src/HOL/Finite_Set.thy
 author wenzelm Tue Sep 03 01:12:40 2013 +0200 (2013-09-03) changeset 53374 a14d2a854c02 parent 53015 a1119cf551e8 child 53820 9c7e97d67b45 permissions -rw-r--r--
tuned proofs -- clarified flow of facts wrt. calculation;
```     1 (*  Title:      HOL/Finite_Set.thy
```
```     2     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
```
```     3                 with contributions by Jeremy Avigad
```
```     4 *)
```
```     5
```
```     6 header {* Finite sets *}
```
```     7
```
```     8 theory Finite_Set
```
```     9 imports Option Power
```
```    10 begin
```
```    11
```
```    12 subsection {* Predicate for finite sets *}
```
```    13
```
```    14 inductive finite :: "'a set \<Rightarrow> bool"
```
```    15   where
```
```    16     emptyI [simp, intro!]: "finite {}"
```
```    17   | insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)"
```
```    18
```
```    19 simproc_setup finite_Collect ("finite (Collect P)") = {* K Set_Comprehension_Pointfree.simproc *}
```
```    20
```
```    21 lemma finite_induct [case_names empty insert, induct set: finite]:
```
```    22   -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
```
```    23   assumes "finite F"
```
```    24   assumes "P {}"
```
```    25     and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
```
```    26   shows "P F"
```
```    27 using `finite F`
```
```    28 proof induct
```
```    29   show "P {}" by fact
```
```    30   fix x F assume F: "finite F" and P: "P F"
```
```    31   show "P (insert x F)"
```
```    32   proof cases
```
```    33     assume "x \<in> F"
```
```    34     hence "insert x F = F" by (rule insert_absorb)
```
```    35     with P show ?thesis by (simp only:)
```
```    36   next
```
```    37     assume "x \<notin> F"
```
```    38     from F this P show ?thesis by (rule insert)
```
```    39   qed
```
```    40 qed
```
```    41
```
```    42 lemma infinite_finite_induct [case_names infinite empty insert]:
```
```    43   assumes infinite: "\<And>A. \<not> finite A \<Longrightarrow> P A"
```
```    44   assumes empty: "P {}"
```
```    45   assumes insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
```
```    46   shows "P A"
```
```    47 proof (cases "finite A")
```
```    48   case False with infinite show ?thesis .
```
```    49 next
```
```    50   case True then show ?thesis by (induct A) (fact empty insert)+
```
```    51 qed
```
```    52
```
```    53
```
```    54 subsubsection {* Choice principles *}
```
```    55
```
```    56 lemma ex_new_if_finite: -- "does not depend on def of finite at all"
```
```    57   assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
```
```    58   shows "\<exists>a::'a. a \<notin> A"
```
```    59 proof -
```
```    60   from assms have "A \<noteq> UNIV" by blast
```
```    61   then show ?thesis by blast
```
```    62 qed
```
```    63
```
```    64 text {* A finite choice principle. Does not need the SOME choice operator. *}
```
```    65
```
```    66 lemma finite_set_choice:
```
```    67   "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
```
```    68 proof (induct rule: finite_induct)
```
```    69   case empty then show ?case by simp
```
```    70 next
```
```    71   case (insert a A)
```
```    72   then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
```
```    73   show ?case (is "EX f. ?P f")
```
```    74   proof
```
```    75     show "?P(%x. if x = a then b else f x)" using f ab by auto
```
```    76   qed
```
```    77 qed
```
```    78
```
```    79
```
```    80 subsubsection {* Finite sets are the images of initial segments of natural numbers *}
```
```    81
```
```    82 lemma finite_imp_nat_seg_image_inj_on:
```
```    83   assumes "finite A"
```
```    84   shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
```
```    85 using assms
```
```    86 proof induct
```
```    87   case empty
```
```    88   show ?case
```
```    89   proof
```
```    90     show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp
```
```    91   qed
```
```    92 next
```
```    93   case (insert a A)
```
```    94   have notinA: "a \<notin> A" by fact
```
```    95   from insert.hyps obtain n f
```
```    96     where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
```
```    97   hence "insert a A = f(n:=a) ` {i. i < Suc n}"
```
```    98         "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
```
```    99     by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
```
```   100   thus ?case by blast
```
```   101 qed
```
```   102
```
```   103 lemma nat_seg_image_imp_finite:
```
```   104   "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
```
```   105 proof (induct n arbitrary: A)
```
```   106   case 0 thus ?case by simp
```
```   107 next
```
```   108   case (Suc n)
```
```   109   let ?B = "f ` {i. i < n}"
```
```   110   have finB: "finite ?B" by(rule Suc.hyps[OF refl])
```
```   111   show ?case
```
```   112   proof cases
```
```   113     assume "\<exists>k<n. f n = f k"
```
```   114     hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   115     thus ?thesis using finB by simp
```
```   116   next
```
```   117     assume "\<not>(\<exists> k<n. f n = f k)"
```
```   118     hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   119     thus ?thesis using finB by simp
```
```   120   qed
```
```   121 qed
```
```   122
```
```   123 lemma finite_conv_nat_seg_image:
```
```   124   "finite A \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"
```
```   125   by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
```
```   126
```
```   127 lemma finite_imp_inj_to_nat_seg:
```
```   128   assumes "finite A"
```
```   129   shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> inj_on f A"
```
```   130 proof -
```
```   131   from finite_imp_nat_seg_image_inj_on[OF `finite A`]
```
```   132   obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
```
```   133     by (auto simp:bij_betw_def)
```
```   134   let ?f = "the_inv_into {i. i<n} f"
```
```   135   have "inj_on ?f A & ?f ` A = {i. i<n}"
```
```   136     by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
```
```   137   thus ?thesis by blast
```
```   138 qed
```
```   139
```
```   140 lemma finite_Collect_less_nat [iff]:
```
```   141   "finite {n::nat. n < k}"
```
```   142   by (fastforce simp: finite_conv_nat_seg_image)
```
```   143
```
```   144 lemma finite_Collect_le_nat [iff]:
```
```   145   "finite {n::nat. n \<le> k}"
```
```   146   by (simp add: le_eq_less_or_eq Collect_disj_eq)
```
```   147
```
```   148
```
```   149 subsubsection {* Finiteness and common set operations *}
```
```   150
```
```   151 lemma rev_finite_subset:
```
```   152   "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
```
```   153 proof (induct arbitrary: A rule: finite_induct)
```
```   154   case empty
```
```   155   then show ?case by simp
```
```   156 next
```
```   157   case (insert x F A)
```
```   158   have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+
```
```   159   show "finite A"
```
```   160   proof cases
```
```   161     assume x: "x \<in> A"
```
```   162     with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
```
```   163     with r have "finite (A - {x})" .
```
```   164     hence "finite (insert x (A - {x}))" ..
```
```   165     also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
```
```   166     finally show ?thesis .
```
```   167   next
```
```   168     show "A \<subseteq> F ==> ?thesis" by fact
```
```   169     assume "x \<notin> A"
```
```   170     with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
```
```   171   qed
```
```   172 qed
```
```   173
```
```   174 lemma finite_subset:
```
```   175   "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
```
```   176   by (rule rev_finite_subset)
```
```   177
```
```   178 lemma finite_UnI:
```
```   179   assumes "finite F" and "finite G"
```
```   180   shows "finite (F \<union> G)"
```
```   181   using assms by induct simp_all
```
```   182
```
```   183 lemma finite_Un [iff]:
```
```   184   "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
```
```   185   by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])
```
```   186
```
```   187 lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"
```
```   188 proof -
```
```   189   have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp
```
```   190   then have "finite ({a} \<union> A) \<longleftrightarrow> finite A" by (simp only: finite_Un)
```
```   191   then show ?thesis by simp
```
```   192 qed
```
```   193
```
```   194 lemma finite_Int [simp, intro]:
```
```   195   "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
```
```   196   by (blast intro: finite_subset)
```
```   197
```
```   198 lemma finite_Collect_conjI [simp, intro]:
```
```   199   "finite {x. P x} \<or> finite {x. Q x} \<Longrightarrow> finite {x. P x \<and> Q x}"
```
```   200   by (simp add: Collect_conj_eq)
```
```   201
```
```   202 lemma finite_Collect_disjI [simp]:
```
```   203   "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"
```
```   204   by (simp add: Collect_disj_eq)
```
```   205
```
```   206 lemma finite_Diff [simp, intro]:
```
```   207   "finite A \<Longrightarrow> finite (A - B)"
```
```   208   by (rule finite_subset, rule Diff_subset)
```
```   209
```
```   210 lemma finite_Diff2 [simp]:
```
```   211   assumes "finite B"
```
```   212   shows "finite (A - B) \<longleftrightarrow> finite A"
```
```   213 proof -
```
```   214   have "finite A \<longleftrightarrow> finite((A - B) \<union> (A \<inter> B))" by (simp add: Un_Diff_Int)
```
```   215   also have "\<dots> \<longleftrightarrow> finite (A - B)" using `finite B` by simp
```
```   216   finally show ?thesis ..
```
```   217 qed
```
```   218
```
```   219 lemma finite_Diff_insert [iff]:
```
```   220   "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
```
```   221 proof -
```
```   222   have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
```
```   223   moreover have "A - insert a B = A - B - {a}" by auto
```
```   224   ultimately show ?thesis by simp
```
```   225 qed
```
```   226
```
```   227 lemma finite_compl[simp]:
```
```   228   "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   229   by (simp add: Compl_eq_Diff_UNIV)
```
```   230
```
```   231 lemma finite_Collect_not[simp]:
```
```   232   "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   233   by (simp add: Collect_neg_eq)
```
```   234
```
```   235 lemma finite_Union [simp, intro]:
```
```   236   "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite(\<Union>A)"
```
```   237   by (induct rule: finite_induct) simp_all
```
```   238
```
```   239 lemma finite_UN_I [intro]:
```
```   240   "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"
```
```   241   by (induct rule: finite_induct) simp_all
```
```   242
```
```   243 lemma finite_UN [simp]:
```
```   244   "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
```
```   245   by (blast intro: finite_subset)
```
```   246
```
```   247 lemma finite_Inter [intro]:
```
```   248   "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
```
```   249   by (blast intro: Inter_lower finite_subset)
```
```   250
```
```   251 lemma finite_INT [intro]:
```
```   252   "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
```
```   253   by (blast intro: INT_lower finite_subset)
```
```   254
```
```   255 lemma finite_imageI [simp, intro]:
```
```   256   "finite F \<Longrightarrow> finite (h ` F)"
```
```   257   by (induct rule: finite_induct) simp_all
```
```   258
```
```   259 lemma finite_image_set [simp]:
```
```   260   "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
```
```   261   by (simp add: image_Collect [symmetric])
```
```   262
```
```   263 lemma finite_imageD:
```
```   264   assumes "finite (f ` A)" and "inj_on f A"
```
```   265   shows "finite A"
```
```   266 using assms
```
```   267 proof (induct "f ` A" arbitrary: A)
```
```   268   case empty then show ?case by simp
```
```   269 next
```
```   270   case (insert x B)
```
```   271   then have B_A: "insert x B = f ` A" by simp
```
```   272   then obtain y where "x = f y" and "y \<in> A" by blast
```
```   273   from B_A `x \<notin> B` have "B = f ` A - {x}" by blast
```
```   274   with B_A `x \<notin> B` `x = f y` `inj_on f A` `y \<in> A` have "B = f ` (A - {y})" by (simp add: inj_on_image_set_diff)
```
```   275   moreover from `inj_on f A` have "inj_on f (A - {y})" by (rule inj_on_diff)
```
```   276   ultimately have "finite (A - {y})" by (rule insert.hyps)
```
```   277   then show "finite A" by simp
```
```   278 qed
```
```   279
```
```   280 lemma finite_surj:
```
```   281   "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
```
```   282   by (erule finite_subset) (rule finite_imageI)
```
```   283
```
```   284 lemma finite_range_imageI:
```
```   285   "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
```
```   286   by (drule finite_imageI) (simp add: range_composition)
```
```   287
```
```   288 lemma finite_subset_image:
```
```   289   assumes "finite B"
```
```   290   shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
```
```   291 using assms
```
```   292 proof induct
```
```   293   case empty then show ?case by simp
```
```   294 next
```
```   295   case insert then show ?case
```
```   296     by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
```
```   297        blast
```
```   298 qed
```
```   299
```
```   300 lemma finite_vimage_IntI:
```
```   301   "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
```
```   302   apply (induct rule: finite_induct)
```
```   303    apply simp_all
```
```   304   apply (subst vimage_insert)
```
```   305   apply (simp add: finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2)
```
```   306   done
```
```   307
```
```   308 lemma finite_vimageI:
```
```   309   "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
```
```   310   using finite_vimage_IntI[of F h UNIV] by auto
```
```   311
```
```   312 lemma finite_vimageD:
```
```   313   assumes fin: "finite (h -` F)" and surj: "surj h"
```
```   314   shows "finite F"
```
```   315 proof -
```
```   316   have "finite (h ` (h -` F))" using fin by (rule finite_imageI)
```
```   317   also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)
```
```   318   finally show "finite F" .
```
```   319 qed
```
```   320
```
```   321 lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
```
```   322   unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
```
```   323
```
```   324 lemma finite_Collect_bex [simp]:
```
```   325   assumes "finite A"
```
```   326   shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"
```
```   327 proof -
```
```   328   have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto
```
```   329   with assms show ?thesis by simp
```
```   330 qed
```
```   331
```
```   332 lemma finite_Collect_bounded_ex [simp]:
```
```   333   assumes "finite {y. P y}"
```
```   334   shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
```
```   335 proof -
```
```   336   have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
```
```   337   with assms show ?thesis by simp
```
```   338 qed
```
```   339
```
```   340 lemma finite_Plus:
```
```   341   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
```
```   342   by (simp add: Plus_def)
```
```   343
```
```   344 lemma finite_PlusD:
```
```   345   fixes A :: "'a set" and B :: "'b set"
```
```   346   assumes fin: "finite (A <+> B)"
```
```   347   shows "finite A" "finite B"
```
```   348 proof -
```
```   349   have "Inl ` A \<subseteq> A <+> B" by auto
```
```   350   then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
```
```   351   then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
```
```   352 next
```
```   353   have "Inr ` B \<subseteq> A <+> B" by auto
```
```   354   then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
```
```   355   then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
```
```   356 qed
```
```   357
```
```   358 lemma finite_Plus_iff [simp]:
```
```   359   "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
```
```   360   by (auto intro: finite_PlusD finite_Plus)
```
```   361
```
```   362 lemma finite_Plus_UNIV_iff [simp]:
```
```   363   "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
```
```   364   by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
```
```   365
```
```   366 lemma finite_SigmaI [simp, intro]:
```
```   367   "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
```
```   368   by (unfold Sigma_def) blast
```
```   369
```
```   370 lemma finite_SigmaI2:
```
```   371   assumes "finite {x\<in>A. B x \<noteq> {}}"
```
```   372   and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)"
```
```   373   shows "finite (Sigma A B)"
```
```   374 proof -
```
```   375   from assms have "finite (Sigma {x\<in>A. B x \<noteq> {}} B)" by auto
```
```   376   also have "Sigma {x:A. B x \<noteq> {}} B = Sigma A B" by auto
```
```   377   finally show ?thesis .
```
```   378 qed
```
```   379
```
```   380 lemma finite_cartesian_product:
```
```   381   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
```
```   382   by (rule finite_SigmaI)
```
```   383
```
```   384 lemma finite_Prod_UNIV:
```
```   385   "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
```
```   386   by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
```
```   387
```
```   388 lemma finite_cartesian_productD1:
```
```   389   assumes "finite (A \<times> B)" and "B \<noteq> {}"
```
```   390   shows "finite A"
```
```   391 proof -
```
```   392   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
```
```   393     by (auto simp add: finite_conv_nat_seg_image)
```
```   394   then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
```
```   395   with `B \<noteq> {}` have "A = (fst \<circ> f) ` {i::nat. i < n}"
```
```   396     by (simp add: image_compose)
```
```   397   then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
```
```   398   then show ?thesis
```
```   399     by (auto simp add: finite_conv_nat_seg_image)
```
```   400 qed
```
```   401
```
```   402 lemma finite_cartesian_productD2:
```
```   403   assumes "finite (A \<times> B)" and "A \<noteq> {}"
```
```   404   shows "finite B"
```
```   405 proof -
```
```   406   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
```
```   407     by (auto simp add: finite_conv_nat_seg_image)
```
```   408   then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
```
```   409   with `A \<noteq> {}` have "B = (snd \<circ> f) ` {i::nat. i < n}"
```
```   410     by (simp add: image_compose)
```
```   411   then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
```
```   412   then show ?thesis
```
```   413     by (auto simp add: finite_conv_nat_seg_image)
```
```   414 qed
```
```   415
```
```   416 lemma finite_prod:
```
```   417   "finite (UNIV :: ('a \<times> 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
```
```   418 by(auto simp add: UNIV_Times_UNIV[symmetric] simp del: UNIV_Times_UNIV
```
```   419    dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```   420
```
```   421 lemma finite_Pow_iff [iff]:
```
```   422   "finite (Pow A) \<longleftrightarrow> finite A"
```
```   423 proof
```
```   424   assume "finite (Pow A)"
```
```   425   then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
```
```   426   then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
```
```   427 next
```
```   428   assume "finite A"
```
```   429   then show "finite (Pow A)"
```
```   430     by induct (simp_all add: Pow_insert)
```
```   431 qed
```
```   432
```
```   433 corollary finite_Collect_subsets [simp, intro]:
```
```   434   "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
```
```   435   by (simp add: Pow_def [symmetric])
```
```   436
```
```   437 lemma finite_set: "finite (UNIV :: 'a set set) \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   438 by(simp only: finite_Pow_iff Pow_UNIV[symmetric])
```
```   439
```
```   440 lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
```
```   441   by (blast intro: finite_subset [OF subset_Pow_Union])
```
```   442
```
```   443
```
```   444 subsubsection {* Further induction rules on finite sets *}
```
```   445
```
```   446 lemma finite_ne_induct [case_names singleton insert, consumes 2]:
```
```   447   assumes "finite F" and "F \<noteq> {}"
```
```   448   assumes "\<And>x. P {x}"
```
```   449     and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
```
```   450   shows "P F"
```
```   451 using assms
```
```   452 proof induct
```
```   453   case empty then show ?case by simp
```
```   454 next
```
```   455   case (insert x F) then show ?case by cases auto
```
```   456 qed
```
```   457
```
```   458 lemma finite_subset_induct [consumes 2, case_names empty insert]:
```
```   459   assumes "finite F" and "F \<subseteq> A"
```
```   460   assumes empty: "P {}"
```
```   461     and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
```
```   462   shows "P F"
```
```   463 using `finite F` `F \<subseteq> A`
```
```   464 proof induct
```
```   465   show "P {}" by fact
```
```   466 next
```
```   467   fix x F
```
```   468   assume "finite F" and "x \<notin> F" and
```
```   469     P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
```
```   470   show "P (insert x F)"
```
```   471   proof (rule insert)
```
```   472     from i show "x \<in> A" by blast
```
```   473     from i have "F \<subseteq> A" by blast
```
```   474     with P show "P F" .
```
```   475     show "finite F" by fact
```
```   476     show "x \<notin> F" by fact
```
```   477   qed
```
```   478 qed
```
```   479
```
```   480 lemma finite_empty_induct:
```
```   481   assumes "finite A"
```
```   482   assumes "P A"
```
```   483     and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
```
```   484   shows "P {}"
```
```   485 proof -
```
```   486   have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
```
```   487   proof -
```
```   488     fix B :: "'a set"
```
```   489     assume "B \<subseteq> A"
```
```   490     with `finite A` have "finite B" by (rule rev_finite_subset)
```
```   491     from this `B \<subseteq> A` show "P (A - B)"
```
```   492     proof induct
```
```   493       case empty
```
```   494       from `P A` show ?case by simp
```
```   495     next
```
```   496       case (insert b B)
```
```   497       have "P (A - B - {b})"
```
```   498       proof (rule remove)
```
```   499         from `finite A` show "finite (A - B)" by induct auto
```
```   500         from insert show "b \<in> A - B" by simp
```
```   501         from insert show "P (A - B)" by simp
```
```   502       qed
```
```   503       also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
```
```   504       finally show ?case .
```
```   505     qed
```
```   506   qed
```
```   507   then have "P (A - A)" by blast
```
```   508   then show ?thesis by simp
```
```   509 qed
```
```   510
```
```   511
```
```   512 subsection {* Class @{text finite}  *}
```
```   513
```
```   514 class finite =
```
```   515   assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
```
```   516 begin
```
```   517
```
```   518 lemma finite [simp]: "finite (A \<Colon> 'a set)"
```
```   519   by (rule subset_UNIV finite_UNIV finite_subset)+
```
```   520
```
```   521 lemma finite_code [code]: "finite (A \<Colon> 'a set) \<longleftrightarrow> True"
```
```   522   by simp
```
```   523
```
```   524 end
```
```   525
```
```   526 instance prod :: (finite, finite) finite
```
```   527   by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
```
```   528
```
```   529 lemma inj_graph: "inj (%f. {(x, y). y = f x})"
```
```   530   by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
```
```   531
```
```   532 instance "fun" :: (finite, finite) finite
```
```   533 proof
```
```   534   show "finite (UNIV :: ('a => 'b) set)"
```
```   535   proof (rule finite_imageD)
```
```   536     let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
```
```   537     have "range ?graph \<subseteq> Pow UNIV" by simp
```
```   538     moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
```
```   539       by (simp only: finite_Pow_iff finite)
```
```   540     ultimately show "finite (range ?graph)"
```
```   541       by (rule finite_subset)
```
```   542     show "inj ?graph" by (rule inj_graph)
```
```   543   qed
```
```   544 qed
```
```   545
```
```   546 instance bool :: finite
```
```   547   by default (simp add: UNIV_bool)
```
```   548
```
```   549 instance set :: (finite) finite
```
```   550   by default (simp only: Pow_UNIV [symmetric] finite_Pow_iff finite)
```
```   551
```
```   552 instance unit :: finite
```
```   553   by default (simp add: UNIV_unit)
```
```   554
```
```   555 instance sum :: (finite, finite) finite
```
```   556   by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
```
```   557
```
```   558 lemma finite_option_UNIV [simp]:
```
```   559   "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
```
```   560   by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
```
```   561
```
```   562 instance option :: (finite) finite
```
```   563   by default (simp add: UNIV_option_conv)
```
```   564
```
```   565
```
```   566 subsection {* A basic fold functional for finite sets *}
```
```   567
```
```   568 text {* The intended behaviour is
```
```   569 @{text "fold f z {x\<^sub>1, ..., x\<^sub>n} = f x\<^sub>1 (\<dots> (f x\<^sub>n z)\<dots>)"}
```
```   570 if @{text f} is ``left-commutative'':
```
```   571 *}
```
```   572
```
```   573 locale comp_fun_commute =
```
```   574   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
```
```   575   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
```
```   576 begin
```
```   577
```
```   578 lemma fun_left_comm: "f y (f x z) = f x (f y z)"
```
```   579   using comp_fun_commute by (simp add: fun_eq_iff)
```
```   580
```
```   581 lemma commute_left_comp:
```
```   582   "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
```
```   583   by (simp add: o_assoc comp_fun_commute)
```
```   584
```
```   585 end
```
```   586
```
```   587 inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
```
```   588 for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
```
```   589   emptyI [intro]: "fold_graph f z {} z" |
```
```   590   insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
```
```   591       \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
```
```   592
```
```   593 inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
```
```   594
```
```   595 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
```
```   596   "fold f z A = (if finite A then (THE y. fold_graph f z A y) else z)"
```
```   597
```
```   598 text{*A tempting alternative for the definiens is
```
```   599 @{term "if finite A then THE y. fold_graph f z A y else e"}.
```
```   600 It allows the removal of finiteness assumptions from the theorems
```
```   601 @{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
```
```   602 The proofs become ugly. It is not worth the effort. (???) *}
```
```   603
```
```   604 lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
```
```   605 by (induct rule: finite_induct) auto
```
```   606
```
```   607
```
```   608 subsubsection{*From @{const fold_graph} to @{term fold}*}
```
```   609
```
```   610 context comp_fun_commute
```
```   611 begin
```
```   612
```
```   613 lemma fold_graph_finite:
```
```   614   assumes "fold_graph f z A y"
```
```   615   shows "finite A"
```
```   616   using assms by induct simp_all
```
```   617
```
```   618 lemma fold_graph_insertE_aux:
```
```   619   "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'"
```
```   620 proof (induct set: fold_graph)
```
```   621   case (insertI x A y) show ?case
```
```   622   proof (cases "x = a")
```
```   623     assume "x = a" with insertI show ?case by auto
```
```   624   next
```
```   625     assume "x \<noteq> a"
```
```   626     then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
```
```   627       using insertI by auto
```
```   628     have "f x y = f a (f x y')"
```
```   629       unfolding y by (rule fun_left_comm)
```
```   630     moreover have "fold_graph f z (insert x A - {a}) (f x y')"
```
```   631       using y' and `x \<noteq> a` and `x \<notin> A`
```
```   632       by (simp add: insert_Diff_if fold_graph.insertI)
```
```   633     ultimately show ?case by fast
```
```   634   qed
```
```   635 qed simp
```
```   636
```
```   637 lemma fold_graph_insertE:
```
```   638   assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
```
```   639   obtains y where "v = f x y" and "fold_graph f z A y"
```
```   640 using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
```
```   641
```
```   642 lemma fold_graph_determ:
```
```   643   "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
```
```   644 proof (induct arbitrary: y set: fold_graph)
```
```   645   case (insertI x A y v)
```
```   646   from `fold_graph f z (insert x A) v` and `x \<notin> A`
```
```   647   obtain y' where "v = f x y'" and "fold_graph f z A y'"
```
```   648     by (rule fold_graph_insertE)
```
```   649   from `fold_graph f z A y'` have "y' = y" by (rule insertI)
```
```   650   with `v = f x y'` show "v = f x y" by simp
```
```   651 qed fast
```
```   652
```
```   653 lemma fold_equality:
```
```   654   "fold_graph f z A y \<Longrightarrow> fold f z A = y"
```
```   655   by (cases "finite A") (auto simp add: fold_def intro: fold_graph_determ dest: fold_graph_finite)
```
```   656
```
```   657 lemma fold_graph_fold:
```
```   658   assumes "finite A"
```
```   659   shows "fold_graph f z A (fold f z A)"
```
```   660 proof -
```
```   661   from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
```
```   662   moreover note fold_graph_determ
```
```   663   ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
```
```   664   then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
```
```   665   with assms show ?thesis by (simp add: fold_def)
```
```   666 qed
```
```   667
```
```   668 text {* The base case for @{text fold}: *}
```
```   669
```
```   670 lemma (in -) fold_infinite [simp]:
```
```   671   assumes "\<not> finite A"
```
```   672   shows "fold f z A = z"
```
```   673   using assms by (auto simp add: fold_def)
```
```   674
```
```   675 lemma (in -) fold_empty [simp]:
```
```   676   "fold f z {} = z"
```
```   677   by (auto simp add: fold_def)
```
```   678
```
```   679 text{* The various recursion equations for @{const fold}: *}
```
```   680
```
```   681 lemma fold_insert [simp]:
```
```   682   assumes "finite A" and "x \<notin> A"
```
```   683   shows "fold f z (insert x A) = f x (fold f z A)"
```
```   684 proof (rule fold_equality)
```
```   685   fix z
```
```   686   from `finite A` have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
```
```   687   with `x \<notin> A` have "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)
```
```   688   then show "fold_graph f z (insert x A) (f x (fold f z A))" by simp
```
```   689 qed
```
```   690
```
```   691 declare (in -) empty_fold_graphE [rule del] fold_graph.intros [rule del]
```
```   692   -- {* No more proofs involve these. *}
```
```   693
```
```   694 lemma fold_fun_left_comm:
```
```   695   "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
```
```   696 proof (induct rule: finite_induct)
```
```   697   case empty then show ?case by simp
```
```   698 next
```
```   699   case (insert y A) then show ?case
```
```   700     by (simp add: fun_left_comm [of x])
```
```   701 qed
```
```   702
```
```   703 lemma fold_insert2:
```
```   704   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A)  = fold f (f x z) A"
```
```   705   by (simp add: fold_fun_left_comm)
```
```   706
```
```   707 lemma fold_rec:
```
```   708   assumes "finite A" and "x \<in> A"
```
```   709   shows "fold f z A = f x (fold f z (A - {x}))"
```
```   710 proof -
```
```   711   have A: "A = insert x (A - {x})" using `x \<in> A` by blast
```
```   712   then have "fold f z A = fold f z (insert x (A - {x}))" by simp
```
```   713   also have "\<dots> = f x (fold f z (A - {x}))"
```
```   714     by (rule fold_insert) (simp add: `finite A`)+
```
```   715   finally show ?thesis .
```
```   716 qed
```
```   717
```
```   718 lemma fold_insert_remove:
```
```   719   assumes "finite A"
```
```   720   shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
```
```   721 proof -
```
```   722   from `finite A` have "finite (insert x A)" by auto
```
```   723   moreover have "x \<in> insert x A" by auto
```
```   724   ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
```
```   725     by (rule fold_rec)
```
```   726   then show ?thesis by simp
```
```   727 qed
```
```   728
```
```   729 end
```
```   730
```
```   731 text{* Other properties of @{const fold}: *}
```
```   732
```
```   733 lemma fold_image:
```
```   734   assumes "inj_on g A"
```
```   735   shows "fold f z (g ` A) = fold (f \<circ> g) z A"
```
```   736 proof (cases "finite A")
```
```   737   case False with assms show ?thesis by (auto dest: finite_imageD simp add: fold_def)
```
```   738 next
```
```   739   case True
```
```   740   have "fold_graph f z (g ` A) = fold_graph (f \<circ> g) z A"
```
```   741   proof
```
```   742     fix w
```
```   743     show "fold_graph f z (g ` A) w \<longleftrightarrow> fold_graph (f \<circ> g) z A w" (is "?P \<longleftrightarrow> ?Q")
```
```   744     proof
```
```   745       assume ?P then show ?Q using assms
```
```   746       proof (induct "g ` A" w arbitrary: A)
```
```   747         case emptyI then show ?case by (auto intro: fold_graph.emptyI)
```
```   748       next
```
```   749         case (insertI x A r B)
```
```   750         from `inj_on g B` `x \<notin> A` `insert x A = image g B` obtain x' A' where
```
```   751           "x' \<notin> A'" and [simp]: "B = insert x' A'" "x = g x'" "A = g ` A'"
```
```   752           by (rule inj_img_insertE)
```
```   753         from insertI.prems have "fold_graph (f o g) z A' r"
```
```   754           by (auto intro: insertI.hyps)
```
```   755         with `x' \<notin> A'` have "fold_graph (f \<circ> g) z (insert x' A') ((f \<circ> g) x' r)"
```
```   756           by (rule fold_graph.insertI)
```
```   757         then show ?case by simp
```
```   758       qed
```
```   759     next
```
```   760       assume ?Q then show ?P using assms
```
```   761       proof induct
```
```   762         case emptyI thus ?case by (auto intro: fold_graph.emptyI)
```
```   763       next
```
```   764         case (insertI x A r)
```
```   765         from `x \<notin> A` insertI.prems have "g x \<notin> g ` A" by auto
```
```   766         moreover from insertI have "fold_graph f z (g ` A) r" by simp
```
```   767         ultimately have "fold_graph f z (insert (g x) (g ` A)) (f (g x) r)"
```
```   768           by (rule fold_graph.insertI)
```
```   769         then show ?case by simp
```
```   770       qed
```
```   771     qed
```
```   772   qed
```
```   773   with True assms show ?thesis by (auto simp add: fold_def)
```
```   774 qed
```
```   775
```
```   776 lemma fold_cong:
```
```   777   assumes "comp_fun_commute f" "comp_fun_commute g"
```
```   778   assumes "finite A" and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
```
```   779     and "s = t" and "A = B"
```
```   780   shows "fold f s A = fold g t B"
```
```   781 proof -
```
```   782   have "fold f s A = fold g s A"
```
```   783   using `finite A` cong proof (induct A)
```
```   784     case empty then show ?case by simp
```
```   785   next
```
```   786     case (insert x A)
```
```   787     interpret f: comp_fun_commute f by (fact `comp_fun_commute f`)
```
```   788     interpret g: comp_fun_commute g by (fact `comp_fun_commute g`)
```
```   789     from insert show ?case by simp
```
```   790   qed
```
```   791   with assms show ?thesis by simp
```
```   792 qed
```
```   793
```
```   794
```
```   795 text {* A simplified version for idempotent functions: *}
```
```   796
```
```   797 locale comp_fun_idem = comp_fun_commute +
```
```   798   assumes comp_fun_idem: "f x \<circ> f x = f x"
```
```   799 begin
```
```   800
```
```   801 lemma fun_left_idem: "f x (f x z) = f x z"
```
```   802   using comp_fun_idem by (simp add: fun_eq_iff)
```
```   803
```
```   804 lemma fold_insert_idem:
```
```   805   assumes fin: "finite A"
```
```   806   shows "fold f z (insert x A)  = f x (fold f z A)"
```
```   807 proof cases
```
```   808   assume "x \<in> A"
```
```   809   then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
```
```   810   then show ?thesis using assms by (simp add: comp_fun_idem fun_left_idem)
```
```   811 next
```
```   812   assume "x \<notin> A" then show ?thesis using assms by simp
```
```   813 qed
```
```   814
```
```   815 declare fold_insert [simp del] fold_insert_idem [simp]
```
```   816
```
```   817 lemma fold_insert_idem2:
```
```   818   "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
```
```   819   by (simp add: fold_fun_left_comm)
```
```   820
```
```   821 end
```
```   822
```
```   823
```
```   824 subsubsection {* Liftings to @{text comp_fun_commute} etc. *}
```
```   825
```
```   826 lemma (in comp_fun_commute) comp_comp_fun_commute:
```
```   827   "comp_fun_commute (f \<circ> g)"
```
```   828 proof
```
```   829 qed (simp_all add: comp_fun_commute)
```
```   830
```
```   831 lemma (in comp_fun_idem) comp_comp_fun_idem:
```
```   832   "comp_fun_idem (f \<circ> g)"
```
```   833   by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
```
```   834     (simp_all add: comp_fun_idem)
```
```   835
```
```   836 lemma (in comp_fun_commute) comp_fun_commute_funpow:
```
```   837   "comp_fun_commute (\<lambda>x. f x ^^ g x)"
```
```   838 proof
```
```   839   fix y x
```
```   840   show "f y ^^ g y \<circ> f x ^^ g x = f x ^^ g x \<circ> f y ^^ g y"
```
```   841   proof (cases "x = y")
```
```   842     case True then show ?thesis by simp
```
```   843   next
```
```   844     case False show ?thesis
```
```   845     proof (induct "g x" arbitrary: g)
```
```   846       case 0 then show ?case by simp
```
```   847     next
```
```   848       case (Suc n g)
```
```   849       have hyp1: "f y ^^ g y \<circ> f x = f x \<circ> f y ^^ g y"
```
```   850       proof (induct "g y" arbitrary: g)
```
```   851         case 0 then show ?case by simp
```
```   852       next
```
```   853         case (Suc n g)
```
```   854         def h \<equiv> "\<lambda>z. g z - 1"
```
```   855         with Suc have "n = h y" by simp
```
```   856         with Suc have hyp: "f y ^^ h y \<circ> f x = f x \<circ> f y ^^ h y"
```
```   857           by auto
```
```   858         from Suc h_def have "g y = Suc (h y)" by simp
```
```   859         then show ?case by (simp add: comp_assoc hyp)
```
```   860           (simp add: o_assoc comp_fun_commute)
```
```   861       qed
```
```   862       def h \<equiv> "\<lambda>z. if z = x then g x - 1 else g z"
```
```   863       with Suc have "n = h x" by simp
```
```   864       with Suc have "f y ^^ h y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ h y"
```
```   865         by auto
```
```   866       with False h_def have hyp2: "f y ^^ g y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ g y" by simp
```
```   867       from Suc h_def have "g x = Suc (h x)" by simp
```
```   868       then show ?case by (simp del: funpow.simps add: funpow_Suc_right o_assoc hyp2)
```
```   869         (simp add: comp_assoc hyp1)
```
```   870     qed
```
```   871   qed
```
```   872 qed
```
```   873
```
```   874
```
```   875 subsubsection {* Expressing set operations via @{const fold} *}
```
```   876
```
```   877 lemma comp_fun_commute_const:
```
```   878   "comp_fun_commute (\<lambda>_. f)"
```
```   879 proof
```
```   880 qed rule
```
```   881
```
```   882 lemma comp_fun_idem_insert:
```
```   883   "comp_fun_idem insert"
```
```   884 proof
```
```   885 qed auto
```
```   886
```
```   887 lemma comp_fun_idem_remove:
```
```   888   "comp_fun_idem Set.remove"
```
```   889 proof
```
```   890 qed auto
```
```   891
```
```   892 lemma (in semilattice_inf) comp_fun_idem_inf:
```
```   893   "comp_fun_idem inf"
```
```   894 proof
```
```   895 qed (auto simp add: inf_left_commute)
```
```   896
```
```   897 lemma (in semilattice_sup) comp_fun_idem_sup:
```
```   898   "comp_fun_idem sup"
```
```   899 proof
```
```   900 qed (auto simp add: sup_left_commute)
```
```   901
```
```   902 lemma union_fold_insert:
```
```   903   assumes "finite A"
```
```   904   shows "A \<union> B = fold insert B A"
```
```   905 proof -
```
```   906   interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
```
```   907   from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
```
```   908 qed
```
```   909
```
```   910 lemma minus_fold_remove:
```
```   911   assumes "finite A"
```
```   912   shows "B - A = fold Set.remove B A"
```
```   913 proof -
```
```   914   interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove)
```
```   915   from `finite A` have "fold Set.remove B A = B - A" by (induct A arbitrary: B) auto
```
```   916   then show ?thesis ..
```
```   917 qed
```
```   918
```
```   919 lemma comp_fun_commute_filter_fold:
```
```   920   "comp_fun_commute (\<lambda>x A'. if P x then Set.insert x A' else A')"
```
```   921 proof -
```
```   922   interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
```
```   923   show ?thesis by default (auto simp: fun_eq_iff)
```
```   924 qed
```
```   925
```
```   926 lemma Set_filter_fold:
```
```   927   assumes "finite A"
```
```   928   shows "Set.filter P A = fold (\<lambda>x A'. if P x then Set.insert x A' else A') {} A"
```
```   929 using assms
```
```   930 by (induct A)
```
```   931   (auto simp add: Set.filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold])
```
```   932
```
```   933 lemma inter_Set_filter:
```
```   934   assumes "finite B"
```
```   935   shows "A \<inter> B = Set.filter (\<lambda>x. x \<in> A) B"
```
```   936 using assms
```
```   937 by (induct B) (auto simp: Set.filter_def)
```
```   938
```
```   939 lemma image_fold_insert:
```
```   940   assumes "finite A"
```
```   941   shows "image f A = fold (\<lambda>k A. Set.insert (f k) A) {} A"
```
```   942 using assms
```
```   943 proof -
```
```   944   interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A" by default auto
```
```   945   show ?thesis using assms by (induct A) auto
```
```   946 qed
```
```   947
```
```   948 lemma Ball_fold:
```
```   949   assumes "finite A"
```
```   950   shows "Ball A P = fold (\<lambda>k s. s \<and> P k) True A"
```
```   951 using assms
```
```   952 proof -
```
```   953   interpret comp_fun_commute "\<lambda>k s. s \<and> P k" by default auto
```
```   954   show ?thesis using assms by (induct A) auto
```
```   955 qed
```
```   956
```
```   957 lemma Bex_fold:
```
```   958   assumes "finite A"
```
```   959   shows "Bex A P = fold (\<lambda>k s. s \<or> P k) False A"
```
```   960 using assms
```
```   961 proof -
```
```   962   interpret comp_fun_commute "\<lambda>k s. s \<or> P k" by default auto
```
```   963   show ?thesis using assms by (induct A) auto
```
```   964 qed
```
```   965
```
```   966 lemma comp_fun_commute_Pow_fold:
```
```   967   "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)"
```
```   968   by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast
```
```   969
```
```   970 lemma Pow_fold:
```
```   971   assumes "finite A"
```
```   972   shows "Pow A = fold (\<lambda>x A. A \<union> Set.insert x ` A) {{}} A"
```
```   973 using assms
```
```   974 proof -
```
```   975   interpret comp_fun_commute "\<lambda>x A. A \<union> Set.insert x ` A" by (rule comp_fun_commute_Pow_fold)
```
```   976   show ?thesis using assms by (induct A) (auto simp: Pow_insert)
```
```   977 qed
```
```   978
```
```   979 lemma fold_union_pair:
```
```   980   assumes "finite B"
```
```   981   shows "(\<Union>y\<in>B. {(x, y)}) \<union> A = fold (\<lambda>y. Set.insert (x, y)) A B"
```
```   982 proof -
```
```   983   interpret comp_fun_commute "\<lambda>y. Set.insert (x, y)" by default auto
```
```   984   show ?thesis using assms  by (induct B arbitrary: A) simp_all
```
```   985 qed
```
```   986
```
```   987 lemma comp_fun_commute_product_fold:
```
```   988   assumes "finite B"
```
```   989   shows "comp_fun_commute (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B)"
```
```   990 by default (auto simp: fold_union_pair[symmetric] assms)
```
```   991
```
```   992 lemma product_fold:
```
```   993   assumes "finite A"
```
```   994   assumes "finite B"
```
```   995   shows "A \<times> B = fold (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B) {} A"
```
```   996 using assms unfolding Sigma_def
```
```   997 by (induct A)
```
```   998   (simp_all add: comp_fun_commute.fold_insert[OF comp_fun_commute_product_fold] fold_union_pair)
```
```   999
```
```  1000
```
```  1001 context complete_lattice
```
```  1002 begin
```
```  1003
```
```  1004 lemma inf_Inf_fold_inf:
```
```  1005   assumes "finite A"
```
```  1006   shows "inf (Inf A) B = fold inf B A"
```
```  1007 proof -
```
```  1008   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
```
```  1009   from `finite A` fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
```
```  1010     (simp_all add: inf_commute fun_eq_iff)
```
```  1011 qed
```
```  1012
```
```  1013 lemma sup_Sup_fold_sup:
```
```  1014   assumes "finite A"
```
```  1015   shows "sup (Sup A) B = fold sup B A"
```
```  1016 proof -
```
```  1017   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
```
```  1018   from `finite A` fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
```
```  1019     (simp_all add: sup_commute fun_eq_iff)
```
```  1020 qed
```
```  1021
```
```  1022 lemma Inf_fold_inf:
```
```  1023   assumes "finite A"
```
```  1024   shows "Inf A = fold inf top A"
```
```  1025   using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
```
```  1026
```
```  1027 lemma Sup_fold_sup:
```
```  1028   assumes "finite A"
```
```  1029   shows "Sup A = fold sup bot A"
```
```  1030   using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
```
```  1031
```
```  1032 lemma inf_INF_fold_inf:
```
```  1033   assumes "finite A"
```
```  1034   shows "inf B (INFI A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold")
```
```  1035 proof (rule sym)
```
```  1036   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
```
```  1037   interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
```
```  1038   from `finite A` show "?fold = ?inf"
```
```  1039     by (induct A arbitrary: B)
```
```  1040       (simp_all add: INF_def inf_left_commute)
```
```  1041 qed
```
```  1042
```
```  1043 lemma sup_SUP_fold_sup:
```
```  1044   assumes "finite A"
```
```  1045   shows "sup B (SUPR A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold")
```
```  1046 proof (rule sym)
```
```  1047   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
```
```  1048   interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
```
```  1049   from `finite A` show "?fold = ?sup"
```
```  1050     by (induct A arbitrary: B)
```
```  1051       (simp_all add: SUP_def sup_left_commute)
```
```  1052 qed
```
```  1053
```
```  1054 lemma INF_fold_inf:
```
```  1055   assumes "finite A"
```
```  1056   shows "INFI A f = fold (inf \<circ> f) top A"
```
```  1057   using assms inf_INF_fold_inf [of A top] by simp
```
```  1058
```
```  1059 lemma SUP_fold_sup:
```
```  1060   assumes "finite A"
```
```  1061   shows "SUPR A f = fold (sup \<circ> f) bot A"
```
```  1062   using assms sup_SUP_fold_sup [of A bot] by simp
```
```  1063
```
```  1064 end
```
```  1065
```
```  1066
```
```  1067 subsection {* Locales as mini-packages for fold operations *}
```
```  1068
```
```  1069 subsubsection {* The natural case *}
```
```  1070
```
```  1071 locale folding =
```
```  1072   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
```
```  1073   fixes z :: "'b"
```
```  1074   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
```
```  1075 begin
```
```  1076
```
```  1077 definition F :: "'a set \<Rightarrow> 'b"
```
```  1078 where
```
```  1079   eq_fold: "F A = fold f z A"
```
```  1080
```
```  1081 lemma empty [simp]:
```
```  1082   "F {} = z"
```
```  1083   by (simp add: eq_fold)
```
```  1084
```
```  1085 lemma infinite [simp]:
```
```  1086   "\<not> finite A \<Longrightarrow> F A = z"
```
```  1087   by (simp add: eq_fold)
```
```  1088
```
```  1089 lemma insert [simp]:
```
```  1090   assumes "finite A" and "x \<notin> A"
```
```  1091   shows "F (insert x A) = f x (F A)"
```
```  1092 proof -
```
```  1093   interpret comp_fun_commute f
```
```  1094     by default (insert comp_fun_commute, simp add: fun_eq_iff)
```
```  1095   from fold_insert assms
```
```  1096   have "fold f z (insert x A) = f x (fold f z A)" by simp
```
```  1097   with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
```
```  1098 qed
```
```  1099
```
```  1100 lemma remove:
```
```  1101   assumes "finite A" and "x \<in> A"
```
```  1102   shows "F A = f x (F (A - {x}))"
```
```  1103 proof -
```
```  1104   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
```
```  1105     by (auto dest: mk_disjoint_insert)
```
```  1106   moreover from `finite A` A have "finite B" by simp
```
```  1107   ultimately show ?thesis by simp
```
```  1108 qed
```
```  1109
```
```  1110 lemma insert_remove:
```
```  1111   assumes "finite A"
```
```  1112   shows "F (insert x A) = f x (F (A - {x}))"
```
```  1113   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
```
```  1114
```
```  1115 end
```
```  1116
```
```  1117
```
```  1118 subsubsection {* With idempotency *}
```
```  1119
```
```  1120 locale folding_idem = folding +
```
```  1121   assumes comp_fun_idem: "f x \<circ> f x = f x"
```
```  1122 begin
```
```  1123
```
```  1124 declare insert [simp del]
```
```  1125
```
```  1126 lemma insert_idem [simp]:
```
```  1127   assumes "finite A"
```
```  1128   shows "F (insert x A) = f x (F A)"
```
```  1129 proof -
```
```  1130   interpret comp_fun_idem f
```
```  1131     by default (insert comp_fun_commute comp_fun_idem, simp add: fun_eq_iff)
```
```  1132   from fold_insert_idem assms
```
```  1133   have "fold f z (insert x A) = f x (fold f z A)" by simp
```
```  1134   with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
```
```  1135 qed
```
```  1136
```
```  1137 end
```
```  1138
```
```  1139
```
```  1140 subsection {* Finite cardinality *}
```
```  1141
```
```  1142 text {*
```
```  1143   The traditional definition
```
```  1144   @{prop "card A \<equiv> LEAST n. EX f. A = {f i | i. i < n}"}
```
```  1145   is ugly to work with.
```
```  1146   But now that we have @{const fold} things are easy:
```
```  1147 *}
```
```  1148
```
```  1149 definition card :: "'a set \<Rightarrow> nat" where
```
```  1150   "card = folding.F (\<lambda>_. Suc) 0"
```
```  1151
```
```  1152 interpretation card!: folding "\<lambda>_. Suc" 0
```
```  1153 where
```
```  1154   "folding.F (\<lambda>_. Suc) 0 = card"
```
```  1155 proof -
```
```  1156   show "folding (\<lambda>_. Suc)" by default rule
```
```  1157   then interpret card!: folding "\<lambda>_. Suc" 0 .
```
```  1158   from card_def show "folding.F (\<lambda>_. Suc) 0 = card" by rule
```
```  1159 qed
```
```  1160
```
```  1161 lemma card_infinite:
```
```  1162   "\<not> finite A \<Longrightarrow> card A = 0"
```
```  1163   by (fact card.infinite)
```
```  1164
```
```  1165 lemma card_empty:
```
```  1166   "card {} = 0"
```
```  1167   by (fact card.empty)
```
```  1168
```
```  1169 lemma card_insert_disjoint:
```
```  1170   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> card (insert x A) = Suc (card A)"
```
```  1171   by (fact card.insert)
```
```  1172
```
```  1173 lemma card_insert_if:
```
```  1174   "finite A \<Longrightarrow> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
```
```  1175   by auto (simp add: card.insert_remove card.remove)
```
```  1176
```
```  1177 lemma card_ge_0_finite:
```
```  1178   "card A > 0 \<Longrightarrow> finite A"
```
```  1179   by (rule ccontr) simp
```
```  1180
```
```  1181 lemma card_0_eq [simp, no_atp]:
```
```  1182   "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
```
```  1183   by (auto dest: mk_disjoint_insert)
```
```  1184
```
```  1185 lemma finite_UNIV_card_ge_0:
```
```  1186   "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
```
```  1187   by (rule ccontr) simp
```
```  1188
```
```  1189 lemma card_eq_0_iff:
```
```  1190   "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
```
```  1191   by auto
```
```  1192
```
```  1193 lemma card_gt_0_iff:
```
```  1194   "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
```
```  1195   by (simp add: neq0_conv [symmetric] card_eq_0_iff)
```
```  1196
```
```  1197 lemma card_Suc_Diff1:
```
```  1198   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> Suc (card (A - {x})) = card A"
```
```  1199 apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
```
```  1200 apply(simp del:insert_Diff_single)
```
```  1201 done
```
```  1202
```
```  1203 lemma card_Diff_singleton:
```
```  1204   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) = card A - 1"
```
```  1205   by (simp add: card_Suc_Diff1 [symmetric])
```
```  1206
```
```  1207 lemma card_Diff_singleton_if:
```
```  1208   "finite A \<Longrightarrow> card (A - {x}) = (if x \<in> A then card A - 1 else card A)"
```
```  1209   by (simp add: card_Diff_singleton)
```
```  1210
```
```  1211 lemma card_Diff_insert[simp]:
```
```  1212   assumes "finite A" and "a \<in> A" and "a \<notin> B"
```
```  1213   shows "card (A - insert a B) = card (A - B) - 1"
```
```  1214 proof -
```
```  1215   have "A - insert a B = (A - B) - {a}" using assms by blast
```
```  1216   then show ?thesis using assms by(simp add: card_Diff_singleton)
```
```  1217 qed
```
```  1218
```
```  1219 lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
```
```  1220   by (fact card.insert_remove)
```
```  1221
```
```  1222 lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
```
```  1223 by (simp add: card_insert_if)
```
```  1224
```
```  1225 lemma card_Collect_less_nat[simp]: "card{i::nat. i < n} = n"
```
```  1226 by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
```
```  1227
```
```  1228 lemma card_Collect_le_nat[simp]: "card{i::nat. i <= n} = Suc n"
```
```  1229 using card_Collect_less_nat[of "Suc n"] by(simp add: less_Suc_eq_le)
```
```  1230
```
```  1231 lemma card_mono:
```
```  1232   assumes "finite B" and "A \<subseteq> B"
```
```  1233   shows "card A \<le> card B"
```
```  1234 proof -
```
```  1235   from assms have "finite A" by (auto intro: finite_subset)
```
```  1236   then show ?thesis using assms proof (induct A arbitrary: B)
```
```  1237     case empty then show ?case by simp
```
```  1238   next
```
```  1239     case (insert x A)
```
```  1240     then have "x \<in> B" by simp
```
```  1241     from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
```
```  1242     with insert.hyps have "card A \<le> card (B - {x})" by auto
```
```  1243     with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
```
```  1244   qed
```
```  1245 qed
```
```  1246
```
```  1247 lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
```
```  1248 apply (induct rule: finite_induct)
```
```  1249 apply simp
```
```  1250 apply clarify
```
```  1251 apply (subgoal_tac "finite A & A - {x} <= F")
```
```  1252  prefer 2 apply (blast intro: finite_subset, atomize)
```
```  1253 apply (drule_tac x = "A - {x}" in spec)
```
```  1254 apply (simp add: card_Diff_singleton_if split add: split_if_asm)
```
```  1255 apply (case_tac "card A", auto)
```
```  1256 done
```
```  1257
```
```  1258 lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
```
```  1259 apply (simp add: psubset_eq linorder_not_le [symmetric])
```
```  1260 apply (blast dest: card_seteq)
```
```  1261 done
```
```  1262
```
```  1263 lemma card_Un_Int:
```
```  1264   assumes "finite A" and "finite B"
```
```  1265   shows "card A + card B = card (A \<union> B) + card (A \<inter> B)"
```
```  1266 using assms proof (induct A)
```
```  1267   case empty then show ?case by simp
```
```  1268 next
```
```  1269  case (insert x A) then show ?case
```
```  1270     by (auto simp add: insert_absorb Int_insert_left)
```
```  1271 qed
```
```  1272
```
```  1273 lemma card_Un_disjoint:
```
```  1274   assumes "finite A" and "finite B"
```
```  1275   assumes "A \<inter> B = {}"
```
```  1276   shows "card (A \<union> B) = card A + card B"
```
```  1277 using assms card_Un_Int [of A B] by simp
```
```  1278
```
```  1279 lemma card_Diff_subset:
```
```  1280   assumes "finite B" and "B \<subseteq> A"
```
```  1281   shows "card (A - B) = card A - card B"
```
```  1282 proof (cases "finite A")
```
```  1283   case False with assms show ?thesis by simp
```
```  1284 next
```
```  1285   case True with assms show ?thesis by (induct B arbitrary: A) simp_all
```
```  1286 qed
```
```  1287
```
```  1288 lemma card_Diff_subset_Int:
```
```  1289   assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
```
```  1290 proof -
```
```  1291   have "A - B = A - A \<inter> B" by auto
```
```  1292   thus ?thesis
```
```  1293     by (simp add: card_Diff_subset AB)
```
```  1294 qed
```
```  1295
```
```  1296 lemma diff_card_le_card_Diff:
```
```  1297 assumes "finite B" shows "card A - card B \<le> card(A - B)"
```
```  1298 proof-
```
```  1299   have "card A - card B \<le> card A - card (A \<inter> B)"
```
```  1300     using card_mono[OF assms Int_lower2, of A] by arith
```
```  1301   also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
```
```  1302   finally show ?thesis .
```
```  1303 qed
```
```  1304
```
```  1305 lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
```
```  1306 apply (rule Suc_less_SucD)
```
```  1307 apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
```
```  1308 done
```
```  1309
```
```  1310 lemma card_Diff2_less:
```
```  1311   "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
```
```  1312 apply (case_tac "x = y")
```
```  1313  apply (simp add: card_Diff1_less del:card_Diff_insert)
```
```  1314 apply (rule less_trans)
```
```  1315  prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
```
```  1316 done
```
```  1317
```
```  1318 lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
```
```  1319 apply (case_tac "x : A")
```
```  1320  apply (simp_all add: card_Diff1_less less_imp_le)
```
```  1321 done
```
```  1322
```
```  1323 lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
```
```  1324 by (erule psubsetI, blast)
```
```  1325
```
```  1326 lemma insert_partition:
```
```  1327   "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
```
```  1328   \<Longrightarrow> x \<inter> \<Union> F = {}"
```
```  1329 by auto
```
```  1330
```
```  1331 lemma finite_psubset_induct[consumes 1, case_names psubset]:
```
```  1332   assumes fin: "finite A"
```
```  1333   and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A"
```
```  1334   shows "P A"
```
```  1335 using fin
```
```  1336 proof (induct A taking: card rule: measure_induct_rule)
```
```  1337   case (less A)
```
```  1338   have fin: "finite A" by fact
```
```  1339   have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
```
```  1340   { fix B
```
```  1341     assume asm: "B \<subset> A"
```
```  1342     from asm have "card B < card A" using psubset_card_mono fin by blast
```
```  1343     moreover
```
```  1344     from asm have "B \<subseteq> A" by auto
```
```  1345     then have "finite B" using fin finite_subset by blast
```
```  1346     ultimately
```
```  1347     have "P B" using ih by simp
```
```  1348   }
```
```  1349   with fin show "P A" using major by blast
```
```  1350 qed
```
```  1351
```
```  1352 text{* main cardinality theorem *}
```
```  1353 lemma card_partition [rule_format]:
```
```  1354   "finite C ==>
```
```  1355      finite (\<Union> C) -->
```
```  1356      (\<forall>c\<in>C. card c = k) -->
```
```  1357      (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
```
```  1358      k * card(C) = card (\<Union> C)"
```
```  1359 apply (erule finite_induct, simp)
```
```  1360 apply (simp add: card_Un_disjoint insert_partition
```
```  1361        finite_subset [of _ "\<Union> (insert x F)"])
```
```  1362 done
```
```  1363
```
```  1364 lemma card_eq_UNIV_imp_eq_UNIV:
```
```  1365   assumes fin: "finite (UNIV :: 'a set)"
```
```  1366   and card: "card A = card (UNIV :: 'a set)"
```
```  1367   shows "A = (UNIV :: 'a set)"
```
```  1368 proof
```
```  1369   show "A \<subseteq> UNIV" by simp
```
```  1370   show "UNIV \<subseteq> A"
```
```  1371   proof
```
```  1372     fix x
```
```  1373     show "x \<in> A"
```
```  1374     proof (rule ccontr)
```
```  1375       assume "x \<notin> A"
```
```  1376       then have "A \<subset> UNIV" by auto
```
```  1377       with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
```
```  1378       with card show False by simp
```
```  1379     qed
```
```  1380   qed
```
```  1381 qed
```
```  1382
```
```  1383 text{*The form of a finite set of given cardinality*}
```
```  1384
```
```  1385 lemma card_eq_SucD:
```
```  1386 assumes "card A = Suc k"
```
```  1387 shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
```
```  1388 proof -
```
```  1389   have fin: "finite A" using assms by (auto intro: ccontr)
```
```  1390   moreover have "card A \<noteq> 0" using assms by auto
```
```  1391   ultimately obtain b where b: "b \<in> A" by auto
```
```  1392   show ?thesis
```
```  1393   proof (intro exI conjI)
```
```  1394     show "A = insert b (A-{b})" using b by blast
```
```  1395     show "b \<notin> A - {b}" by blast
```
```  1396     show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
```
```  1397       using assms b fin by(fastforce dest:mk_disjoint_insert)+
```
```  1398   qed
```
```  1399 qed
```
```  1400
```
```  1401 lemma card_Suc_eq:
```
```  1402   "(card A = Suc k) =
```
```  1403    (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
```
```  1404 apply(rule iffI)
```
```  1405  apply(erule card_eq_SucD)
```
```  1406 apply(auto)
```
```  1407 apply(subst card.insert)
```
```  1408  apply(auto intro:ccontr)
```
```  1409 done
```
```  1410
```
```  1411 lemma card_le_Suc_iff: "finite A \<Longrightarrow>
```
```  1412   Suc n \<le> card A = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
```
```  1413 by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
```
```  1414   dest: subset_singletonD split: nat.splits if_splits)
```
```  1415
```
```  1416 lemma finite_fun_UNIVD2:
```
```  1417   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
```
```  1418   shows "finite (UNIV :: 'b set)"
```
```  1419 proof -
```
```  1420   from fin have "\<And>arbitrary. finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
```
```  1421     by (rule finite_imageI)
```
```  1422   moreover have "\<And>arbitrary. UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
```
```  1423     by (rule UNIV_eq_I) auto
```
```  1424   ultimately show "finite (UNIV :: 'b set)" by simp
```
```  1425 qed
```
```  1426
```
```  1427 lemma card_UNIV_unit [simp]: "card (UNIV :: unit set) = 1"
```
```  1428   unfolding UNIV_unit by simp
```
```  1429
```
```  1430 lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
```
```  1431   unfolding UNIV_bool by simp
```
```  1432
```
```  1433
```
```  1434 subsubsection {* Cardinality of image *}
```
```  1435
```
```  1436 lemma card_image_le: "finite A ==> card (f ` A) <= card A"
```
```  1437 apply (induct rule: finite_induct)
```
```  1438  apply simp
```
```  1439 apply (simp add: le_SucI card_insert_if)
```
```  1440 done
```
```  1441
```
```  1442 lemma card_image:
```
```  1443   assumes "inj_on f A"
```
```  1444   shows "card (f ` A) = card A"
```
```  1445 proof (cases "finite A")
```
```  1446   case True then show ?thesis using assms by (induct A) simp_all
```
```  1447 next
```
```  1448   case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
```
```  1449   with False show ?thesis by simp
```
```  1450 qed
```
```  1451
```
```  1452 lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
```
```  1453 by(auto simp: card_image bij_betw_def)
```
```  1454
```
```  1455 lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
```
```  1456 by (simp add: card_seteq card_image)
```
```  1457
```
```  1458 lemma eq_card_imp_inj_on:
```
```  1459   "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
```
```  1460 apply (induct rule:finite_induct)
```
```  1461 apply simp
```
```  1462 apply(frule card_image_le[where f = f])
```
```  1463 apply(simp add:card_insert_if split:if_splits)
```
```  1464 done
```
```  1465
```
```  1466 lemma inj_on_iff_eq_card:
```
```  1467   "finite A ==> inj_on f A = (card(f ` A) = card A)"
```
```  1468 by(blast intro: card_image eq_card_imp_inj_on)
```
```  1469
```
```  1470
```
```  1471 lemma card_inj_on_le:
```
```  1472   "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
```
```  1473 apply (subgoal_tac "finite A")
```
```  1474  apply (force intro: card_mono simp add: card_image [symmetric])
```
```  1475 apply (blast intro: finite_imageD dest: finite_subset)
```
```  1476 done
```
```  1477
```
```  1478 lemma card_bij_eq:
```
```  1479   "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
```
```  1480      finite A; finite B |] ==> card A = card B"
```
```  1481 by (auto intro: le_antisym card_inj_on_le)
```
```  1482
```
```  1483 lemma bij_betw_finite:
```
```  1484   assumes "bij_betw f A B"
```
```  1485   shows "finite A \<longleftrightarrow> finite B"
```
```  1486 using assms unfolding bij_betw_def
```
```  1487 using finite_imageD[of f A] by auto
```
```  1488
```
```  1489
```
```  1490 subsubsection {* Pigeonhole Principles *}
```
```  1491
```
```  1492 lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
```
```  1493 by (auto dest: card_image less_irrefl_nat)
```
```  1494
```
```  1495 lemma pigeonhole_infinite:
```
```  1496 assumes  "~ finite A" and "finite(f`A)"
```
```  1497 shows "EX a0:A. ~finite{a:A. f a = f a0}"
```
```  1498 proof -
```
```  1499   have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
```
```  1500   proof(induct "f`A" arbitrary: A rule: finite_induct)
```
```  1501     case empty thus ?case by simp
```
```  1502   next
```
```  1503     case (insert b F)
```
```  1504     show ?case
```
```  1505     proof cases
```
```  1506       assume "finite{a:A. f a = b}"
```
```  1507       hence "~ finite(A - {a:A. f a = b})" using `\<not> finite A` by simp
```
```  1508       also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
```
```  1509       finally have "~ finite({a:A. f a \<noteq> b})" .
```
```  1510       from insert(3)[OF _ this]
```
```  1511       show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
```
```  1512     next
```
```  1513       assume 1: "~finite{a:A. f a = b}"
```
```  1514       hence "{a \<in> A. f a = b} \<noteq> {}" by force
```
```  1515       thus ?thesis using 1 by blast
```
```  1516     qed
```
```  1517   qed
```
```  1518   from this[OF assms(2,1)] show ?thesis .
```
```  1519 qed
```
```  1520
```
```  1521 lemma pigeonhole_infinite_rel:
```
```  1522 assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
```
```  1523 shows "EX b:B. ~finite{a:A. R a b}"
```
```  1524 proof -
```
```  1525    let ?F = "%a. {b:B. R a b}"
```
```  1526    from finite_Pow_iff[THEN iffD2, OF `finite B`]
```
```  1527    have "finite(?F ` A)" by(blast intro: rev_finite_subset)
```
```  1528    from pigeonhole_infinite[where f = ?F, OF assms(1) this]
```
```  1529    obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
```
```  1530    obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast
```
```  1531    { assume "finite{a:A. R a b0}"
```
```  1532      then have "finite {a\<in>A. ?F a = ?F a0}"
```
```  1533        using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)
```
```  1534    }
```
```  1535    with 1 `b0 : B` show ?thesis by blast
```
```  1536 qed
```
```  1537
```
```  1538
```
```  1539 subsubsection {* Cardinality of sums *}
```
```  1540
```
```  1541 lemma card_Plus:
```
```  1542   assumes "finite A" and "finite B"
```
```  1543   shows "card (A <+> B) = card A + card B"
```
```  1544 proof -
```
```  1545   have "Inl`A \<inter> Inr`B = {}" by fast
```
```  1546   with assms show ?thesis
```
```  1547     unfolding Plus_def
```
```  1548     by (simp add: card_Un_disjoint card_image)
```
```  1549 qed
```
```  1550
```
```  1551 lemma card_Plus_conv_if:
```
```  1552   "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
```
```  1553   by (auto simp add: card_Plus)
```
```  1554
```
```  1555
```
```  1556 subsubsection {* Cardinality of the Powerset *}
```
```  1557
```
```  1558 lemma card_Pow: "finite A ==> card (Pow A) = 2 ^ card A"
```
```  1559 apply (induct rule: finite_induct)
```
```  1560  apply (simp_all add: Pow_insert)
```
```  1561 apply (subst card_Un_disjoint, blast)
```
```  1562   apply (blast, blast)
```
```  1563 apply (subgoal_tac "inj_on (insert x) (Pow F)")
```
```  1564  apply (subst mult_2)
```
```  1565  apply (simp add: card_image Pow_insert)
```
```  1566 apply (unfold inj_on_def)
```
```  1567 apply (blast elim!: equalityE)
```
```  1568 done
```
```  1569
```
```  1570 text {* Relates to equivalence classes.  Based on a theorem of F. Kamm\"uller.  *}
```
```  1571
```
```  1572 lemma dvd_partition:
```
```  1573   "finite (Union C) ==>
```
```  1574     ALL c : C. k dvd card c ==>
```
```  1575     (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
```
```  1576   k dvd card (Union C)"
```
```  1577 apply (frule finite_UnionD)
```
```  1578 apply (rotate_tac -1)
```
```  1579 apply (induct rule: finite_induct)
```
```  1580 apply simp_all
```
```  1581 apply clarify
```
```  1582 apply (subst card_Un_disjoint)
```
```  1583    apply (auto simp add: disjoint_eq_subset_Compl)
```
```  1584 done
```
```  1585
```
```  1586
```
```  1587 subsubsection {* Relating injectivity and surjectivity *}
```
```  1588
```
```  1589 lemma finite_surj_inj: "finite A \<Longrightarrow> A \<subseteq> f ` A \<Longrightarrow> inj_on f A"
```
```  1590 apply(rule eq_card_imp_inj_on, assumption)
```
```  1591 apply(frule finite_imageI)
```
```  1592 apply(drule (1) card_seteq)
```
```  1593  apply(erule card_image_le)
```
```  1594 apply simp
```
```  1595 done
```
```  1596
```
```  1597 lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
```
```  1598 shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
```
```  1599 by (blast intro: finite_surj_inj subset_UNIV)
```
```  1600
```
```  1601 lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
```
```  1602 shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
```
```  1603 by(fastforce simp:surj_def dest!: endo_inj_surj)
```
```  1604
```
```  1605 corollary infinite_UNIV_nat [iff]:
```
```  1606   "\<not> finite (UNIV :: nat set)"
```
```  1607 proof
```
```  1608   assume "finite (UNIV :: nat set)"
```
```  1609   with finite_UNIV_inj_surj [of Suc]
```
```  1610   show False by simp (blast dest: Suc_neq_Zero surjD)
```
```  1611 qed
```
```  1612
```
```  1613 (* Often leads to bogus ATP proofs because of reduced type information, hence no_atp *)
```
```  1614 lemma infinite_UNIV_char_0 [no_atp]:
```
```  1615   "\<not> finite (UNIV :: 'a::semiring_char_0 set)"
```
```  1616 proof
```
```  1617   assume "finite (UNIV :: 'a set)"
```
```  1618   with subset_UNIV have "finite (range of_nat :: 'a set)"
```
```  1619     by (rule finite_subset)
```
```  1620   moreover have "inj (of_nat :: nat \<Rightarrow> 'a)"
```
```  1621     by (simp add: inj_on_def)
```
```  1622   ultimately have "finite (UNIV :: nat set)"
```
```  1623     by (rule finite_imageD)
```
```  1624   then show False
```
```  1625     by simp
```
```  1626 qed
```
```  1627
```
```  1628 hide_const (open) Finite_Set.fold
```
```  1629
```
```  1630 end
```
```  1631
```