src/HOL/Finite_Set.thy
 author haftmann Tue Mar 26 21:53:56 2013 +0100 (2013-03-26) changeset 51546 2e26df807dc7 parent 51489 f738e6dbd844 child 51598 5dbe537087aa permissions -rw-r--r--
more uniform style for interpretation and sublocale declarations
```     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
```
```    43 subsubsection {* Choice principles *}
```
```    44
```
```    45 lemma ex_new_if_finite: -- "does not depend on def of finite at all"
```
```    46   assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
```
```    47   shows "\<exists>a::'a. a \<notin> A"
```
```    48 proof -
```
```    49   from assms have "A \<noteq> UNIV" by blast
```
```    50   then show ?thesis by blast
```
```    51 qed
```
```    52
```
```    53 text {* A finite choice principle. Does not need the SOME choice operator. *}
```
```    54
```
```    55 lemma finite_set_choice:
```
```    56   "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
```
```    57 proof (induct rule: finite_induct)
```
```    58   case empty then show ?case by simp
```
```    59 next
```
```    60   case (insert a A)
```
```    61   then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
```
```    62   show ?case (is "EX f. ?P f")
```
```    63   proof
```
```    64     show "?P(%x. if x = a then b else f x)" using f ab by auto
```
```    65   qed
```
```    66 qed
```
```    67
```
```    68
```
```    69 subsubsection {* Finite sets are the images of initial segments of natural numbers *}
```
```    70
```
```    71 lemma finite_imp_nat_seg_image_inj_on:
```
```    72   assumes "finite A"
```
```    73   shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
```
```    74 using assms
```
```    75 proof induct
```
```    76   case empty
```
```    77   show ?case
```
```    78   proof
```
```    79     show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp
```
```    80   qed
```
```    81 next
```
```    82   case (insert a A)
```
```    83   have notinA: "a \<notin> A" by fact
```
```    84   from insert.hyps obtain n f
```
```    85     where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
```
```    86   hence "insert a A = f(n:=a) ` {i. i < Suc n}"
```
```    87         "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
```
```    88     by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
```
```    89   thus ?case by blast
```
```    90 qed
```
```    91
```
```    92 lemma nat_seg_image_imp_finite:
```
```    93   "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
```
```    94 proof (induct n arbitrary: A)
```
```    95   case 0 thus ?case by simp
```
```    96 next
```
```    97   case (Suc n)
```
```    98   let ?B = "f ` {i. i < n}"
```
```    99   have finB: "finite ?B" by(rule Suc.hyps[OF refl])
```
```   100   show ?case
```
```   101   proof cases
```
```   102     assume "\<exists>k<n. f n = f k"
```
```   103     hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   104     thus ?thesis using finB by simp
```
```   105   next
```
```   106     assume "\<not>(\<exists> k<n. f n = f k)"
```
```   107     hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   108     thus ?thesis using finB by simp
```
```   109   qed
```
```   110 qed
```
```   111
```
```   112 lemma finite_conv_nat_seg_image:
```
```   113   "finite A \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"
```
```   114   by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
```
```   115
```
```   116 lemma finite_imp_inj_to_nat_seg:
```
```   117   assumes "finite A"
```
```   118   shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> inj_on f A"
```
```   119 proof -
```
```   120   from finite_imp_nat_seg_image_inj_on[OF `finite A`]
```
```   121   obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
```
```   122     by (auto simp:bij_betw_def)
```
```   123   let ?f = "the_inv_into {i. i<n} f"
```
```   124   have "inj_on ?f A & ?f ` A = {i. i<n}"
```
```   125     by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
```
```   126   thus ?thesis by blast
```
```   127 qed
```
```   128
```
```   129 lemma finite_Collect_less_nat [iff]:
```
```   130   "finite {n::nat. n < k}"
```
```   131   by (fastforce simp: finite_conv_nat_seg_image)
```
```   132
```
```   133 lemma finite_Collect_le_nat [iff]:
```
```   134   "finite {n::nat. n \<le> k}"
```
```   135   by (simp add: le_eq_less_or_eq Collect_disj_eq)
```
```   136
```
```   137
```
```   138 subsubsection {* Finiteness and common set operations *}
```
```   139
```
```   140 lemma rev_finite_subset:
```
```   141   "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
```
```   142 proof (induct arbitrary: A rule: finite_induct)
```
```   143   case empty
```
```   144   then show ?case by simp
```
```   145 next
```
```   146   case (insert x F A)
```
```   147   have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+
```
```   148   show "finite A"
```
```   149   proof cases
```
```   150     assume x: "x \<in> A"
```
```   151     with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
```
```   152     with r have "finite (A - {x})" .
```
```   153     hence "finite (insert x (A - {x}))" ..
```
```   154     also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
```
```   155     finally show ?thesis .
```
```   156   next
```
```   157     show "A \<subseteq> F ==> ?thesis" by fact
```
```   158     assume "x \<notin> A"
```
```   159     with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
```
```   160   qed
```
```   161 qed
```
```   162
```
```   163 lemma finite_subset:
```
```   164   "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
```
```   165   by (rule rev_finite_subset)
```
```   166
```
```   167 lemma finite_UnI:
```
```   168   assumes "finite F" and "finite G"
```
```   169   shows "finite (F \<union> G)"
```
```   170   using assms by induct simp_all
```
```   171
```
```   172 lemma finite_Un [iff]:
```
```   173   "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
```
```   174   by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])
```
```   175
```
```   176 lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"
```
```   177 proof -
```
```   178   have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp
```
```   179   then have "finite ({a} \<union> A) \<longleftrightarrow> finite A" by (simp only: finite_Un)
```
```   180   then show ?thesis by simp
```
```   181 qed
```
```   182
```
```   183 lemma finite_Int [simp, intro]:
```
```   184   "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
```
```   185   by (blast intro: finite_subset)
```
```   186
```
```   187 lemma finite_Collect_conjI [simp, intro]:
```
```   188   "finite {x. P x} \<or> finite {x. Q x} \<Longrightarrow> finite {x. P x \<and> Q x}"
```
```   189   by (simp add: Collect_conj_eq)
```
```   190
```
```   191 lemma finite_Collect_disjI [simp]:
```
```   192   "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"
```
```   193   by (simp add: Collect_disj_eq)
```
```   194
```
```   195 lemma finite_Diff [simp, intro]:
```
```   196   "finite A \<Longrightarrow> finite (A - B)"
```
```   197   by (rule finite_subset, rule Diff_subset)
```
```   198
```
```   199 lemma finite_Diff2 [simp]:
```
```   200   assumes "finite B"
```
```   201   shows "finite (A - B) \<longleftrightarrow> finite A"
```
```   202 proof -
```
```   203   have "finite A \<longleftrightarrow> finite((A - B) \<union> (A \<inter> B))" by (simp add: Un_Diff_Int)
```
```   204   also have "\<dots> \<longleftrightarrow> finite (A - B)" using `finite B` by simp
```
```   205   finally show ?thesis ..
```
```   206 qed
```
```   207
```
```   208 lemma finite_Diff_insert [iff]:
```
```   209   "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
```
```   210 proof -
```
```   211   have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
```
```   212   moreover have "A - insert a B = A - B - {a}" by auto
```
```   213   ultimately show ?thesis by simp
```
```   214 qed
```
```   215
```
```   216 lemma finite_compl[simp]:
```
```   217   "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   218   by (simp add: Compl_eq_Diff_UNIV)
```
```   219
```
```   220 lemma finite_Collect_not[simp]:
```
```   221   "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   222   by (simp add: Collect_neg_eq)
```
```   223
```
```   224 lemma finite_Union [simp, intro]:
```
```   225   "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite(\<Union>A)"
```
```   226   by (induct rule: finite_induct) simp_all
```
```   227
```
```   228 lemma finite_UN_I [intro]:
```
```   229   "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"
```
```   230   by (induct rule: finite_induct) simp_all
```
```   231
```
```   232 lemma finite_UN [simp]:
```
```   233   "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
```
```   234   by (blast intro: finite_subset)
```
```   235
```
```   236 lemma finite_Inter [intro]:
```
```   237   "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
```
```   238   by (blast intro: Inter_lower finite_subset)
```
```   239
```
```   240 lemma finite_INT [intro]:
```
```   241   "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
```
```   242   by (blast intro: INT_lower finite_subset)
```
```   243
```
```   244 lemma finite_imageI [simp, intro]:
```
```   245   "finite F \<Longrightarrow> finite (h ` F)"
```
```   246   by (induct rule: finite_induct) simp_all
```
```   247
```
```   248 lemma finite_image_set [simp]:
```
```   249   "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
```
```   250   by (simp add: image_Collect [symmetric])
```
```   251
```
```   252 lemma finite_imageD:
```
```   253   assumes "finite (f ` A)" and "inj_on f A"
```
```   254   shows "finite A"
```
```   255 using assms
```
```   256 proof (induct "f ` A" arbitrary: A)
```
```   257   case empty then show ?case by simp
```
```   258 next
```
```   259   case (insert x B)
```
```   260   then have B_A: "insert x B = f ` A" by simp
```
```   261   then obtain y where "x = f y" and "y \<in> A" by blast
```
```   262   from B_A `x \<notin> B` have "B = f ` A - {x}" by blast
```
```   263   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)
```
```   264   moreover from `inj_on f A` have "inj_on f (A - {y})" by (rule inj_on_diff)
```
```   265   ultimately have "finite (A - {y})" by (rule insert.hyps)
```
```   266   then show "finite A" by simp
```
```   267 qed
```
```   268
```
```   269 lemma finite_surj:
```
```   270   "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
```
```   271   by (erule finite_subset) (rule finite_imageI)
```
```   272
```
```   273 lemma finite_range_imageI:
```
```   274   "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
```
```   275   by (drule finite_imageI) (simp add: range_composition)
```
```   276
```
```   277 lemma finite_subset_image:
```
```   278   assumes "finite B"
```
```   279   shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
```
```   280 using assms
```
```   281 proof induct
```
```   282   case empty then show ?case by simp
```
```   283 next
```
```   284   case insert then show ?case
```
```   285     by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
```
```   286        blast
```
```   287 qed
```
```   288
```
```   289 lemma finite_vimage_IntI:
```
```   290   "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
```
```   291   apply (induct rule: finite_induct)
```
```   292    apply simp_all
```
```   293   apply (subst vimage_insert)
```
```   294   apply (simp add: finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2)
```
```   295   done
```
```   296
```
```   297 lemma finite_vimageI:
```
```   298   "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
```
```   299   using finite_vimage_IntI[of F h UNIV] by auto
```
```   300
```
```   301 lemma finite_vimageD:
```
```   302   assumes fin: "finite (h -` F)" and surj: "surj h"
```
```   303   shows "finite F"
```
```   304 proof -
```
```   305   have "finite (h ` (h -` F))" using fin by (rule finite_imageI)
```
```   306   also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)
```
```   307   finally show "finite F" .
```
```   308 qed
```
```   309
```
```   310 lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
```
```   311   unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
```
```   312
```
```   313 lemma finite_Collect_bex [simp]:
```
```   314   assumes "finite A"
```
```   315   shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"
```
```   316 proof -
```
```   317   have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto
```
```   318   with assms show ?thesis by simp
```
```   319 qed
```
```   320
```
```   321 lemma finite_Collect_bounded_ex [simp]:
```
```   322   assumes "finite {y. P y}"
```
```   323   shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
```
```   324 proof -
```
```   325   have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
```
```   326   with assms show ?thesis by simp
```
```   327 qed
```
```   328
```
```   329 lemma finite_Plus:
```
```   330   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
```
```   331   by (simp add: Plus_def)
```
```   332
```
```   333 lemma finite_PlusD:
```
```   334   fixes A :: "'a set" and B :: "'b set"
```
```   335   assumes fin: "finite (A <+> B)"
```
```   336   shows "finite A" "finite B"
```
```   337 proof -
```
```   338   have "Inl ` A \<subseteq> A <+> B" by auto
```
```   339   then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
```
```   340   then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
```
```   341 next
```
```   342   have "Inr ` B \<subseteq> A <+> B" by auto
```
```   343   then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
```
```   344   then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
```
```   345 qed
```
```   346
```
```   347 lemma finite_Plus_iff [simp]:
```
```   348   "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
```
```   349   by (auto intro: finite_PlusD finite_Plus)
```
```   350
```
```   351 lemma finite_Plus_UNIV_iff [simp]:
```
```   352   "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
```
```   353   by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
```
```   354
```
```   355 lemma finite_SigmaI [simp, intro]:
```
```   356   "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
```
```   357   by (unfold Sigma_def) blast
```
```   358
```
```   359 lemma finite_SigmaI2:
```
```   360   assumes "finite {x\<in>A. B x \<noteq> {}}"
```
```   361   and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)"
```
```   362   shows "finite (Sigma A B)"
```
```   363 proof -
```
```   364   from assms have "finite (Sigma {x\<in>A. B x \<noteq> {}} B)" by auto
```
```   365   also have "Sigma {x:A. B x \<noteq> {}} B = Sigma A B" by auto
```
```   366   finally show ?thesis .
```
```   367 qed
```
```   368
```
```   369 lemma finite_cartesian_product:
```
```   370   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
```
```   371   by (rule finite_SigmaI)
```
```   372
```
```   373 lemma finite_Prod_UNIV:
```
```   374   "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
```
```   375   by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
```
```   376
```
```   377 lemma finite_cartesian_productD1:
```
```   378   assumes "finite (A \<times> B)" and "B \<noteq> {}"
```
```   379   shows "finite A"
```
```   380 proof -
```
```   381   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
```
```   382     by (auto simp add: finite_conv_nat_seg_image)
```
```   383   then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
```
```   384   with `B \<noteq> {}` have "A = (fst \<circ> f) ` {i::nat. i < n}"
```
```   385     by (simp add: image_compose)
```
```   386   then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
```
```   387   then show ?thesis
```
```   388     by (auto simp add: finite_conv_nat_seg_image)
```
```   389 qed
```
```   390
```
```   391 lemma finite_cartesian_productD2:
```
```   392   assumes "finite (A \<times> B)" and "A \<noteq> {}"
```
```   393   shows "finite B"
```
```   394 proof -
```
```   395   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
```
```   396     by (auto simp add: finite_conv_nat_seg_image)
```
```   397   then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
```
```   398   with `A \<noteq> {}` have "B = (snd \<circ> f) ` {i::nat. i < n}"
```
```   399     by (simp add: image_compose)
```
```   400   then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
```
```   401   then show ?thesis
```
```   402     by (auto simp add: finite_conv_nat_seg_image)
```
```   403 qed
```
```   404
```
```   405 lemma finite_prod:
```
```   406   "finite (UNIV :: ('a \<times> 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
```
```   407 by(auto simp add: UNIV_Times_UNIV[symmetric] simp del: UNIV_Times_UNIV
```
```   408    dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```   409
```
```   410 lemma finite_Pow_iff [iff]:
```
```   411   "finite (Pow A) \<longleftrightarrow> finite A"
```
```   412 proof
```
```   413   assume "finite (Pow A)"
```
```   414   then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
```
```   415   then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
```
```   416 next
```
```   417   assume "finite A"
```
```   418   then show "finite (Pow A)"
```
```   419     by induct (simp_all add: Pow_insert)
```
```   420 qed
```
```   421
```
```   422 corollary finite_Collect_subsets [simp, intro]:
```
```   423   "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
```
```   424   by (simp add: Pow_def [symmetric])
```
```   425
```
```   426 lemma finite_set: "finite (UNIV :: 'a set set) \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   427 by(simp only: finite_Pow_iff Pow_UNIV[symmetric])
```
```   428
```
```   429 lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
```
```   430   by (blast intro: finite_subset [OF subset_Pow_Union])
```
```   431
```
```   432
```
```   433 subsubsection {* Further induction rules on finite sets *}
```
```   434
```
```   435 lemma finite_ne_induct [case_names singleton insert, consumes 2]:
```
```   436   assumes "finite F" and "F \<noteq> {}"
```
```   437   assumes "\<And>x. P {x}"
```
```   438     and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
```
```   439   shows "P F"
```
```   440 using assms
```
```   441 proof induct
```
```   442   case empty then show ?case by simp
```
```   443 next
```
```   444   case (insert x F) then show ?case by cases auto
```
```   445 qed
```
```   446
```
```   447 lemma finite_subset_induct [consumes 2, case_names empty insert]:
```
```   448   assumes "finite F" and "F \<subseteq> A"
```
```   449   assumes empty: "P {}"
```
```   450     and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
```
```   451   shows "P F"
```
```   452 using `finite F` `F \<subseteq> A`
```
```   453 proof induct
```
```   454   show "P {}" by fact
```
```   455 next
```
```   456   fix x F
```
```   457   assume "finite F" and "x \<notin> F" and
```
```   458     P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
```
```   459   show "P (insert x F)"
```
```   460   proof (rule insert)
```
```   461     from i show "x \<in> A" by blast
```
```   462     from i have "F \<subseteq> A" by blast
```
```   463     with P show "P F" .
```
```   464     show "finite F" by fact
```
```   465     show "x \<notin> F" by fact
```
```   466   qed
```
```   467 qed
```
```   468
```
```   469 lemma finite_empty_induct:
```
```   470   assumes "finite A"
```
```   471   assumes "P A"
```
```   472     and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
```
```   473   shows "P {}"
```
```   474 proof -
```
```   475   have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
```
```   476   proof -
```
```   477     fix B :: "'a set"
```
```   478     assume "B \<subseteq> A"
```
```   479     with `finite A` have "finite B" by (rule rev_finite_subset)
```
```   480     from this `B \<subseteq> A` show "P (A - B)"
```
```   481     proof induct
```
```   482       case empty
```
```   483       from `P A` show ?case by simp
```
```   484     next
```
```   485       case (insert b B)
```
```   486       have "P (A - B - {b})"
```
```   487       proof (rule remove)
```
```   488         from `finite A` show "finite (A - B)" by induct auto
```
```   489         from insert show "b \<in> A - B" by simp
```
```   490         from insert show "P (A - B)" by simp
```
```   491       qed
```
```   492       also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
```
```   493       finally show ?case .
```
```   494     qed
```
```   495   qed
```
```   496   then have "P (A - A)" by blast
```
```   497   then show ?thesis by simp
```
```   498 qed
```
```   499
```
```   500
```
```   501 subsection {* Class @{text finite}  *}
```
```   502
```
```   503 class finite =
```
```   504   assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
```
```   505 begin
```
```   506
```
```   507 lemma finite [simp]: "finite (A \<Colon> 'a set)"
```
```   508   by (rule subset_UNIV finite_UNIV finite_subset)+
```
```   509
```
```   510 lemma finite_code [code]: "finite (A \<Colon> 'a set) \<longleftrightarrow> True"
```
```   511   by simp
```
```   512
```
```   513 end
```
```   514
```
```   515 instance prod :: (finite, finite) finite
```
```   516   by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
```
```   517
```
```   518 lemma inj_graph: "inj (%f. {(x, y). y = f x})"
```
```   519   by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
```
```   520
```
```   521 instance "fun" :: (finite, finite) finite
```
```   522 proof
```
```   523   show "finite (UNIV :: ('a => 'b) set)"
```
```   524   proof (rule finite_imageD)
```
```   525     let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
```
```   526     have "range ?graph \<subseteq> Pow UNIV" by simp
```
```   527     moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
```
```   528       by (simp only: finite_Pow_iff finite)
```
```   529     ultimately show "finite (range ?graph)"
```
```   530       by (rule finite_subset)
```
```   531     show "inj ?graph" by (rule inj_graph)
```
```   532   qed
```
```   533 qed
```
```   534
```
```   535 instance bool :: finite
```
```   536   by default (simp add: UNIV_bool)
```
```   537
```
```   538 instance set :: (finite) finite
```
```   539   by default (simp only: Pow_UNIV [symmetric] finite_Pow_iff finite)
```
```   540
```
```   541 instance unit :: finite
```
```   542   by default (simp add: UNIV_unit)
```
```   543
```
```   544 instance sum :: (finite, finite) finite
```
```   545   by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
```
```   546
```
```   547 lemma finite_option_UNIV [simp]:
```
```   548   "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
```
```   549   by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
```
```   550
```
```   551 instance option :: (finite) finite
```
```   552   by default (simp add: UNIV_option_conv)
```
```   553
```
```   554
```
```   555 subsection {* A basic fold functional for finite sets *}
```
```   556
```
```   557 text {* The intended behaviour is
```
```   558 @{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
```
```   559 if @{text f} is ``left-commutative'':
```
```   560 *}
```
```   561
```
```   562 locale comp_fun_commute =
```
```   563   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
```
```   564   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
```
```   565 begin
```
```   566
```
```   567 lemma fun_left_comm: "f y (f x z) = f x (f y z)"
```
```   568   using comp_fun_commute by (simp add: fun_eq_iff)
```
```   569
```
```   570 lemma commute_left_comp:
```
```   571   "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
```
```   572   by (simp add: o_assoc comp_fun_commute)
```
```   573
```
```   574 end
```
```   575
```
```   576 inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
```
```   577 for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
```
```   578   emptyI [intro]: "fold_graph f z {} z" |
```
```   579   insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
```
```   580       \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
```
```   581
```
```   582 inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
```
```   583
```
```   584 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
```
```   585   "fold f z A = (if finite A then (THE y. fold_graph f z A y) else z)"
```
```   586
```
```   587 text{*A tempting alternative for the definiens is
```
```   588 @{term "if finite A then THE y. fold_graph f z A y else e"}.
```
```   589 It allows the removal of finiteness assumptions from the theorems
```
```   590 @{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
```
```   591 The proofs become ugly. It is not worth the effort. (???) *}
```
```   592
```
```   593 lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
```
```   594 by (induct rule: finite_induct) auto
```
```   595
```
```   596
```
```   597 subsubsection{*From @{const fold_graph} to @{term fold}*}
```
```   598
```
```   599 context comp_fun_commute
```
```   600 begin
```
```   601
```
```   602 lemma fold_graph_finite:
```
```   603   assumes "fold_graph f z A y"
```
```   604   shows "finite A"
```
```   605   using assms by induct simp_all
```
```   606
```
```   607 lemma fold_graph_insertE_aux:
```
```   608   "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'"
```
```   609 proof (induct set: fold_graph)
```
```   610   case (insertI x A y) show ?case
```
```   611   proof (cases "x = a")
```
```   612     assume "x = a" with insertI show ?case by auto
```
```   613   next
```
```   614     assume "x \<noteq> a"
```
```   615     then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
```
```   616       using insertI by auto
```
```   617     have "f x y = f a (f x y')"
```
```   618       unfolding y by (rule fun_left_comm)
```
```   619     moreover have "fold_graph f z (insert x A - {a}) (f x y')"
```
```   620       using y' and `x \<noteq> a` and `x \<notin> A`
```
```   621       by (simp add: insert_Diff_if fold_graph.insertI)
```
```   622     ultimately show ?case by fast
```
```   623   qed
```
```   624 qed simp
```
```   625
```
```   626 lemma fold_graph_insertE:
```
```   627   assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
```
```   628   obtains y where "v = f x y" and "fold_graph f z A y"
```
```   629 using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
```
```   630
```
```   631 lemma fold_graph_determ:
```
```   632   "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
```
```   633 proof (induct arbitrary: y set: fold_graph)
```
```   634   case (insertI x A y v)
```
```   635   from `fold_graph f z (insert x A) v` and `x \<notin> A`
```
```   636   obtain y' where "v = f x y'" and "fold_graph f z A y'"
```
```   637     by (rule fold_graph_insertE)
```
```   638   from `fold_graph f z A y'` have "y' = y" by (rule insertI)
```
```   639   with `v = f x y'` show "v = f x y" by simp
```
```   640 qed fast
```
```   641
```
```   642 lemma fold_equality:
```
```   643   "fold_graph f z A y \<Longrightarrow> fold f z A = y"
```
```   644   by (cases "finite A") (auto simp add: fold_def intro: fold_graph_determ dest: fold_graph_finite)
```
```   645
```
```   646 lemma fold_graph_fold:
```
```   647   assumes "finite A"
```
```   648   shows "fold_graph f z A (fold f z A)"
```
```   649 proof -
```
```   650   from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
```
```   651   moreover note fold_graph_determ
```
```   652   ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
```
```   653   then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
```
```   654   with assms show ?thesis by (simp add: fold_def)
```
```   655 qed
```
```   656
```
```   657 text {* The base case for @{text fold}: *}
```
```   658
```
```   659 lemma (in -) fold_infinite [simp]:
```
```   660   assumes "\<not> finite A"
```
```   661   shows "fold f z A = z"
```
```   662   using assms by (auto simp add: fold_def)
```
```   663
```
```   664 lemma (in -) fold_empty [simp]:
```
```   665   "fold f z {} = z"
```
```   666   by (auto simp add: fold_def)
```
```   667
```
```   668 text{* The various recursion equations for @{const fold}: *}
```
```   669
```
```   670 lemma fold_insert [simp]:
```
```   671   assumes "finite A" and "x \<notin> A"
```
```   672   shows "fold f z (insert x A) = f x (fold f z A)"
```
```   673 proof (rule fold_equality)
```
```   674   fix z
```
```   675   from `finite A` have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
```
```   676   with `x \<notin> A` have "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)
```
```   677   then show "fold_graph f z (insert x A) (f x (fold f z A))" by simp
```
```   678 qed
```
```   679
```
```   680 declare (in -) empty_fold_graphE [rule del] fold_graph.intros [rule del]
```
```   681   -- {* No more proofs involve these. *}
```
```   682
```
```   683 lemma fold_fun_left_comm:
```
```   684   "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
```
```   685 proof (induct rule: finite_induct)
```
```   686   case empty then show ?case by simp
```
```   687 next
```
```   688   case (insert y A) then show ?case
```
```   689     by (simp add: fun_left_comm [of x])
```
```   690 qed
```
```   691
```
```   692 lemma fold_insert2:
```
```   693   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A)  = fold f (f x z) A"
```
```   694   by (simp add: fold_fun_left_comm)
```
```   695
```
```   696 lemma fold_rec:
```
```   697   assumes "finite A" and "x \<in> A"
```
```   698   shows "fold f z A = f x (fold f z (A - {x}))"
```
```   699 proof -
```
```   700   have A: "A = insert x (A - {x})" using `x \<in> A` by blast
```
```   701   then have "fold f z A = fold f z (insert x (A - {x}))" by simp
```
```   702   also have "\<dots> = f x (fold f z (A - {x}))"
```
```   703     by (rule fold_insert) (simp add: `finite A`)+
```
```   704   finally show ?thesis .
```
```   705 qed
```
```   706
```
```   707 lemma fold_insert_remove:
```
```   708   assumes "finite A"
```
```   709   shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
```
```   710 proof -
```
```   711   from `finite A` have "finite (insert x A)" by auto
```
```   712   moreover have "x \<in> insert x A" by auto
```
```   713   ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
```
```   714     by (rule fold_rec)
```
```   715   then show ?thesis by simp
```
```   716 qed
```
```   717
```
```   718 text{* Other properties of @{const fold}: *}
```
```   719
```
```   720 lemma fold_image:
```
```   721   assumes "finite A" and "inj_on g A"
```
```   722   shows "fold f z (g ` A) = fold (f \<circ> g) z A"
```
```   723 using assms
```
```   724 proof induction
```
```   725   case (insert a F)
```
```   726     interpret comp_fun_commute "\<lambda>x. f (g x)" by default (simp add: comp_fun_commute)
```
```   727     from insert show ?case by auto
```
```   728 qed simp
```
```   729
```
```   730 end
```
```   731
```
```   732 lemma fold_cong:
```
```   733   assumes "comp_fun_commute f" "comp_fun_commute g"
```
```   734   assumes "finite A" and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
```
```   735     and "s = t" and "A = B"
```
```   736   shows "fold f s A = fold g t B"
```
```   737 proof -
```
```   738   have "fold f s A = fold g s A"
```
```   739   using `finite A` cong proof (induct A)
```
```   740     case empty then show ?case by simp
```
```   741   next
```
```   742     case (insert x A)
```
```   743     interpret f: comp_fun_commute f by (fact `comp_fun_commute f`)
```
```   744     interpret g: comp_fun_commute g by (fact `comp_fun_commute g`)
```
```   745     from insert show ?case by simp
```
```   746   qed
```
```   747   with assms show ?thesis by simp
```
```   748 qed
```
```   749
```
```   750
```
```   751 text {* A simplified version for idempotent functions: *}
```
```   752
```
```   753 locale comp_fun_idem = comp_fun_commute +
```
```   754   assumes comp_fun_idem: "f x \<circ> f x = f x"
```
```   755 begin
```
```   756
```
```   757 lemma fun_left_idem: "f x (f x z) = f x z"
```
```   758   using comp_fun_idem by (simp add: fun_eq_iff)
```
```   759
```
```   760 lemma fold_insert_idem:
```
```   761   assumes fin: "finite A"
```
```   762   shows "fold f z (insert x A)  = f x (fold f z A)"
```
```   763 proof cases
```
```   764   assume "x \<in> A"
```
```   765   then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
```
```   766   then show ?thesis using assms by (simp add: comp_fun_idem fun_left_idem)
```
```   767 next
```
```   768   assume "x \<notin> A" then show ?thesis using assms by simp
```
```   769 qed
```
```   770
```
```   771 declare fold_insert [simp del] fold_insert_idem [simp]
```
```   772
```
```   773 lemma fold_insert_idem2:
```
```   774   "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
```
```   775   by (simp add: fold_fun_left_comm)
```
```   776
```
```   777 end
```
```   778
```
```   779
```
```   780 subsubsection {* Liftings to @{text comp_fun_commute} etc. *}
```
```   781
```
```   782 lemma (in comp_fun_commute) comp_comp_fun_commute:
```
```   783   "comp_fun_commute (f \<circ> g)"
```
```   784 proof
```
```   785 qed (simp_all add: comp_fun_commute)
```
```   786
```
```   787 lemma (in comp_fun_idem) comp_comp_fun_idem:
```
```   788   "comp_fun_idem (f \<circ> g)"
```
```   789   by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
```
```   790     (simp_all add: comp_fun_idem)
```
```   791
```
```   792 lemma (in comp_fun_commute) comp_fun_commute_funpow:
```
```   793   "comp_fun_commute (\<lambda>x. f x ^^ g x)"
```
```   794 proof
```
```   795   fix y x
```
```   796   show "f y ^^ g y \<circ> f x ^^ g x = f x ^^ g x \<circ> f y ^^ g y"
```
```   797   proof (cases "x = y")
```
```   798     case True then show ?thesis by simp
```
```   799   next
```
```   800     case False show ?thesis
```
```   801     proof (induct "g x" arbitrary: g)
```
```   802       case 0 then show ?case by simp
```
```   803     next
```
```   804       case (Suc n g)
```
```   805       have hyp1: "f y ^^ g y \<circ> f x = f x \<circ> f y ^^ g y"
```
```   806       proof (induct "g y" arbitrary: g)
```
```   807         case 0 then show ?case by simp
```
```   808       next
```
```   809         case (Suc n g)
```
```   810         def h \<equiv> "\<lambda>z. g z - 1"
```
```   811         with Suc have "n = h y" by simp
```
```   812         with Suc have hyp: "f y ^^ h y \<circ> f x = f x \<circ> f y ^^ h y"
```
```   813           by auto
```
```   814         from Suc h_def have "g y = Suc (h y)" by simp
```
```   815         then show ?case by (simp add: comp_assoc hyp)
```
```   816           (simp add: o_assoc comp_fun_commute)
```
```   817       qed
```
```   818       def h \<equiv> "\<lambda>z. if z = x then g x - 1 else g z"
```
```   819       with Suc have "n = h x" by simp
```
```   820       with Suc have "f y ^^ h y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ h y"
```
```   821         by auto
```
```   822       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
```
```   823       from Suc h_def have "g x = Suc (h x)" by simp
```
```   824       then show ?case by (simp del: funpow.simps add: funpow_Suc_right o_assoc hyp2)
```
```   825         (simp add: comp_assoc hyp1)
```
```   826     qed
```
```   827   qed
```
```   828 qed
```
```   829
```
```   830
```
```   831 subsubsection {* Expressing set operations via @{const fold} *}
```
```   832
```
```   833 lemma comp_fun_commute_const:
```
```   834   "comp_fun_commute (\<lambda>_. f)"
```
```   835 proof
```
```   836 qed rule
```
```   837
```
```   838 lemma comp_fun_idem_insert:
```
```   839   "comp_fun_idem insert"
```
```   840 proof
```
```   841 qed auto
```
```   842
```
```   843 lemma comp_fun_idem_remove:
```
```   844   "comp_fun_idem Set.remove"
```
```   845 proof
```
```   846 qed auto
```
```   847
```
```   848 lemma (in semilattice_inf) comp_fun_idem_inf:
```
```   849   "comp_fun_idem inf"
```
```   850 proof
```
```   851 qed (auto simp add: inf_left_commute)
```
```   852
```
```   853 lemma (in semilattice_sup) comp_fun_idem_sup:
```
```   854   "comp_fun_idem sup"
```
```   855 proof
```
```   856 qed (auto simp add: sup_left_commute)
```
```   857
```
```   858 lemma union_fold_insert:
```
```   859   assumes "finite A"
```
```   860   shows "A \<union> B = fold insert B A"
```
```   861 proof -
```
```   862   interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
```
```   863   from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
```
```   864 qed
```
```   865
```
```   866 lemma minus_fold_remove:
```
```   867   assumes "finite A"
```
```   868   shows "B - A = fold Set.remove B A"
```
```   869 proof -
```
```   870   interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove)
```
```   871   from `finite A` have "fold Set.remove B A = B - A" by (induct A arbitrary: B) auto
```
```   872   then show ?thesis ..
```
```   873 qed
```
```   874
```
```   875 lemma comp_fun_commute_filter_fold:
```
```   876   "comp_fun_commute (\<lambda>x A'. if P x then Set.insert x A' else A')"
```
```   877 proof -
```
```   878   interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
```
```   879   show ?thesis by default (auto simp: fun_eq_iff)
```
```   880 qed
```
```   881
```
```   882 lemma Set_filter_fold:
```
```   883   assumes "finite A"
```
```   884   shows "Set.filter P A = fold (\<lambda>x A'. if P x then Set.insert x A' else A') {} A"
```
```   885 using assms
```
```   886 by (induct A)
```
```   887   (auto simp add: Set.filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold])
```
```   888
```
```   889 lemma inter_Set_filter:
```
```   890   assumes "finite B"
```
```   891   shows "A \<inter> B = Set.filter (\<lambda>x. x \<in> A) B"
```
```   892 using assms
```
```   893 by (induct B) (auto simp: Set.filter_def)
```
```   894
```
```   895 lemma image_fold_insert:
```
```   896   assumes "finite A"
```
```   897   shows "image f A = fold (\<lambda>k A. Set.insert (f k) A) {} A"
```
```   898 using assms
```
```   899 proof -
```
```   900   interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A" by default auto
```
```   901   show ?thesis using assms by (induct A) auto
```
```   902 qed
```
```   903
```
```   904 lemma Ball_fold:
```
```   905   assumes "finite A"
```
```   906   shows "Ball A P = fold (\<lambda>k s. s \<and> P k) True A"
```
```   907 using assms
```
```   908 proof -
```
```   909   interpret comp_fun_commute "\<lambda>k s. s \<and> P k" by default auto
```
```   910   show ?thesis using assms by (induct A) auto
```
```   911 qed
```
```   912
```
```   913 lemma Bex_fold:
```
```   914   assumes "finite A"
```
```   915   shows "Bex A P = fold (\<lambda>k s. s \<or> P k) False A"
```
```   916 using assms
```
```   917 proof -
```
```   918   interpret comp_fun_commute "\<lambda>k s. s \<or> P k" by default auto
```
```   919   show ?thesis using assms by (induct A) auto
```
```   920 qed
```
```   921
```
```   922 lemma comp_fun_commute_Pow_fold:
```
```   923   "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)"
```
```   924   by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast
```
```   925
```
```   926 lemma Pow_fold:
```
```   927   assumes "finite A"
```
```   928   shows "Pow A = fold (\<lambda>x A. A \<union> Set.insert x ` A) {{}} A"
```
```   929 using assms
```
```   930 proof -
```
```   931   interpret comp_fun_commute "\<lambda>x A. A \<union> Set.insert x ` A" by (rule comp_fun_commute_Pow_fold)
```
```   932   show ?thesis using assms by (induct A) (auto simp: Pow_insert)
```
```   933 qed
```
```   934
```
```   935 lemma fold_union_pair:
```
```   936   assumes "finite B"
```
```   937   shows "(\<Union>y\<in>B. {(x, y)}) \<union> A = fold (\<lambda>y. Set.insert (x, y)) A B"
```
```   938 proof -
```
```   939   interpret comp_fun_commute "\<lambda>y. Set.insert (x, y)" by default auto
```
```   940   show ?thesis using assms  by (induct B arbitrary: A) simp_all
```
```   941 qed
```
```   942
```
```   943 lemma comp_fun_commute_product_fold:
```
```   944   assumes "finite B"
```
```   945   shows "comp_fun_commute (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B)"
```
```   946 by default (auto simp: fold_union_pair[symmetric] assms)
```
```   947
```
```   948 lemma product_fold:
```
```   949   assumes "finite A"
```
```   950   assumes "finite B"
```
```   951   shows "A \<times> B = fold (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B) {} A"
```
```   952 using assms unfolding Sigma_def
```
```   953 by (induct A)
```
```   954   (simp_all add: comp_fun_commute.fold_insert[OF comp_fun_commute_product_fold] fold_union_pair)
```
```   955
```
```   956
```
```   957 context complete_lattice
```
```   958 begin
```
```   959
```
```   960 lemma inf_Inf_fold_inf:
```
```   961   assumes "finite A"
```
```   962   shows "inf (Inf A) B = fold inf B A"
```
```   963 proof -
```
```   964   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
```
```   965   from `finite A` fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
```
```   966     (simp_all add: inf_commute fun_eq_iff)
```
```   967 qed
```
```   968
```
```   969 lemma sup_Sup_fold_sup:
```
```   970   assumes "finite A"
```
```   971   shows "sup (Sup A) B = fold sup B A"
```
```   972 proof -
```
```   973   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
```
```   974   from `finite A` fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
```
```   975     (simp_all add: sup_commute fun_eq_iff)
```
```   976 qed
```
```   977
```
```   978 lemma Inf_fold_inf:
```
```   979   assumes "finite A"
```
```   980   shows "Inf A = fold inf top A"
```
```   981   using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
```
```   982
```
```   983 lemma Sup_fold_sup:
```
```   984   assumes "finite A"
```
```   985   shows "Sup A = fold sup bot A"
```
```   986   using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
```
```   987
```
```   988 lemma inf_INF_fold_inf:
```
```   989   assumes "finite A"
```
```   990   shows "inf B (INFI A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold")
```
```   991 proof (rule sym)
```
```   992   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
```
```   993   interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
```
```   994   from `finite A` show "?fold = ?inf"
```
```   995     by (induct A arbitrary: B)
```
```   996       (simp_all add: INF_def inf_left_commute)
```
```   997 qed
```
```   998
```
```   999 lemma sup_SUP_fold_sup:
```
```  1000   assumes "finite A"
```
```  1001   shows "sup B (SUPR A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold")
```
```  1002 proof (rule sym)
```
```  1003   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
```
```  1004   interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
```
```  1005   from `finite A` show "?fold = ?sup"
```
```  1006     by (induct A arbitrary: B)
```
```  1007       (simp_all add: SUP_def sup_left_commute)
```
```  1008 qed
```
```  1009
```
```  1010 lemma INF_fold_inf:
```
```  1011   assumes "finite A"
```
```  1012   shows "INFI A f = fold (inf \<circ> f) top A"
```
```  1013   using assms inf_INF_fold_inf [of A top] by simp
```
```  1014
```
```  1015 lemma SUP_fold_sup:
```
```  1016   assumes "finite A"
```
```  1017   shows "SUPR A f = fold (sup \<circ> f) bot A"
```
```  1018   using assms sup_SUP_fold_sup [of A bot] by simp
```
```  1019
```
```  1020 end
```
```  1021
```
```  1022
```
```  1023 subsection {* Locales as mini-packages for fold operations *}
```
```  1024
```
```  1025 subsubsection {* The natural case *}
```
```  1026
```
```  1027 locale folding =
```
```  1028   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
```
```  1029   fixes z :: "'b"
```
```  1030   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
```
```  1031 begin
```
```  1032
```
```  1033 definition F :: "'a set \<Rightarrow> 'b"
```
```  1034 where
```
```  1035   eq_fold: "F A = fold f z A"
```
```  1036
```
```  1037 lemma empty [simp]:
```
```  1038   "F {} = z"
```
```  1039   by (simp add: eq_fold)
```
```  1040
```
```  1041 lemma infinite [simp]:
```
```  1042   "\<not> finite A \<Longrightarrow> F A = z"
```
```  1043   by (simp add: eq_fold)
```
```  1044
```
```  1045 lemma insert [simp]:
```
```  1046   assumes "finite A" and "x \<notin> A"
```
```  1047   shows "F (insert x A) = f x (F A)"
```
```  1048 proof -
```
```  1049   interpret comp_fun_commute f
```
```  1050     by default (insert comp_fun_commute, simp add: fun_eq_iff)
```
```  1051   from fold_insert assms
```
```  1052   have "fold f z (insert x A) = f x (fold f z A)" by simp
```
```  1053   with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
```
```  1054 qed
```
```  1055
```
```  1056 lemma remove:
```
```  1057   assumes "finite A" and "x \<in> A"
```
```  1058   shows "F A = f x (F (A - {x}))"
```
```  1059 proof -
```
```  1060   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
```
```  1061     by (auto dest: mk_disjoint_insert)
```
```  1062   moreover from `finite A` this have "finite B" by simp
```
```  1063   ultimately show ?thesis by simp
```
```  1064 qed
```
```  1065
```
```  1066 lemma insert_remove:
```
```  1067   assumes "finite A"
```
```  1068   shows "F (insert x A) = f x (F (A - {x}))"
```
```  1069   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
```
```  1070
```
```  1071 end
```
```  1072
```
```  1073
```
```  1074 subsubsection {* With idempotency *}
```
```  1075
```
```  1076 locale folding_idem = folding +
```
```  1077   assumes comp_fun_idem: "f x \<circ> f x = f x"
```
```  1078 begin
```
```  1079
```
```  1080 declare insert [simp del]
```
```  1081
```
```  1082 lemma insert_idem [simp]:
```
```  1083   assumes "finite A"
```
```  1084   shows "F (insert x A) = f x (F A)"
```
```  1085 proof -
```
```  1086   interpret comp_fun_idem f
```
```  1087     by default (insert comp_fun_commute comp_fun_idem, simp add: fun_eq_iff)
```
```  1088   from fold_insert_idem assms
```
```  1089   have "fold f z (insert x A) = f x (fold f z A)" by simp
```
```  1090   with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
```
```  1091 qed
```
```  1092
```
```  1093 end
```
```  1094
```
```  1095
```
```  1096 subsection {* Finite cardinality *}
```
```  1097
```
```  1098 text {*
```
```  1099   The traditional definition
```
```  1100   @{prop "card A \<equiv> LEAST n. EX f. A = {f i | i. i < n}"}
```
```  1101   is ugly to work with.
```
```  1102   But now that we have @{const fold} things are easy:
```
```  1103 *}
```
```  1104
```
```  1105 definition card :: "'a set \<Rightarrow> nat" where
```
```  1106   "card = folding.F (\<lambda>_. Suc) 0"
```
```  1107
```
```  1108 interpretation card!: folding "\<lambda>_. Suc" 0
```
```  1109 where
```
```  1110   "folding.F (\<lambda>_. Suc) 0 = card"
```
```  1111 proof -
```
```  1112   show "folding (\<lambda>_. Suc)" by default rule
```
```  1113   then interpret card!: folding "\<lambda>_. Suc" 0 .
```
```  1114   from card_def show "folding.F (\<lambda>_. Suc) 0 = card" by rule
```
```  1115 qed
```
```  1116
```
```  1117 lemma card_infinite:
```
```  1118   "\<not> finite A \<Longrightarrow> card A = 0"
```
```  1119   by (fact card.infinite)
```
```  1120
```
```  1121 lemma card_empty:
```
```  1122   "card {} = 0"
```
```  1123   by (fact card.empty)
```
```  1124
```
```  1125 lemma card_insert_disjoint:
```
```  1126   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> card (insert x A) = Suc (card A)"
```
```  1127   by (fact card.insert)
```
```  1128
```
```  1129 lemma card_insert_if:
```
```  1130   "finite A \<Longrightarrow> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
```
```  1131   by auto (simp add: card.insert_remove card.remove)
```
```  1132
```
```  1133 lemma card_ge_0_finite:
```
```  1134   "card A > 0 \<Longrightarrow> finite A"
```
```  1135   by (rule ccontr) simp
```
```  1136
```
```  1137 lemma card_0_eq [simp, no_atp]:
```
```  1138   "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
```
```  1139   by (auto dest: mk_disjoint_insert)
```
```  1140
```
```  1141 lemma finite_UNIV_card_ge_0:
```
```  1142   "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
```
```  1143   by (rule ccontr) simp
```
```  1144
```
```  1145 lemma card_eq_0_iff:
```
```  1146   "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
```
```  1147   by auto
```
```  1148
```
```  1149 lemma card_gt_0_iff:
```
```  1150   "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
```
```  1151   by (simp add: neq0_conv [symmetric] card_eq_0_iff)
```
```  1152
```
```  1153 lemma card_Suc_Diff1:
```
```  1154   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> Suc (card (A - {x})) = card A"
```
```  1155 apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
```
```  1156 apply(simp del:insert_Diff_single)
```
```  1157 done
```
```  1158
```
```  1159 lemma card_Diff_singleton:
```
```  1160   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) = card A - 1"
```
```  1161   by (simp add: card_Suc_Diff1 [symmetric])
```
```  1162
```
```  1163 lemma card_Diff_singleton_if:
```
```  1164   "finite A \<Longrightarrow> card (A - {x}) = (if x \<in> A then card A - 1 else card A)"
```
```  1165   by (simp add: card_Diff_singleton)
```
```  1166
```
```  1167 lemma card_Diff_insert[simp]:
```
```  1168   assumes "finite A" and "a \<in> A" and "a \<notin> B"
```
```  1169   shows "card (A - insert a B) = card (A - B) - 1"
```
```  1170 proof -
```
```  1171   have "A - insert a B = (A - B) - {a}" using assms by blast
```
```  1172   then show ?thesis using assms by(simp add: card_Diff_singleton)
```
```  1173 qed
```
```  1174
```
```  1175 lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
```
```  1176   by (fact card.insert_remove)
```
```  1177
```
```  1178 lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
```
```  1179 by (simp add: card_insert_if)
```
```  1180
```
```  1181 lemma card_Collect_less_nat[simp]: "card{i::nat. i < n} = n"
```
```  1182 by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
```
```  1183
```
```  1184 lemma card_Collect_le_nat[simp]: "card{i::nat. i <= n} = Suc n"
```
```  1185 using card_Collect_less_nat[of "Suc n"] by(simp add: less_Suc_eq_le)
```
```  1186
```
```  1187 lemma card_mono:
```
```  1188   assumes "finite B" and "A \<subseteq> B"
```
```  1189   shows "card A \<le> card B"
```
```  1190 proof -
```
```  1191   from assms have "finite A" by (auto intro: finite_subset)
```
```  1192   then show ?thesis using assms proof (induct A arbitrary: B)
```
```  1193     case empty then show ?case by simp
```
```  1194   next
```
```  1195     case (insert x A)
```
```  1196     then have "x \<in> B" by simp
```
```  1197     from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
```
```  1198     with insert.hyps have "card A \<le> card (B - {x})" by auto
```
```  1199     with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
```
```  1200   qed
```
```  1201 qed
```
```  1202
```
```  1203 lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
```
```  1204 apply (induct rule: finite_induct)
```
```  1205 apply simp
```
```  1206 apply clarify
```
```  1207 apply (subgoal_tac "finite A & A - {x} <= F")
```
```  1208  prefer 2 apply (blast intro: finite_subset, atomize)
```
```  1209 apply (drule_tac x = "A - {x}" in spec)
```
```  1210 apply (simp add: card_Diff_singleton_if split add: split_if_asm)
```
```  1211 apply (case_tac "card A", auto)
```
```  1212 done
```
```  1213
```
```  1214 lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
```
```  1215 apply (simp add: psubset_eq linorder_not_le [symmetric])
```
```  1216 apply (blast dest: card_seteq)
```
```  1217 done
```
```  1218
```
```  1219 lemma card_Un_Int:
```
```  1220   assumes "finite A" and "finite B"
```
```  1221   shows "card A + card B = card (A \<union> B) + card (A \<inter> B)"
```
```  1222 using assms proof (induct A)
```
```  1223   case empty then show ?case by simp
```
```  1224 next
```
```  1225  case (insert x A) then show ?case
```
```  1226     by (auto simp add: insert_absorb Int_insert_left)
```
```  1227 qed
```
```  1228
```
```  1229 lemma card_Un_disjoint:
```
```  1230   assumes "finite A" and "finite B"
```
```  1231   assumes "A \<inter> B = {}"
```
```  1232   shows "card (A \<union> B) = card A + card B"
```
```  1233 using assms card_Un_Int [of A B] by simp
```
```  1234
```
```  1235 lemma card_Diff_subset:
```
```  1236   assumes "finite B" and "B \<subseteq> A"
```
```  1237   shows "card (A - B) = card A - card B"
```
```  1238 proof (cases "finite A")
```
```  1239   case False with assms show ?thesis by simp
```
```  1240 next
```
```  1241   case True with assms show ?thesis by (induct B arbitrary: A) simp_all
```
```  1242 qed
```
```  1243
```
```  1244 lemma card_Diff_subset_Int:
```
```  1245   assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
```
```  1246 proof -
```
```  1247   have "A - B = A - A \<inter> B" by auto
```
```  1248   thus ?thesis
```
```  1249     by (simp add: card_Diff_subset AB)
```
```  1250 qed
```
```  1251
```
```  1252 lemma diff_card_le_card_Diff:
```
```  1253 assumes "finite B" shows "card A - card B \<le> card(A - B)"
```
```  1254 proof-
```
```  1255   have "card A - card B \<le> card A - card (A \<inter> B)"
```
```  1256     using card_mono[OF assms Int_lower2, of A] by arith
```
```  1257   also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
```
```  1258   finally show ?thesis .
```
```  1259 qed
```
```  1260
```
```  1261 lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
```
```  1262 apply (rule Suc_less_SucD)
```
```  1263 apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
```
```  1264 done
```
```  1265
```
```  1266 lemma card_Diff2_less:
```
```  1267   "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
```
```  1268 apply (case_tac "x = y")
```
```  1269  apply (simp add: card_Diff1_less del:card_Diff_insert)
```
```  1270 apply (rule less_trans)
```
```  1271  prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
```
```  1272 done
```
```  1273
```
```  1274 lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
```
```  1275 apply (case_tac "x : A")
```
```  1276  apply (simp_all add: card_Diff1_less less_imp_le)
```
```  1277 done
```
```  1278
```
```  1279 lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
```
```  1280 by (erule psubsetI, blast)
```
```  1281
```
```  1282 lemma insert_partition:
```
```  1283   "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
```
```  1284   \<Longrightarrow> x \<inter> \<Union> F = {}"
```
```  1285 by auto
```
```  1286
```
```  1287 lemma finite_psubset_induct[consumes 1, case_names psubset]:
```
```  1288   assumes fin: "finite A"
```
```  1289   and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A"
```
```  1290   shows "P A"
```
```  1291 using fin
```
```  1292 proof (induct A taking: card rule: measure_induct_rule)
```
```  1293   case (less A)
```
```  1294   have fin: "finite A" by fact
```
```  1295   have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
```
```  1296   { fix B
```
```  1297     assume asm: "B \<subset> A"
```
```  1298     from asm have "card B < card A" using psubset_card_mono fin by blast
```
```  1299     moreover
```
```  1300     from asm have "B \<subseteq> A" by auto
```
```  1301     then have "finite B" using fin finite_subset by blast
```
```  1302     ultimately
```
```  1303     have "P B" using ih by simp
```
```  1304   }
```
```  1305   with fin show "P A" using major by blast
```
```  1306 qed
```
```  1307
```
```  1308 text{* main cardinality theorem *}
```
```  1309 lemma card_partition [rule_format]:
```
```  1310   "finite C ==>
```
```  1311      finite (\<Union> C) -->
```
```  1312      (\<forall>c\<in>C. card c = k) -->
```
```  1313      (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
```
```  1314      k * card(C) = card (\<Union> C)"
```
```  1315 apply (erule finite_induct, simp)
```
```  1316 apply (simp add: card_Un_disjoint insert_partition
```
```  1317        finite_subset [of _ "\<Union> (insert x F)"])
```
```  1318 done
```
```  1319
```
```  1320 lemma card_eq_UNIV_imp_eq_UNIV:
```
```  1321   assumes fin: "finite (UNIV :: 'a set)"
```
```  1322   and card: "card A = card (UNIV :: 'a set)"
```
```  1323   shows "A = (UNIV :: 'a set)"
```
```  1324 proof
```
```  1325   show "A \<subseteq> UNIV" by simp
```
```  1326   show "UNIV \<subseteq> A"
```
```  1327   proof
```
```  1328     fix x
```
```  1329     show "x \<in> A"
```
```  1330     proof (rule ccontr)
```
```  1331       assume "x \<notin> A"
```
```  1332       then have "A \<subset> UNIV" by auto
```
```  1333       with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
```
```  1334       with card show False by simp
```
```  1335     qed
```
```  1336   qed
```
```  1337 qed
```
```  1338
```
```  1339 text{*The form of a finite set of given cardinality*}
```
```  1340
```
```  1341 lemma card_eq_SucD:
```
```  1342 assumes "card A = Suc k"
```
```  1343 shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
```
```  1344 proof -
```
```  1345   have fin: "finite A" using assms by (auto intro: ccontr)
```
```  1346   moreover have "card A \<noteq> 0" using assms by auto
```
```  1347   ultimately obtain b where b: "b \<in> A" by auto
```
```  1348   show ?thesis
```
```  1349   proof (intro exI conjI)
```
```  1350     show "A = insert b (A-{b})" using b by blast
```
```  1351     show "b \<notin> A - {b}" by blast
```
```  1352     show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
```
```  1353       using assms b fin by(fastforce dest:mk_disjoint_insert)+
```
```  1354   qed
```
```  1355 qed
```
```  1356
```
```  1357 lemma card_Suc_eq:
```
```  1358   "(card A = Suc k) =
```
```  1359    (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
```
```  1360 apply(rule iffI)
```
```  1361  apply(erule card_eq_SucD)
```
```  1362 apply(auto)
```
```  1363 apply(subst card.insert)
```
```  1364  apply(auto intro:ccontr)
```
```  1365 done
```
```  1366
```
```  1367 lemma card_le_Suc_iff: "finite A \<Longrightarrow>
```
```  1368   Suc n \<le> card A = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
```
```  1369 by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
```
```  1370   dest: subset_singletonD split: nat.splits if_splits)
```
```  1371
```
```  1372 lemma finite_fun_UNIVD2:
```
```  1373   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
```
```  1374   shows "finite (UNIV :: 'b set)"
```
```  1375 proof -
```
```  1376   from fin have "\<And>arbitrary. finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
```
```  1377     by (rule finite_imageI)
```
```  1378   moreover have "\<And>arbitrary. UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
```
```  1379     by (rule UNIV_eq_I) auto
```
```  1380   ultimately show "finite (UNIV :: 'b set)" by simp
```
```  1381 qed
```
```  1382
```
```  1383 lemma card_UNIV_unit [simp]: "card (UNIV :: unit set) = 1"
```
```  1384   unfolding UNIV_unit by simp
```
```  1385
```
```  1386 lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
```
```  1387   unfolding UNIV_bool by simp
```
```  1388
```
```  1389
```
```  1390 subsubsection {* Cardinality of image *}
```
```  1391
```
```  1392 lemma card_image_le: "finite A ==> card (f ` A) <= card A"
```
```  1393 apply (induct rule: finite_induct)
```
```  1394  apply simp
```
```  1395 apply (simp add: le_SucI card_insert_if)
```
```  1396 done
```
```  1397
```
```  1398 lemma card_image:
```
```  1399   assumes "inj_on f A"
```
```  1400   shows "card (f ` A) = card A"
```
```  1401 proof (cases "finite A")
```
```  1402   case True then show ?thesis using assms by (induct A) simp_all
```
```  1403 next
```
```  1404   case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
```
```  1405   with False show ?thesis by simp
```
```  1406 qed
```
```  1407
```
```  1408 lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
```
```  1409 by(auto simp: card_image bij_betw_def)
```
```  1410
```
```  1411 lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
```
```  1412 by (simp add: card_seteq card_image)
```
```  1413
```
```  1414 lemma eq_card_imp_inj_on:
```
```  1415   "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
```
```  1416 apply (induct rule:finite_induct)
```
```  1417 apply simp
```
```  1418 apply(frule card_image_le[where f = f])
```
```  1419 apply(simp add:card_insert_if split:if_splits)
```
```  1420 done
```
```  1421
```
```  1422 lemma inj_on_iff_eq_card:
```
```  1423   "finite A ==> inj_on f A = (card(f ` A) = card A)"
```
```  1424 by(blast intro: card_image eq_card_imp_inj_on)
```
```  1425
```
```  1426
```
```  1427 lemma card_inj_on_le:
```
```  1428   "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
```
```  1429 apply (subgoal_tac "finite A")
```
```  1430  apply (force intro: card_mono simp add: card_image [symmetric])
```
```  1431 apply (blast intro: finite_imageD dest: finite_subset)
```
```  1432 done
```
```  1433
```
```  1434 lemma card_bij_eq:
```
```  1435   "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
```
```  1436      finite A; finite B |] ==> card A = card B"
```
```  1437 by (auto intro: le_antisym card_inj_on_le)
```
```  1438
```
```  1439 lemma bij_betw_finite:
```
```  1440   assumes "bij_betw f A B"
```
```  1441   shows "finite A \<longleftrightarrow> finite B"
```
```  1442 using assms unfolding bij_betw_def
```
```  1443 using finite_imageD[of f A] by auto
```
```  1444
```
```  1445
```
```  1446 subsubsection {* Pigeonhole Principles *}
```
```  1447
```
```  1448 lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
```
```  1449 by (auto dest: card_image less_irrefl_nat)
```
```  1450
```
```  1451 lemma pigeonhole_infinite:
```
```  1452 assumes  "~ finite A" and "finite(f`A)"
```
```  1453 shows "EX a0:A. ~finite{a:A. f a = f a0}"
```
```  1454 proof -
```
```  1455   have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
```
```  1456   proof(induct "f`A" arbitrary: A rule: finite_induct)
```
```  1457     case empty thus ?case by simp
```
```  1458   next
```
```  1459     case (insert b F)
```
```  1460     show ?case
```
```  1461     proof cases
```
```  1462       assume "finite{a:A. f a = b}"
```
```  1463       hence "~ finite(A - {a:A. f a = b})" using `\<not> finite A` by simp
```
```  1464       also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
```
```  1465       finally have "~ finite({a:A. f a \<noteq> b})" .
```
```  1466       from insert(3)[OF _ this]
```
```  1467       show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
```
```  1468     next
```
```  1469       assume 1: "~finite{a:A. f a = b}"
```
```  1470       hence "{a \<in> A. f a = b} \<noteq> {}" by force
```
```  1471       thus ?thesis using 1 by blast
```
```  1472     qed
```
```  1473   qed
```
```  1474   from this[OF assms(2,1)] show ?thesis .
```
```  1475 qed
```
```  1476
```
```  1477 lemma pigeonhole_infinite_rel:
```
```  1478 assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
```
```  1479 shows "EX b:B. ~finite{a:A. R a b}"
```
```  1480 proof -
```
```  1481    let ?F = "%a. {b:B. R a b}"
```
```  1482    from finite_Pow_iff[THEN iffD2, OF `finite B`]
```
```  1483    have "finite(?F ` A)" by(blast intro: rev_finite_subset)
```
```  1484    from pigeonhole_infinite[where f = ?F, OF assms(1) this]
```
```  1485    obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
```
```  1486    obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast
```
```  1487    { assume "finite{a:A. R a b0}"
```
```  1488      then have "finite {a\<in>A. ?F a = ?F a0}"
```
```  1489        using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)
```
```  1490    }
```
```  1491    with 1 `b0 : B` show ?thesis by blast
```
```  1492 qed
```
```  1493
```
```  1494
```
```  1495 subsubsection {* Cardinality of sums *}
```
```  1496
```
```  1497 lemma card_Plus:
```
```  1498   assumes "finite A" and "finite B"
```
```  1499   shows "card (A <+> B) = card A + card B"
```
```  1500 proof -
```
```  1501   have "Inl`A \<inter> Inr`B = {}" by fast
```
```  1502   with assms show ?thesis
```
```  1503     unfolding Plus_def
```
```  1504     by (simp add: card_Un_disjoint card_image)
```
```  1505 qed
```
```  1506
```
```  1507 lemma card_Plus_conv_if:
```
```  1508   "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
```
```  1509   by (auto simp add: card_Plus)
```
```  1510
```
```  1511
```
```  1512 subsubsection {* Cardinality of the Powerset *}
```
```  1513
```
```  1514 lemma card_Pow: "finite A ==> card (Pow A) = 2 ^ card A"
```
```  1515 apply (induct rule: finite_induct)
```
```  1516  apply (simp_all add: Pow_insert)
```
```  1517 apply (subst card_Un_disjoint, blast)
```
```  1518   apply (blast, blast)
```
```  1519 apply (subgoal_tac "inj_on (insert x) (Pow F)")
```
```  1520  apply (subst mult_2)
```
```  1521  apply (simp add: card_image Pow_insert)
```
```  1522 apply (unfold inj_on_def)
```
```  1523 apply (blast elim!: equalityE)
```
```  1524 done
```
```  1525
```
```  1526 text {* Relates to equivalence classes.  Based on a theorem of F. Kamm\"uller.  *}
```
```  1527
```
```  1528 lemma dvd_partition:
```
```  1529   "finite (Union C) ==>
```
```  1530     ALL c : C. k dvd card c ==>
```
```  1531     (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
```
```  1532   k dvd card (Union C)"
```
```  1533 apply (frule finite_UnionD)
```
```  1534 apply (rotate_tac -1)
```
```  1535 apply (induct rule: finite_induct)
```
```  1536 apply simp_all
```
```  1537 apply clarify
```
```  1538 apply (subst card_Un_disjoint)
```
```  1539    apply (auto simp add: disjoint_eq_subset_Compl)
```
```  1540 done
```
```  1541
```
```  1542
```
```  1543 subsubsection {* Relating injectivity and surjectivity *}
```
```  1544
```
```  1545 lemma finite_surj_inj: "finite A \<Longrightarrow> A \<subseteq> f ` A \<Longrightarrow> inj_on f A"
```
```  1546 apply(rule eq_card_imp_inj_on, assumption)
```
```  1547 apply(frule finite_imageI)
```
```  1548 apply(drule (1) card_seteq)
```
```  1549  apply(erule card_image_le)
```
```  1550 apply simp
```
```  1551 done
```
```  1552
```
```  1553 lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
```
```  1554 shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
```
```  1555 by (blast intro: finite_surj_inj subset_UNIV)
```
```  1556
```
```  1557 lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
```
```  1558 shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
```
```  1559 by(fastforce simp:surj_def dest!: endo_inj_surj)
```
```  1560
```
```  1561 corollary infinite_UNIV_nat [iff]:
```
```  1562   "\<not> finite (UNIV :: nat set)"
```
```  1563 proof
```
```  1564   assume "finite (UNIV :: nat set)"
```
```  1565   with finite_UNIV_inj_surj [of Suc]
```
```  1566   show False by simp (blast dest: Suc_neq_Zero surjD)
```
```  1567 qed
```
```  1568
```
```  1569 (* Often leads to bogus ATP proofs because of reduced type information, hence no_atp *)
```
```  1570 lemma infinite_UNIV_char_0 [no_atp]:
```
```  1571   "\<not> finite (UNIV :: 'a::semiring_char_0 set)"
```
```  1572 proof
```
```  1573   assume "finite (UNIV :: 'a set)"
```
```  1574   with subset_UNIV have "finite (range of_nat :: 'a set)"
```
```  1575     by (rule finite_subset)
```
```  1576   moreover have "inj (of_nat :: nat \<Rightarrow> 'a)"
```
```  1577     by (simp add: inj_on_def)
```
```  1578   ultimately have "finite (UNIV :: nat set)"
```
```  1579     by (rule finite_imageD)
```
```  1580   then show False
```
```  1581     by simp
```
```  1582 qed
```
```  1583
```
```  1584 hide_const (open) Finite_Set.fold
```
```  1585
```
```  1586 end
```
```  1587
```