src/HOL/Finite_Set.thy
 author huffman Fri Mar 30 12:32:35 2012 +0200 (2012-03-30) changeset 47220 52426c62b5d0 parent 47210 b1dd32b2a505 child 47221 7205eb4a0a05 permissions -rw-r--r--
replace lemmas eval_nat_numeral with a simpler reformulation
```     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 lemma finite_induct [case_names empty insert, induct set: finite]:
```
```    20   -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
```
```    21   assumes "finite F"
```
```    22   assumes "P {}"
```
```    23     and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
```
```    24   shows "P F"
```
```    25 using `finite F`
```
```    26 proof induct
```
```    27   show "P {}" by fact
```
```    28   fix x F assume F: "finite F" and P: "P F"
```
```    29   show "P (insert x F)"
```
```    30   proof cases
```
```    31     assume "x \<in> F"
```
```    32     hence "insert x F = F" by (rule insert_absorb)
```
```    33     with P show ?thesis by (simp only:)
```
```    34   next
```
```    35     assume "x \<notin> F"
```
```    36     from F this P show ?thesis by (rule insert)
```
```    37   qed
```
```    38 qed
```
```    39
```
```    40
```
```    41 subsubsection {* Choice principles *}
```
```    42
```
```    43 lemma ex_new_if_finite: -- "does not depend on def of finite at all"
```
```    44   assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
```
```    45   shows "\<exists>a::'a. a \<notin> A"
```
```    46 proof -
```
```    47   from assms have "A \<noteq> UNIV" by blast
```
```    48   then show ?thesis by blast
```
```    49 qed
```
```    50
```
```    51 text {* A finite choice principle. Does not need the SOME choice operator. *}
```
```    52
```
```    53 lemma finite_set_choice:
```
```    54   "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
```
```    55 proof (induct rule: finite_induct)
```
```    56   case empty then show ?case by simp
```
```    57 next
```
```    58   case (insert a A)
```
```    59   then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
```
```    60   show ?case (is "EX f. ?P f")
```
```    61   proof
```
```    62     show "?P(%x. if x = a then b else f x)" using f ab by auto
```
```    63   qed
```
```    64 qed
```
```    65
```
```    66
```
```    67 subsubsection {* Finite sets are the images of initial segments of natural numbers *}
```
```    68
```
```    69 lemma finite_imp_nat_seg_image_inj_on:
```
```    70   assumes "finite A"
```
```    71   shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
```
```    72 using assms
```
```    73 proof induct
```
```    74   case empty
```
```    75   show ?case
```
```    76   proof
```
```    77     show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp
```
```    78   qed
```
```    79 next
```
```    80   case (insert a A)
```
```    81   have notinA: "a \<notin> A" by fact
```
```    82   from insert.hyps obtain n f
```
```    83     where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
```
```    84   hence "insert a A = f(n:=a) ` {i. i < Suc n}"
```
```    85         "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
```
```    86     by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
```
```    87   thus ?case by blast
```
```    88 qed
```
```    89
```
```    90 lemma nat_seg_image_imp_finite:
```
```    91   "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
```
```    92 proof (induct n arbitrary: A)
```
```    93   case 0 thus ?case by simp
```
```    94 next
```
```    95   case (Suc n)
```
```    96   let ?B = "f ` {i. i < n}"
```
```    97   have finB: "finite ?B" by(rule Suc.hyps[OF refl])
```
```    98   show ?case
```
```    99   proof cases
```
```   100     assume "\<exists>k<n. f n = f k"
```
```   101     hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   102     thus ?thesis using finB by simp
```
```   103   next
```
```   104     assume "\<not>(\<exists> k<n. f n = f k)"
```
```   105     hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   106     thus ?thesis using finB by simp
```
```   107   qed
```
```   108 qed
```
```   109
```
```   110 lemma finite_conv_nat_seg_image:
```
```   111   "finite A \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"
```
```   112   by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
```
```   113
```
```   114 lemma finite_imp_inj_to_nat_seg:
```
```   115   assumes "finite A"
```
```   116   shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> inj_on f A"
```
```   117 proof -
```
```   118   from finite_imp_nat_seg_image_inj_on[OF `finite A`]
```
```   119   obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
```
```   120     by (auto simp:bij_betw_def)
```
```   121   let ?f = "the_inv_into {i. i<n} f"
```
```   122   have "inj_on ?f A & ?f ` A = {i. i<n}"
```
```   123     by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
```
```   124   thus ?thesis by blast
```
```   125 qed
```
```   126
```
```   127 lemma finite_Collect_less_nat [iff]:
```
```   128   "finite {n::nat. n < k}"
```
```   129   by (fastforce simp: finite_conv_nat_seg_image)
```
```   130
```
```   131 lemma finite_Collect_le_nat [iff]:
```
```   132   "finite {n::nat. n \<le> k}"
```
```   133   by (simp add: le_eq_less_or_eq Collect_disj_eq)
```
```   134
```
```   135
```
```   136 subsubsection {* Finiteness and common set operations *}
```
```   137
```
```   138 lemma rev_finite_subset:
```
```   139   "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
```
```   140 proof (induct arbitrary: A rule: finite_induct)
```
```   141   case empty
```
```   142   then show ?case by simp
```
```   143 next
```
```   144   case (insert x F A)
```
```   145   have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+
```
```   146   show "finite A"
```
```   147   proof cases
```
```   148     assume x: "x \<in> A"
```
```   149     with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
```
```   150     with r have "finite (A - {x})" .
```
```   151     hence "finite (insert x (A - {x}))" ..
```
```   152     also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
```
```   153     finally show ?thesis .
```
```   154   next
```
```   155     show "A \<subseteq> F ==> ?thesis" by fact
```
```   156     assume "x \<notin> A"
```
```   157     with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
```
```   158   qed
```
```   159 qed
```
```   160
```
```   161 lemma finite_subset:
```
```   162   "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
```
```   163   by (rule rev_finite_subset)
```
```   164
```
```   165 lemma finite_UnI:
```
```   166   assumes "finite F" and "finite G"
```
```   167   shows "finite (F \<union> G)"
```
```   168   using assms by induct simp_all
```
```   169
```
```   170 lemma finite_Un [iff]:
```
```   171   "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
```
```   172   by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])
```
```   173
```
```   174 lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"
```
```   175 proof -
```
```   176   have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp
```
```   177   then have "finite ({a} \<union> A) \<longleftrightarrow> finite A" by (simp only: finite_Un)
```
```   178   then show ?thesis by simp
```
```   179 qed
```
```   180
```
```   181 lemma finite_Int [simp, intro]:
```
```   182   "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
```
```   183   by (blast intro: finite_subset)
```
```   184
```
```   185 lemma finite_Collect_conjI [simp, intro]:
```
```   186   "finite {x. P x} \<or> finite {x. Q x} \<Longrightarrow> finite {x. P x \<and> Q x}"
```
```   187   by (simp add: Collect_conj_eq)
```
```   188
```
```   189 lemma finite_Collect_disjI [simp]:
```
```   190   "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"
```
```   191   by (simp add: Collect_disj_eq)
```
```   192
```
```   193 lemma finite_Diff [simp, intro]:
```
```   194   "finite A \<Longrightarrow> finite (A - B)"
```
```   195   by (rule finite_subset, rule Diff_subset)
```
```   196
```
```   197 lemma finite_Diff2 [simp]:
```
```   198   assumes "finite B"
```
```   199   shows "finite (A - B) \<longleftrightarrow> finite A"
```
```   200 proof -
```
```   201   have "finite A \<longleftrightarrow> finite((A - B) \<union> (A \<inter> B))" by (simp add: Un_Diff_Int)
```
```   202   also have "\<dots> \<longleftrightarrow> finite (A - B)" using `finite B` by simp
```
```   203   finally show ?thesis ..
```
```   204 qed
```
```   205
```
```   206 lemma finite_Diff_insert [iff]:
```
```   207   "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
```
```   208 proof -
```
```   209   have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
```
```   210   moreover have "A - insert a B = A - B - {a}" by auto
```
```   211   ultimately show ?thesis by simp
```
```   212 qed
```
```   213
```
```   214 lemma finite_compl[simp]:
```
```   215   "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   216   by (simp add: Compl_eq_Diff_UNIV)
```
```   217
```
```   218 lemma finite_Collect_not[simp]:
```
```   219   "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   220   by (simp add: Collect_neg_eq)
```
```   221
```
```   222 lemma finite_Union [simp, intro]:
```
```   223   "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite(\<Union>A)"
```
```   224   by (induct rule: finite_induct) simp_all
```
```   225
```
```   226 lemma finite_UN_I [intro]:
```
```   227   "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"
```
```   228   by (induct rule: finite_induct) simp_all
```
```   229
```
```   230 lemma finite_UN [simp]:
```
```   231   "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
```
```   232   by (blast intro: finite_subset)
```
```   233
```
```   234 lemma finite_Inter [intro]:
```
```   235   "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
```
```   236   by (blast intro: Inter_lower finite_subset)
```
```   237
```
```   238 lemma finite_INT [intro]:
```
```   239   "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
```
```   240   by (blast intro: INT_lower finite_subset)
```
```   241
```
```   242 lemma finite_imageI [simp, intro]:
```
```   243   "finite F \<Longrightarrow> finite (h ` F)"
```
```   244   by (induct rule: finite_induct) simp_all
```
```   245
```
```   246 lemma finite_image_set [simp]:
```
```   247   "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
```
```   248   by (simp add: image_Collect [symmetric])
```
```   249
```
```   250 lemma finite_imageD:
```
```   251   assumes "finite (f ` A)" and "inj_on f A"
```
```   252   shows "finite A"
```
```   253 using assms
```
```   254 proof (induct "f ` A" arbitrary: A)
```
```   255   case empty then show ?case by simp
```
```   256 next
```
```   257   case (insert x B)
```
```   258   then have B_A: "insert x B = f ` A" by simp
```
```   259   then obtain y where "x = f y" and "y \<in> A" by blast
```
```   260   from B_A `x \<notin> B` have "B = f ` A - {x}" by blast
```
```   261   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)
```
```   262   moreover from `inj_on f A` have "inj_on f (A - {y})" by (rule inj_on_diff)
```
```   263   ultimately have "finite (A - {y})" by (rule insert.hyps)
```
```   264   then show "finite A" by simp
```
```   265 qed
```
```   266
```
```   267 lemma finite_surj:
```
```   268   "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
```
```   269   by (erule finite_subset) (rule finite_imageI)
```
```   270
```
```   271 lemma finite_range_imageI:
```
```   272   "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
```
```   273   by (drule finite_imageI) (simp add: range_composition)
```
```   274
```
```   275 lemma finite_subset_image:
```
```   276   assumes "finite B"
```
```   277   shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
```
```   278 using assms
```
```   279 proof induct
```
```   280   case empty then show ?case by simp
```
```   281 next
```
```   282   case insert then show ?case
```
```   283     by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
```
```   284        blast
```
```   285 qed
```
```   286
```
```   287 lemma finite_vimage_IntI:
```
```   288   "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
```
```   289   apply (induct rule: finite_induct)
```
```   290    apply simp_all
```
```   291   apply (subst vimage_insert)
```
```   292   apply (simp add: finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2)
```
```   293   done
```
```   294
```
```   295 lemma finite_vimageI:
```
```   296   "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
```
```   297   using finite_vimage_IntI[of F h UNIV] by auto
```
```   298
```
```   299 lemma finite_vimageD:
```
```   300   assumes fin: "finite (h -` F)" and surj: "surj h"
```
```   301   shows "finite F"
```
```   302 proof -
```
```   303   have "finite (h ` (h -` F))" using fin by (rule finite_imageI)
```
```   304   also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)
```
```   305   finally show "finite F" .
```
```   306 qed
```
```   307
```
```   308 lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
```
```   309   unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
```
```   310
```
```   311 lemma finite_Collect_bex [simp]:
```
```   312   assumes "finite A"
```
```   313   shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"
```
```   314 proof -
```
```   315   have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto
```
```   316   with assms show ?thesis by simp
```
```   317 qed
```
```   318
```
```   319 lemma finite_Collect_bounded_ex [simp]:
```
```   320   assumes "finite {y. P y}"
```
```   321   shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
```
```   322 proof -
```
```   323   have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
```
```   324   with assms show ?thesis by simp
```
```   325 qed
```
```   326
```
```   327 lemma finite_Plus:
```
```   328   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
```
```   329   by (simp add: Plus_def)
```
```   330
```
```   331 lemma finite_PlusD:
```
```   332   fixes A :: "'a set" and B :: "'b set"
```
```   333   assumes fin: "finite (A <+> B)"
```
```   334   shows "finite A" "finite B"
```
```   335 proof -
```
```   336   have "Inl ` A \<subseteq> A <+> B" by auto
```
```   337   then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
```
```   338   then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
```
```   339 next
```
```   340   have "Inr ` B \<subseteq> A <+> B" by auto
```
```   341   then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
```
```   342   then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
```
```   343 qed
```
```   344
```
```   345 lemma finite_Plus_iff [simp]:
```
```   346   "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
```
```   347   by (auto intro: finite_PlusD finite_Plus)
```
```   348
```
```   349 lemma finite_Plus_UNIV_iff [simp]:
```
```   350   "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
```
```   351   by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
```
```   352
```
```   353 lemma finite_SigmaI [simp, intro]:
```
```   354   "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
```
```   355   by (unfold Sigma_def) blast
```
```   356
```
```   357 lemma finite_cartesian_product:
```
```   358   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
```
```   359   by (rule finite_SigmaI)
```
```   360
```
```   361 lemma finite_Prod_UNIV:
```
```   362   "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
```
```   363   by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
```
```   364
```
```   365 lemma finite_cartesian_productD1:
```
```   366   assumes "finite (A \<times> B)" and "B \<noteq> {}"
```
```   367   shows "finite A"
```
```   368 proof -
```
```   369   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
```
```   370     by (auto simp add: finite_conv_nat_seg_image)
```
```   371   then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
```
```   372   with `B \<noteq> {}` have "A = (fst \<circ> f) ` {i::nat. i < n}"
```
```   373     by (simp add: image_compose)
```
```   374   then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
```
```   375   then show ?thesis
```
```   376     by (auto simp add: finite_conv_nat_seg_image)
```
```   377 qed
```
```   378
```
```   379 lemma finite_cartesian_productD2:
```
```   380   assumes "finite (A \<times> B)" and "A \<noteq> {}"
```
```   381   shows "finite B"
```
```   382 proof -
```
```   383   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
```
```   384     by (auto simp add: finite_conv_nat_seg_image)
```
```   385   then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
```
```   386   with `A \<noteq> {}` have "B = (snd \<circ> f) ` {i::nat. i < n}"
```
```   387     by (simp add: image_compose)
```
```   388   then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
```
```   389   then show ?thesis
```
```   390     by (auto simp add: finite_conv_nat_seg_image)
```
```   391 qed
```
```   392
```
```   393 lemma finite_Pow_iff [iff]:
```
```   394   "finite (Pow A) \<longleftrightarrow> finite A"
```
```   395 proof
```
```   396   assume "finite (Pow A)"
```
```   397   then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
```
```   398   then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
```
```   399 next
```
```   400   assume "finite A"
```
```   401   then show "finite (Pow A)"
```
```   402     by induct (simp_all add: Pow_insert)
```
```   403 qed
```
```   404
```
```   405 corollary finite_Collect_subsets [simp, intro]:
```
```   406   "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
```
```   407   by (simp add: Pow_def [symmetric])
```
```   408
```
```   409 lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
```
```   410   by (blast intro: finite_subset [OF subset_Pow_Union])
```
```   411
```
```   412
```
```   413 subsubsection {* Further induction rules on finite sets *}
```
```   414
```
```   415 lemma finite_ne_induct [case_names singleton insert, consumes 2]:
```
```   416   assumes "finite F" and "F \<noteq> {}"
```
```   417   assumes "\<And>x. P {x}"
```
```   418     and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
```
```   419   shows "P F"
```
```   420 using assms
```
```   421 proof induct
```
```   422   case empty then show ?case by simp
```
```   423 next
```
```   424   case (insert x F) then show ?case by cases auto
```
```   425 qed
```
```   426
```
```   427 lemma finite_subset_induct [consumes 2, case_names empty insert]:
```
```   428   assumes "finite F" and "F \<subseteq> A"
```
```   429   assumes empty: "P {}"
```
```   430     and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
```
```   431   shows "P F"
```
```   432 using `finite F` `F \<subseteq> A`
```
```   433 proof induct
```
```   434   show "P {}" by fact
```
```   435 next
```
```   436   fix x F
```
```   437   assume "finite F" and "x \<notin> F" and
```
```   438     P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
```
```   439   show "P (insert x F)"
```
```   440   proof (rule insert)
```
```   441     from i show "x \<in> A" by blast
```
```   442     from i have "F \<subseteq> A" by blast
```
```   443     with P show "P F" .
```
```   444     show "finite F" by fact
```
```   445     show "x \<notin> F" by fact
```
```   446   qed
```
```   447 qed
```
```   448
```
```   449 lemma finite_empty_induct:
```
```   450   assumes "finite A"
```
```   451   assumes "P A"
```
```   452     and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
```
```   453   shows "P {}"
```
```   454 proof -
```
```   455   have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
```
```   456   proof -
```
```   457     fix B :: "'a set"
```
```   458     assume "B \<subseteq> A"
```
```   459     with `finite A` have "finite B" by (rule rev_finite_subset)
```
```   460     from this `B \<subseteq> A` show "P (A - B)"
```
```   461     proof induct
```
```   462       case empty
```
```   463       from `P A` show ?case by simp
```
```   464     next
```
```   465       case (insert b B)
```
```   466       have "P (A - B - {b})"
```
```   467       proof (rule remove)
```
```   468         from `finite A` show "finite (A - B)" by induct auto
```
```   469         from insert show "b \<in> A - B" by simp
```
```   470         from insert show "P (A - B)" by simp
```
```   471       qed
```
```   472       also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
```
```   473       finally show ?case .
```
```   474     qed
```
```   475   qed
```
```   476   then have "P (A - A)" by blast
```
```   477   then show ?thesis by simp
```
```   478 qed
```
```   479
```
```   480
```
```   481 subsection {* Class @{text finite}  *}
```
```   482
```
```   483 class finite =
```
```   484   assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
```
```   485 begin
```
```   486
```
```   487 lemma finite [simp]: "finite (A \<Colon> 'a set)"
```
```   488   by (rule subset_UNIV finite_UNIV finite_subset)+
```
```   489
```
```   490 lemma finite_code [code]: "finite (A \<Colon> 'a set) \<longleftrightarrow> True"
```
```   491   by simp
```
```   492
```
```   493 end
```
```   494
```
```   495 instance prod :: (finite, finite) finite
```
```   496   by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
```
```   497
```
```   498 lemma inj_graph: "inj (%f. {(x, y). y = f x})"
```
```   499   by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
```
```   500
```
```   501 instance "fun" :: (finite, finite) finite
```
```   502 proof
```
```   503   show "finite (UNIV :: ('a => 'b) set)"
```
```   504   proof (rule finite_imageD)
```
```   505     let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
```
```   506     have "range ?graph \<subseteq> Pow UNIV" by simp
```
```   507     moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
```
```   508       by (simp only: finite_Pow_iff finite)
```
```   509     ultimately show "finite (range ?graph)"
```
```   510       by (rule finite_subset)
```
```   511     show "inj ?graph" by (rule inj_graph)
```
```   512   qed
```
```   513 qed
```
```   514
```
```   515 instance bool :: finite
```
```   516   by default (simp add: UNIV_bool)
```
```   517
```
```   518 instance set :: (finite) finite
```
```   519   by default (simp only: Pow_UNIV [symmetric] finite_Pow_iff finite)
```
```   520
```
```   521 instance unit :: finite
```
```   522   by default (simp add: UNIV_unit)
```
```   523
```
```   524 instance sum :: (finite, finite) finite
```
```   525   by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
```
```   526
```
```   527 lemma finite_option_UNIV [simp]:
```
```   528   "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
```
```   529   by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
```
```   530
```
```   531 instance option :: (finite) finite
```
```   532   by default (simp add: UNIV_option_conv)
```
```   533
```
```   534
```
```   535 subsection {* A basic fold functional for finite sets *}
```
```   536
```
```   537 text {* The intended behaviour is
```
```   538 @{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
```
```   539 if @{text f} is ``left-commutative'':
```
```   540 *}
```
```   541
```
```   542 locale comp_fun_commute =
```
```   543   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
```
```   544   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
```
```   545 begin
```
```   546
```
```   547 lemma fun_left_comm: "f x (f y z) = f y (f x z)"
```
```   548   using comp_fun_commute by (simp add: fun_eq_iff)
```
```   549
```
```   550 end
```
```   551
```
```   552 inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
```
```   553 for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
```
```   554   emptyI [intro]: "fold_graph f z {} z" |
```
```   555   insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
```
```   556       \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
```
```   557
```
```   558 inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
```
```   559
```
```   560 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
```
```   561   "fold f z A = (THE y. fold_graph f z A y)"
```
```   562
```
```   563 text{*A tempting alternative for the definiens is
```
```   564 @{term "if finite A then THE y. fold_graph f z A y else e"}.
```
```   565 It allows the removal of finiteness assumptions from the theorems
```
```   566 @{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
```
```   567 The proofs become ugly. It is not worth the effort. (???) *}
```
```   568
```
```   569 lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
```
```   570 by (induct rule: finite_induct) auto
```
```   571
```
```   572
```
```   573 subsubsection{*From @{const fold_graph} to @{term fold}*}
```
```   574
```
```   575 context comp_fun_commute
```
```   576 begin
```
```   577
```
```   578 lemma fold_graph_insertE_aux:
```
```   579   "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'"
```
```   580 proof (induct set: fold_graph)
```
```   581   case (insertI x A y) show ?case
```
```   582   proof (cases "x = a")
```
```   583     assume "x = a" with insertI show ?case by auto
```
```   584   next
```
```   585     assume "x \<noteq> a"
```
```   586     then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
```
```   587       using insertI by auto
```
```   588     have "f x y = f a (f x y')"
```
```   589       unfolding y by (rule fun_left_comm)
```
```   590     moreover have "fold_graph f z (insert x A - {a}) (f x y')"
```
```   591       using y' and `x \<noteq> a` and `x \<notin> A`
```
```   592       by (simp add: insert_Diff_if fold_graph.insertI)
```
```   593     ultimately show ?case by fast
```
```   594   qed
```
```   595 qed simp
```
```   596
```
```   597 lemma fold_graph_insertE:
```
```   598   assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
```
```   599   obtains y where "v = f x y" and "fold_graph f z A y"
```
```   600 using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
```
```   601
```
```   602 lemma fold_graph_determ:
```
```   603   "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
```
```   604 proof (induct arbitrary: y set: fold_graph)
```
```   605   case (insertI x A y v)
```
```   606   from `fold_graph f z (insert x A) v` and `x \<notin> A`
```
```   607   obtain y' where "v = f x y'" and "fold_graph f z A y'"
```
```   608     by (rule fold_graph_insertE)
```
```   609   from `fold_graph f z A y'` have "y' = y" by (rule insertI)
```
```   610   with `v = f x y'` show "v = f x y" by simp
```
```   611 qed fast
```
```   612
```
```   613 lemma fold_equality:
```
```   614   "fold_graph f z A y \<Longrightarrow> fold f z A = y"
```
```   615 by (unfold fold_def) (blast intro: fold_graph_determ)
```
```   616
```
```   617 lemma fold_graph_fold:
```
```   618   assumes "finite A"
```
```   619   shows "fold_graph f z A (fold f z A)"
```
```   620 proof -
```
```   621   from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
```
```   622   moreover note fold_graph_determ
```
```   623   ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
```
```   624   then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
```
```   625   then show ?thesis by (unfold fold_def)
```
```   626 qed
```
```   627
```
```   628 text{* The base case for @{text fold}: *}
```
```   629
```
```   630 lemma (in -) fold_empty [simp]: "fold f z {} = z"
```
```   631 by (unfold fold_def) blast
```
```   632
```
```   633 text{* The various recursion equations for @{const fold}: *}
```
```   634
```
```   635 lemma fold_insert [simp]:
```
```   636   assumes "finite A" and "x \<notin> A"
```
```   637   shows "fold f z (insert x A) = f x (fold f z A)"
```
```   638 proof (rule fold_equality)
```
```   639   from `finite A` have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
```
```   640   with `x \<notin> A`show "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)
```
```   641 qed
```
```   642
```
```   643 lemma fold_fun_comm:
```
```   644   "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
```
```   645 proof (induct rule: finite_induct)
```
```   646   case empty then show ?case by simp
```
```   647 next
```
```   648   case (insert y A) then show ?case
```
```   649     by (simp add: fun_left_comm[of x])
```
```   650 qed
```
```   651
```
```   652 lemma fold_insert2:
```
```   653   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
```
```   654 by (simp add: fold_fun_comm)
```
```   655
```
```   656 lemma fold_rec:
```
```   657   assumes "finite A" and "x \<in> A"
```
```   658   shows "fold f z A = f x (fold f z (A - {x}))"
```
```   659 proof -
```
```   660   have A: "A = insert x (A - {x})" using `x \<in> A` by blast
```
```   661   then have "fold f z A = fold f z (insert x (A - {x}))" by simp
```
```   662   also have "\<dots> = f x (fold f z (A - {x}))"
```
```   663     by (rule fold_insert) (simp add: `finite A`)+
```
```   664   finally show ?thesis .
```
```   665 qed
```
```   666
```
```   667 lemma fold_insert_remove:
```
```   668   assumes "finite A"
```
```   669   shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
```
```   670 proof -
```
```   671   from `finite A` have "finite (insert x A)" by auto
```
```   672   moreover have "x \<in> insert x A" by auto
```
```   673   ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
```
```   674     by (rule fold_rec)
```
```   675   then show ?thesis by simp
```
```   676 qed
```
```   677
```
```   678 end
```
```   679
```
```   680 text{* A simplified version for idempotent functions: *}
```
```   681
```
```   682 locale comp_fun_idem = comp_fun_commute +
```
```   683   assumes comp_fun_idem: "f x o f x = f x"
```
```   684 begin
```
```   685
```
```   686 lemma fun_left_idem: "f x (f x z) = f x z"
```
```   687   using comp_fun_idem by (simp add: fun_eq_iff)
```
```   688
```
```   689 lemma fold_insert_idem:
```
```   690   assumes fin: "finite A"
```
```   691   shows "fold f z (insert x A) = f x (fold f z A)"
```
```   692 proof cases
```
```   693   assume "x \<in> A"
```
```   694   then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
```
```   695   then show ?thesis using assms by (simp add:fun_left_idem)
```
```   696 next
```
```   697   assume "x \<notin> A" then show ?thesis using assms by simp
```
```   698 qed
```
```   699
```
```   700 declare fold_insert[simp del] fold_insert_idem[simp]
```
```   701
```
```   702 lemma fold_insert_idem2:
```
```   703   "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
```
```   704 by(simp add:fold_fun_comm)
```
```   705
```
```   706 end
```
```   707
```
```   708
```
```   709 subsubsection {* Expressing set operations via @{const fold} *}
```
```   710
```
```   711 lemma (in comp_fun_commute) comp_comp_fun_commute:
```
```   712   "comp_fun_commute (f \<circ> g)"
```
```   713 proof
```
```   714 qed (simp_all add: comp_fun_commute)
```
```   715
```
```   716 lemma (in comp_fun_idem) comp_comp_fun_idem:
```
```   717   "comp_fun_idem (f \<circ> g)"
```
```   718   by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
```
```   719     (simp_all add: comp_fun_idem)
```
```   720
```
```   721 lemma comp_fun_idem_insert:
```
```   722   "comp_fun_idem insert"
```
```   723 proof
```
```   724 qed auto
```
```   725
```
```   726 lemma comp_fun_idem_remove:
```
```   727   "comp_fun_idem Set.remove"
```
```   728 proof
```
```   729 qed auto
```
```   730
```
```   731 lemma (in semilattice_inf) comp_fun_idem_inf:
```
```   732   "comp_fun_idem inf"
```
```   733 proof
```
```   734 qed (auto simp add: inf_left_commute)
```
```   735
```
```   736 lemma (in semilattice_sup) comp_fun_idem_sup:
```
```   737   "comp_fun_idem sup"
```
```   738 proof
```
```   739 qed (auto simp add: sup_left_commute)
```
```   740
```
```   741 lemma union_fold_insert:
```
```   742   assumes "finite A"
```
```   743   shows "A \<union> B = fold insert B A"
```
```   744 proof -
```
```   745   interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
```
```   746   from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
```
```   747 qed
```
```   748
```
```   749 lemma minus_fold_remove:
```
```   750   assumes "finite A"
```
```   751   shows "B - A = fold Set.remove B A"
```
```   752 proof -
```
```   753   interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove)
```
```   754   from `finite A` have "fold Set.remove B A = B - A" by (induct A arbitrary: B) auto
```
```   755   then show ?thesis ..
```
```   756 qed
```
```   757
```
```   758 context complete_lattice
```
```   759 begin
```
```   760
```
```   761 lemma inf_Inf_fold_inf:
```
```   762   assumes "finite A"
```
```   763   shows "inf B (Inf A) = fold inf B A"
```
```   764 proof -
```
```   765   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
```
```   766   from `finite A` show ?thesis by (induct A arbitrary: B)
```
```   767     (simp_all add: inf_commute fold_fun_comm)
```
```   768 qed
```
```   769
```
```   770 lemma sup_Sup_fold_sup:
```
```   771   assumes "finite A"
```
```   772   shows "sup B (Sup A) = fold sup B A"
```
```   773 proof -
```
```   774   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
```
```   775   from `finite A` show ?thesis by (induct A arbitrary: B)
```
```   776     (simp_all add: sup_commute fold_fun_comm)
```
```   777 qed
```
```   778
```
```   779 lemma Inf_fold_inf:
```
```   780   assumes "finite A"
```
```   781   shows "Inf A = fold inf top A"
```
```   782   using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
```
```   783
```
```   784 lemma Sup_fold_sup:
```
```   785   assumes "finite A"
```
```   786   shows "Sup A = fold sup bot A"
```
```   787   using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
```
```   788
```
```   789 lemma inf_INF_fold_inf:
```
```   790   assumes "finite A"
```
```   791   shows "inf B (INFI A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold")
```
```   792 proof (rule sym)
```
```   793   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
```
```   794   interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
```
```   795   from `finite A` show "?fold = ?inf"
```
```   796     by (induct A arbitrary: B)
```
```   797       (simp_all add: INF_def inf_left_commute)
```
```   798 qed
```
```   799
```
```   800 lemma sup_SUP_fold_sup:
```
```   801   assumes "finite A"
```
```   802   shows "sup B (SUPR A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold")
```
```   803 proof (rule sym)
```
```   804   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
```
```   805   interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
```
```   806   from `finite A` show "?fold = ?sup"
```
```   807     by (induct A arbitrary: B)
```
```   808       (simp_all add: SUP_def sup_left_commute)
```
```   809 qed
```
```   810
```
```   811 lemma INF_fold_inf:
```
```   812   assumes "finite A"
```
```   813   shows "INFI A f = fold (inf \<circ> f) top A"
```
```   814   using assms inf_INF_fold_inf [of A top] by simp
```
```   815
```
```   816 lemma SUP_fold_sup:
```
```   817   assumes "finite A"
```
```   818   shows "SUPR A f = fold (sup \<circ> f) bot A"
```
```   819   using assms sup_SUP_fold_sup [of A bot] by simp
```
```   820
```
```   821 end
```
```   822
```
```   823
```
```   824 subsection {* The derived combinator @{text fold_image} *}
```
```   825
```
```   826 definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
```
```   827   where "fold_image f g = fold (\<lambda>x y. f (g x) y)"
```
```   828
```
```   829 lemma fold_image_empty[simp]: "fold_image f g z {} = z"
```
```   830   by (simp add:fold_image_def)
```
```   831
```
```   832 context ab_semigroup_mult
```
```   833 begin
```
```   834
```
```   835 lemma fold_image_insert[simp]:
```
```   836   assumes "finite A" and "a \<notin> A"
```
```   837   shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
```
```   838 proof -
```
```   839   interpret comp_fun_commute "%x y. (g x) * y"
```
```   840     by default (simp add: fun_eq_iff mult_ac)
```
```   841   from assms show ?thesis by (simp add: fold_image_def)
```
```   842 qed
```
```   843
```
```   844 lemma fold_image_reindex:
```
```   845   assumes "finite A"
```
```   846   shows "inj_on h A \<Longrightarrow> fold_image times g z (h ` A) = fold_image times (g \<circ> h) z A"
```
```   847   using assms by induct auto
```
```   848
```
```   849 lemma fold_image_cong:
```
```   850   assumes "finite A" and g_h: "\<And>x. x\<in>A \<Longrightarrow> g x = h x"
```
```   851   shows "fold_image times g z A = fold_image times h z A"
```
```   852 proof -
```
```   853   from `finite A`
```
```   854   have "\<And>C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C"
```
```   855   proof (induct arbitrary: C)
```
```   856     case empty then show ?case by simp
```
```   857   next
```
```   858     case (insert x F) then show ?case apply -
```
```   859     apply (simp add: subset_insert_iff, clarify)
```
```   860     apply (subgoal_tac "finite C")
```
```   861       prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
```
```   862     apply (subgoal_tac "C = insert x (C - {x})")
```
```   863       prefer 2 apply blast
```
```   864     apply (erule ssubst)
```
```   865     apply (simp add: Ball_def del: insert_Diff_single)
```
```   866     done
```
```   867   qed
```
```   868   with g_h show ?thesis by simp
```
```   869 qed
```
```   870
```
```   871 end
```
```   872
```
```   873 context comm_monoid_mult
```
```   874 begin
```
```   875
```
```   876 lemma fold_image_1:
```
```   877   "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
```
```   878   apply (induct rule: finite_induct)
```
```   879   apply simp by auto
```
```   880
```
```   881 lemma fold_image_Un_Int:
```
```   882   "finite A ==> finite B ==>
```
```   883     fold_image times g 1 A * fold_image times g 1 B =
```
```   884     fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
```
```   885   apply (induct rule: finite_induct)
```
```   886 by (induct set: finite)
```
```   887    (auto simp add: mult_ac insert_absorb Int_insert_left)
```
```   888
```
```   889 lemma fold_image_Un_one:
```
```   890   assumes fS: "finite S" and fT: "finite T"
```
```   891   and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
```
```   892   shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
```
```   893 proof-
```
```   894   have "fold_image op * f 1 (S \<inter> T) = 1"
```
```   895     apply (rule fold_image_1)
```
```   896     using fS fT I0 by auto
```
```   897   with fold_image_Un_Int[OF fS fT] show ?thesis by simp
```
```   898 qed
```
```   899
```
```   900 corollary fold_Un_disjoint:
```
```   901   "finite A ==> finite B ==> A Int B = {} ==>
```
```   902    fold_image times g 1 (A Un B) =
```
```   903    fold_image times g 1 A * fold_image times g 1 B"
```
```   904 by (simp add: fold_image_Un_Int)
```
```   905
```
```   906 lemma fold_image_UN_disjoint:
```
```   907   "\<lbrakk> finite I; ALL i:I. finite (A i);
```
```   908      ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
```
```   909    \<Longrightarrow> fold_image times g 1 (UNION I A) =
```
```   910        fold_image times (%i. fold_image times g 1 (A i)) 1 I"
```
```   911 apply (induct rule: finite_induct)
```
```   912 apply simp
```
```   913 apply atomize
```
```   914 apply (subgoal_tac "ALL i:F. x \<noteq> i")
```
```   915  prefer 2 apply blast
```
```   916 apply (subgoal_tac "A x Int UNION F A = {}")
```
```   917  prefer 2 apply blast
```
```   918 apply (simp add: fold_Un_disjoint)
```
```   919 done
```
```   920
```
```   921 lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```   922   fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A =
```
```   923   fold_image times (split g) 1 (SIGMA x:A. B x)"
```
```   924 apply (subst Sigma_def)
```
```   925 apply (subst fold_image_UN_disjoint, assumption, simp)
```
```   926  apply blast
```
```   927 apply (erule fold_image_cong)
```
```   928 apply (subst fold_image_UN_disjoint, simp, simp)
```
```   929  apply blast
```
```   930 apply simp
```
```   931 done
```
```   932
```
```   933 lemma fold_image_distrib: "finite A \<Longrightarrow>
```
```   934    fold_image times (%x. g x * h x) 1 A =
```
```   935    fold_image times g 1 A *  fold_image times h 1 A"
```
```   936 by (erule finite_induct) (simp_all add: mult_ac)
```
```   937
```
```   938 lemma fold_image_related:
```
```   939   assumes Re: "R e e"
```
```   940   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
```
```   941   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
```
```   942   shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
```
```   943   using fS by (rule finite_subset_induct) (insert assms, auto)
```
```   944
```
```   945 lemma  fold_image_eq_general:
```
```   946   assumes fS: "finite S"
```
```   947   and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y"
```
```   948   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
```
```   949   shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
```
```   950 proof-
```
```   951   from h f12 have hS: "h ` S = S'" by auto
```
```   952   {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
```
```   953     from f12 h H  have "x = y" by auto }
```
```   954   hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
```
```   955   from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
```
```   956   from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
```
```   957   also have "\<dots> = fold_image (op *) (f2 o h) e S"
```
```   958     using fold_image_reindex[OF fS hinj, of f2 e] .
```
```   959   also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
```
```   960     by blast
```
```   961   finally show ?thesis ..
```
```   962 qed
```
```   963
```
```   964 lemma fold_image_eq_general_inverses:
```
```   965   assumes fS: "finite S"
```
```   966   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
```
```   967   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
```
```   968   shows "fold_image (op *) f e S = fold_image (op *) g e T"
```
```   969   (* metis solves it, but not yet available here *)
```
```   970   apply (rule fold_image_eq_general[OF fS, of T h g f e])
```
```   971   apply (rule ballI)
```
```   972   apply (frule kh)
```
```   973   apply (rule ex1I[])
```
```   974   apply blast
```
```   975   apply clarsimp
```
```   976   apply (drule hk) apply simp
```
```   977   apply (rule sym)
```
```   978   apply (erule conjunct1[OF conjunct2[OF hk]])
```
```   979   apply (rule ballI)
```
```   980   apply (drule  hk)
```
```   981   apply blast
```
```   982   done
```
```   983
```
```   984 end
```
```   985
```
```   986
```
```   987 subsection {* A fold functional for non-empty sets *}
```
```   988
```
```   989 text{* Does not require start value. *}
```
```   990
```
```   991 inductive
```
```   992   fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
```
```   993   for f :: "'a => 'a => 'a"
```
```   994 where
```
```   995   fold1Set_insertI [intro]:
```
```   996    "\<lbrakk> fold_graph f a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
```
```   997
```
```   998 definition fold1 :: "('a => 'a => 'a) => 'a set => 'a" where
```
```   999   "fold1 f A == THE x. fold1Set f A x"
```
```  1000
```
```  1001 lemma fold1Set_nonempty:
```
```  1002   "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
```
```  1003 by(erule fold1Set.cases, simp_all)
```
```  1004
```
```  1005 inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
```
```  1006
```
```  1007 inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
```
```  1008
```
```  1009
```
```  1010 lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
```
```  1011 by (blast elim: fold_graph.cases)
```
```  1012
```
```  1013 lemma fold1_singleton [simp]: "fold1 f {a} = a"
```
```  1014 by (unfold fold1_def) blast
```
```  1015
```
```  1016 lemma finite_nonempty_imp_fold1Set:
```
```  1017   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
```
```  1018 apply (induct A rule: finite_induct)
```
```  1019 apply (auto dest: finite_imp_fold_graph [of _ f])
```
```  1020 done
```
```  1021
```
```  1022 text{*First, some lemmas about @{const fold_graph}.*}
```
```  1023
```
```  1024 context ab_semigroup_mult
```
```  1025 begin
```
```  1026
```
```  1027 lemma comp_fun_commute: "comp_fun_commute (op *)"
```
```  1028   by default (simp add: fun_eq_iff mult_ac)
```
```  1029
```
```  1030 lemma fold_graph_insert_swap:
```
```  1031 assumes fold: "fold_graph times (b::'a) A y" and "b \<notin> A"
```
```  1032 shows "fold_graph times z (insert b A) (z * y)"
```
```  1033 proof -
```
```  1034   interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
```
```  1035 from assms show ?thesis
```
```  1036 proof (induct rule: fold_graph.induct)
```
```  1037   case emptyI show ?case by (subst mult_commute [of z b], fast)
```
```  1038 next
```
```  1039   case (insertI x A y)
```
```  1040     have "fold_graph times z (insert x (insert b A)) (x * (z * y))"
```
```  1041       using insertI by force  --{*how does @{term id} get unfolded?*}
```
```  1042     thus ?case by (simp add: insert_commute mult_ac)
```
```  1043 qed
```
```  1044 qed
```
```  1045
```
```  1046 lemma fold_graph_permute_diff:
```
```  1047 assumes fold: "fold_graph times b A x"
```
```  1048 shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> fold_graph times a (insert b (A-{a})) x"
```
```  1049 using fold
```
```  1050 proof (induct rule: fold_graph.induct)
```
```  1051   case emptyI thus ?case by simp
```
```  1052 next
```
```  1053   case (insertI x A y)
```
```  1054   have "a = x \<or> a \<in> A" using insertI by simp
```
```  1055   thus ?case
```
```  1056   proof
```
```  1057     assume "a = x"
```
```  1058     with insertI show ?thesis
```
```  1059       by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap)
```
```  1060   next
```
```  1061     assume ainA: "a \<in> A"
```
```  1062     hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)"
```
```  1063       using insertI by force
```
```  1064     moreover
```
```  1065     have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
```
```  1066       using ainA insertI by blast
```
```  1067     ultimately show ?thesis by simp
```
```  1068   qed
```
```  1069 qed
```
```  1070
```
```  1071 lemma fold1_eq_fold:
```
```  1072 assumes "finite A" "a \<notin> A" shows "fold1 times (insert a A) = fold times a A"
```
```  1073 proof -
```
```  1074   interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
```
```  1075   from assms show ?thesis
```
```  1076 apply (simp add: fold1_def fold_def)
```
```  1077 apply (rule the_equality)
```
```  1078 apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times])
```
```  1079 apply (rule sym, clarify)
```
```  1080 apply (case_tac "Aa=A")
```
```  1081  apply (best intro: fold_graph_determ)
```
```  1082 apply (subgoal_tac "fold_graph times a A x")
```
```  1083  apply (best intro: fold_graph_determ)
```
```  1084 apply (subgoal_tac "insert aa (Aa - {a}) = A")
```
```  1085  prefer 2 apply (blast elim: equalityE)
```
```  1086 apply (auto dest: fold_graph_permute_diff [where a=a])
```
```  1087 done
```
```  1088 qed
```
```  1089
```
```  1090 lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
```
```  1091 apply safe
```
```  1092  apply simp
```
```  1093  apply (drule_tac x=x in spec)
```
```  1094  apply (drule_tac x="A-{x}" in spec, auto)
```
```  1095 done
```
```  1096
```
```  1097 lemma fold1_insert:
```
```  1098   assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
```
```  1099   shows "fold1 times (insert x A) = x * fold1 times A"
```
```  1100 proof -
```
```  1101   interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
```
```  1102   from nonempty obtain a A' where "A = insert a A' & a ~: A'"
```
```  1103     by (auto simp add: nonempty_iff)
```
```  1104   with A show ?thesis
```
```  1105     by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
```
```  1106 qed
```
```  1107
```
```  1108 end
```
```  1109
```
```  1110 context ab_semigroup_idem_mult
```
```  1111 begin
```
```  1112
```
```  1113 lemma comp_fun_idem: "comp_fun_idem (op *)"
```
```  1114   by default (simp_all add: fun_eq_iff mult_left_commute)
```
```  1115
```
```  1116 lemma fold1_insert_idem [simp]:
```
```  1117   assumes nonempty: "A \<noteq> {}" and A: "finite A"
```
```  1118   shows "fold1 times (insert x A) = x * fold1 times A"
```
```  1119 proof -
```
```  1120   interpret comp_fun_idem "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```  1121     by (rule comp_fun_idem)
```
```  1122   from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
```
```  1123     by (auto simp add: nonempty_iff)
```
```  1124   show ?thesis
```
```  1125   proof cases
```
```  1126     assume a: "a = x"
```
```  1127     show ?thesis
```
```  1128     proof cases
```
```  1129       assume "A' = {}"
```
```  1130       with A' a show ?thesis by simp
```
```  1131     next
```
```  1132       assume "A' \<noteq> {}"
```
```  1133       with A A' a show ?thesis
```
```  1134         by (simp add: fold1_insert mult_assoc [symmetric])
```
```  1135     qed
```
```  1136   next
```
```  1137     assume "a \<noteq> x"
```
```  1138     with A A' show ?thesis
```
```  1139       by (simp add: insert_commute fold1_eq_fold)
```
```  1140   qed
```
```  1141 qed
```
```  1142
```
```  1143 lemma hom_fold1_commute:
```
```  1144 assumes hom: "!!x y. h (x * y) = h x * h y"
```
```  1145 and N: "finite N" "N \<noteq> {}" shows "h (fold1 times N) = fold1 times (h ` N)"
```
```  1146 using N
```
```  1147 proof (induct rule: finite_ne_induct)
```
```  1148   case singleton thus ?case by simp
```
```  1149 next
```
```  1150   case (insert n N)
```
```  1151   then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp
```
```  1152   also have "\<dots> = h n * h (fold1 times N)" by(rule hom)
```
```  1153   also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert)
```
```  1154   also have "times (h n) \<dots> = fold1 times (insert (h n) (h ` N))"
```
```  1155     using insert by(simp)
```
```  1156   also have "insert (h n) (h ` N) = h ` insert n N" by simp
```
```  1157   finally show ?case .
```
```  1158 qed
```
```  1159
```
```  1160 lemma fold1_eq_fold_idem:
```
```  1161   assumes "finite A"
```
```  1162   shows "fold1 times (insert a A) = fold times a A"
```
```  1163 proof (cases "a \<in> A")
```
```  1164   case False
```
```  1165   with assms show ?thesis by (simp add: fold1_eq_fold)
```
```  1166 next
```
```  1167   interpret comp_fun_idem times by (fact comp_fun_idem)
```
```  1168   case True then obtain b B
```
```  1169     where A: "A = insert a B" and "a \<notin> B" by (rule set_insert)
```
```  1170   with assms have "finite B" by auto
```
```  1171   then have "fold times a (insert a B) = fold times (a * a) B"
```
```  1172     using `a \<notin> B` by (rule fold_insert2)
```
```  1173   then show ?thesis
```
```  1174     using `a \<notin> B` `finite B` by (simp add: fold1_eq_fold A)
```
```  1175 qed
```
```  1176
```
```  1177 end
```
```  1178
```
```  1179
```
```  1180 text{* Now the recursion rules for definitions: *}
```
```  1181
```
```  1182 lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
```
```  1183 by simp
```
```  1184
```
```  1185 lemma (in ab_semigroup_mult) fold1_insert_def:
```
```  1186   "\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
```
```  1187 by (simp add:fold1_insert)
```
```  1188
```
```  1189 lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:
```
```  1190   "\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
```
```  1191 by simp
```
```  1192
```
```  1193 subsubsection{* Determinacy for @{term fold1Set} *}
```
```  1194
```
```  1195 (*Not actually used!!*)
```
```  1196 (*
```
```  1197 context ab_semigroup_mult
```
```  1198 begin
```
```  1199
```
```  1200 lemma fold_graph_permute:
```
```  1201   "[|fold_graph times id b (insert a A) x; a \<notin> A; b \<notin> A|]
```
```  1202    ==> fold_graph times id a (insert b A) x"
```
```  1203 apply (cases "a=b")
```
```  1204 apply (auto dest: fold_graph_permute_diff)
```
```  1205 done
```
```  1206
```
```  1207 lemma fold1Set_determ:
```
```  1208   "fold1Set times A x ==> fold1Set times A y ==> y = x"
```
```  1209 proof (clarify elim!: fold1Set.cases)
```
```  1210   fix A x B y a b
```
```  1211   assume Ax: "fold_graph times id a A x"
```
```  1212   assume By: "fold_graph times id b B y"
```
```  1213   assume anotA:  "a \<notin> A"
```
```  1214   assume bnotB:  "b \<notin> B"
```
```  1215   assume eq: "insert a A = insert b B"
```
```  1216   show "y=x"
```
```  1217   proof cases
```
```  1218     assume same: "a=b"
```
```  1219     hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
```
```  1220     thus ?thesis using Ax By same by (blast intro: fold_graph_determ)
```
```  1221   next
```
```  1222     assume diff: "a\<noteq>b"
```
```  1223     let ?D = "B - {a}"
```
```  1224     have B: "B = insert a ?D" and A: "A = insert b ?D"
```
```  1225      and aB: "a \<in> B" and bA: "b \<in> A"
```
```  1226       using eq anotA bnotB diff by (blast elim!:equalityE)+
```
```  1227     with aB bnotB By
```
```  1228     have "fold_graph times id a (insert b ?D) y"
```
```  1229       by (auto intro: fold_graph_permute simp add: insert_absorb)
```
```  1230     moreover
```
```  1231     have "fold_graph times id a (insert b ?D) x"
```
```  1232       by (simp add: A [symmetric] Ax)
```
```  1233     ultimately show ?thesis by (blast intro: fold_graph_determ)
```
```  1234   qed
```
```  1235 qed
```
```  1236
```
```  1237 lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y"
```
```  1238   by (unfold fold1_def) (blast intro: fold1Set_determ)
```
```  1239
```
```  1240 end
```
```  1241 *)
```
```  1242
```
```  1243 declare
```
```  1244   empty_fold_graphE [rule del]  fold_graph.intros [rule del]
```
```  1245   empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
```
```  1246   -- {* No more proofs involve these relations. *}
```
```  1247
```
```  1248 subsubsection {* Lemmas about @{text fold1} *}
```
```  1249
```
```  1250 context ab_semigroup_mult
```
```  1251 begin
```
```  1252
```
```  1253 lemma fold1_Un:
```
```  1254 assumes A: "finite A" "A \<noteq> {}"
```
```  1255 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
```
```  1256        fold1 times (A Un B) = fold1 times A * fold1 times B"
```
```  1257 using A by (induct rule: finite_ne_induct)
```
```  1258   (simp_all add: fold1_insert mult_assoc)
```
```  1259
```
```  1260 lemma fold1_in:
```
```  1261   assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x,y}"
```
```  1262   shows "fold1 times A \<in> A"
```
```  1263 using A
```
```  1264 proof (induct rule:finite_ne_induct)
```
```  1265   case singleton thus ?case by simp
```
```  1266 next
```
```  1267   case insert thus ?case using elem by (force simp add:fold1_insert)
```
```  1268 qed
```
```  1269
```
```  1270 end
```
```  1271
```
```  1272 lemma (in ab_semigroup_idem_mult) fold1_Un2:
```
```  1273 assumes A: "finite A" "A \<noteq> {}"
```
```  1274 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
```
```  1275        fold1 times (A Un B) = fold1 times A * fold1 times B"
```
```  1276 using A
```
```  1277 proof(induct rule:finite_ne_induct)
```
```  1278   case singleton thus ?case by simp
```
```  1279 next
```
```  1280   case insert thus ?case by (simp add: mult_assoc)
```
```  1281 qed
```
```  1282
```
```  1283
```
```  1284 subsection {* Locales as mini-packages for fold operations *}
```
```  1285
```
```  1286 subsubsection {* The natural case *}
```
```  1287
```
```  1288 locale folding =
```
```  1289   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
```
```  1290   fixes F :: "'a set \<Rightarrow> 'b \<Rightarrow> 'b"
```
```  1291   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
```
```  1292   assumes eq_fold: "finite A \<Longrightarrow> F A s = fold f s A"
```
```  1293 begin
```
```  1294
```
```  1295 lemma empty [simp]:
```
```  1296   "F {} = id"
```
```  1297   by (simp add: eq_fold fun_eq_iff)
```
```  1298
```
```  1299 lemma insert [simp]:
```
```  1300   assumes "finite A" and "x \<notin> A"
```
```  1301   shows "F (insert x A) = F A \<circ> f x"
```
```  1302 proof -
```
```  1303   interpret comp_fun_commute f
```
```  1304     by default (insert comp_fun_commute, simp add: fun_eq_iff)
```
```  1305   from fold_insert2 assms
```
```  1306   have "\<And>s. fold f s (insert x A) = fold f (f x s) A" .
```
```  1307   with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
```
```  1308 qed
```
```  1309
```
```  1310 lemma remove:
```
```  1311   assumes "finite A" and "x \<in> A"
```
```  1312   shows "F A = F (A - {x}) \<circ> f x"
```
```  1313 proof -
```
```  1314   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
```
```  1315     by (auto dest: mk_disjoint_insert)
```
```  1316   moreover from `finite A` this have "finite B" by simp
```
```  1317   ultimately show ?thesis by simp
```
```  1318 qed
```
```  1319
```
```  1320 lemma insert_remove:
```
```  1321   assumes "finite A"
```
```  1322   shows "F (insert x A) = F (A - {x}) \<circ> f x"
```
```  1323   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
```
```  1324
```
```  1325 lemma commute_left_comp:
```
```  1326   "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
```
```  1327   by (simp add: o_assoc comp_fun_commute)
```
```  1328
```
```  1329 lemma comp_fun_commute':
```
```  1330   assumes "finite A"
```
```  1331   shows "f x \<circ> F A = F A \<circ> f x"
```
```  1332   using assms by (induct A)
```
```  1333     (simp, simp del: o_apply add: o_assoc, simp del: o_apply add: o_assoc [symmetric] comp_fun_commute)
```
```  1334
```
```  1335 lemma commute_left_comp':
```
```  1336   assumes "finite A"
```
```  1337   shows "f x \<circ> (F A \<circ> g) = F A \<circ> (f x \<circ> g)"
```
```  1338   using assms by (simp add: o_assoc comp_fun_commute')
```
```  1339
```
```  1340 lemma comp_fun_commute'':
```
```  1341   assumes "finite A" and "finite B"
```
```  1342   shows "F B \<circ> F A = F A \<circ> F B"
```
```  1343   using assms by (induct A)
```
```  1344     (simp_all add: o_assoc, simp add: o_assoc [symmetric] comp_fun_commute')
```
```  1345
```
```  1346 lemma commute_left_comp'':
```
```  1347   assumes "finite A" and "finite B"
```
```  1348   shows "F B \<circ> (F A \<circ> g) = F A \<circ> (F B \<circ> g)"
```
```  1349   using assms by (simp add: o_assoc comp_fun_commute'')
```
```  1350
```
```  1351 lemmas comp_fun_commutes = o_assoc [symmetric] comp_fun_commute commute_left_comp
```
```  1352   comp_fun_commute' commute_left_comp' comp_fun_commute'' commute_left_comp''
```
```  1353
```
```  1354 lemma union_inter:
```
```  1355   assumes "finite A" and "finite B"
```
```  1356   shows "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B"
```
```  1357   using assms by (induct A)
```
```  1358     (simp_all del: o_apply add: insert_absorb Int_insert_left comp_fun_commutes,
```
```  1359       simp add: o_assoc)
```
```  1360
```
```  1361 lemma union:
```
```  1362   assumes "finite A" and "finite B"
```
```  1363   and "A \<inter> B = {}"
```
```  1364   shows "F (A \<union> B) = F A \<circ> F B"
```
```  1365 proof -
```
```  1366   from union_inter `finite A` `finite B` have "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B" .
```
```  1367   with `A \<inter> B = {}` show ?thesis by simp
```
```  1368 qed
```
```  1369
```
```  1370 end
```
```  1371
```
```  1372
```
```  1373 subsubsection {* The natural case with idempotency *}
```
```  1374
```
```  1375 locale folding_idem = folding +
```
```  1376   assumes idem_comp: "f x \<circ> f x = f x"
```
```  1377 begin
```
```  1378
```
```  1379 lemma idem_left_comp:
```
```  1380   "f x \<circ> (f x \<circ> g) = f x \<circ> g"
```
```  1381   by (simp add: o_assoc idem_comp)
```
```  1382
```
```  1383 lemma in_comp_idem:
```
```  1384   assumes "finite A" and "x \<in> A"
```
```  1385   shows "F A \<circ> f x = F A"
```
```  1386 using assms by (induct A)
```
```  1387   (auto simp add: comp_fun_commutes idem_comp, simp add: commute_left_comp' [symmetric] comp_fun_commute')
```
```  1388
```
```  1389 lemma subset_comp_idem:
```
```  1390   assumes "finite A" and "B \<subseteq> A"
```
```  1391   shows "F A \<circ> F B = F A"
```
```  1392 proof -
```
```  1393   from assms have "finite B" by (blast dest: finite_subset)
```
```  1394   then show ?thesis using `B \<subseteq> A` by (induct B)
```
```  1395     (simp_all add: o_assoc in_comp_idem `finite A`)
```
```  1396 qed
```
```  1397
```
```  1398 declare insert [simp del]
```
```  1399
```
```  1400 lemma insert_idem [simp]:
```
```  1401   assumes "finite A"
```
```  1402   shows "F (insert x A) = F A \<circ> f x"
```
```  1403   using assms by (cases "x \<in> A") (simp_all add: insert in_comp_idem insert_absorb)
```
```  1404
```
```  1405 lemma union_idem:
```
```  1406   assumes "finite A" and "finite B"
```
```  1407   shows "F (A \<union> B) = F A \<circ> F B"
```
```  1408 proof -
```
```  1409   from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
```
```  1410   then have "F (A \<union> B) \<circ> F (A \<inter> B) = F (A \<union> B)" by (rule subset_comp_idem)
```
```  1411   with assms show ?thesis by (simp add: union_inter)
```
```  1412 qed
```
```  1413
```
```  1414 end
```
```  1415
```
```  1416
```
```  1417 subsubsection {* The image case with fixed function *}
```
```  1418
```
```  1419 no_notation times (infixl "*" 70)
```
```  1420 no_notation Groups.one ("1")
```
```  1421
```
```  1422 locale folding_image_simple = comm_monoid +
```
```  1423   fixes g :: "('b \<Rightarrow> 'a)"
```
```  1424   fixes F :: "'b set \<Rightarrow> 'a"
```
```  1425   assumes eq_fold_g: "finite A \<Longrightarrow> F A = fold_image f g 1 A"
```
```  1426 begin
```
```  1427
```
```  1428 lemma empty [simp]:
```
```  1429   "F {} = 1"
```
```  1430   by (simp add: eq_fold_g)
```
```  1431
```
```  1432 lemma insert [simp]:
```
```  1433   assumes "finite A" and "x \<notin> A"
```
```  1434   shows "F (insert x A) = g x * F A"
```
```  1435 proof -
```
```  1436   interpret comp_fun_commute "%x y. (g x) * y"
```
```  1437     by default (simp add: ac_simps fun_eq_iff)
```
```  1438   from assms have "fold_image (op *) g 1 (insert x A) = g x * fold_image (op *) g 1 A"
```
```  1439     by (simp add: fold_image_def)
```
```  1440   with `finite A` show ?thesis by (simp add: eq_fold_g)
```
```  1441 qed
```
```  1442
```
```  1443 lemma remove:
```
```  1444   assumes "finite A" and "x \<in> A"
```
```  1445   shows "F A = g x * F (A - {x})"
```
```  1446 proof -
```
```  1447   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
```
```  1448     by (auto dest: mk_disjoint_insert)
```
```  1449   moreover from `finite A` this have "finite B" by simp
```
```  1450   ultimately show ?thesis by simp
```
```  1451 qed
```
```  1452
```
```  1453 lemma insert_remove:
```
```  1454   assumes "finite A"
```
```  1455   shows "F (insert x A) = g x * F (A - {x})"
```
```  1456   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
```
```  1457
```
```  1458 lemma neutral:
```
```  1459   assumes "finite A" and "\<forall>x\<in>A. g x = 1"
```
```  1460   shows "F A = 1"
```
```  1461   using assms by (induct A) simp_all
```
```  1462
```
```  1463 lemma union_inter:
```
```  1464   assumes "finite A" and "finite B"
```
```  1465   shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
```
```  1466 using assms proof (induct A)
```
```  1467   case empty then show ?case by simp
```
```  1468 next
```
```  1469   case (insert x A) then show ?case
```
```  1470     by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
```
```  1471 qed
```
```  1472
```
```  1473 corollary union_inter_neutral:
```
```  1474   assumes "finite A" and "finite B"
```
```  1475   and I0: "\<forall>x \<in> A\<inter>B. g x = 1"
```
```  1476   shows "F (A \<union> B) = F A * F B"
```
```  1477   using assms by (simp add: union_inter [symmetric] neutral)
```
```  1478
```
```  1479 corollary union_disjoint:
```
```  1480   assumes "finite A" and "finite B"
```
```  1481   assumes "A \<inter> B = {}"
```
```  1482   shows "F (A \<union> B) = F A * F B"
```
```  1483   using assms by (simp add: union_inter_neutral)
```
```  1484
```
```  1485 end
```
```  1486
```
```  1487
```
```  1488 subsubsection {* The image case with flexible function *}
```
```  1489
```
```  1490 locale folding_image = comm_monoid +
```
```  1491   fixes F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
```
```  1492   assumes eq_fold: "\<And>g. finite A \<Longrightarrow> F g A = fold_image f g 1 A"
```
```  1493
```
```  1494 sublocale folding_image < folding_image_simple "op *" 1 g "F g" proof
```
```  1495 qed (fact eq_fold)
```
```  1496
```
```  1497 context folding_image
```
```  1498 begin
```
```  1499
```
```  1500 lemma reindex: (* FIXME polymorhism *)
```
```  1501   assumes "finite A" and "inj_on h A"
```
```  1502   shows "F g (h ` A) = F (g \<circ> h) A"
```
```  1503   using assms by (induct A) auto
```
```  1504
```
```  1505 lemma cong:
```
```  1506   assumes "finite A" and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"
```
```  1507   shows "F g A = F h A"
```
```  1508 proof -
```
```  1509   from assms have "ALL C. C <= A --> (ALL x:C. g x = h x) --> F g C = F h C"
```
```  1510   apply - apply (erule finite_induct) apply simp
```
```  1511   apply (simp add: subset_insert_iff, clarify)
```
```  1512   apply (subgoal_tac "finite C")
```
```  1513   prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
```
```  1514   apply (subgoal_tac "C = insert x (C - {x})")
```
```  1515   prefer 2 apply blast
```
```  1516   apply (erule ssubst)
```
```  1517   apply (drule spec)
```
```  1518   apply (erule (1) notE impE)
```
```  1519   apply (simp add: Ball_def del: insert_Diff_single)
```
```  1520   done
```
```  1521   with assms show ?thesis by simp
```
```  1522 qed
```
```  1523
```
```  1524 lemma UNION_disjoint:
```
```  1525   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
```
```  1526   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
```
```  1527   shows "F g (UNION I A) = F (F g \<circ> A) I"
```
```  1528 apply (insert assms)
```
```  1529 apply (induct rule: finite_induct)
```
```  1530 apply simp
```
```  1531 apply atomize
```
```  1532 apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
```
```  1533  prefer 2 apply blast
```
```  1534 apply (subgoal_tac "A x Int UNION Fa A = {}")
```
```  1535  prefer 2 apply blast
```
```  1536 apply (simp add: union_disjoint)
```
```  1537 done
```
```  1538
```
```  1539 lemma distrib:
```
```  1540   assumes "finite A"
```
```  1541   shows "F (\<lambda>x. g x * h x) A = F g A * F h A"
```
```  1542   using assms by (rule finite_induct) (simp_all add: assoc commute left_commute)
```
```  1543
```
```  1544 lemma related:
```
```  1545   assumes Re: "R 1 1"
```
```  1546   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
```
```  1547   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
```
```  1548   shows "R (F h S) (F g S)"
```
```  1549   using fS by (rule finite_subset_induct) (insert assms, auto)
```
```  1550
```
```  1551 lemma eq_general:
```
```  1552   assumes fS: "finite S"
```
```  1553   and h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y"
```
```  1554   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
```
```  1555   shows "F f1 S = F f2 S'"
```
```  1556 proof-
```
```  1557   from h f12 have hS: "h ` S = S'" by blast
```
```  1558   {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
```
```  1559     from f12 h H  have "x = y" by auto }
```
```  1560   hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
```
```  1561   from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
```
```  1562   from hS have "F f2 S' = F f2 (h ` S)" by simp
```
```  1563   also have "\<dots> = F (f2 o h) S" using reindex [OF fS hinj, of f2] .
```
```  1564   also have "\<dots> = F f1 S " using th cong [OF fS, of "f2 o h" f1]
```
```  1565     by blast
```
```  1566   finally show ?thesis ..
```
```  1567 qed
```
```  1568
```
```  1569 lemma eq_general_inverses:
```
```  1570   assumes fS: "finite S"
```
```  1571   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
```
```  1572   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"
```
```  1573   shows "F j S = F g T"
```
```  1574   (* metis solves it, but not yet available here *)
```
```  1575   apply (rule eq_general [OF fS, of T h g j])
```
```  1576   apply (rule ballI)
```
```  1577   apply (frule kh)
```
```  1578   apply (rule ex1I[])
```
```  1579   apply blast
```
```  1580   apply clarsimp
```
```  1581   apply (drule hk) apply simp
```
```  1582   apply (rule sym)
```
```  1583   apply (erule conjunct1[OF conjunct2[OF hk]])
```
```  1584   apply (rule ballI)
```
```  1585   apply (drule hk)
```
```  1586   apply blast
```
```  1587   done
```
```  1588
```
```  1589 end
```
```  1590
```
```  1591
```
```  1592 subsubsection {* The image case with fixed function and idempotency *}
```
```  1593
```
```  1594 locale folding_image_simple_idem = folding_image_simple +
```
```  1595   assumes idem: "x * x = x"
```
```  1596
```
```  1597 sublocale folding_image_simple_idem < semilattice proof
```
```  1598 qed (fact idem)
```
```  1599
```
```  1600 context folding_image_simple_idem
```
```  1601 begin
```
```  1602
```
```  1603 lemma in_idem:
```
```  1604   assumes "finite A" and "x \<in> A"
```
```  1605   shows "g x * F A = F A"
```
```  1606   using assms by (induct A) (auto simp add: left_commute)
```
```  1607
```
```  1608 lemma subset_idem:
```
```  1609   assumes "finite A" and "B \<subseteq> A"
```
```  1610   shows "F B * F A = F A"
```
```  1611 proof -
```
```  1612   from assms have "finite B" by (blast dest: finite_subset)
```
```  1613   then show ?thesis using `B \<subseteq> A` by (induct B)
```
```  1614     (auto simp add: assoc in_idem `finite A`)
```
```  1615 qed
```
```  1616
```
```  1617 declare insert [simp del]
```
```  1618
```
```  1619 lemma insert_idem [simp]:
```
```  1620   assumes "finite A"
```
```  1621   shows "F (insert x A) = g x * F A"
```
```  1622   using assms by (cases "x \<in> A") (simp_all add: insert in_idem insert_absorb)
```
```  1623
```
```  1624 lemma union_idem:
```
```  1625   assumes "finite A" and "finite B"
```
```  1626   shows "F (A \<union> B) = F A * F B"
```
```  1627 proof -
```
```  1628   from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
```
```  1629   then have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (rule subset_idem)
```
```  1630   with assms show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
```
```  1631 qed
```
```  1632
```
```  1633 end
```
```  1634
```
```  1635
```
```  1636 subsubsection {* The image case with flexible function and idempotency *}
```
```  1637
```
```  1638 locale folding_image_idem = folding_image +
```
```  1639   assumes idem: "x * x = x"
```
```  1640
```
```  1641 sublocale folding_image_idem < folding_image_simple_idem "op *" 1 g "F g" proof
```
```  1642 qed (fact idem)
```
```  1643
```
```  1644
```
```  1645 subsubsection {* The neutral-less case *}
```
```  1646
```
```  1647 locale folding_one = abel_semigroup +
```
```  1648   fixes F :: "'a set \<Rightarrow> 'a"
```
```  1649   assumes eq_fold: "finite A \<Longrightarrow> F A = fold1 f A"
```
```  1650 begin
```
```  1651
```
```  1652 lemma singleton [simp]:
```
```  1653   "F {x} = x"
```
```  1654   by (simp add: eq_fold)
```
```  1655
```
```  1656 lemma eq_fold':
```
```  1657   assumes "finite A" and "x \<notin> A"
```
```  1658   shows "F (insert x A) = fold (op *) x A"
```
```  1659 proof -
```
```  1660   interpret ab_semigroup_mult "op *" by default (simp_all add: ac_simps)
```
```  1661   from assms show ?thesis by (simp add: eq_fold fold1_eq_fold)
```
```  1662 qed
```
```  1663
```
```  1664 lemma insert [simp]:
```
```  1665   assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
```
```  1666   shows "F (insert x A) = x * F A"
```
```  1667 proof -
```
```  1668   from `A \<noteq> {}` obtain b where "b \<in> A" by blast
```
```  1669   then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
```
```  1670   with `finite A` have "finite B" by simp
```
```  1671   interpret fold: folding "op *" "\<lambda>a b. fold (op *) b a" proof
```
```  1672   qed (simp_all add: fun_eq_iff ac_simps)
```
```  1673   thm fold.comp_fun_commute' [of B b, simplified fun_eq_iff, simplified]
```
```  1674   from `finite B` fold.comp_fun_commute' [of B x]
```
```  1675     have "op * x \<circ> (\<lambda>b. fold op * b B) = (\<lambda>b. fold op * b B) \<circ> op * x" by simp
```
```  1676   then have A: "x * fold op * b B = fold op * (b * x) B" by (simp add: fun_eq_iff commute)
```
```  1677   from `finite B` * fold.insert [of B b]
```
```  1678     have "(\<lambda>x. fold op * x (insert b B)) = (\<lambda>x. fold op * x B) \<circ> op * b" by simp
```
```  1679   then have B: "fold op * x (insert b B) = fold op * (b * x) B" by (simp add: fun_eq_iff)
```
```  1680   from A B assms * show ?thesis by (simp add: eq_fold' del: fold.insert)
```
```  1681 qed
```
```  1682
```
```  1683 lemma remove:
```
```  1684   assumes "finite A" and "x \<in> A"
```
```  1685   shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
```
```  1686 proof -
```
```  1687   from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
```
```  1688   with assms show ?thesis by simp
```
```  1689 qed
```
```  1690
```
```  1691 lemma insert_remove:
```
```  1692   assumes "finite A"
```
```  1693   shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
```
```  1694   using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
```
```  1695
```
```  1696 lemma union_disjoint:
```
```  1697   assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}" and "A \<inter> B = {}"
```
```  1698   shows "F (A \<union> B) = F A * F B"
```
```  1699   using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
```
```  1700
```
```  1701 lemma union_inter:
```
```  1702   assumes "finite A" and "finite B" and "A \<inter> B \<noteq> {}"
```
```  1703   shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
```
```  1704 proof -
```
```  1705   from assms have "A \<noteq> {}" and "B \<noteq> {}" by auto
```
```  1706   from `finite A` `A \<noteq> {}` `A \<inter> B \<noteq> {}` show ?thesis proof (induct A rule: finite_ne_induct)
```
```  1707     case (singleton x) then show ?case by (simp add: insert_absorb ac_simps)
```
```  1708   next
```
```  1709     case (insert x A) show ?case proof (cases "x \<in> B")
```
```  1710       case True then have "B \<noteq> {}" by auto
```
```  1711       with insert True `finite B` show ?thesis by (cases "A \<inter> B = {}")
```
```  1712         (simp_all add: insert_absorb ac_simps union_disjoint)
```
```  1713     next
```
```  1714       case False with insert have "F (A \<union> B) * F (A \<inter> B) = F A * F B" by simp
```
```  1715       moreover from False `finite B` insert have "finite (A \<union> B)" "x \<notin> A \<union> B" "A \<union> B \<noteq> {}"
```
```  1716         by auto
```
```  1717       ultimately show ?thesis using False `finite A` `x \<notin> A` `A \<noteq> {}` by (simp add: assoc)
```
```  1718     qed
```
```  1719   qed
```
```  1720 qed
```
```  1721
```
```  1722 lemma closed:
```
```  1723   assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
```
```  1724   shows "F A \<in> A"
```
```  1725 using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
```
```  1726   case singleton then show ?case by simp
```
```  1727 next
```
```  1728   case insert with elem show ?case by force
```
```  1729 qed
```
```  1730
```
```  1731 end
```
```  1732
```
```  1733
```
```  1734 subsubsection {* The neutral-less case with idempotency *}
```
```  1735
```
```  1736 locale folding_one_idem = folding_one +
```
```  1737   assumes idem: "x * x = x"
```
```  1738
```
```  1739 sublocale folding_one_idem < semilattice proof
```
```  1740 qed (fact idem)
```
```  1741
```
```  1742 context folding_one_idem
```
```  1743 begin
```
```  1744
```
```  1745 lemma in_idem:
```
```  1746   assumes "finite A" and "x \<in> A"
```
```  1747   shows "x * F A = F A"
```
```  1748 proof -
```
```  1749   from assms have "A \<noteq> {}" by auto
```
```  1750   with `finite A` show ?thesis using `x \<in> A` by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
```
```  1751 qed
```
```  1752
```
```  1753 lemma subset_idem:
```
```  1754   assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
```
```  1755   shows "F B * F A = F A"
```
```  1756 proof -
```
```  1757   from assms have "finite B" by (blast dest: finite_subset)
```
```  1758   then show ?thesis using `B \<noteq> {}` `B \<subseteq> A` by (induct B rule: finite_ne_induct)
```
```  1759     (simp_all add: assoc in_idem `finite A`)
```
```  1760 qed
```
```  1761
```
```  1762 lemma eq_fold_idem':
```
```  1763   assumes "finite A"
```
```  1764   shows "F (insert a A) = fold (op *) a A"
```
```  1765 proof -
```
```  1766   interpret ab_semigroup_idem_mult "op *" by default (simp_all add: ac_simps)
```
```  1767   from assms show ?thesis by (simp add: eq_fold fold1_eq_fold_idem)
```
```  1768 qed
```
```  1769
```
```  1770 lemma insert_idem [simp]:
```
```  1771   assumes "finite A" and "A \<noteq> {}"
```
```  1772   shows "F (insert x A) = x * F A"
```
```  1773 proof (cases "x \<in> A")
```
```  1774   case False from `finite A` `x \<notin> A` `A \<noteq> {}` show ?thesis by (rule insert)
```
```  1775 next
```
```  1776   case True
```
```  1777   from `finite A` `A \<noteq> {}` show ?thesis by (simp add: in_idem insert_absorb True)
```
```  1778 qed
```
```  1779
```
```  1780 lemma union_idem:
```
```  1781   assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
```
```  1782   shows "F (A \<union> B) = F A * F B"
```
```  1783 proof (cases "A \<inter> B = {}")
```
```  1784   case True with assms show ?thesis by (simp add: union_disjoint)
```
```  1785 next
```
```  1786   case False
```
```  1787   from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
```
```  1788   with False have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (auto intro: subset_idem)
```
```  1789   with assms False show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
```
```  1790 qed
```
```  1791
```
```  1792 lemma hom_commute:
```
```  1793   assumes hom: "\<And>x y. h (x * y) = h x * h y"
```
```  1794   and N: "finite N" "N \<noteq> {}" shows "h (F N) = F (h ` N)"
```
```  1795 using N proof (induct rule: finite_ne_induct)
```
```  1796   case singleton thus ?case by simp
```
```  1797 next
```
```  1798   case (insert n N)
```
```  1799   then have "h (F (insert n N)) = h (n * F N)" by simp
```
```  1800   also have "\<dots> = h n * h (F N)" by (rule hom)
```
```  1801   also have "h (F N) = F (h ` N)" by(rule insert)
```
```  1802   also have "h n * \<dots> = F (insert (h n) (h ` N))"
```
```  1803     using insert by(simp)
```
```  1804   also have "insert (h n) (h ` N) = h ` insert n N" by simp
```
```  1805   finally show ?case .
```
```  1806 qed
```
```  1807
```
```  1808 end
```
```  1809
```
```  1810 notation times (infixl "*" 70)
```
```  1811 notation Groups.one ("1")
```
```  1812
```
```  1813
```
```  1814 subsection {* Finite cardinality *}
```
```  1815
```
```  1816 text {* This definition, although traditional, is ugly to work with:
```
```  1817 @{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
```
```  1818 But now that we have @{text fold_image} things are easy:
```
```  1819 *}
```
```  1820
```
```  1821 definition card :: "'a set \<Rightarrow> nat" where
```
```  1822   "card A = (if finite A then fold_image (op +) (\<lambda>x. 1) 0 A else 0)"
```
```  1823
```
```  1824 interpretation card: folding_image_simple "op +" 0 "\<lambda>x. 1" card proof
```
```  1825 qed (simp add: card_def)
```
```  1826
```
```  1827 lemma card_infinite [simp]:
```
```  1828   "\<not> finite A \<Longrightarrow> card A = 0"
```
```  1829   by (simp add: card_def)
```
```  1830
```
```  1831 lemma card_empty:
```
```  1832   "card {} = 0"
```
```  1833   by (fact card.empty)
```
```  1834
```
```  1835 lemma card_insert_disjoint:
```
```  1836   "finite A ==> x \<notin> A ==> card (insert x A) = Suc (card A)"
```
```  1837   by simp
```
```  1838
```
```  1839 lemma card_insert_if:
```
```  1840   "finite A ==> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
```
```  1841   by auto (simp add: card.insert_remove card.remove)
```
```  1842
```
```  1843 lemma card_ge_0_finite:
```
```  1844   "card A > 0 \<Longrightarrow> finite A"
```
```  1845   by (rule ccontr) simp
```
```  1846
```
```  1847 lemma card_0_eq [simp, no_atp]:
```
```  1848   "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
```
```  1849   by (auto dest: mk_disjoint_insert)
```
```  1850
```
```  1851 lemma finite_UNIV_card_ge_0:
```
```  1852   "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
```
```  1853   by (rule ccontr) simp
```
```  1854
```
```  1855 lemma card_eq_0_iff:
```
```  1856   "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
```
```  1857   by auto
```
```  1858
```
```  1859 lemma card_gt_0_iff:
```
```  1860   "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
```
```  1861   by (simp add: neq0_conv [symmetric] card_eq_0_iff)
```
```  1862
```
```  1863 lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
```
```  1864 apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
```
```  1865 apply(simp del:insert_Diff_single)
```
```  1866 done
```
```  1867
```
```  1868 lemma card_Diff_singleton:
```
```  1869   "finite A ==> x: A ==> card (A - {x}) = card A - 1"
```
```  1870 by (simp add: card_Suc_Diff1 [symmetric])
```
```  1871
```
```  1872 lemma card_Diff_singleton_if:
```
```  1873   "finite A ==> card (A - {x}) = (if x : A then card A - 1 else card A)"
```
```  1874 by (simp add: card_Diff_singleton)
```
```  1875
```
```  1876 lemma card_Diff_insert[simp]:
```
```  1877 assumes "finite A" and "a:A" and "a ~: B"
```
```  1878 shows "card(A - insert a B) = card(A - B) - 1"
```
```  1879 proof -
```
```  1880   have "A - insert a B = (A - B) - {a}" using assms by blast
```
```  1881   then show ?thesis using assms by(simp add:card_Diff_singleton)
```
```  1882 qed
```
```  1883
```
```  1884 lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
```
```  1885 by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert)
```
```  1886
```
```  1887 lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
```
```  1888 by (simp add: card_insert_if)
```
```  1889
```
```  1890 lemma card_Collect_less_nat[simp]: "card{i::nat. i < n} = n"
```
```  1891 by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
```
```  1892
```
```  1893 lemma card_Collect_le_nat[simp]: "card{i::nat. i <= n} = Suc n"
```
```  1894 using card_Collect_less_nat[of "Suc n"] by(simp add: less_Suc_eq_le)
```
```  1895
```
```  1896 lemma card_mono:
```
```  1897   assumes "finite B" and "A \<subseteq> B"
```
```  1898   shows "card A \<le> card B"
```
```  1899 proof -
```
```  1900   from assms have "finite A" by (auto intro: finite_subset)
```
```  1901   then show ?thesis using assms proof (induct A arbitrary: B)
```
```  1902     case empty then show ?case by simp
```
```  1903   next
```
```  1904     case (insert x A)
```
```  1905     then have "x \<in> B" by simp
```
```  1906     from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
```
```  1907     with insert.hyps have "card A \<le> card (B - {x})" by auto
```
```  1908     with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
```
```  1909   qed
```
```  1910 qed
```
```  1911
```
```  1912 lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
```
```  1913 apply (induct rule: finite_induct)
```
```  1914 apply simp
```
```  1915 apply clarify
```
```  1916 apply (subgoal_tac "finite A & A - {x} <= F")
```
```  1917  prefer 2 apply (blast intro: finite_subset, atomize)
```
```  1918 apply (drule_tac x = "A - {x}" in spec)
```
```  1919 apply (simp add: card_Diff_singleton_if split add: split_if_asm)
```
```  1920 apply (case_tac "card A", auto)
```
```  1921 done
```
```  1922
```
```  1923 lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
```
```  1924 apply (simp add: psubset_eq linorder_not_le [symmetric])
```
```  1925 apply (blast dest: card_seteq)
```
```  1926 done
```
```  1927
```
```  1928 lemma card_Un_Int: "finite A ==> finite B
```
```  1929     ==> card A + card B = card (A Un B) + card (A Int B)"
```
```  1930   by (fact card.union_inter [symmetric])
```
```  1931
```
```  1932 lemma card_Un_disjoint: "finite A ==> finite B
```
```  1933     ==> A Int B = {} ==> card (A Un B) = card A + card B"
```
```  1934   by (fact card.union_disjoint)
```
```  1935
```
```  1936 lemma card_Diff_subset:
```
```  1937   assumes "finite B" and "B \<subseteq> A"
```
```  1938   shows "card (A - B) = card A - card B"
```
```  1939 proof (cases "finite A")
```
```  1940   case False with assms show ?thesis by simp
```
```  1941 next
```
```  1942   case True with assms show ?thesis by (induct B arbitrary: A) simp_all
```
```  1943 qed
```
```  1944
```
```  1945 lemma card_Diff_subset_Int:
```
```  1946   assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
```
```  1947 proof -
```
```  1948   have "A - B = A - A \<inter> B" by auto
```
```  1949   thus ?thesis
```
```  1950     by (simp add: card_Diff_subset AB)
```
```  1951 qed
```
```  1952
```
```  1953 lemma diff_card_le_card_Diff:
```
```  1954 assumes "finite B" shows "card A - card B \<le> card(A - B)"
```
```  1955 proof-
```
```  1956   have "card A - card B \<le> card A - card (A \<inter> B)"
```
```  1957     using card_mono[OF assms Int_lower2, of A] by arith
```
```  1958   also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
```
```  1959   finally show ?thesis .
```
```  1960 qed
```
```  1961
```
```  1962 lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
```
```  1963 apply (rule Suc_less_SucD)
```
```  1964 apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
```
```  1965 done
```
```  1966
```
```  1967 lemma card_Diff2_less:
```
```  1968   "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
```
```  1969 apply (case_tac "x = y")
```
```  1970  apply (simp add: card_Diff1_less del:card_Diff_insert)
```
```  1971 apply (rule less_trans)
```
```  1972  prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
```
```  1973 done
```
```  1974
```
```  1975 lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
```
```  1976 apply (case_tac "x : A")
```
```  1977  apply (simp_all add: card_Diff1_less less_imp_le)
```
```  1978 done
```
```  1979
```
```  1980 lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
```
```  1981 by (erule psubsetI, blast)
```
```  1982
```
```  1983 lemma insert_partition:
```
```  1984   "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
```
```  1985   \<Longrightarrow> x \<inter> \<Union> F = {}"
```
```  1986 by auto
```
```  1987
```
```  1988 lemma finite_psubset_induct[consumes 1, case_names psubset]:
```
```  1989   assumes fin: "finite A"
```
```  1990   and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A"
```
```  1991   shows "P A"
```
```  1992 using fin
```
```  1993 proof (induct A taking: card rule: measure_induct_rule)
```
```  1994   case (less A)
```
```  1995   have fin: "finite A" by fact
```
```  1996   have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
```
```  1997   { fix B
```
```  1998     assume asm: "B \<subset> A"
```
```  1999     from asm have "card B < card A" using psubset_card_mono fin by blast
```
```  2000     moreover
```
```  2001     from asm have "B \<subseteq> A" by auto
```
```  2002     then have "finite B" using fin finite_subset by blast
```
```  2003     ultimately
```
```  2004     have "P B" using ih by simp
```
```  2005   }
```
```  2006   with fin show "P A" using major by blast
```
```  2007 qed
```
```  2008
```
```  2009 text{* main cardinality theorem *}
```
```  2010 lemma card_partition [rule_format]:
```
```  2011   "finite C ==>
```
```  2012      finite (\<Union> C) -->
```
```  2013      (\<forall>c\<in>C. card c = k) -->
```
```  2014      (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
```
```  2015      k * card(C) = card (\<Union> C)"
```
```  2016 apply (erule finite_induct, simp)
```
```  2017 apply (simp add: card_Un_disjoint insert_partition
```
```  2018        finite_subset [of _ "\<Union> (insert x F)"])
```
```  2019 done
```
```  2020
```
```  2021 lemma card_eq_UNIV_imp_eq_UNIV:
```
```  2022   assumes fin: "finite (UNIV :: 'a set)"
```
```  2023   and card: "card A = card (UNIV :: 'a set)"
```
```  2024   shows "A = (UNIV :: 'a set)"
```
```  2025 proof
```
```  2026   show "A \<subseteq> UNIV" by simp
```
```  2027   show "UNIV \<subseteq> A"
```
```  2028   proof
```
```  2029     fix x
```
```  2030     show "x \<in> A"
```
```  2031     proof (rule ccontr)
```
```  2032       assume "x \<notin> A"
```
```  2033       then have "A \<subset> UNIV" by auto
```
```  2034       with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
```
```  2035       with card show False by simp
```
```  2036     qed
```
```  2037   qed
```
```  2038 qed
```
```  2039
```
```  2040 text{*The form of a finite set of given cardinality*}
```
```  2041
```
```  2042 lemma card_eq_SucD:
```
```  2043 assumes "card A = Suc k"
```
```  2044 shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
```
```  2045 proof -
```
```  2046   have fin: "finite A" using assms by (auto intro: ccontr)
```
```  2047   moreover have "card A \<noteq> 0" using assms by auto
```
```  2048   ultimately obtain b where b: "b \<in> A" by auto
```
```  2049   show ?thesis
```
```  2050   proof (intro exI conjI)
```
```  2051     show "A = insert b (A-{b})" using b by blast
```
```  2052     show "b \<notin> A - {b}" by blast
```
```  2053     show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
```
```  2054       using assms b fin by(fastforce dest:mk_disjoint_insert)+
```
```  2055   qed
```
```  2056 qed
```
```  2057
```
```  2058 lemma card_Suc_eq:
```
```  2059   "(card A = Suc k) =
```
```  2060    (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
```
```  2061 apply(rule iffI)
```
```  2062  apply(erule card_eq_SucD)
```
```  2063 apply(auto)
```
```  2064 apply(subst card_insert)
```
```  2065  apply(auto intro:ccontr)
```
```  2066 done
```
```  2067
```
```  2068 lemma card_le_Suc_iff: "finite A \<Longrightarrow>
```
```  2069   Suc n \<le> card A = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
```
```  2070 by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
```
```  2071   dest: subset_singletonD split: nat.splits if_splits)
```
```  2072
```
```  2073 lemma finite_fun_UNIVD2:
```
```  2074   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
```
```  2075   shows "finite (UNIV :: 'b set)"
```
```  2076 proof -
```
```  2077   from fin have "\<And>arbitrary. finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
```
```  2078     by (rule finite_imageI)
```
```  2079   moreover have "\<And>arbitrary. UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
```
```  2080     by (rule UNIV_eq_I) auto
```
```  2081   ultimately show "finite (UNIV :: 'b set)" by simp
```
```  2082 qed
```
```  2083
```
```  2084 lemma card_UNIV_unit: "card (UNIV :: unit set) = 1"
```
```  2085   unfolding UNIV_unit by simp
```
```  2086
```
```  2087 lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
```
```  2088   unfolding UNIV_bool by simp
```
```  2089
```
```  2090
```
```  2091 subsubsection {* Cardinality of image *}
```
```  2092
```
```  2093 lemma card_image_le: "finite A ==> card (f ` A) <= card A"
```
```  2094 apply (induct rule: finite_induct)
```
```  2095  apply simp
```
```  2096 apply (simp add: le_SucI card_insert_if)
```
```  2097 done
```
```  2098
```
```  2099 lemma card_image:
```
```  2100   assumes "inj_on f A"
```
```  2101   shows "card (f ` A) = card A"
```
```  2102 proof (cases "finite A")
```
```  2103   case True then show ?thesis using assms by (induct A) simp_all
```
```  2104 next
```
```  2105   case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
```
```  2106   with False show ?thesis by simp
```
```  2107 qed
```
```  2108
```
```  2109 lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
```
```  2110 by(auto simp: card_image bij_betw_def)
```
```  2111
```
```  2112 lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
```
```  2113 by (simp add: card_seteq card_image)
```
```  2114
```
```  2115 lemma eq_card_imp_inj_on:
```
```  2116   "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
```
```  2117 apply (induct rule:finite_induct)
```
```  2118 apply simp
```
```  2119 apply(frule card_image_le[where f = f])
```
```  2120 apply(simp add:card_insert_if split:if_splits)
```
```  2121 done
```
```  2122
```
```  2123 lemma inj_on_iff_eq_card:
```
```  2124   "finite A ==> inj_on f A = (card(f ` A) = card A)"
```
```  2125 by(blast intro: card_image eq_card_imp_inj_on)
```
```  2126
```
```  2127
```
```  2128 lemma card_inj_on_le:
```
```  2129   "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
```
```  2130 apply (subgoal_tac "finite A")
```
```  2131  apply (force intro: card_mono simp add: card_image [symmetric])
```
```  2132 apply (blast intro: finite_imageD dest: finite_subset)
```
```  2133 done
```
```  2134
```
```  2135 lemma card_bij_eq:
```
```  2136   "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
```
```  2137      finite A; finite B |] ==> card A = card B"
```
```  2138 by (auto intro: le_antisym card_inj_on_le)
```
```  2139
```
```  2140 lemma bij_betw_finite:
```
```  2141   assumes "bij_betw f A B"
```
```  2142   shows "finite A \<longleftrightarrow> finite B"
```
```  2143 using assms unfolding bij_betw_def
```
```  2144 using finite_imageD[of f A] by auto
```
```  2145
```
```  2146
```
```  2147 subsubsection {* Pigeonhole Principles *}
```
```  2148
```
```  2149 lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
```
```  2150 by (auto dest: card_image less_irrefl_nat)
```
```  2151
```
```  2152 lemma pigeonhole_infinite:
```
```  2153 assumes  "~ finite A" and "finite(f`A)"
```
```  2154 shows "EX a0:A. ~finite{a:A. f a = f a0}"
```
```  2155 proof -
```
```  2156   have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
```
```  2157   proof(induct "f`A" arbitrary: A rule: finite_induct)
```
```  2158     case empty thus ?case by simp
```
```  2159   next
```
```  2160     case (insert b F)
```
```  2161     show ?case
```
```  2162     proof cases
```
```  2163       assume "finite{a:A. f a = b}"
```
```  2164       hence "~ finite(A - {a:A. f a = b})" using `\<not> finite A` by simp
```
```  2165       also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
```
```  2166       finally have "~ finite({a:A. f a \<noteq> b})" .
```
```  2167       from insert(3)[OF _ this]
```
```  2168       show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
```
```  2169     next
```
```  2170       assume 1: "~finite{a:A. f a = b}"
```
```  2171       hence "{a \<in> A. f a = b} \<noteq> {}" by force
```
```  2172       thus ?thesis using 1 by blast
```
```  2173     qed
```
```  2174   qed
```
```  2175   from this[OF assms(2,1)] show ?thesis .
```
```  2176 qed
```
```  2177
```
```  2178 lemma pigeonhole_infinite_rel:
```
```  2179 assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
```
```  2180 shows "EX b:B. ~finite{a:A. R a b}"
```
```  2181 proof -
```
```  2182    let ?F = "%a. {b:B. R a b}"
```
```  2183    from finite_Pow_iff[THEN iffD2, OF `finite B`]
```
```  2184    have "finite(?F ` A)" by(blast intro: rev_finite_subset)
```
```  2185    from pigeonhole_infinite[where f = ?F, OF assms(1) this]
```
```  2186    obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
```
```  2187    obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast
```
```  2188    { assume "finite{a:A. R a b0}"
```
```  2189      then have "finite {a\<in>A. ?F a = ?F a0}"
```
```  2190        using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)
```
```  2191    }
```
```  2192    with 1 `b0 : B` show ?thesis by blast
```
```  2193 qed
```
```  2194
```
```  2195
```
```  2196 subsubsection {* Cardinality of sums *}
```
```  2197
```
```  2198 lemma card_Plus:
```
```  2199   assumes "finite A" and "finite B"
```
```  2200   shows "card (A <+> B) = card A + card B"
```
```  2201 proof -
```
```  2202   have "Inl`A \<inter> Inr`B = {}" by fast
```
```  2203   with assms show ?thesis
```
```  2204     unfolding Plus_def
```
```  2205     by (simp add: card_Un_disjoint card_image)
```
```  2206 qed
```
```  2207
```
```  2208 lemma card_Plus_conv_if:
```
```  2209   "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
```
```  2210   by (auto simp add: card_Plus)
```
```  2211
```
```  2212
```
```  2213 subsubsection {* Cardinality of the Powerset *}
```
```  2214
```
```  2215 lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
```
```  2216 apply (induct rule: finite_induct)
```
```  2217  apply (simp_all add: Pow_insert)
```
```  2218 apply (subst card_Un_disjoint, blast)
```
```  2219   apply (blast, blast)
```
```  2220 apply (subgoal_tac "inj_on (insert x) (Pow F)")
```
```  2221  apply (simp add: card_image Pow_insert)
```
```  2222 apply (unfold inj_on_def)
```
```  2223 apply (blast elim!: equalityE)
```
```  2224 done
```
```  2225
```
```  2226 text {* Relates to equivalence classes.  Based on a theorem of F. Kamm\"uller.  *}
```
```  2227
```
```  2228 lemma dvd_partition:
```
```  2229   "finite (Union C) ==>
```
```  2230     ALL c : C. k dvd card c ==>
```
```  2231     (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
```
```  2232   k dvd card (Union C)"
```
```  2233 apply (frule finite_UnionD)
```
```  2234 apply (rotate_tac -1)
```
```  2235 apply (induct rule: finite_induct)
```
```  2236 apply simp_all
```
```  2237 apply clarify
```
```  2238 apply (subst card_Un_disjoint)
```
```  2239    apply (auto simp add: disjoint_eq_subset_Compl)
```
```  2240 done
```
```  2241
```
```  2242
```
```  2243 subsubsection {* Relating injectivity and surjectivity *}
```
```  2244
```
```  2245 lemma finite_surj_inj: "finite A \<Longrightarrow> A \<subseteq> f ` A \<Longrightarrow> inj_on f A"
```
```  2246 apply(rule eq_card_imp_inj_on, assumption)
```
```  2247 apply(frule finite_imageI)
```
```  2248 apply(drule (1) card_seteq)
```
```  2249  apply(erule card_image_le)
```
```  2250 apply simp
```
```  2251 done
```
```  2252
```
```  2253 lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
```
```  2254 shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
```
```  2255 by (blast intro: finite_surj_inj subset_UNIV)
```
```  2256
```
```  2257 lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
```
```  2258 shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
```
```  2259 by(fastforce simp:surj_def dest!: endo_inj_surj)
```
```  2260
```
```  2261 corollary infinite_UNIV_nat[iff]: "~finite(UNIV::nat set)"
```
```  2262 proof
```
```  2263   assume "finite(UNIV::nat set)"
```
```  2264   with finite_UNIV_inj_surj[of Suc]
```
```  2265   show False by simp (blast dest: Suc_neq_Zero surjD)
```
```  2266 qed
```
```  2267
```
```  2268 (* Often leads to bogus ATP proofs because of reduced type information, hence no_atp *)
```
```  2269 lemma infinite_UNIV_char_0[no_atp]:
```
```  2270   "\<not> finite (UNIV::'a::semiring_char_0 set)"
```
```  2271 proof
```
```  2272   assume "finite (UNIV::'a set)"
```
```  2273   with subset_UNIV have "finite (range of_nat::'a set)"
```
```  2274     by (rule finite_subset)
```
```  2275   moreover have "inj (of_nat::nat \<Rightarrow> 'a)"
```
```  2276     by (simp add: inj_on_def)
```
```  2277   ultimately have "finite (UNIV::nat set)"
```
```  2278     by (rule finite_imageD)
```
```  2279   then show "False"
```
```  2280     by simp
```
```  2281 qed
```
```  2282
```
```  2283 hide_const (open) Finite_Set.fold
```
```  2284
```
```  2285 end
```