372
```
```   373 lemma finite_Plus: "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
```
```   374   by (simp add: Plus_def)
```
```   375
```
```   376 lemma finite_PlusD:
```
```   377   fixes A :: "'a set" and B :: "'b set"
```
```   378   assumes fin: "finite (A <+> B)"
```
```   379   shows "finite A" "finite B"
```
```   380 proof -
```
```   381   have "Inl ` A \<subseteq> A <+> B"
```
```   382     by auto
```
```   383   then have "finite (Inl ` A :: ('a + 'b) set)"
```
```   384     using fin by (rule finite_subset)
```
```   385   then show "finite A"
```
```   386     by (rule finite_imageD) (auto intro: inj_onI)
```
```   387 next
```
```   388   have "Inr ` B \<subseteq> A <+> B"
```
```   389     by auto
```
```   390   then have "finite (Inr ` B :: ('a + 'b) set)"
```
```   391     using fin by (rule finite_subset)
```
```   392   then show "finite B"
```
```   393     by (rule finite_imageD) (auto intro: inj_onI)
```
```   394 qed
```
```   395
```
```   396 lemma finite_Plus_iff [simp]: "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
```
```   397   by (auto intro: finite_PlusD finite_Plus)
```
```   398
```
```   399 lemma finite_Plus_UNIV_iff [simp]:
```
```   400   "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
```
```   401   by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
```
```   402
```
```   403 lemma finite_SigmaI [simp, intro]:
```
```   404   "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (SIGMA a:A. B a)"
```
```   405   unfolding Sigma_def by blast
```
```   406
```
```   407 lemma finite_SigmaI2:
```
```   408   assumes "finite {x\<in>A. B x \<noteq> {}}"
```
```   409   and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)"
```
```   410   shows "finite (Sigma A B)"
```
```   411 proof -
```
```   412   from assms have "finite (Sigma {x\<in>A. B x \<noteq> {}} B)"
```
```   413     by auto
```
```   414   also have "Sigma {x:A. B x \<noteq> {}} B = Sigma A B"
```
```   415     by auto
```
```   416   finally show ?thesis .
```
```   417 qed
```
```   418
```
```   419 lemma finite_cartesian_product: "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
```
```   420   by (rule finite_SigmaI)
```
```   421
```
```   422 lemma finite_Prod_UNIV:
```
```   423   "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
```
```   424   by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
```
```   425
```
```   426 lemma finite_cartesian_productD1:
```
```   427   assumes "finite (A \<times> B)" and "B \<noteq> {}"
```
```   428   shows "finite A"
```
```   429 proof -
```
```   430   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
```
```   431     by (auto simp add: finite_conv_nat_seg_image)
```
```   432   then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}"
```
```   433     by simp
```
```   434   with \<open>B \<noteq> {}\<close> have "A = (fst \<circ> f) ` {i::nat. i < n}"
```
```   435     by (simp add: image_comp)
```
```   436   then have "\<exists>n f. A = f ` {i::nat. i < n}"
```
```   437     by blast
```
```   438   then show ?thesis
```
```   439     by (auto simp add: finite_conv_nat_seg_image)
```
```   440 qed
```
```   441
```
```   442 lemma finite_cartesian_productD2:
```
```   443   assumes "finite (A \<times> B)" and "A \<noteq> {}"
```
```   444   shows "finite B"
```
```   445 proof -
```
```   446   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
```
```   447     by (auto simp add: finite_conv_nat_seg_image)
```
```   448   then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}"
```
```   449     by simp
```
```   450   with \<open>A \<noteq> {}\<close> have "B = (snd \<circ> f) ` {i::nat. i < n}"
```
```   451     by (simp add: image_comp)
```
```   452   then have "\<exists>n f. B = f ` {i::nat. i < n}"
```
```   453     by blast
```
```   454   then show ?thesis
```
```   455     by (auto simp add: finite_conv_nat_seg_image)
```
```   456 qed
```
```   457
```
```   458 lemma finite_cartesian_product_iff:
```
```   459   "finite (A \<times> B) \<longleftrightarrow> (A = {} \<or> B = {} \<or> (finite A \<and> finite B))"
```
```   460   by (auto dest: finite_cartesian_productD1 finite_cartesian_productD2 finite_cartesian_product)
```
```   461
```
```   462 lemma finite_prod:
```
```   463   "finite (UNIV :: ('a \<times> 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
```
```   464   using finite_cartesian_product_iff[of UNIV UNIV] by simp
```
```   465
```
```   466 lemma finite_Pow_iff [iff]: "finite (Pow A) \<longleftrightarrow> finite A"
```
```   467 proof
```
```   468   assume "finite (Pow A)"
```
```   469   then have "finite ((\<lambda>x. {x}) ` A)"
```
```   470     by (blast intro: finite_subset)  (* somewhat slow *)
```
```   471   then show "finite A"
```
```   472     by (rule finite_imageD [unfolded inj_on_def]) simp
```
```   473 next
```
```   474   assume "finite A"
```
```   475   then show "finite (Pow A)"
```
```   476     by induct (simp_all add: Pow_insert)
```
```   477 qed
```
```   478
```
```   479 corollary finite_Collect_subsets [simp, intro]: "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
```
```   480   by (simp add: Pow_def [symmetric])
```
```   481
```
```   482 lemma finite_set: "finite (UNIV :: 'a set set) \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   483   by (simp only: finite_Pow_iff Pow_UNIV[symmetric])
```
```   484
```
```   485 lemma finite_UnionD: "finite (\<Union>A) \<Longrightarrow> finite A"
```
```   486   by (blast intro: finite_subset [OF subset_Pow_Union])
```
```   487
```
```   488 lemma finite_set_of_finite_funs:
```
```   489   assumes "finite A" "finite B"
```
```   490   shows "finite {f. \<forall>x. (x \<in> A \<longrightarrow> f x \<in> B) \<and> (x \<notin> A \<longrightarrow> f x = d)}" (is "finite ?S")
```
```   491 proof -
```
```   492   let ?F = "\<lambda>f. {(a,b). a \<in> A \<and> b = f a}"
```
```   493   have "?F ` ?S \<subseteq> Pow(A \<times> B)"
```
```   494     by auto
```
```   495   from finite_subset[OF this] assms have 1: "finite (?F ` ?S)"
```
```   496     by simp
```
```   497   have 2: "inj_on ?F ?S"
```
```   498     by (fastforce simp add: inj_on_def set_eq_iff fun_eq_iff)  (* somewhat slow *)
```
```   499   show ?thesis
```
```   500     by (rule finite_imageD [OF 1 2])
```
```   501 qed
```
```   502
```
```   503 lemma not_finite_existsD:
```
```   504   assumes "\<not> finite {a. P a}"
```
```   505   shows "\<exists>a. P a"
```
```   506 proof (rule classical)
```
```   507   assume "\<not> ?thesis"
```
```   508   with assms show ?thesis by auto
```
```   509 qed
```
```   510
```
```   511
```
```   512 subsubsection \<open>Further induction rules on finite sets\<close>
```
```   513
```
```   514 lemma finite_ne_induct [case_names singleton insert, consumes 2]:
```
```   515   assumes "finite F" and "F \<noteq> {}"
```
```   516   assumes "\<And>x. P {x}"
```
```   517     and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
```
```   518   shows "P F"
```
```   519   using assms
```
```   520 proof induct
```
```   521   case empty
```
```   522   then show ?case by simp
```
```   523 next
```
```   524   case (insert x F)
```
```   525   then show ?case by cases auto
```
```   526 qed
```
```   527
```
```   528 lemma finite_subset_induct [consumes 2, case_names empty insert]:
```
```   529   assumes "finite F" and "F \<subseteq> A"
```
```   530     and empty: "P {}"
```
```   531     and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
```
```   532   shows "P F"
```
```   533   using \<open>finite F\<close> \<open>F \<subseteq> A\<close>
```
```   534 proof induct
```
```   535   show "P {}" by fact
```
```   536 next
```
```   537   fix x F
```
```   538   assume "finite F" and "x \<notin> F" and P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
```
```   539   show "P (insert x F)"
```
```   540   proof (rule insert)
```
```   541     from i show "x \<in> A" by blast
```
```   542     from i have "F \<subseteq> A" by blast
```
```   543     with P show "P F" .
```
```   544     show "finite F" by fact
```
```   545     show "x \<notin> F" by fact
```
```   546   qed
```
```   547 qed
```
```   548
```
```   549 lemma finite_empty_induct:
```
```   550   assumes "finite A"
```
```   551     and "P A"
```
```   552     and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
```
```   553   shows "P {}"
```
```   554 proof -
```
```   555   have "P (A - B)" if "B \<subseteq> A" for B :: "'a set"
```
```   556   proof -
```
```   557     from \<open>finite A\<close> that have "finite B"
```
```   558       by (rule rev_finite_subset)
```
```   559     from this \<open>B \<subseteq> A\<close> show "P (A - B)"
```
```   560     proof induct
```
```   561       case empty
```
```   562       from \<open>P A\<close> show ?case by simp
```
```   563     next
```
```   564       case (insert b B)
```
```   565       have "P (A - B - {b})"
```
```   566       proof (rule remove)
```
```   567         from \<open>finite A\<close> show "finite (A - B)"
```
```   568           by induct auto
```
```   569         from insert show "b \<in> A - B"
```
```   570           by simp
```
```   571         from insert show "P (A - B)"
```
```   572           by simp
```
```   573       qed
```
```   574       also have "A - B - {b} = A - insert b B"
```
```   575         by (rule Diff_insert [symmetric])
```
```   576       finally show ?case .
```
```   577     qed
```
```   578   qed
```
```   579   then have "P (A - A)" by blast
```
```   580   then show ?thesis by simp
```
```   581 qed
```
```   582
```
```   583 lemma finite_update_induct [consumes 1, case_names const update]:
```
```   584   assumes finite: "finite {a. f a \<noteq> c}"
```
```   585     and const: "P (\<lambda>a. c)"
```
```   586     and 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))"
```
```   587   shows "P f"
```
```   588   using finite
```
```   589 proof (induct "{a. f a \<noteq> c}" arbitrary: f)
```
```   590   case empty
```
```   591   with const show ?case by simp
```
```   592 next
```
```   593   case (insert a A)
```
```   594   then have "A = {a'. (f(a := c)) a' \<noteq> c}" and "f a \<noteq> c"
```
```   595     by auto
```
```   596   with \<open>finite A\<close> have "finite {a'. (f(a := c)) a' \<noteq> c}"
```
```   597     by simp
```
```   598   have "(f(a := c)) a = c"
```
```   599     by simp
```
```   600   from insert \<open>A = {a'. (f(a := c)) a' \<noteq> c}\<close> have "P (f(a := c))"
```
```   601     by simp
```
```   602   with \<open>finite {a'. (f(a := c)) a' \<noteq> c}\<close> \<open>(f(a := c)) a = c\<close> \<open>f a \<noteq> c\<close>
```
```   603   have "P ((f(a := c))(a := f a))"
```
```   604     by (rule update)
```
```   605   then show ?case by simp
```
```   606 qed
```
```   607
```
```   608 lemma finite_subset_induct' [consumes 2, case_names empty insert]:
```
```   609   assumes "finite F" and "F \<subseteq> A"
```
```   610     and empty: "P {}"
```
```   611     and insert: "\<And>a F. \<lbrakk>finite F; a \<in> A; F \<subseteq> A; a \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert a F)"
```
```   612   shows "P F"
```
```   613   using assms(1,2)
```
```   614 proof induct
```
```   615   show "P {}" by fact
```
```   616 next
```
```   617   fix x F
```
```   618   assume "finite F" and "x \<notin> F" and
```
```   619     P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
```
```   620   show "P (insert x F)"
```
```   621   proof (rule insert)
```
```   622     from i show "x \<in> A" by blast
```
```   623     from i have "F \<subseteq> A" by blast
```
```   624     with P show "P F" .
```
```   625     show "finite F" by fact
```
```   626     show "x \<notin> F" by fact
```
```   627     show "F \<subseteq> A" by fact
```
```   628   qed
```
```   629 qed
```
```   630
```
```   631
```
```   632 subsection \<open>Class \<open>finite\<close>\<close>
```
```   633
```
```   634 class finite =
```
```   635   assumes finite_UNIV: "finite (UNIV :: 'a set)"
```
```   636 begin
```
```   637
```
```   638 lemma finite [simp]: "finite (A :: 'a set)"
```
```   639   by (rule subset_UNIV finite_UNIV finite_subset)+
```
```   640
```
```   641 lemma finite_code [code]: "finite (A :: 'a set) \<longleftrightarrow> True"
```
```   642   by simp
```
```   643
```
```   644 end
```
```   645
```
```   646 instance prod :: (finite, finite) finite
```
```   647   by standard (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
```
```   648
```
```   649 lemma inj_graph: "inj (\<lambda>f. {(x, y). y = f x})"
```
```   650   by (rule inj_onI) (auto simp add: set_eq_iff fun_eq_iff)
```
```   651
```
```   652 instance "fun" :: (finite, finite) finite
```
```   653 proof
```
```   654   show "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
```
```   655   proof (rule finite_imageD)
```
```   656     let ?graph = "\<lambda>f::'a \<Rightarrow> 'b. {(x, y). y = f x}"
```
```   657     have "range ?graph \<subseteq> Pow UNIV"
```
```   658       by simp
```
```   659     moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
```
```   660       by (simp only: finite_Pow_iff finite)
```
```   661     ultimately show "finite (range ?graph)"
```
```   662       by (rule finite_subset)
```
```   663     show "inj ?graph"
```
```   664       by (rule inj_graph)
```
```   665   qed
```
```   666 qed
```
```   667
```
```   668 instance bool :: finite
```
```   669   by standard (simp add: UNIV_bool)
```
```   670
```
```   671 instance set :: (finite) finite
```
```   672   by standard (simp only: Pow_UNIV [symmetric] finite_Pow_iff finite)
```
```   673
```
```   674 instance unit :: finite
```
```   675   by standard (simp add: UNIV_unit)
```
```   676
```
```   677 instance sum :: (finite, finite) finite
```
```   678   by standard (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
```
```   679
```
```   680
```
```   681 subsection \<open>A basic fold functional for finite sets\<close>
```
```   682
```
```   683 text \<open>The intended behaviour is
```
```   684   \<open>fold f z {x\<^sub>1, \<dots>, x\<^sub>n} = f x\<^sub>1 (\<dots> (f x\<^sub>n z)\<dots>)\<close>
```
```   685   if \<open>f\<close> is ``left-commutative'':
```
```   686 \<close>
```
```   687
```
```   688 locale comp_fun_commute =
```
```   689   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
```
```   690   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
```
```   691 begin
```
```   692
```
```   693 lemma fun_left_comm: "f y (f x z) = f x (f y z)"
```
```   694   using comp_fun_commute by (simp add: fun_eq_iff)
```
```   695
```
```   696 lemma commute_left_comp: "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
```
```   697   by (simp add: o_assoc comp_fun_commute)
```
```   698
```
```   699 end
```
```   700
```
```   701 inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
```
```   702   for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b
```
```   703   where
```
```   704     emptyI [intro]: "fold_graph f z {} z"
```
```   705   | insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
```
```   706
```
```   707 inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
```
```   708
```
```   709 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
```
```   710   where "fold f z A = (if finite A then (THE y. fold_graph f z A y) else z)"
```
```   711
```
```   712 text \<open>
```
```   713   A tempting alternative for the definiens is
```
```   714   @{term "if finite A then THE y. fold_graph f z A y else e"}.
```
```   715   It allows the removal of finiteness assumptions from the theorems
```
```   716   \<open>fold_comm\<close>, \<open>fold_reindex\<close> and \<open>fold_distrib\<close>.
```
```   717   The proofs become ugly. It is not worth the effort. (???)
```
```   718 \<close>
```
```   719
```
```   720 lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
```
```   721   by (induct rule: finite_induct) auto
```
```   722
```
```   723
```
```   724 subsubsection \<open>From @{const fold_graph} to @{term fold}\<close>
```
```   725
```
```   726 context comp_fun_commute
```
```   727 begin
```
```   728
```
```   729 lemma fold_graph_finite:
```
```   730   assumes "fold_graph f z A y"
```
```   731   shows "finite A"
```
```   732   using assms by induct simp_all
```
```   733
```
```   734 lemma fold_graph_insertE_aux:
```
```   735   "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'"
```
```   736 proof (induct set: fold_graph)
```
```   737   case emptyI
```
```   738   then show ?case by simp
```
```   739 next
```
```   740   case (insertI x A y)
```
```   741   show ?case
```
```   742   proof (cases "x = a")
```
```   743     case True
```
```   744     with insertI show ?thesis by auto
```
```   745   next
```
```   746     case False
```
```   747     then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
```
```   748       using insertI by auto
```
```   749     have "f x y = f a (f x y')"
```
```   750       unfolding y by (rule fun_left_comm)
```
```   751     moreover have "fold_graph f z (insert x A - {a}) (f x y')"
```
```   752       using y' and \<open>x \<noteq> a\<close> and \<open>x \<notin> A\<close>
```
```   753       by (simp add: insert_Diff_if fold_graph.insertI)
```
```   754     ultimately show ?thesis
```
```   755       by fast
```
```   756   qed
```
```   757 qed
```
```   758
```
```   759 lemma fold_graph_insertE:
```
```   760   assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
```
```   761   obtains y where "v = f x y" and "fold_graph f z A y"
```
```   762   using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
```
```   763
```
```   764 lemma fold_graph_determ: "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
```
```   765 proof (induct arbitrary: y set: fold_graph)
```
```   766   case emptyI
```
```   767   then show ?case by fast
```
```   768 next
```
```   769   case (insertI x A y v)
```
```   770   from \<open>fold_graph f z (insert x A) v\<close> and \<open>x \<notin> A\<close>
```
```   771   obtain y' where "v = f x y'" and "fold_graph f z A y'"
```
```   772     by (rule fold_graph_insertE)
```
```   773   from \<open>fold_graph f z A y'\<close> have "y' = y"
```
```   774     by (rule insertI)
```
```   775   with \<open>v = f x y'\<close> show "v = f x y"
```
```   776     by simp
```
```   777 qed
```
```   778
```
```   779 lemma fold_equality: "fold_graph f z A y \<Longrightarrow> fold f z A = y"
```
```   780   by (cases "finite A") (auto simp add: fold_def intro: fold_graph_determ dest: fold_graph_finite)
```
```   781
```
```   782 lemma fold_graph_fold:
```
```   783   assumes "finite A"
```
```   784   shows "fold_graph f z A (fold f z A)"
```
```   785 proof -
```
```   786   from assms have "\<exists>x. fold_graph f z A x"
```
```   787     by (rule finite_imp_fold_graph)
```
```   788   moreover note fold_graph_determ
```
```   789   ultimately have "\<exists>!x. fold_graph f z A x"
```
```   790     by (rule ex_ex1I)
```
```   791   then have "fold_graph f z A (The (fold_graph f z A))"
```
```   792     by (rule theI')
```
```   793   with assms show ?thesis
```
```   794     by (simp add: fold_def)
```
```   795 qed
```
```   796
```
```   797 text \<open>The base case for \<open>fold\<close>:\<close>
```
```   798
```
```   799 lemma (in -) fold_infinite [simp]: "\<not> finite A \<Longrightarrow> fold f z A = z"
```
```   800   by (auto simp: fold_def)
```
```   801
```
```   802 lemma (in -) fold_empty [simp]: "fold f z {} = z"
```
```   803   by (auto simp: fold_def)
```
```   804
```
```   805 text \<open>The various recursion equations for @{const fold}:\<close>
```
```   806
```
```   807 lemma fold_insert [simp]:
```
```   808   assumes "finite A" and "x \<notin> A"
```
```   809   shows "fold f z (insert x A) = f x (fold f z A)"
```
```   810 proof (rule fold_equality)
```
```   811   fix z
```
```   812   from \<open>finite A\<close> have "fold_graph f z A (fold f z A)"
```
```   813     by (rule fold_graph_fold)
```
```   814   with \<open>x \<notin> A\<close> have "fold_graph f z (insert x A) (f x (fold f z A))"
```
```   815     by (rule fold_graph.insertI)
```
```   816   then show "fold_graph f z (insert x A) (f x (fold f z A))"
```
```   817     by simp
```
```   818 qed
```
```   819
```
```   820 declare (in -) empty_fold_graphE [rule del] fold_graph.intros [rule del]
```
```   821   \<comment> \<open>No more proofs involve these.\<close>
```
```   822
```
```   823 lemma fold_fun_left_comm: "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
```
```   824 proof (induct rule: finite_induct)
```
```   825   case empty
```
```   826   then show ?case by simp
```
```   827 next
```
```   828   case insert
```
```   829   then show ?case
```
```   830     by (simp add: fun_left_comm [of x])
```
```   831 qed
```
```   832
```
```   833 lemma fold_insert2: "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A)  = fold f (f x z) A"
```
```   834   by (simp add: fold_fun_left_comm)
```
```   835
```
```   836 lemma fold_rec:
```
```   837   assumes "finite A" and "x \<in> A"
```
```   838   shows "fold f z A = f x (fold f z (A - {x}))"
```
```   839 proof -
```
```   840   have A: "A = insert x (A - {x})"
```
```   841     using \<open>x \<in> A\<close> by blast
```
```   842   then have "fold f z A = fold f z (insert x (A - {x}))"
```
```   843     by simp
```
```   844   also have "\<dots> = f x (fold f z (A - {x}))"
```
```   845     by (rule fold_insert) (simp add: \<open>finite A\<close>)+
```
```   846   finally show ?thesis .
```
```   847 qed
```
```   848
```
```   849 lemma fold_insert_remove:
```
```   850   assumes "finite A"
```
```   851   shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
```
```   852 proof -
```
```   853   from \<open>finite A\<close> have "finite (insert x A)"
```
```   854     by auto
```
```   855   moreover have "x \<in> insert x A"
```
```   856     by auto
```
```   857   ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
```
```   858     by (rule fold_rec)
```
```   859   then show ?thesis
```
```   860     by simp
```
```   861 qed
```
```   862
```
```   863 lemma fold_set_union_disj:
```
```   864   assumes "finite A" "finite B" "A \<inter> B = {}"
```
```   865   shows "Finite_Set.fold f z (A \<union> B) = Finite_Set.fold f (Finite_Set.fold f z A) B"
```
```   866   using assms(2,1,3) by induct simp_all
```
```   867
```
```   868 end
```
```   869
```
```   870 text \<open>Other properties of @{const fold}:\<close>
```
```   871
```
```   872 lemma fold_image:
```
```   873   assumes "inj_on g A"
```
```   874   shows "fold f z (g ` A) = fold (f \<circ> g) z A"
```
```   875 proof (cases "finite A")
```
```   876   case False
```
```   877   with assms show ?thesis
```
```   878     by (auto dest: finite_imageD simp add: fold_def)
```
```   879 next
```
```   880   case True
```
```   881   have "fold_graph f z (g ` A) = fold_graph (f \<circ> g) z A"
```
```   882   proof
```
```   883     fix w
```
```   884     show "fold_graph f z (g ` A) w \<longleftrightarrow> fold_graph (f \<circ> g) z A w" (is "?P \<longleftrightarrow> ?Q")
```
```   885     proof
```
```   886       assume ?P
```
```   887       then show ?Q
```
```   888         using assms
```
```   889       proof (induct "g ` A" w arbitrary: A)
```
```   890         case emptyI
```
```   891         then show ?case by (auto intro: fold_graph.emptyI)
```
```   892       next
```
```   893         case (insertI x A r B)
```
```   894         from \<open>inj_on g B\<close> \<open>x \<notin> A\<close> \<open>insert x A = image g B\<close> obtain x' A'
```
```   895           where "x' \<notin> A'" and [simp]: "B = insert x' A'" "x = g x'" "A = g ` A'"
```
```   896           by (rule inj_img_insertE)
```
```   897         from insertI.prems have "fold_graph (f \<circ> g) z A' r"
```
```   898           by (auto intro: insertI.hyps)
```
```   899         with \<open>x' \<notin> A'\<close> have "fold_graph (f \<circ> g) z (insert x' A') ((f \<circ> g) x' r)"
```
```   900           by (rule fold_graph.insertI)
```
```   901         then show ?case
```
```   902           by simp
```
```   903       qed
```
```   904     next
```
```   905       assume ?Q
```
```   906       then show ?P
```
```   907         using assms
```
```   908       proof induct
```
```   909         case emptyI
```
```   910         then show ?case
```
```   911           by (auto intro: fold_graph.emptyI)
```
```   912       next
```
```   913         case (insertI x A r)
```
```   914         from \<open>x \<notin> A\<close> insertI.prems have "g x \<notin> g ` A"
```
```   915           by auto
```
```   916         moreover from insertI have "fold_graph f z (g ` A) r"
```
```   917           by simp
```
```   918         ultimately have "fold_graph f z (insert (g x) (g ` A)) (f (g x) r)"
```
```   919           by (rule fold_graph.insertI)
```
```   920         then show ?case
```
```   921           by simp
```
```   922       qed
```
```   923     qed
```
```   924   qed
```
```   925   with True assms show ?thesis
```
```   926     by (auto simp add: fold_def)
```
```   927 qed
```
```   928
```
```   929 lemma fold_cong:
```
```   930   assumes "comp_fun_commute f" "comp_fun_commute g"
```
```   931     and "finite A"
```
```   932     and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
```
```   933     and "s = t" and "A = B"
```
```   934   shows "fold f s A = fold g t B"
```
```   935 proof -
```
```   936   have "fold f s A = fold g s A"
```
```   937     using \<open>finite A\<close> cong
```
```   938   proof (induct A)
```
```   939     case empty
```
```   940     then show ?case by simp
```
```   941   next
```
```   942     case insert
```
```   943     interpret f: comp_fun_commute f by (fact \<open>comp_fun_commute f\<close>)
```
```   944     interpret g: comp_fun_commute g by (fact \<open>comp_fun_commute g\<close>)
```
```   945     from insert show ?case by simp
```
```   946   qed
```
```   947   with assms show ?thesis by simp
```
```   948 qed
```
```   949
```
```   950
```
```   951 text \<open>A simplified version for idempotent functions:\<close>
```
```   952
```
```   953 locale comp_fun_idem = comp_fun_commute +
```
```   954   assumes comp_fun_idem: "f x \<circ> f x = f x"
```
```   955 begin
```
```   956
```
```   957 lemma fun_left_idem: "f x (f x z) = f x z"
```
```   958   using comp_fun_idem by (simp add: fun_eq_iff)
```
```   959
```
```   960 lemma fold_insert_idem:
```
```   961   assumes fin: "finite A"
```
```   962   shows "fold f z (insert x A)  = f x (fold f z A)"
```
```   963 proof cases
```
```   964   assume "x \<in> A"
```
```   965   then obtain B where "A = insert x B" and "x \<notin> B"
```
```   966     by (rule set_insert)
```
```   967   then show ?thesis
```
```   968     using assms by (simp add: comp_fun_idem fun_left_idem)
```
```   969 next
```
```   970   assume "x \<notin> A"
```
```   971   then show ?thesis
```
```   972     using assms by simp
```
```   973 qed
```
```   974
```
```   975 declare fold_insert [simp del] fold_insert_idem [simp]
```
```   976
```
```   977 lemma fold_insert_idem2: "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
```
```   978   by (simp add: fold_fun_left_comm)
```
```   979
```
```   980 end
```
```   981
```
```   982
```
```   983 subsubsection \<open>Liftings to \<open>comp_fun_commute\<close> etc.\<close>
```
```   984
```
```   985 lemma (in comp_fun_commute) comp_comp_fun_commute: "comp_fun_commute (f \<circ> g)"
```
```   986   by standard (simp_all add: comp_fun_commute)
```
```   987
```
```   988 lemma (in comp_fun_idem) comp_comp_fun_idem: "comp_fun_idem (f \<circ> g)"
```
```   989   by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
```
```   990     (simp_all add: comp_fun_idem)
```
```   991
```
```   992 lemma (in comp_fun_commute) comp_fun_commute_funpow: "comp_fun_commute (\<lambda>x. f x ^^ g x)"
```
```   993 proof
```
```   994   show "f y ^^ g y \<circ> f x ^^ g x = f x ^^ g x \<circ> f y ^^ g y" for x y
```
```   995   proof (cases "x = y")
```
```   996     case True
```
```   997     then show ?thesis by simp
```
```   998   next
```
```   999     case False
```
```  1000     show ?thesis
```
```  1001     proof (induct "g x" arbitrary: g)
```
```  1002       case 0
```
```  1003       then show ?case by simp
```
```  1004     next
```
```  1005       case (Suc n g)
```
```  1006       have hyp1: "f y ^^ g y \<circ> f x = f x \<circ> f y ^^ g y"
```
```  1007       proof (induct "g y" arbitrary: g)
```
```  1008         case 0
```
```  1009         then show ?case by simp
```
```  1010       next
```
```  1011         case (Suc n g)
```
```  1012         define h where "h z = g z - 1" for z
```
```  1013         with Suc have "n = h y"
```
```  1014           by simp
```
```  1015         with Suc have hyp: "f y ^^ h y \<circ> f x = f x \<circ> f y ^^ h y"
```
```  1016           by auto
```
```  1017         from Suc h_def have "g y = Suc (h y)"
```
```  1018           by simp
```
```  1019         then show ?case
```
```  1020           by (simp add: comp_assoc hyp) (simp add: o_assoc comp_fun_commute)
```
```  1021       qed
```
```  1022       define h where "h z = (if z = x then g x - 1 else g z)" for z
```
```  1023       with Suc have "n = h x"
```
```  1024         by simp
```
```  1025       with Suc have "f y ^^ h y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ h y"
```
```  1026         by auto
```
```  1027       with False h_def have hyp2: "f y ^^ g y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ g y"
```
```  1028         by simp
```
```  1029       from Suc h_def have "g x = Suc (h x)"
```
```  1030         by simp
```
```  1031       then show ?case
```
```  1032         by (simp del: funpow.simps add: funpow_Suc_right o_assoc hyp2) (simp add: comp_assoc hyp1)
```
```  1033     qed
```
```  1034   qed
```
```  1035 qed
```
```  1036
```
```  1037
```
```  1038 subsubsection \<open>Expressing set operations via @{const fold}\<close>
```
```  1039
```
```  1040 lemma comp_fun_commute_const: "comp_fun_commute (\<lambda>_. f)"
```
```  1041   by standard rule
```
```  1042
```
```  1043 lemma comp_fun_idem_insert: "comp_fun_idem insert"
```
```  1044   by standard auto
```
```  1045
```
```  1046 lemma comp_fun_idem_remove: "comp_fun_idem Set.remove"
```
```  1047   by standard auto
```
```  1048
```
```  1049 lemma (in semilattice_inf) comp_fun_idem_inf: "comp_fun_idem inf"
```
```  1050   by standard (auto simp add: inf_left_commute)
```
```  1051
```
```  1052 lemma (in semilattice_sup) comp_fun_idem_sup: "comp_fun_idem sup"
```
```  1053   by standard (auto simp add: sup_left_commute)
```
```  1054
```
```  1055 lemma union_fold_insert:
```
```  1056   assumes "finite A"
```
```  1057   shows "A \<union> B = fold insert B A"
```
```  1058 proof -
```
```  1059   interpret comp_fun_idem insert
```
```  1060     by (fact comp_fun_idem_insert)
```
```  1061   from \<open>finite A\<close> show ?thesis
```
```  1062     by (induct A arbitrary: B) simp_all
```
```  1063 qed
```
```  1064
```
```  1065 lemma minus_fold_remove:
```
```  1066   assumes "finite A"
```
```  1067   shows "B - A = fold Set.remove B A"
```
```  1068 proof -
```
```  1069   interpret comp_fun_idem Set.remove
```
```  1070     by (fact comp_fun_idem_remove)
```
```  1071   from \<open>finite A\<close> have "fold Set.remove B A = B - A"
```
```  1072     by (induct A arbitrary: B) auto  (* slow *)
```
```  1073   then show ?thesis ..
```
```  1074 qed
```
```  1075
```
```  1076 lemma comp_fun_commute_filter_fold:
```
```  1077   "comp_fun_commute (\<lambda>x A'. if P x then Set.insert x A' else A')"
```
```  1078 proof -
```
```  1079   interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
```
```  1080   show ?thesis by standard (auto simp: fun_eq_iff)
```
```  1081 qed
```
```  1082
```
```  1083 lemma Set_filter_fold:
```
```  1084   assumes "finite A"
```
```  1085   shows "Set.filter P A = fold (\<lambda>x A'. if P x then Set.insert x A' else A') {} A"
```
```  1086   using assms
```
```  1087   by induct
```
```  1088     (auto simp add: Set.filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold])
```
```  1089
```
```  1090 lemma inter_Set_filter:
```
```  1091   assumes "finite B"
```
```  1092   shows "A \<inter> B = Set.filter (\<lambda>x. x \<in> A) B"
```
```  1093   using assms
```
```  1094   by induct (auto simp: Set.filter_def)
```
```  1095
```
```  1096 lemma image_fold_insert:
```
```  1097   assumes "finite A"
```
```  1098   shows "image f A = fold (\<lambda>k A. Set.insert (f k) A) {} A"
```
```  1099 proof -
```
```  1100   interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A"
```
```  1101     by standard auto
```
```  1102   show ?thesis
```
```  1103     using assms by (induct A) auto
```
```  1104 qed
```
```  1105
```
```  1106 lemma Ball_fold:
```
```  1107   assumes "finite A"
```
```  1108   shows "Ball A P = fold (\<lambda>k s. s \<and> P k) True A"
```
```  1109 proof -
```
```  1110   interpret comp_fun_commute "\<lambda>k s. s \<and> P k"
```
```  1111     by standard auto
```
```  1112   show ?thesis
```
```  1113     using assms by (induct A) auto
```
```  1114 qed
```
```  1115
```
```  1116 lemma Bex_fold:
```
```  1117   assumes "finite A"
```
```  1118   shows "Bex A P = fold (\<lambda>k s. s \<or> P k) False A"
```
```  1119 proof -
```
```  1120   interpret comp_fun_commute "\<lambda>k s. s \<or> P k"
```
```  1121     by standard auto
```
```  1122   show ?thesis
```
```  1123     using assms by (induct A) auto
```
```  1124 qed
```
```  1125
```
```  1126 lemma comp_fun_commute_Pow_fold: "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)"
```
```  1127   by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast  (* somewhat slow *)
```
```  1128
```
```  1129 lemma Pow_fold:
```
```  1130   assumes "finite A"
```
```  1131   shows "Pow A = fold (\<lambda>x A. A \<union> Set.insert x ` A) {{}} A"
```
```  1132 proof -
```
```  1133   interpret comp_fun_commute "\<lambda>x A. A \<union> Set.insert x ` A"
```
```  1134     by (rule comp_fun_commute_Pow_fold)
```
```  1135   show ?thesis
```
```  1136     using assms by (induct A) (auto simp: Pow_insert)
```
```  1137 qed
```
```  1138
```
```  1139 lemma fold_union_pair:
```
```  1140   assumes "finite B"
```
```  1141   shows "(\<Union>y\<in>B. {(x, y)}) \<union> A = fold (\<lambda>y. Set.insert (x, y)) A B"
```
```  1142 proof -
```
```  1143   interpret comp_fun_commute "\<lambda>y. Set.insert (x, y)"
```
```  1144     by standard auto
```
```  1145   show ?thesis
```
```  1146     using assms by (induct arbitrary: A) simp_all
```
```  1147 qed
```
```  1148
```
```  1149 lemma comp_fun_commute_product_fold:
```
```  1150   "finite B \<Longrightarrow> comp_fun_commute (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B)"
```
```  1151   by standard (auto simp: fold_union_pair [symmetric])
```
```  1152
```
```  1153 lemma product_fold:
```
```  1154   assumes "finite A" "finite B"
```
```  1155   shows "A \<times> B = fold (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B) {} A"
```
```  1156   using assms unfolding Sigma_def
```
```  1157   by (induct A)
```
```  1158     (simp_all add: comp_fun_commute.fold_insert[OF comp_fun_commute_product_fold] fold_union_pair)
```
```  1159
```
```  1160 context complete_lattice
```
```  1161 begin
```
```  1162
```
```  1163 lemma inf_Inf_fold_inf:
```
```  1164   assumes "finite A"
```
```  1165   shows "inf (Inf A) B = fold inf B A"
```
```  1166 proof -
```
```  1167   interpret comp_fun_idem inf
```
```  1168     by (fact comp_fun_idem_inf)
```
```  1169   from \<open>finite A\<close> fold_fun_left_comm show ?thesis
```
```  1170     by (induct A arbitrary: B) (simp_all add: inf_commute fun_eq_iff)
```
```  1171 qed
```
```  1172
```
```  1173 lemma sup_Sup_fold_sup:
```
```  1174   assumes "finite A"
```
```  1175   shows "sup (Sup A) B = fold sup B A"
```
```  1176 proof -
```
```  1177   interpret comp_fun_idem sup
```
```  1178     by (fact comp_fun_idem_sup)
```
```  1179   from \<open>finite A\<close> fold_fun_left_comm show ?thesis
```
```  1180     by (induct A arbitrary: B) (simp_all add: sup_commute fun_eq_iff)
```
```  1181 qed
```
```  1182
```
```  1183 lemma Inf_fold_inf: "finite A \<Longrightarrow> Inf A = fold inf top A"
```
```  1184   using inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
```
```  1185
```
```  1186 lemma Sup_fold_sup: "finite A \<Longrightarrow> Sup A = fold sup bot A"
```
```  1187   using sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
```
```  1188
```
```  1189 lemma inf_INF_fold_inf:
```
```  1190   assumes "finite A"
```
```  1191   shows "inf B (INFIMUM A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold")
```
```  1192 proof -
```
```  1193   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
```
```  1194   interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
```
```  1195   from \<open>finite A\<close> have "?fold = ?inf"
```
```  1196     by (induct A arbitrary: B) (simp_all add: inf_left_commute)
```
```  1197   then show ?thesis ..
```
```  1198 qed
```
```  1199
```
```  1200 lemma sup_SUP_fold_sup:
```
```  1201   assumes "finite A"
```
```  1202   shows "sup B (SUPREMUM A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold")
```
```  1203 proof -
```
```  1204   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
```
```  1205   interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
```
```  1206   from \<open>finite A\<close> have "?fold = ?sup"
```
```  1207     by (induct A arbitrary: B) (simp_all add: sup_left_commute)
```
```  1208   then show ?thesis ..
```
```  1209 qed
```
```  1210
```
```  1211 lemma INF_fold_inf: "finite A \<Longrightarrow> INFIMUM A f = fold (inf \<circ> f) top A"
```
```  1212   using inf_INF_fold_inf [of A top] by simp
```
```  1213
```
```  1214 lemma SUP_fold_sup: "finite A \<Longrightarrow> SUPREMUM A f = fold (sup \<circ> f) bot A"
```
```  1215   using sup_SUP_fold_sup [of A bot] by simp
```
```  1216
```
```  1217 end
```
```  1218
```
```  1219
```
```  1220 subsection \<open>Locales as mini-packages for fold operations\<close>
```
```  1221
```
```  1222 subsubsection \<open>The natural case\<close>
```
```  1223
```
```  1224 locale folding =
```
```  1225   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: "'b"
```
```  1226   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
```
```  1227 begin
```
```  1228
```
```  1229 interpretation fold?: comp_fun_commute f
```
```  1230   by standard (use comp_fun_commute in \<open>simp add: fun_eq_iff\<close>)
```
```  1231
```
```  1232 definition F :: "'a set \<Rightarrow> 'b"
```
```  1233   where eq_fold: "F A = fold f z A"
```
```  1234
```
```  1235 lemma empty [simp]:"F {} = z"
```
```  1236   by (simp add: eq_fold)
```
```  1237
```
```  1238 lemma infinite [simp]: "\<not> finite A \<Longrightarrow> F A = z"
```
```  1239   by (simp add: eq_fold)
```
```  1240
```
```  1241 lemma insert [simp]:
```
```  1242   assumes "finite A" and "x \<notin> A"
```
```  1243   shows "F (insert x A) = f x (F A)"
```
```  1244 proof -
```
```  1245   from fold_insert assms
```
```  1246   have "fold f z (insert x A) = f x (fold f z A)" by simp
```
```  1247   with \<open>finite A\<close> show ?thesis by (simp add: eq_fold fun_eq_iff)
```
```  1248 qed
```
```  1249
```
```  1250 lemma remove:
```
```  1251   assumes "finite A" and "x \<in> A"
```
```  1252   shows "F A = f x (F (A - {x}))"
```
```  1253 proof -
```
```  1254   from \<open>x \<in> A\<close> obtain B where A: "A = insert x B" and "x \<notin> B"
```
```  1255     by (auto dest: mk_disjoint_insert)
```
```  1256   moreover from \<open>finite A\<close> A have "finite B" by simp
```
```  1257   ultimately show ?thesis by simp
```
```  1258 qed
```
```  1259
```
```  1260 lemma insert_remove: "finite A \<Longrightarrow> F (insert x A) = f x (F (A - {x}))"
```
```  1261   by (cases "x \<in> A") (simp_all add: remove insert_absorb)
```
```  1262
```
```  1263 end
```
```  1264
```
```  1265
```
```  1266 subsubsection \<open>With idempotency\<close>
```
```  1267
```
```  1268 locale folding_idem = folding +
```
```  1269   assumes comp_fun_idem: "f x \<circ> f x = f x"
```
```  1270 begin
```
```  1271
```
```  1272 declare insert [simp del]
```
```  1273
```
```  1274 interpretation fold?: comp_fun_idem f
```
```  1275   by standard (insert comp_fun_commute comp_fun_idem, simp add: fun_eq_iff)
```
```  1276
```
```  1277 lemma insert_idem [simp]:
```
```  1278   assumes "finite A"
```
```  1279   shows "F (insert x A) = f x (F A)"
```
```  1280 proof -
```
```  1281   from fold_insert_idem assms
```
```  1282   have "fold f z (insert x A) = f x (fold f z A)" by simp
```
```  1283   with \<open>finite A\<close> show ?thesis by (simp add: eq_fold fun_eq_iff)
```
```  1284 qed
```
```  1285
```
```  1286 end
```
```  1287
```
```  1288
```
```  1289 subsection \<open>Finite cardinality\<close>
```
```  1290
```
```  1291 text \<open>
```
```  1292   The traditional definition
```
```  1293   @{prop "card A \<equiv> LEAST n. \<exists>f. A = {f i |i. i < n}"}
```
```  1294   is ugly to work with.
```
```  1295   But now that we have @{const fold} things are easy:
```
```  1296 \<close>
```
```  1297
```
```  1298 global_interpretation card: folding "\<lambda>_. Suc" 0
```
```  1299   defines card = "folding.F (\<lambda>_. Suc) 0"
```
```  1300   by standard rule
```
```  1301
```
```  1302 lemma card_infinite: "\<not> finite A \<Longrightarrow> card A = 0"
```
```  1303   by (fact card.infinite)
```
```  1304
```
```  1305 lemma card_empty: "card {} = 0"
```
```  1306   by (fact card.empty)
```
```  1307
```
```  1308 lemma card_insert_disjoint: "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> card (insert x A) = Suc (card A)"
```
```  1309   by (fact card.insert)
```
```  1310
```
```  1311 lemma card_insert_if: "finite A \<Longrightarrow> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
```
```  1312   by auto (simp add: card.insert_remove card.remove)
```
```  1313
```
```  1314 lemma card_ge_0_finite: "card A > 0 \<Longrightarrow> finite A"
```
```  1315   by (rule ccontr) simp
```
```  1316
```
```  1317 lemma card_0_eq [simp]: "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
```
```  1318   by (auto dest: mk_disjoint_insert)
```
```  1319
```
```  1320 lemma finite_UNIV_card_ge_0: "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
```
```  1321   by (rule ccontr) simp
```
```  1322
```
```  1323 lemma card_eq_0_iff: "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
```
```  1324   by auto
```
```  1325
```
```  1326 lemma card_range_greater_zero: "finite (range f) \<Longrightarrow> card (range f) > 0"
```
```  1327   by (rule ccontr) (simp add: card_eq_0_iff)
```
```  1328
```
```  1329 lemma card_gt_0_iff: "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
```
```  1330   by (simp add: neq0_conv [symmetric] card_eq_0_iff)
```
```  1331
```
```  1332 lemma card_Suc_Diff1: "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> Suc (card (A - {x})) = card A"
```
```  1333   apply (rule insert_Diff [THEN subst, where t = A])
```
```  1334    apply assumption
```
```  1335   apply (simp del: insert_Diff_single)
```
```  1336   done
```
```  1337
```
```  1338 lemma card_insert_le_m1: "n > 0 \<Longrightarrow> card y \<le> n - 1 \<Longrightarrow> card (insert x y) \<le> n"
```
```  1339   apply (cases "finite y")
```
```  1340    apply (cases "x \<in> y")
```
```  1341     apply (auto simp: insert_absorb)
```
```  1342   done
```
```  1343
```
```  1344 lemma card_Diff_singleton: "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) = card A - 1"
```
```  1345   by (simp add: card_Suc_Diff1 [symmetric])
```
```  1346
```
```  1347 lemma card_Diff_singleton_if:
```
```  1348   "finite A \<Longrightarrow> card (A - {x}) = (if x \<in> A then card A - 1 else card A)"
```
```  1349   by (simp add: card_Diff_singleton)
```
```  1350
```
```  1351 lemma card_Diff_insert[simp]:
```
```  1352   assumes "finite A" and "a \<in> A" and "a \<notin> B"
```
```  1353   shows "card (A - insert a B) = card (A - B) - 1"
```
```  1354 proof -
```
```  1355   have "A - insert a B = (A - B) - {a}"
```
```  1356     using assms by blast
```
```  1357   then show ?thesis
```
```  1358     using assms by (simp add: card_Diff_singleton)
```
```  1359 qed
```
```  1360
```
```  1361 lemma card_insert: "finite A \<Longrightarrow> card (insert x A) = Suc (card (A - {x}))"
```
```  1362   by (fact card.insert_remove)
```
```  1363
```
```  1364 lemma card_insert_le: "finite A \<Longrightarrow> card A \<le> card (insert x A)"
```
```  1365   by (simp add: card_insert_if)
```
```  1366
```
```  1367 lemma card_Collect_less_nat[simp]: "card {i::nat. i < n} = n"
```
```  1368   by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
```
```  1369
```
```  1370 lemma card_Collect_le_nat[simp]: "card {i::nat. i \<le> n} = Suc n"
```
```  1371   using card_Collect_less_nat[of "Suc n"] by (simp add: less_Suc_eq_le)
```
```  1372
```
```  1373 lemma card_mono:
```
```  1374   assumes "finite B" and "A \<subseteq> B"
```
```  1375   shows "card A \<le> card B"
```
```  1376 proof -
```
```  1377   from assms have "finite A"
```
```  1378     by (auto intro: finite_subset)
```
```  1379   then show ?thesis
```
```  1380     using assms
```
```  1381   proof (induct A arbitrary: B)
```
```  1382     case empty
```
```  1383     then show ?case by simp
```
```  1384   next
```
```  1385     case (insert x A)
```
```  1386     then have "x \<in> B"
```
```  1387       by simp
```
```  1388     from insert have "A \<subseteq> B - {x}" and "finite (B - {x})"
```
```  1389       by auto
```
```  1390     with insert.hyps have "card A \<le> card (B - {x})"
```
```  1391       by auto
```
```  1392     with \<open>finite A\<close> \<open>x \<notin> A\<close> \<open>finite B\<close> \<open>x \<in> B\<close> show ?case
```
```  1393       by simp (simp only: card.remove)
```
```  1394   qed
```
```  1395 qed
```
```  1396
```
```  1397 lemma card_seteq: "finite B \<Longrightarrow> (\<And>A. A \<subseteq> B \<Longrightarrow> card B \<le> card A \<Longrightarrow> A = B)"
```
```  1398   apply (induct rule: finite_induct)
```
```  1399    apply simp
```
```  1400   apply clarify
```
```  1401   apply (subgoal_tac "finite A \<and> A - {x} \<subseteq> F")
```
```  1402    prefer 2 apply (blast intro: finite_subset, atomize)
```
```  1403   apply (drule_tac x = "A - {x}" in spec)
```
```  1404   apply (simp add: card_Diff_singleton_if split: if_split_asm)
```
```  1405   apply (case_tac "card A", auto)
```
```  1406   done
```
```  1407
```
```  1408 lemma psubset_card_mono: "finite B \<Longrightarrow> A < B \<Longrightarrow> card A < card B"
```
```  1409   apply (simp add: psubset_eq linorder_not_le [symmetric])
```
```  1410   apply (blast dest: card_seteq)
```
```  1411   done
```
```  1412
```
```  1413 lemma card_Un_Int:
```
```  1414   assumes "finite A" "finite B"
```
```  1415   shows "card A + card B = card (A \<union> B) + card (A \<inter> B)"
```
```  1416   using assms
```
```  1417 proof (induct A)
```
```  1418   case empty
```
```  1419   then show ?case by simp
```
```  1420 next
```
```  1421   case insert
```
```  1422   then show ?case
```
```  1423     by (auto simp add: insert_absorb Int_insert_left)
```
```  1424 qed
```
```  1425
```
```  1426 lemma card_Un_disjoint: "finite A \<Longrightarrow> finite B \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> card (A \<union> B) = card A + card B"
```
```  1427   using card_Un_Int [of A B] by simp
```
```  1428
```
```  1429 lemma card_Un_le: "card (A \<union> B) \<le> card A + card B"
```
```  1430   apply (cases "finite A")
```
```  1431    apply (cases "finite B")
```
```  1432     apply (use le_iff_add card_Un_Int in blast)
```
```  1433    apply simp
```
```  1434   apply simp
```
```  1435   done
```
```  1436
```
```  1437 lemma card_Diff_subset:
```
```  1438   assumes "finite B"
```
```  1439     and "B \<subseteq> A"
```
```  1440   shows "card (A - B) = card A - card B"
```
```  1441   using assms
```
```  1442 proof (cases "finite A")
```
```  1443   case False
```
```  1444   with assms show ?thesis
```
```  1445     by simp
```
```  1446 next
```
```  1447   case True
```
```  1448   with assms show ?thesis
```
```  1449     by (induct B arbitrary: A) simp_all
```
```  1450 qed
```
```  1451
```
```  1452 lemma card_Diff_subset_Int:
```
```  1453   assumes "finite (A \<inter> B)"
```
```  1454   shows "card (A - B) = card A - card (A \<inter> B)"
```
```  1455 proof -
```
```  1456   have "A - B = A - A \<inter> B" by auto
```
```  1457   with assms show ?thesis
```
```  1458     by (simp add: card_Diff_subset)
```
```  1459 qed
```
```  1460
```
```  1461 lemma diff_card_le_card_Diff:
```
```  1462   assumes "finite B"
```
```  1463   shows "card A - card B \<le> card (A - B)"
```
```  1464 proof -
```
```  1465   have "card A - card B \<le> card A - card (A \<inter> B)"
```
```  1466     using card_mono[OF assms Int_lower2, of A] by arith
```
```  1467   also have "\<dots> = card (A - B)"
```
```  1468     using assms by (simp add: card_Diff_subset_Int)
```
```  1469   finally show ?thesis .
```
```  1470 qed
```
```  1471
```
```  1472 lemma card_Diff1_less: "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) < card A"
```
```  1473   by (rule Suc_less_SucD) (simp add: card_Suc_Diff1 del: card_Diff_insert)
```
```  1474
```
```  1475 lemma card_Diff2_less: "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> card (A - {x} - {y}) < card A"
```
```  1476   apply (cases "x = y")
```
```  1477    apply (simp add: card_Diff1_less del:card_Diff_insert)
```
```  1478   apply (rule less_trans)
```
```  1479    prefer 2 apply (auto intro!: card_Diff1_less simp del: card_Diff_insert)
```
```  1480   done
```
```  1481
```
```  1482 lemma card_Diff1_le: "finite A \<Longrightarrow> card (A - {x}) \<le> card A"
```
```  1483   by (cases "x \<in> A") (simp_all add: card_Diff1_less less_imp_le)
```
```  1484
```
```  1485 lemma card_psubset: "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> card A < card B \<Longrightarrow> A < B"
```
```  1486   by (erule psubsetI) blast
```
```  1487
```
```  1488 lemma card_le_inj:
```
```  1489   assumes fA: "finite A"
```
```  1490     and fB: "finite B"
```
```  1491     and c: "card A \<le> card B"
```
```  1492   shows "\<exists>f. f ` A \<subseteq> B \<and> inj_on f A"
```
```  1493   using fA fB c
```
```  1494 proof (induct arbitrary: B rule: finite_induct)
```
```  1495   case empty
```
```  1496   then show ?case by simp
```
```  1497 next
```
```  1498   case (insert x s t)
```
```  1499   then show ?case
```
```  1500   proof (induct rule: finite_induct [OF insert.prems(1)])
```
```  1501     case 1
```
```  1502     then show ?case by simp
```
```  1503   next
```
```  1504     case (2 y t)
```
```  1505     from "2.prems"(1,2,5) "2.hyps"(1,2) have cst: "card s \<le> card t"
```
```  1506       by simp
```
```  1507     from "2.prems"(3) [OF "2.hyps"(1) cst]
```
```  1508     obtain f where "f ` s \<subseteq> t" "inj_on f s"
```
```  1509       by blast
```
```  1510     with "2.prems"(2) "2.hyps"(2) show ?case
```
```  1511       apply -
```
```  1512       apply (rule exI[where x = "\<lambda>z. if z = x then y else f z"])
```
```  1513       apply (auto simp add: inj_on_def)
```
```  1514       done
```
```  1515   qed
```
```  1516 qed
```
```  1517
```
```  1518 lemma card_subset_eq:
```
```  1519   assumes fB: "finite B"
```
```  1520     and AB: "A \<subseteq> B"
```
```  1521     and c: "card A = card B"
```
```  1522   shows "A = B"
```
```  1523 proof -
```
```  1524   from fB AB have fA: "finite A"
```
```  1525     by (auto intro: finite_subset)
```
```  1526   from fA fB have fBA: "finite (B - A)"
```
```  1527     by auto
```
```  1528   have e: "A \<inter> (B - A) = {}"
```
```  1529     by blast
```
```  1530   have eq: "A \<union> (B - A) = B"
```
```  1531     using AB by blast
```
```  1532   from card_Un_disjoint[OF fA fBA e, unfolded eq c] have "card (B - A) = 0"
```
```  1533     by arith
```
```  1534   then have "B - A = {}"
```
```  1535     unfolding card_eq_0_iff using fA fB by simp
```
```  1536   with AB show "A = B"
```
```  1537     by blast
```
```  1538 qed
```
```  1539
```
```  1540 lemma insert_partition:
```
```  1541   "x \<notin> F \<Longrightarrow> \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<Longrightarrow> x \<inter> \<Union>F = {}"
```
```  1542   by auto  (* somewhat slow *)
```
```  1543
```
```  1544 lemma finite_psubset_induct [consumes 1, case_names psubset]:
```
```  1545   assumes finite: "finite A"
```
```  1546     and major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A"
```
```  1547   shows "P A"
```
```  1548   using finite
```
```  1549 proof (induct A taking: card rule: measure_induct_rule)
```
```  1550   case (less A)
```
```  1551   have fin: "finite A" by fact
```
```  1552   have ih: "card B < card A \<Longrightarrow> finite B \<Longrightarrow> P B" for B by fact
```
```  1553   have "P B" if "B \<subset> A" for B
```
```  1554   proof -
```
```  1555     from that have "card B < card A"
```
```  1556       using psubset_card_mono fin by blast
```
```  1557     moreover
```
```  1558     from that have "B \<subseteq> A"
```
```  1559       by auto
```
```  1560     then have "finite B"
```
```  1561       using fin finite_subset by blast
```
```  1562     ultimately show ?thesis using ih by simp
```
```  1563   qed
```
```  1564   with fin show "P A" using major by blast
```
```  1565 qed
```
```  1566
```
```  1567 lemma finite_induct_select [consumes 1, case_names empty select]:
```
```  1568   assumes "finite S"
```
```  1569     and "P {}"
```
```  1570     and select: "\<And>T. T \<subset> S \<Longrightarrow> P T \<Longrightarrow> \<exists>s\<in>S - T. P (insert s T)"
```
```  1571   shows "P S"
```
```  1572 proof -
```
```  1573   have "0 \<le> card S" by simp
```
```  1574   then have "\<exists>T \<subseteq> S. card T = card S \<and> P T"
```
```  1575   proof (induct rule: dec_induct)
```
```  1576     case base with \<open>P {}\<close>
```
```  1577     show ?case
```
```  1578       by (intro exI[of _ "{}"]) auto
```
```  1579   next
```
```  1580     case (step n)
```
```  1581     then obtain T where T: "T \<subseteq> S" "card T = n" "P T"
```
```  1582       by auto
```
```  1583     with \<open>n < card S\<close> have "T \<subset> S" "P T"
```
```  1584       by auto
```
```  1585     with select[of T] obtain s where "s \<in> S" "s \<notin> T" "P (insert s T)"
```
```  1586       by auto
```
```  1587     with step(2) T \<open>finite S\<close> show ?case
```
```  1588       by (intro exI[of _ "insert s T"]) (auto dest: finite_subset)
```
```  1589   qed
```
```  1590   with \<open>finite S\<close> show "P S"
```
```  1591     by (auto dest: card_subset_eq)
```
```  1592 qed
```
```  1593
```
```  1594 lemma remove_induct [case_names empty infinite remove]:
```
```  1595   assumes empty: "P ({} :: 'a set)"
```
```  1596     and infinite: "\<not> finite B \<Longrightarrow> P B"
```
```  1597     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"
```
```  1598   shows "P B"
```
```  1599 proof (cases "finite B")
```
```  1600   case False
```
```  1601   then show ?thesis by (rule infinite)
```
```  1602 next
```
```  1603   case True
```
```  1604   define A where "A = B"
```
```  1605   with True have "finite A" "A \<subseteq> B"
```
```  1606     by simp_all
```
```  1607   then show "P A"
```
```  1608   proof (induct "card A" arbitrary: A)
```
```  1609     case 0
```
```  1610     then have "A = {}" by auto
```
```  1611     with empty show ?case by simp
```
```  1612   next
```
```  1613     case (Suc n A)
```
```  1614     from \<open>A \<subseteq> B\<close> and \<open>finite B\<close> have "finite A"
```
```  1615       by (rule finite_subset)
```
```  1616     moreover from Suc.hyps have "A \<noteq> {}" by auto
```
```  1617     moreover note \<open>A \<subseteq> B\<close>
```
```  1618     moreover have "P (A - {x})" if x: "x \<in> A" for x
```
```  1619       using x Suc.prems \<open>Suc n = card A\<close> by (intro Suc) auto
```
```  1620     ultimately show ?case by (rule remove)
```
```  1621   qed
```
```  1622 qed
```
```  1623
```
```  1624 lemma finite_remove_induct [consumes 1, case_names empty remove]:
```
```  1625   fixes P :: "'a set \<Rightarrow> bool"
```
```  1626   assumes "finite B"
```
```  1627     and "P {}"
```
```  1628     and "\<And>A. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P (A - {x})) \<Longrightarrow> P A"
```
```  1629   defines "B' \<equiv> B"
```
```  1630   shows "P B'"
```
```  1631   by (induct B' rule: remove_induct) (simp_all add: assms)
```
```  1632
```
```  1633
```
```  1634 text \<open>Main cardinality theorem.\<close>
```
```  1635 lemma card_partition [rule_format]:
```
```  1636   "finite C \<Longrightarrow> finite (\<Union>C) \<Longrightarrow> (\<forall>c\<in>C. card c = k) \<Longrightarrow>
```
```  1637     (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {}) \<Longrightarrow>
```
```  1638     k * card C = card (\<Union>C)"
```
```  1639 proof (induct rule: finite_induct)
```
```  1640   case empty
```
```  1641   then show ?case by simp
```
```  1642 next
```
```  1643   case (insert x F)
```
```  1644   then show ?case
```
```  1645     by (simp add: card_Un_disjoint insert_partition finite_subset [of _ "\<Union>(insert _ _)"])
```
```  1646 qed
```
```  1647
```
```  1648 lemma card_eq_UNIV_imp_eq_UNIV:
```
```  1649   assumes fin: "finite (UNIV :: 'a set)"
```
```  1650     and card: "card A = card (UNIV :: 'a set)"
```
```  1651   shows "A = (UNIV :: 'a set)"
```
```  1652 proof
```
```  1653   show "A \<subseteq> UNIV" by simp
```
```  1654   show "UNIV \<subseteq> A"
```
```  1655   proof
```
```  1656     show "x \<in> A" for x
```
```  1657     proof (rule ccontr)
```
```  1658       assume "x \<notin> A"
```
```  1659       then have "A \<subset> UNIV" by auto
```
```  1660       with fin have "card A < card (UNIV :: 'a set)"
```
```  1661         by (fact psubset_card_mono)
```
```  1662       with card show False by simp
```
```  1663     qed
```
```  1664   qed
```
```  1665 qed
```
```  1666
```
```  1667 text \<open>The form of a finite set of given cardinality\<close>
```
```  1668
```
```  1669 lemma card_eq_SucD:
```
```  1670   assumes "card A = Suc k"
```
```  1671   shows "\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {})"
```
```  1672 proof -
```
```  1673   have fin: "finite A"
```
```  1674     using assms by (auto intro: ccontr)
```
```  1675   moreover have "card A \<noteq> 0"
```
```  1676     using assms by auto
```
```  1677   ultimately obtain b where b: "b \<in> A"
```
```  1678     by auto
```
```  1679   show ?thesis
```
```  1680   proof (intro exI conjI)
```
```  1681     show "A = insert b (A - {b})"
```
```  1682       using b by blast
```
```  1683     show "b \<notin> A - {b}"
```
```  1684       by blast
```
```  1685     show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
```
```  1686       using assms b fin by (fastforce dest: mk_disjoint_insert)+
```
```  1687   qed
```
```  1688 qed
```
```  1689
```
```  1690 lemma card_Suc_eq:
```
```  1691   "card A = Suc k \<longleftrightarrow>
```
```  1692     (\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {}))"
```
```  1693   apply (auto elim!: card_eq_SucD)
```
```  1694   apply (subst card.insert)
```
```  1695     apply (auto simp add: intro:ccontr)
```
```  1696   done
```
```  1697
```
```  1698 lemma card_1_singletonE:
```
```  1699   assumes "card A = 1"
```
```  1700   obtains x where "A = {x}"
```
```  1701   using assms by (auto simp: card_Suc_eq)
```
```  1702
```
```  1703 lemma is_singleton_altdef: "is_singleton A \<longleftrightarrow> card A = 1"
```
```  1704   unfolding is_singleton_def
```
```  1705   by (auto elim!: card_1_singletonE is_singletonE simp del: One_nat_def)
```
```  1706
```
```  1707 lemma card_le_Suc_iff:
```
```  1708   "finite A \<Longrightarrow> Suc n \<le> card A = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
```
```  1709   by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
```
```  1710     dest: subset_singletonD split: nat.splits if_splits)
```
```  1711
```
```  1712 lemma finite_fun_UNIVD2:
```
```  1713   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
```
```  1714   shows "finite (UNIV :: 'b set)"
```
```  1715 proof -
```
```  1716   from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))" for arbitrary
```
```  1717     by (rule finite_imageI)
```
```  1718   moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)" for arbitrary
```
```  1719     by (rule UNIV_eq_I) auto
```
```  1720   ultimately show "finite (UNIV :: 'b set)"
```
```  1721     by simp
```
```  1722 qed
```
```  1723
```
```  1724 lemma card_UNIV_unit [simp]: "card (UNIV :: unit set) = 1"
```
```  1725   unfolding UNIV_unit by simp
```
```  1726
```
```  1727 lemma infinite_arbitrarily_large:
```
```  1728   assumes "\<not> finite A"
```
```  1729   shows "\<exists>B. finite B \<and> card B = n \<and> B \<subseteq> A"
```
```  1730 proof (induction n)
```
```  1731   case 0
```
```  1732   show ?case by (intro exI[of _ "{}"]) auto
```
```  1733 next
```
```  1734   case (Suc n)
```
```  1735   then obtain B where B: "finite B \<and> card B = n \<and> B \<subseteq> A" ..
```
```  1736   with \<open>\<not> finite A\<close> have "A \<noteq> B" by auto
```
```  1737   with B have "B \<subset> A" by auto
```
```  1738   then have "\<exists>x. x \<in> A - B"
```
```  1739     by (elim psubset_imp_ex_mem)
```
```  1740   then obtain x where x: "x \<in> A - B" ..
```
```  1741   with B have "finite (insert x B) \<and> card (insert x B) = Suc n \<and> insert x B \<subseteq> A"
```
```  1742     by auto
```
```  1743   then show "\<exists>B. finite B \<and> card B = Suc n \<and> B \<subseteq> A" ..
```
```  1744 qed
```
```  1745
```
```  1746
```
```  1747 subsubsection \<open>Cardinality of image\<close>
```
```  1748
```
```  1749 lemma card_image_le: "finite A \<Longrightarrow> card (f ` A) \<le> card A"
```
```  1750   by (induct rule: finite_induct) (simp_all add: le_SucI card_insert_if)
```
```  1751
```
```  1752 lemma card_image: "inj_on f A \<Longrightarrow> card (f ` A) = card A"
```
```  1753 proof (induct A rule: infinite_finite_induct)
```
```  1754   case (infinite A)
```
```  1755   then have "\<not> finite (f ` A)" by (auto dest: finite_imageD)
```
```  1756   with infinite show ?case by simp
```
```  1757 qed simp_all
```
```  1758
```
```  1759 lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
```
```  1760   by (auto simp: card_image bij_betw_def)
```
```  1761
```
```  1762 lemma endo_inj_surj: "finite A \<Longrightarrow> f ` A \<subseteq> A \<Longrightarrow> inj_on f A \<Longrightarrow> f ` A = A"
```
```  1763   by (simp add: card_seteq card_image)
```
```  1764
```
```  1765 lemma eq_card_imp_inj_on:
```
```  1766   assumes "finite A" "card(f ` A) = card A"
```
```  1767   shows "inj_on f A"
```
```  1768   using assms
```
```  1769 proof (induct rule:finite_induct)
```
```  1770   case empty
```
```  1771   show ?case by simp
```
```  1772 next
```
```  1773   case (insert x A)
```
```  1774   then show ?case
```
```  1775     using card_image_le [of A f] by (simp add: card_insert_if split: if_splits)
```
```  1776 qed
```
```  1777
```
```  1778 lemma inj_on_iff_eq_card: "finite A \<Longrightarrow> inj_on f A \<longleftrightarrow> card (f ` A) = card A"
```
```  1779   by (blast intro: card_image eq_card_imp_inj_on)
```
```  1780
```
```  1781 lemma card_inj_on_le:
```
```  1782   assumes "inj_on f A" "f ` A \<subseteq> B" "finite B"
```
```  1783   shows "card A \<le> card B"
```
```  1784 proof -
```
```  1785   have "finite A"
```
```  1786     using assms by (blast intro: finite_imageD dest: finite_subset)
```
```  1787   then show ?thesis
```
```  1788     using assms by (force intro: card_mono simp: card_image [symmetric])
```
```  1789 qed
```
```  1790
```
```  1791 lemma surj_card_le: "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> card B \<le> card A"
```
```  1792   by (blast intro: card_image_le card_mono le_trans)
```
```  1793
```
```  1794 lemma card_bij_eq:
```
```  1795   "inj_on f A \<Longrightarrow> f ` A \<subseteq> B \<Longrightarrow> inj_on g B \<Longrightarrow> g ` B \<subseteq> A \<Longrightarrow> finite A \<Longrightarrow> finite B
```
```  1796     \<Longrightarrow> card A = card B"
```
```  1797   by (auto intro: le_antisym card_inj_on_le)
```
```  1798
```
```  1799 lemma bij_betw_finite: "bij_betw f A B \<Longrightarrow> finite A \<longleftrightarrow> finite B"
```
```  1800   unfolding bij_betw_def using finite_imageD [of f A] by auto
```
```  1801
```
```  1802 lemma inj_on_finite: "inj_on f A \<Longrightarrow> f ` A \<le> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
```
```  1803   using finite_imageD finite_subset by blast
```
```  1804
```
```  1805 lemma card_vimage_inj: "inj f \<Longrightarrow> A \<subseteq> range f \<Longrightarrow> card (f -` A) = card A"
```
```  1806   by (auto 4 3 simp: subset_image_iff inj_vimage_image_eq
```
```  1807       intro: card_image[symmetric, OF subset_inj_on])
```
```  1808
```
```  1809
```
```  1810 subsubsection \<open>Pigeonhole Principles\<close>
```
```  1811
```
```  1812 lemma pigeonhole: "card A > card (f ` A) \<Longrightarrow> \<not> inj_on f A "
```
```  1813   by (auto dest: card_image less_irrefl_nat)
```
```  1814
```
```  1815 lemma pigeonhole_infinite:
```
```  1816   assumes "\<not> finite A" and "finite (f`A)"
```
```  1817   shows "\<exists>a0\<in>A. \<not> finite {a\<in>A. f a = f a0}"
```
```  1818   using assms(2,1)
```
```  1819 proof (induct "f`A" arbitrary: A rule: finite_induct)
```
```  1820   case empty
```
```  1821   then show ?case by simp
```
```  1822 next
```
```  1823   case (insert b F)
```
```  1824   show ?case
```
```  1825   proof (cases "finite {a\<in>A. f a = b}")
```
```  1826     case True
```
```  1827     with \<open>\<not> finite A\<close> have "\<not> finite (A - {a\<in>A. f a = b})"
```
```  1828       by simp
```
```  1829     also have "A - {a\<in>A. f a = b} = {a\<in>A. f a \<noteq> b}"
```
```  1830       by blast
```
```  1831     finally have "\<not> finite {a\<in>A. f a \<noteq> b}" .
```
```  1832     from insert(3)[OF _ this] insert(2,4) show ?thesis
```
```  1833       by simp (blast intro: rev_finite_subset)
```
```  1834   next
```
```  1835     case False
```
```  1836     then have "{a \<in> A. f a = b} \<noteq> {}" by force
```
```  1837     with False show ?thesis by blast
```
```  1838   qed
```
```  1839 qed
```
```  1840
```
```  1841 lemma pigeonhole_infinite_rel:
```
```  1842   assumes "\<not> finite A"
```
```  1843     and "finite B"
```
```  1844     and "\<forall>a\<in>A. \<exists>b\<in>B. R a b"
```
```  1845   shows "\<exists>b\<in>B. \<not> finite {a:A. R a b}"
```
```  1846 proof -
```
```  1847   let ?F = "\<lambda>a. {b\<in>B. R a b}"
```
```  1848   from finite_Pow_iff[THEN iffD2, OF \<open>finite B\<close>] have "finite (?F ` A)"
```
```  1849     by (blast intro: rev_finite_subset)
```
```  1850   from pigeonhole_infinite [where f = ?F, OF assms(1) this]
```
```  1851   obtain a0 where "a0 \<in> A" and infinite: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
```
```  1852   obtain b0 where "b0 \<in> B" and "R a0 b0"
```
```  1853     using \<open>a0 \<in> A\<close> assms(3) by blast
```
```  1854   have "finite {a\<in>A. ?F a = ?F a0}" if "finite {a\<in>A. R a b0}"
```
```  1855     using \<open>b0 \<in> B\<close> \<open>R a0 b0\<close> that by (blast intro: rev_finite_subset)
```
```  1856   with infinite \<open>b0 \<in> B\<close> show ?thesis
```
```  1857     by blast
```
```  1858 qed
```
```  1859
```
```  1860
```
```  1861 subsubsection \<open>Cardinality of sums\<close>
```
```  1862
```
```  1863 lemma card_Plus:
```
```  1864   assumes "finite A" "finite B"
```
```  1865   shows "card (A <+> B) = card A + card B"
```
```  1866 proof -
```
```  1867   have "Inl`A \<inter> Inr`B = {}" by fast
```
```  1868   with assms show ?thesis
```
```  1869     by (simp add: Plus_def card_Un_disjoint card_image)
```
```  1870 qed
```
```  1871
```
```  1872 lemma card_Plus_conv_if:
```
```  1873   "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
```
```  1874   by (auto simp add: card_Plus)
```
```  1875
```
```  1876 text \<open>Relates to equivalence classes.  Based on a theorem of F. KammÃ¼ller.\<close>
```
```  1877
```
```  1878 lemma dvd_partition:
```
```  1879   assumes f: "finite (\<Union>C)"
```
```  1880     and "\<forall>c\<in>C. k dvd card c" "\<forall>c1\<in>C. \<forall>c2\<in>C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {}"
```
```  1881   shows "k dvd card (\<Union>C)"
```
```  1882 proof -
```
```  1883   have "finite C"
```
```  1884     by (rule finite_UnionD [OF f])
```
```  1885   then show ?thesis
```
```  1886     using assms
```
```  1887   proof (induct rule: finite_induct)
```
```  1888     case empty
```
```  1889     show ?case by simp
```
```  1890   next
```
```  1891     case insert
```
```  1892     then show ?case
```
```  1893       apply simp
```
```  1894       apply (subst card_Un_disjoint)
```
```  1895          apply (auto simp add: disjoint_eq_subset_Compl)
```
```  1896       done
```
```  1897   qed
```
```  1898 qed
```
```  1899
```
```  1900
```
```  1901 subsubsection \<open>Relating injectivity and surjectivity\<close>
```
```  1902
```
```  1903 lemma finite_surj_inj:
```
```  1904   assumes "finite A" "A \<subseteq> f ` A"
```
```  1905   shows "inj_on f A"
```
```  1906 proof -
```
```  1907   have "f ` A = A"
```
```  1908     by (rule card_seteq [THEN sym]) (auto simp add: assms card_image_le)
```
```  1909   then show ?thesis using assms
```
```  1910     by (simp add: eq_card_imp_inj_on)
```
```  1911 qed
```
```  1912
```
```  1913 lemma finite_UNIV_surj_inj: "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
```
```  1914   for f :: "'a \<Rightarrow> 'a"
```
```  1915   by (blast intro: finite_surj_inj subset_UNIV)
```
```  1916
```
```  1917 lemma finite_UNIV_inj_surj: "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
```
```  1918   for f :: "'a \<Rightarrow> 'a"
```
```  1919   by (fastforce simp:surj_def dest!: endo_inj_surj)
```
```  1920
```
```  1921 corollary infinite_UNIV_nat [iff]: "\<not> finite (UNIV :: nat set)"
```
```  1922 proof
```
```  1923   assume "finite (UNIV :: nat set)"
```
```  1924   with finite_UNIV_inj_surj [of Suc] show False
```
```  1925     by simp (blast dest: Suc_neq_Zero surjD)
```
```  1926 qed
```
```  1927
```
```  1928 lemma infinite_UNIV_char_0: "\<not> finite (UNIV :: 'a::semiring_char_0 set)"
```
```  1929 proof
```
```  1930   assume "finite (UNIV :: 'a set)"
```
```  1931   with subset_UNIV have "finite (range of_nat :: 'a set)"
```
```  1932     by (rule finite_subset)
```
```  1933   moreover have "inj (of_nat :: nat \<Rightarrow> 'a)"
```
```  1934     by (simp add: inj_on_def)
```
```  1935   ultimately have "finite (UNIV :: nat set)"
```
```  1936     by (rule finite_imageD)
```
```  1937   then show False
```
```  1938     by simp
```
```  1939 qed
```
```  1940
```
```  1941 hide_const (open) Finite_Set.fold
```
```  1942
```
```  1943
```
```  1944 subsection \<open>Infinite Sets\<close>
```
```  1945
```
```  1946 text \<open>
```
```  1947   Some elementary facts about infinite sets, mostly by Stephan Merz.
```
```  1948   Beware! Because "infinite" merely abbreviates a negation, these
```
```  1949   lemmas may not work well with \<open>blast\<close>.
```
```  1950 \<close>
```
```  1951
```
```  1952 abbreviation infinite :: "'a set \<Rightarrow> bool"
```
```  1953   where "infinite S \<equiv> \<not> finite S"
```
```  1954
```
```  1955 text \<open>
```
```  1956   Infinite sets are non-empty, and if we remove some elements from an
```
```  1957   infinite set, the result is still infinite.
```
```  1958 \<close>
```
```  1959
```
```  1960 lemma infinite_imp_nonempty: "infinite S \<Longrightarrow> S \<noteq> {}"
```
```  1961   by auto
```
```  1962
```
```  1963 lemma infinite_remove: "infinite S \<Longrightarrow> infinite (S - {a})"
```
```  1964   by simp
```
```  1965
```
```  1966 lemma Diff_infinite_finite:
```
```  1967   assumes "finite T" "infinite S"
```
```  1968   shows "infinite (S - T)"
```
```  1969   using \<open>finite T\<close>
```
```  1970 proof induct
```
```  1971   from \<open>infinite S\<close> show "infinite (S - {})"
```
```  1972     by auto
```
```  1973 next
```
```  1974   fix T x
```
```  1975   assume ih: "infinite (S - T)"
```
```  1976   have "S - (insert x T) = (S - T) - {x}"
```
```  1977     by (rule Diff_insert)
```
```  1978   with ih show "infinite (S - (insert x T))"
```
```  1979     by (simp add: infinite_remove)
```
```  1980 qed
```
```  1981
```
```  1982 lemma Un_infinite: "infinite S \<Longrightarrow> infinite (S \<union> T)"
```
```  1983   by simp
```
```  1984
```
```  1985 lemma infinite_Un: "infinite (S \<union> T) \<longleftrightarrow> infinite S \<or> infinite T"
```
```  1986   by simp
```
```  1987
```
```  1988 lemma infinite_super:
```
```  1989   assumes "S \<subseteq> T"
```
```  1990     and "infinite S"
```
```  1991   shows "infinite T"
```
```  1992 proof
```
```  1993   assume "finite T"
```
```  1994   with \<open>S \<subseteq> T\<close> have "finite S" by (simp add: finite_subset)
```
```  1995   with \<open>infinite S\<close> show False by simp
```
```  1996 qed
```
```  1997
```
```  1998 proposition infinite_coinduct [consumes 1, case_names infinite]:
```
```  1999   assumes "X A"
```
```  2000     and step: "\<And>A. X A \<Longrightarrow> \<exists>x\<in>A. X (A - {x}) \<or> infinite (A - {x})"
```
```  2001   shows "infinite A"
```
```  2002 proof
```
```  2003   assume "finite A"
```
```  2004   then show False
```
```  2005     using \<open>X A\<close>
```
```  2006   proof (induction rule: finite_psubset_induct)
```
```  2007     case (psubset A)
```
```  2008     then obtain x where "x \<in> A" "X (A - {x}) \<or> infinite (A - {x})"
```
```  2009       using local.step psubset.prems by blast
```
```  2010     then have "X (A - {x})"
```
```  2011       using psubset.hyps by blast
```
```  2012     show False
```
```  2013       apply (rule psubset.IH [where B = "A - {x}"])
```
```  2014        apply (use \<open>x \<in> A\<close> in blast)
```
```  2015       apply (simp add: \<open>X (A - {x})\<close>)
```
```  2016       done
```
```  2017   qed
```
```  2018 qed
```
```  2019
```
```  2020 text \<open>
```
```  2021   For any function with infinite domain and finite range there is some
```
```  2022   element that is the image of infinitely many domain elements.  In
```
```  2023   particular, any infinite sequence of elements from a finite set
```
```  2024   contains some element that occurs infinitely often.
```
```  2025 \<close>
```
```  2026
```
```  2027 lemma inf_img_fin_dom':
```
```  2028   assumes img: "finite (f ` A)"
```
```  2029     and dom: "infinite A"
```
```  2030   shows "\<exists>y \<in> f ` A. infinite (f -` {y} \<inter> A)"
```
```  2031 proof (rule ccontr)
```
```  2032   have "A \<subseteq> (\<Union>y\<in>f ` A. f -` {y} \<inter> A)" by auto
```
```  2033   moreover assume "\<not> ?thesis"
```
```  2034   with img have "finite (\<Union>y\<in>f ` A. f -` {y} \<inter> A)" by blast
```
```  2035   ultimately have "finite A" by (rule finite_subset)
```
```  2036   with dom show False by contradiction
```
```  2037 qed
```
```  2038
```
```  2039 lemma inf_img_fin_domE':
```
```  2040   assumes "finite (f ` A)" and "infinite A"
```
```  2041   obtains y where "y \<in> f`A" and "infinite (f -` {y} \<inter> A)"
```
```  2042   using assms by (blast dest: inf_img_fin_dom')
```
```  2043
```
```  2044 lemma inf_img_fin_dom:
```
```  2045   assumes img: "finite (f`A)" and dom: "infinite A"
```
```  2046   shows "\<exists>y \<in> f`A. infinite (f -` {y})"
```
```  2047   using inf_img_fin_dom'[OF assms] by auto
```
```  2048
```
```  2049 lemma inf_img_fin_domE:
```
```  2050   assumes "finite (f`A)" and "infinite A"
```
```  2051   obtains y where "y \<in> f`A" and "infinite (f -` {y})"
```
```  2052   using assms by (blast dest: inf_img_fin_dom)
```
```  2053
```
```  2054 proposition finite_image_absD: "finite (abs ` S) \<Longrightarrow> finite S"
```
```  2055   for S :: "'a::linordered_ring set"
```
```  2056   by (rule ccontr) (auto simp: abs_eq_iff vimage_def dest: inf_img_fin_dom)
```
```  2057
```
```  2058 end
```