src/HOL/Finite_Set.thy
 author haftmann Sun Mar 16 18:09:04 2014 +0100 (2014-03-16) changeset 56166 9a241bc276cd parent 56154 f0a927235162 child 56218 1c3f1f2431f9 permissions -rw-r--r--
normalising simp rules for compound operators
```     1 (*  Title:      HOL/Finite_Set.thy
```
```     2     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
```
```     3                 with contributions by Jeremy Avigad and Andrei Popescu
```
```     4 *)
```
```     5
```
```     6 header {* Finite sets *}
```
```     7
```
```     8 theory Finite_Set
```
```     9 imports Product_Type Sum_Type Nat
```
```    10 begin
```
```    11
```
```    12 subsection {* Predicate for finite sets *}
```
```    13
```
```    14 inductive finite :: "'a set \<Rightarrow> bool"
```
```    15   where
```
```    16     emptyI [simp, intro!]: "finite {}"
```
```    17   | insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)"
```
```    18
```
```    19 simproc_setup finite_Collect ("finite (Collect P)") = {* K Set_Comprehension_Pointfree.simproc *}
```
```    20
```
```    21 declare [[simproc del: finite_Collect]]
```
```    22
```
```    23 lemma finite_induct [case_names empty insert, induct set: finite]:
```
```    24   -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
```
```    25   assumes "finite F"
```
```    26   assumes "P {}"
```
```    27     and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
```
```    28   shows "P F"
```
```    29 using `finite F`
```
```    30 proof induct
```
```    31   show "P {}" by fact
```
```    32   fix x F assume F: "finite F" and P: "P F"
```
```    33   show "P (insert x F)"
```
```    34   proof cases
```
```    35     assume "x \<in> F"
```
```    36     hence "insert x F = F" by (rule insert_absorb)
```
```    37     with P show ?thesis by (simp only:)
```
```    38   next
```
```    39     assume "x \<notin> F"
```
```    40     from F this P show ?thesis by (rule insert)
```
```    41   qed
```
```    42 qed
```
```    43
```
```    44 lemma infinite_finite_induct [case_names infinite empty insert]:
```
```    45   assumes infinite: "\<And>A. \<not> finite A \<Longrightarrow> P A"
```
```    46   assumes empty: "P {}"
```
```    47   assumes insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
```
```    48   shows "P A"
```
```    49 proof (cases "finite A")
```
```    50   case False with infinite show ?thesis .
```
```    51 next
```
```    52   case True then show ?thesis by (induct A) (fact empty insert)+
```
```    53 qed
```
```    54
```
```    55
```
```    56 subsubsection {* Choice principles *}
```
```    57
```
```    58 lemma ex_new_if_finite: -- "does not depend on def of finite at all"
```
```    59   assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
```
```    60   shows "\<exists>a::'a. a \<notin> A"
```
```    61 proof -
```
```    62   from assms have "A \<noteq> UNIV" by blast
```
```    63   then show ?thesis by blast
```
```    64 qed
```
```    65
```
```    66 text {* A finite choice principle. Does not need the SOME choice operator. *}
```
```    67
```
```    68 lemma finite_set_choice:
```
```    69   "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
```
```    70 proof (induct rule: finite_induct)
```
```    71   case empty then show ?case by simp
```
```    72 next
```
```    73   case (insert a A)
```
```    74   then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
```
```    75   show ?case (is "EX f. ?P f")
```
```    76   proof
```
```    77     show "?P(%x. if x = a then b else f x)" using f ab by auto
```
```    78   qed
```
```    79 qed
```
```    80
```
```    81
```
```    82 subsubsection {* Finite sets are the images of initial segments of natural numbers *}
```
```    83
```
```    84 lemma finite_imp_nat_seg_image_inj_on:
```
```    85   assumes "finite A"
```
```    86   shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
```
```    87 using assms
```
```    88 proof induct
```
```    89   case empty
```
```    90   show ?case
```
```    91   proof
```
```    92     show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp
```
```    93   qed
```
```    94 next
```
```    95   case (insert a A)
```
```    96   have notinA: "a \<notin> A" by fact
```
```    97   from insert.hyps obtain n f
```
```    98     where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
```
```    99   hence "insert a A = f(n:=a) ` {i. i < Suc n}"
```
```   100         "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
```
```   101     by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
```
```   102   thus ?case by blast
```
```   103 qed
```
```   104
```
```   105 lemma nat_seg_image_imp_finite:
```
```   106   "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
```
```   107 proof (induct n arbitrary: A)
```
```   108   case 0 thus ?case by simp
```
```   109 next
```
```   110   case (Suc n)
```
```   111   let ?B = "f ` {i. i < n}"
```
```   112   have finB: "finite ?B" by(rule Suc.hyps[OF refl])
```
```   113   show ?case
```
```   114   proof cases
```
```   115     assume "\<exists>k<n. f n = f k"
```
```   116     hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   117     thus ?thesis using finB by simp
```
```   118   next
```
```   119     assume "\<not>(\<exists> k<n. f n = f k)"
```
```   120     hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   121     thus ?thesis using finB by simp
```
```   122   qed
```
```   123 qed
```
```   124
```
```   125 lemma finite_conv_nat_seg_image:
```
```   126   "finite A \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"
```
```   127   by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
```
```   128
```
```   129 lemma finite_imp_inj_to_nat_seg:
```
```   130   assumes "finite A"
```
```   131   shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> inj_on f A"
```
```   132 proof -
```
```   133   from finite_imp_nat_seg_image_inj_on[OF `finite A`]
```
```   134   obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
```
```   135     by (auto simp:bij_betw_def)
```
```   136   let ?f = "the_inv_into {i. i<n} f"
```
```   137   have "inj_on ?f A & ?f ` A = {i. i<n}"
```
```   138     by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
```
```   139   thus ?thesis by blast
```
```   140 qed
```
```   141
```
```   142 lemma finite_Collect_less_nat [iff]:
```
```   143   "finite {n::nat. n < k}"
```
```   144   by (fastforce simp: finite_conv_nat_seg_image)
```
```   145
```
```   146 lemma finite_Collect_le_nat [iff]:
```
```   147   "finite {n::nat. n \<le> k}"
```
```   148   by (simp add: le_eq_less_or_eq Collect_disj_eq)
```
```   149
```
```   150
```
```   151 subsubsection {* Finiteness and common set operations *}
```
```   152
```
```   153 lemma rev_finite_subset:
```
```   154   "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
```
```   155 proof (induct arbitrary: A rule: finite_induct)
```
```   156   case empty
```
```   157   then show ?case by simp
```
```   158 next
```
```   159   case (insert x F A)
```
```   160   have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+
```
```   161   show "finite A"
```
```   162   proof cases
```
```   163     assume x: "x \<in> A"
```
```   164     with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
```
```   165     with r have "finite (A - {x})" .
```
```   166     hence "finite (insert x (A - {x}))" ..
```
```   167     also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
```
```   168     finally show ?thesis .
```
```   169   next
```
```   170     show "A \<subseteq> F ==> ?thesis" by fact
```
```   171     assume "x \<notin> A"
```
```   172     with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
```
```   173   qed
```
```   174 qed
```
```   175
```
```   176 lemma finite_subset:
```
```   177   "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
```
```   178   by (rule rev_finite_subset)
```
```   179
```
```   180 lemma finite_UnI:
```
```   181   assumes "finite F" and "finite G"
```
```   182   shows "finite (F \<union> G)"
```
```   183   using assms by induct simp_all
```
```   184
```
```   185 lemma finite_Un [iff]:
```
```   186   "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
```
```   187   by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])
```
```   188
```
```   189 lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"
```
```   190 proof -
```
```   191   have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp
```
```   192   then have "finite ({a} \<union> A) \<longleftrightarrow> finite A" by (simp only: finite_Un)
```
```   193   then show ?thesis by simp
```
```   194 qed
```
```   195
```
```   196 lemma finite_Int [simp, intro]:
```
```   197   "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
```
```   198   by (blast intro: finite_subset)
```
```   199
```
```   200 lemma finite_Collect_conjI [simp, intro]:
```
```   201   "finite {x. P x} \<or> finite {x. Q x} \<Longrightarrow> finite {x. P x \<and> Q x}"
```
```   202   by (simp add: Collect_conj_eq)
```
```   203
```
```   204 lemma finite_Collect_disjI [simp]:
```
```   205   "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"
```
```   206   by (simp add: Collect_disj_eq)
```
```   207
```
```   208 lemma finite_Diff [simp, intro]:
```
```   209   "finite A \<Longrightarrow> finite (A - B)"
```
```   210   by (rule finite_subset, rule Diff_subset)
```
```   211
```
```   212 lemma finite_Diff2 [simp]:
```
```   213   assumes "finite B"
```
```   214   shows "finite (A - B) \<longleftrightarrow> finite A"
```
```   215 proof -
```
```   216   have "finite A \<longleftrightarrow> finite((A - B) \<union> (A \<inter> B))" by (simp add: Un_Diff_Int)
```
```   217   also have "\<dots> \<longleftrightarrow> finite (A - B)" using `finite B` by simp
```
```   218   finally show ?thesis ..
```
```   219 qed
```
```   220
```
```   221 lemma finite_Diff_insert [iff]:
```
```   222   "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
```
```   223 proof -
```
```   224   have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
```
```   225   moreover have "A - insert a B = A - B - {a}" by auto
```
```   226   ultimately show ?thesis by simp
```
```   227 qed
```
```   228
```
```   229 lemma finite_compl[simp]:
```
```   230   "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   231   by (simp add: Compl_eq_Diff_UNIV)
```
```   232
```
```   233 lemma finite_Collect_not[simp]:
```
```   234   "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   235   by (simp add: Collect_neg_eq)
```
```   236
```
```   237 lemma finite_Union [simp, intro]:
```
```   238   "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite(\<Union>A)"
```
```   239   by (induct rule: finite_induct) simp_all
```
```   240
```
```   241 lemma finite_UN_I [intro]:
```
```   242   "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"
```
```   243   by (induct rule: finite_induct) simp_all
```
```   244
```
```   245 lemma finite_UN [simp]:
```
```   246   "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
```
```   247   by (blast intro: finite_subset)
```
```   248
```
```   249 lemma finite_Inter [intro]:
```
```   250   "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
```
```   251   by (blast intro: Inter_lower finite_subset)
```
```   252
```
```   253 lemma finite_INT [intro]:
```
```   254   "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
```
```   255   by (blast intro: INT_lower finite_subset)
```
```   256
```
```   257 lemma finite_imageI [simp, intro]:
```
```   258   "finite F \<Longrightarrow> finite (h ` F)"
```
```   259   by (induct rule: finite_induct) simp_all
```
```   260
```
```   261 lemma finite_image_set [simp]:
```
```   262   "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
```
```   263   by (simp add: image_Collect [symmetric])
```
```   264
```
```   265 lemma finite_imageD:
```
```   266   assumes "finite (f ` A)" and "inj_on f A"
```
```   267   shows "finite A"
```
```   268 using assms
```
```   269 proof (induct "f ` A" arbitrary: A)
```
```   270   case empty then show ?case by simp
```
```   271 next
```
```   272   case (insert x B)
```
```   273   then have B_A: "insert x B = f ` A" by simp
```
```   274   then obtain y where "x = f y" and "y \<in> A" by blast
```
```   275   from B_A `x \<notin> B` have "B = f ` A - {x}" by blast
```
```   276   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)
```
```   277   moreover from `inj_on f A` have "inj_on f (A - {y})" by (rule inj_on_diff)
```
```   278   ultimately have "finite (A - {y})" by (rule insert.hyps)
```
```   279   then show "finite A" by simp
```
```   280 qed
```
```   281
```
```   282 lemma finite_surj:
```
```   283   "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
```
```   284   by (erule finite_subset) (rule finite_imageI)
```
```   285
```
```   286 lemma finite_range_imageI:
```
```   287   "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
```
```   288   by (drule finite_imageI) (simp add: range_composition)
```
```   289
```
```   290 lemma finite_subset_image:
```
```   291   assumes "finite B"
```
```   292   shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
```
```   293 using assms
```
```   294 proof induct
```
```   295   case empty then show ?case by simp
```
```   296 next
```
```   297   case insert then show ?case
```
```   298     by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
```
```   299        blast
```
```   300 qed
```
```   301
```
```   302 lemma finite_vimage_IntI:
```
```   303   "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
```
```   304   apply (induct rule: finite_induct)
```
```   305    apply simp_all
```
```   306   apply (subst vimage_insert)
```
```   307   apply (simp add: finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2)
```
```   308   done
```
```   309
```
```   310 lemma finite_vimageI:
```
```   311   "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
```
```   312   using finite_vimage_IntI[of F h UNIV] by auto
```
```   313
```
```   314 lemma finite_vimageD:
```
```   315   assumes fin: "finite (h -` F)" and surj: "surj h"
```
```   316   shows "finite F"
```
```   317 proof -
```
```   318   have "finite (h ` (h -` F))" using fin by (rule finite_imageI)
```
```   319   also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)
```
```   320   finally show "finite F" .
```
```   321 qed
```
```   322
```
```   323 lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
```
```   324   unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
```
```   325
```
```   326 lemma finite_Collect_bex [simp]:
```
```   327   assumes "finite A"
```
```   328   shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"
```
```   329 proof -
```
```   330   have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto
```
```   331   with assms show ?thesis by simp
```
```   332 qed
```
```   333
```
```   334 lemma finite_Collect_bounded_ex [simp]:
```
```   335   assumes "finite {y. P y}"
```
```   336   shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
```
```   337 proof -
```
```   338   have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
```
```   339   with assms show ?thesis by simp
```
```   340 qed
```
```   341
```
```   342 lemma finite_Plus:
```
```   343   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
```
```   344   by (simp add: Plus_def)
```
```   345
```
```   346 lemma finite_PlusD:
```
```   347   fixes A :: "'a set" and B :: "'b set"
```
```   348   assumes fin: "finite (A <+> B)"
```
```   349   shows "finite A" "finite B"
```
```   350 proof -
```
```   351   have "Inl ` A \<subseteq> A <+> B" by auto
```
```   352   then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
```
```   353   then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
```
```   354 next
```
```   355   have "Inr ` B \<subseteq> A <+> B" by auto
```
```   356   then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
```
```   357   then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
```
```   358 qed
```
```   359
```
```   360 lemma finite_Plus_iff [simp]:
```
```   361   "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
```
```   362   by (auto intro: finite_PlusD finite_Plus)
```
```   363
```
```   364 lemma finite_Plus_UNIV_iff [simp]:
```
```   365   "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
```
```   366   by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
```
```   367
```
```   368 lemma finite_SigmaI [simp, intro]:
```
```   369   "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
```
```   370   by (unfold Sigma_def) blast
```
```   371
```
```   372 lemma finite_SigmaI2:
```
```   373   assumes "finite {x\<in>A. B x \<noteq> {}}"
```
```   374   and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)"
```
```   375   shows "finite (Sigma A B)"
```
```   376 proof -
```
```   377   from assms have "finite (Sigma {x\<in>A. B x \<noteq> {}} B)" by auto
```
```   378   also have "Sigma {x:A. B x \<noteq> {}} B = Sigma A B" by auto
```
```   379   finally show ?thesis .
```
```   380 qed
```
```   381
```
```   382 lemma finite_cartesian_product:
```
```   383   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
```
```   384   by (rule finite_SigmaI)
```
```   385
```
```   386 lemma finite_Prod_UNIV:
```
```   387   "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
```
```   388   by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
```
```   389
```
```   390 lemma finite_cartesian_productD1:
```
```   391   assumes "finite (A \<times> B)" and "B \<noteq> {}"
```
```   392   shows "finite A"
```
```   393 proof -
```
```   394   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
```
```   395     by (auto simp add: finite_conv_nat_seg_image)
```
```   396   then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
```
```   397   with `B \<noteq> {}` have "A = (fst \<circ> f) ` {i::nat. i < n}"
```
```   398     by (simp add: image_comp)
```
```   399   then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
```
```   400   then show ?thesis
```
```   401     by (auto simp add: finite_conv_nat_seg_image)
```
```   402 qed
```
```   403
```
```   404 lemma finite_cartesian_productD2:
```
```   405   assumes "finite (A \<times> B)" and "A \<noteq> {}"
```
```   406   shows "finite B"
```
```   407 proof -
```
```   408   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
```
```   409     by (auto simp add: finite_conv_nat_seg_image)
```
```   410   then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
```
```   411   with `A \<noteq> {}` have "B = (snd \<circ> f) ` {i::nat. i < n}"
```
```   412     by (simp add: image_comp)
```
```   413   then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
```
```   414   then show ?thesis
```
```   415     by (auto simp add: finite_conv_nat_seg_image)
```
```   416 qed
```
```   417
```
```   418 lemma finite_prod:
```
```   419   "finite (UNIV :: ('a \<times> 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
```
```   420 by(auto simp add: UNIV_Times_UNIV[symmetric] simp del: UNIV_Times_UNIV
```
```   421    dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```   422
```
```   423 lemma finite_Pow_iff [iff]:
```
```   424   "finite (Pow A) \<longleftrightarrow> finite A"
```
```   425 proof
```
```   426   assume "finite (Pow A)"
```
```   427   then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
```
```   428   then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
```
```   429 next
```
```   430   assume "finite A"
```
```   431   then show "finite (Pow A)"
```
```   432     by induct (simp_all add: Pow_insert)
```
```   433 qed
```
```   434
```
```   435 corollary finite_Collect_subsets [simp, intro]:
```
```   436   "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
```
```   437   by (simp add: Pow_def [symmetric])
```
```   438
```
```   439 lemma finite_set: "finite (UNIV :: 'a set set) \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   440 by(simp only: finite_Pow_iff Pow_UNIV[symmetric])
```
```   441
```
```   442 lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
```
```   443   by (blast intro: finite_subset [OF subset_Pow_Union])
```
```   444
```
```   445 lemma finite_set_of_finite_funs: assumes "finite A" "finite B"
```
```   446 shows "finite{f. \<forall>x. (x \<in> A \<longrightarrow> f x \<in> B) \<and> (x \<notin> A \<longrightarrow> f x = d)}" (is "finite ?S")
```
```   447 proof-
```
```   448   let ?F = "\<lambda>f. {(a,b). a \<in> A \<and> b = f a}"
```
```   449   have "?F ` ?S \<subseteq> Pow(A \<times> B)" by auto
```
```   450   from finite_subset[OF this] assms have 1: "finite (?F ` ?S)" by simp
```
```   451   have 2: "inj_on ?F ?S"
```
```   452     by(fastforce simp add: inj_on_def set_eq_iff fun_eq_iff)
```
```   453   show ?thesis by(rule finite_imageD[OF 1 2])
```
```   454 qed
```
```   455
```
```   456 subsubsection {* Further induction rules on finite sets *}
```
```   457
```
```   458 lemma finite_ne_induct [case_names singleton insert, consumes 2]:
```
```   459   assumes "finite F" and "F \<noteq> {}"
```
```   460   assumes "\<And>x. P {x}"
```
```   461     and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
```
```   462   shows "P F"
```
```   463 using assms
```
```   464 proof induct
```
```   465   case empty then show ?case by simp
```
```   466 next
```
```   467   case (insert x F) then show ?case by cases auto
```
```   468 qed
```
```   469
```
```   470 lemma finite_subset_induct [consumes 2, case_names empty insert]:
```
```   471   assumes "finite F" and "F \<subseteq> A"
```
```   472   assumes empty: "P {}"
```
```   473     and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
```
```   474   shows "P F"
```
```   475 using `finite F` `F \<subseteq> A`
```
```   476 proof induct
```
```   477   show "P {}" by fact
```
```   478 next
```
```   479   fix x F
```
```   480   assume "finite F" and "x \<notin> F" and
```
```   481     P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
```
```   482   show "P (insert x F)"
```
```   483   proof (rule insert)
```
```   484     from i show "x \<in> A" by blast
```
```   485     from i have "F \<subseteq> A" by blast
```
```   486     with P show "P F" .
```
```   487     show "finite F" by fact
```
```   488     show "x \<notin> F" by fact
```
```   489   qed
```
```   490 qed
```
```   491
```
```   492 lemma finite_empty_induct:
```
```   493   assumes "finite A"
```
```   494   assumes "P A"
```
```   495     and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
```
```   496   shows "P {}"
```
```   497 proof -
```
```   498   have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
```
```   499   proof -
```
```   500     fix B :: "'a set"
```
```   501     assume "B \<subseteq> A"
```
```   502     with `finite A` have "finite B" by (rule rev_finite_subset)
```
```   503     from this `B \<subseteq> A` show "P (A - B)"
```
```   504     proof induct
```
```   505       case empty
```
```   506       from `P A` show ?case by simp
```
```   507     next
```
```   508       case (insert b B)
```
```   509       have "P (A - B - {b})"
```
```   510       proof (rule remove)
```
```   511         from `finite A` show "finite (A - B)" by induct auto
```
```   512         from insert show "b \<in> A - B" by simp
```
```   513         from insert show "P (A - B)" by simp
```
```   514       qed
```
```   515       also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
```
```   516       finally show ?case .
```
```   517     qed
```
```   518   qed
```
```   519   then have "P (A - A)" by blast
```
```   520   then show ?thesis by simp
```
```   521 qed
```
```   522
```
```   523 subsection {* Class @{text finite}  *}
```
```   524
```
```   525 class finite =
```
```   526   assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
```
```   527 begin
```
```   528
```
```   529 lemma finite [simp]: "finite (A \<Colon> 'a set)"
```
```   530   by (rule subset_UNIV finite_UNIV finite_subset)+
```
```   531
```
```   532 lemma finite_code [code]: "finite (A \<Colon> 'a set) \<longleftrightarrow> True"
```
```   533   by simp
```
```   534
```
```   535 end
```
```   536
```
```   537 instance prod :: (finite, finite) finite
```
```   538   by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
```
```   539
```
```   540 lemma inj_graph: "inj (%f. {(x, y). y = f x})"
```
```   541   by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
```
```   542
```
```   543 instance "fun" :: (finite, finite) finite
```
```   544 proof
```
```   545   show "finite (UNIV :: ('a => 'b) set)"
```
```   546   proof (rule finite_imageD)
```
```   547     let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
```
```   548     have "range ?graph \<subseteq> Pow UNIV" by simp
```
```   549     moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
```
```   550       by (simp only: finite_Pow_iff finite)
```
```   551     ultimately show "finite (range ?graph)"
```
```   552       by (rule finite_subset)
```
```   553     show "inj ?graph" by (rule inj_graph)
```
```   554   qed
```
```   555 qed
```
```   556
```
```   557 instance bool :: finite
```
```   558   by default (simp add: UNIV_bool)
```
```   559
```
```   560 instance set :: (finite) finite
```
```   561   by default (simp only: Pow_UNIV [symmetric] finite_Pow_iff finite)
```
```   562
```
```   563 instance unit :: finite
```
```   564   by default (simp add: UNIV_unit)
```
```   565
```
```   566 instance sum :: (finite, finite) finite
```
```   567   by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
```
```   568
```
```   569
```
```   570 subsection {* A basic fold functional for finite sets *}
```
```   571
```
```   572 text {* The intended behaviour is
```
```   573 @{text "fold f z {x\<^sub>1, ..., x\<^sub>n} = f x\<^sub>1 (\<dots> (f x\<^sub>n z)\<dots>)"}
```
```   574 if @{text f} is ``left-commutative'':
```
```   575 *}
```
```   576
```
```   577 locale comp_fun_commute =
```
```   578   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
```
```   579   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
```
```   580 begin
```
```   581
```
```   582 lemma fun_left_comm: "f y (f x z) = f x (f y z)"
```
```   583   using comp_fun_commute by (simp add: fun_eq_iff)
```
```   584
```
```   585 lemma commute_left_comp:
```
```   586   "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
```
```   587   by (simp add: o_assoc comp_fun_commute)
```
```   588
```
```   589 end
```
```   590
```
```   591 inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
```
```   592 for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
```
```   593   emptyI [intro]: "fold_graph f z {} z" |
```
```   594   insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
```
```   595       \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
```
```   596
```
```   597 inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
```
```   598
```
```   599 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
```
```   600   "fold f z A = (if finite A then (THE y. fold_graph f z A y) else z)"
```
```   601
```
```   602 text{*A tempting alternative for the definiens is
```
```   603 @{term "if finite A then THE y. fold_graph f z A y else e"}.
```
```   604 It allows the removal of finiteness assumptions from the theorems
```
```   605 @{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
```
```   606 The proofs become ugly. It is not worth the effort. (???) *}
```
```   607
```
```   608 lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
```
```   609 by (induct rule: finite_induct) auto
```
```   610
```
```   611
```
```   612 subsubsection{*From @{const fold_graph} to @{term fold}*}
```
```   613
```
```   614 context comp_fun_commute
```
```   615 begin
```
```   616
```
```   617 lemma fold_graph_finite:
```
```   618   assumes "fold_graph f z A y"
```
```   619   shows "finite A"
```
```   620   using assms by induct simp_all
```
```   621
```
```   622 lemma fold_graph_insertE_aux:
```
```   623   "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'"
```
```   624 proof (induct set: fold_graph)
```
```   625   case (insertI x A y) show ?case
```
```   626   proof (cases "x = a")
```
```   627     assume "x = a" with insertI show ?case by auto
```
```   628   next
```
```   629     assume "x \<noteq> a"
```
```   630     then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
```
```   631       using insertI by auto
```
```   632     have "f x y = f a (f x y')"
```
```   633       unfolding y by (rule fun_left_comm)
```
```   634     moreover have "fold_graph f z (insert x A - {a}) (f x y')"
```
```   635       using y' and `x \<noteq> a` and `x \<notin> A`
```
```   636       by (simp add: insert_Diff_if fold_graph.insertI)
```
```   637     ultimately show ?case by fast
```
```   638   qed
```
```   639 qed simp
```
```   640
```
```   641 lemma fold_graph_insertE:
```
```   642   assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
```
```   643   obtains y where "v = f x y" and "fold_graph f z A y"
```
```   644 using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
```
```   645
```
```   646 lemma fold_graph_determ:
```
```   647   "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
```
```   648 proof (induct arbitrary: y set: fold_graph)
```
```   649   case (insertI x A y v)
```
```   650   from `fold_graph f z (insert x A) v` and `x \<notin> A`
```
```   651   obtain y' where "v = f x y'" and "fold_graph f z A y'"
```
```   652     by (rule fold_graph_insertE)
```
```   653   from `fold_graph f z A y'` have "y' = y" by (rule insertI)
```
```   654   with `v = f x y'` show "v = f x y" by simp
```
```   655 qed fast
```
```   656
```
```   657 lemma fold_equality:
```
```   658   "fold_graph f z A y \<Longrightarrow> fold f z A = y"
```
```   659   by (cases "finite A") (auto simp add: fold_def intro: fold_graph_determ dest: fold_graph_finite)
```
```   660
```
```   661 lemma fold_graph_fold:
```
```   662   assumes "finite A"
```
```   663   shows "fold_graph f z A (fold f z A)"
```
```   664 proof -
```
```   665   from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
```
```   666   moreover note fold_graph_determ
```
```   667   ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
```
```   668   then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
```
```   669   with assms show ?thesis by (simp add: fold_def)
```
```   670 qed
```
```   671
```
```   672 text {* The base case for @{text fold}: *}
```
```   673
```
```   674 lemma (in -) fold_infinite [simp]:
```
```   675   assumes "\<not> finite A"
```
```   676   shows "fold f z A = z"
```
```   677   using assms by (auto simp add: fold_def)
```
```   678
```
```   679 lemma (in -) fold_empty [simp]:
```
```   680   "fold f z {} = z"
```
```   681   by (auto simp add: fold_def)
```
```   682
```
```   683 text{* The various recursion equations for @{const fold}: *}
```
```   684
```
```   685 lemma fold_insert [simp]:
```
```   686   assumes "finite A" and "x \<notin> A"
```
```   687   shows "fold f z (insert x A) = f x (fold f z A)"
```
```   688 proof (rule fold_equality)
```
```   689   fix z
```
```   690   from `finite A` have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
```
```   691   with `x \<notin> A` have "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)
```
```   692   then show "fold_graph f z (insert x A) (f x (fold f z A))" by simp
```
```   693 qed
```
```   694
```
```   695 declare (in -) empty_fold_graphE [rule del] fold_graph.intros [rule del]
```
```   696   -- {* No more proofs involve these. *}
```
```   697
```
```   698 lemma fold_fun_left_comm:
```
```   699   "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
```
```   700 proof (induct rule: finite_induct)
```
```   701   case empty then show ?case by simp
```
```   702 next
```
```   703   case (insert y A) then show ?case
```
```   704     by (simp add: fun_left_comm [of x])
```
```   705 qed
```
```   706
```
```   707 lemma fold_insert2:
```
```   708   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A)  = fold f (f x z) A"
```
```   709   by (simp add: fold_fun_left_comm)
```
```   710
```
```   711 lemma fold_rec:
```
```   712   assumes "finite A" and "x \<in> A"
```
```   713   shows "fold f z A = f x (fold f z (A - {x}))"
```
```   714 proof -
```
```   715   have A: "A = insert x (A - {x})" using `x \<in> A` by blast
```
```   716   then have "fold f z A = fold f z (insert x (A - {x}))" by simp
```
```   717   also have "\<dots> = f x (fold f z (A - {x}))"
```
```   718     by (rule fold_insert) (simp add: `finite A`)+
```
```   719   finally show ?thesis .
```
```   720 qed
```
```   721
```
```   722 lemma fold_insert_remove:
```
```   723   assumes "finite A"
```
```   724   shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
```
```   725 proof -
```
```   726   from `finite A` have "finite (insert x A)" by auto
```
```   727   moreover have "x \<in> insert x A" by auto
```
```   728   ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
```
```   729     by (rule fold_rec)
```
```   730   then show ?thesis by simp
```
```   731 qed
```
```   732
```
```   733 end
```
```   734
```
```   735 text{* Other properties of @{const fold}: *}
```
```   736
```
```   737 lemma fold_image:
```
```   738   assumes "inj_on g A"
```
```   739   shows "fold f z (g ` A) = fold (f \<circ> g) z A"
```
```   740 proof (cases "finite A")
```
```   741   case False with assms show ?thesis by (auto dest: finite_imageD simp add: fold_def)
```
```   742 next
```
```   743   case True
```
```   744   have "fold_graph f z (g ` A) = fold_graph (f \<circ> g) z A"
```
```   745   proof
```
```   746     fix w
```
```   747     show "fold_graph f z (g ` A) w \<longleftrightarrow> fold_graph (f \<circ> g) z A w" (is "?P \<longleftrightarrow> ?Q")
```
```   748     proof
```
```   749       assume ?P then show ?Q using assms
```
```   750       proof (induct "g ` A" w arbitrary: A)
```
```   751         case emptyI then show ?case by (auto intro: fold_graph.emptyI)
```
```   752       next
```
```   753         case (insertI x A r B)
```
```   754         from `inj_on g B` `x \<notin> A` `insert x A = image g B` obtain x' A' where
```
```   755           "x' \<notin> A'" and [simp]: "B = insert x' A'" "x = g x'" "A = g ` A'"
```
```   756           by (rule inj_img_insertE)
```
```   757         from insertI.prems have "fold_graph (f o g) z A' r"
```
```   758           by (auto intro: insertI.hyps)
```
```   759         with `x' \<notin> A'` have "fold_graph (f \<circ> g) z (insert x' A') ((f \<circ> g) x' r)"
```
```   760           by (rule fold_graph.insertI)
```
```   761         then show ?case by simp
```
```   762       qed
```
```   763     next
```
```   764       assume ?Q then show ?P using assms
```
```   765       proof induct
```
```   766         case emptyI thus ?case by (auto intro: fold_graph.emptyI)
```
```   767       next
```
```   768         case (insertI x A r)
```
```   769         from `x \<notin> A` insertI.prems have "g x \<notin> g ` A" by auto
```
```   770         moreover from insertI have "fold_graph f z (g ` A) r" by simp
```
```   771         ultimately have "fold_graph f z (insert (g x) (g ` A)) (f (g x) r)"
```
```   772           by (rule fold_graph.insertI)
```
```   773         then show ?case by simp
```
```   774       qed
```
```   775     qed
```
```   776   qed
```
```   777   with True assms show ?thesis by (auto simp add: fold_def)
```
```   778 qed
```
```   779
```
```   780 lemma fold_cong:
```
```   781   assumes "comp_fun_commute f" "comp_fun_commute g"
```
```   782   assumes "finite A" and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
```
```   783     and "s = t" and "A = B"
```
```   784   shows "fold f s A = fold g t B"
```
```   785 proof -
```
```   786   have "fold f s A = fold g s A"
```
```   787   using `finite A` cong proof (induct A)
```
```   788     case empty then show ?case by simp
```
```   789   next
```
```   790     case (insert x A)
```
```   791     interpret f: comp_fun_commute f by (fact `comp_fun_commute f`)
```
```   792     interpret g: comp_fun_commute g by (fact `comp_fun_commute g`)
```
```   793     from insert show ?case by simp
```
```   794   qed
```
```   795   with assms show ?thesis by simp
```
```   796 qed
```
```   797
```
```   798
```
```   799 text {* A simplified version for idempotent functions: *}
```
```   800
```
```   801 locale comp_fun_idem = comp_fun_commute +
```
```   802   assumes comp_fun_idem: "f x \<circ> f x = f x"
```
```   803 begin
```
```   804
```
```   805 lemma fun_left_idem: "f x (f x z) = f x z"
```
```   806   using comp_fun_idem by (simp add: fun_eq_iff)
```
```   807
```
```   808 lemma fold_insert_idem:
```
```   809   assumes fin: "finite A"
```
```   810   shows "fold f z (insert x A)  = f x (fold f z A)"
```
```   811 proof cases
```
```   812   assume "x \<in> A"
```
```   813   then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
```
```   814   then show ?thesis using assms by (simp add: comp_fun_idem fun_left_idem)
```
```   815 next
```
```   816   assume "x \<notin> A" then show ?thesis using assms by simp
```
```   817 qed
```
```   818
```
```   819 declare fold_insert [simp del] fold_insert_idem [simp]
```
```   820
```
```   821 lemma fold_insert_idem2:
```
```   822   "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
```
```   823   by (simp add: fold_fun_left_comm)
```
```   824
```
```   825 end
```
```   826
```
```   827
```
```   828 subsubsection {* Liftings to @{text comp_fun_commute} etc. *}
```
```   829
```
```   830 lemma (in comp_fun_commute) comp_comp_fun_commute:
```
```   831   "comp_fun_commute (f \<circ> g)"
```
```   832 proof
```
```   833 qed (simp_all add: comp_fun_commute)
```
```   834
```
```   835 lemma (in comp_fun_idem) comp_comp_fun_idem:
```
```   836   "comp_fun_idem (f \<circ> g)"
```
```   837   by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
```
```   838     (simp_all add: comp_fun_idem)
```
```   839
```
```   840 lemma (in comp_fun_commute) comp_fun_commute_funpow:
```
```   841   "comp_fun_commute (\<lambda>x. f x ^^ g x)"
```
```   842 proof
```
```   843   fix y x
```
```   844   show "f y ^^ g y \<circ> f x ^^ g x = f x ^^ g x \<circ> f y ^^ g y"
```
```   845   proof (cases "x = y")
```
```   846     case True then show ?thesis by simp
```
```   847   next
```
```   848     case False show ?thesis
```
```   849     proof (induct "g x" arbitrary: g)
```
```   850       case 0 then show ?case by simp
```
```   851     next
```
```   852       case (Suc n g)
```
```   853       have hyp1: "f y ^^ g y \<circ> f x = f x \<circ> f y ^^ g y"
```
```   854       proof (induct "g y" arbitrary: g)
```
```   855         case 0 then show ?case by simp
```
```   856       next
```
```   857         case (Suc n g)
```
```   858         def h \<equiv> "\<lambda>z. g z - 1"
```
```   859         with Suc have "n = h y" by simp
```
```   860         with Suc have hyp: "f y ^^ h y \<circ> f x = f x \<circ> f y ^^ h y"
```
```   861           by auto
```
```   862         from Suc h_def have "g y = Suc (h y)" by simp
```
```   863         then show ?case by (simp add: comp_assoc hyp)
```
```   864           (simp add: o_assoc comp_fun_commute)
```
```   865       qed
```
```   866       def h \<equiv> "\<lambda>z. if z = x then g x - 1 else g z"
```
```   867       with Suc have "n = h x" by simp
```
```   868       with Suc have "f y ^^ h y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ h y"
```
```   869         by auto
```
```   870       with False h_def have hyp2: "f y ^^ g y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ g y" by simp
```
```   871       from Suc h_def have "g x = Suc (h x)" by simp
```
```   872       then show ?case by (simp del: funpow.simps add: funpow_Suc_right o_assoc hyp2)
```
```   873         (simp add: comp_assoc hyp1)
```
```   874     qed
```
```   875   qed
```
```   876 qed
```
```   877
```
```   878
```
```   879 subsubsection {* Expressing set operations via @{const fold} *}
```
```   880
```
```   881 lemma comp_fun_commute_const:
```
```   882   "comp_fun_commute (\<lambda>_. f)"
```
```   883 proof
```
```   884 qed rule
```
```   885
```
```   886 lemma comp_fun_idem_insert:
```
```   887   "comp_fun_idem insert"
```
```   888 proof
```
```   889 qed auto
```
```   890
```
```   891 lemma comp_fun_idem_remove:
```
```   892   "comp_fun_idem Set.remove"
```
```   893 proof
```
```   894 qed auto
```
```   895
```
```   896 lemma (in semilattice_inf) comp_fun_idem_inf:
```
```   897   "comp_fun_idem inf"
```
```   898 proof
```
```   899 qed (auto simp add: inf_left_commute)
```
```   900
```
```   901 lemma (in semilattice_sup) comp_fun_idem_sup:
```
```   902   "comp_fun_idem sup"
```
```   903 proof
```
```   904 qed (auto simp add: sup_left_commute)
```
```   905
```
```   906 lemma union_fold_insert:
```
```   907   assumes "finite A"
```
```   908   shows "A \<union> B = fold insert B A"
```
```   909 proof -
```
```   910   interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
```
```   911   from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
```
```   912 qed
```
```   913
```
```   914 lemma minus_fold_remove:
```
```   915   assumes "finite A"
```
```   916   shows "B - A = fold Set.remove B A"
```
```   917 proof -
```
```   918   interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove)
```
```   919   from `finite A` have "fold Set.remove B A = B - A" by (induct A arbitrary: B) auto
```
```   920   then show ?thesis ..
```
```   921 qed
```
```   922
```
```   923 lemma comp_fun_commute_filter_fold:
```
```   924   "comp_fun_commute (\<lambda>x A'. if P x then Set.insert x A' else A')"
```
```   925 proof -
```
```   926   interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
```
```   927   show ?thesis by default (auto simp: fun_eq_iff)
```
```   928 qed
```
```   929
```
```   930 lemma Set_filter_fold:
```
```   931   assumes "finite A"
```
```   932   shows "Set.filter P A = fold (\<lambda>x A'. if P x then Set.insert x A' else A') {} A"
```
```   933 using assms
```
```   934 by (induct A)
```
```   935   (auto simp add: Set.filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold])
```
```   936
```
```   937 lemma inter_Set_filter:
```
```   938   assumes "finite B"
```
```   939   shows "A \<inter> B = Set.filter (\<lambda>x. x \<in> A) B"
```
```   940 using assms
```
```   941 by (induct B) (auto simp: Set.filter_def)
```
```   942
```
```   943 lemma image_fold_insert:
```
```   944   assumes "finite A"
```
```   945   shows "image f A = fold (\<lambda>k A. Set.insert (f k) A) {} A"
```
```   946 using assms
```
```   947 proof -
```
```   948   interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A" by default auto
```
```   949   show ?thesis using assms by (induct A) auto
```
```   950 qed
```
```   951
```
```   952 lemma Ball_fold:
```
```   953   assumes "finite A"
```
```   954   shows "Ball A P = fold (\<lambda>k s. s \<and> P k) True A"
```
```   955 using assms
```
```   956 proof -
```
```   957   interpret comp_fun_commute "\<lambda>k s. s \<and> P k" by default auto
```
```   958   show ?thesis using assms by (induct A) auto
```
```   959 qed
```
```   960
```
```   961 lemma Bex_fold:
```
```   962   assumes "finite A"
```
```   963   shows "Bex A P = fold (\<lambda>k s. s \<or> P k) False A"
```
```   964 using assms
```
```   965 proof -
```
```   966   interpret comp_fun_commute "\<lambda>k s. s \<or> P k" by default auto
```
```   967   show ?thesis using assms by (induct A) auto
```
```   968 qed
```
```   969
```
```   970 lemma comp_fun_commute_Pow_fold:
```
```   971   "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)"
```
```   972   by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast
```
```   973
```
```   974 lemma Pow_fold:
```
```   975   assumes "finite A"
```
```   976   shows "Pow A = fold (\<lambda>x A. A \<union> Set.insert x ` A) {{}} A"
```
```   977 using assms
```
```   978 proof -
```
```   979   interpret comp_fun_commute "\<lambda>x A. A \<union> Set.insert x ` A" by (rule comp_fun_commute_Pow_fold)
```
```   980   show ?thesis using assms by (induct A) (auto simp: Pow_insert)
```
```   981 qed
```
```   982
```
```   983 lemma fold_union_pair:
```
```   984   assumes "finite B"
```
```   985   shows "(\<Union>y\<in>B. {(x, y)}) \<union> A = fold (\<lambda>y. Set.insert (x, y)) A B"
```
```   986 proof -
```
```   987   interpret comp_fun_commute "\<lambda>y. Set.insert (x, y)" by default auto
```
```   988   show ?thesis using assms  by (induct B arbitrary: A) simp_all
```
```   989 qed
```
```   990
```
```   991 lemma comp_fun_commute_product_fold:
```
```   992   assumes "finite B"
```
```   993   shows "comp_fun_commute (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B)"
```
```   994 by default (auto simp: fold_union_pair[symmetric] assms)
```
```   995
```
```   996 lemma product_fold:
```
```   997   assumes "finite A"
```
```   998   assumes "finite B"
```
```   999   shows "A \<times> B = fold (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B) {} A"
```
```  1000 using assms unfolding Sigma_def
```
```  1001 by (induct A)
```
```  1002   (simp_all add: comp_fun_commute.fold_insert[OF comp_fun_commute_product_fold] fold_union_pair)
```
```  1003
```
```  1004
```
```  1005 context complete_lattice
```
```  1006 begin
```
```  1007
```
```  1008 lemma inf_Inf_fold_inf:
```
```  1009   assumes "finite A"
```
```  1010   shows "inf (Inf A) B = fold inf B A"
```
```  1011 proof -
```
```  1012   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
```
```  1013   from `finite A` fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
```
```  1014     (simp_all add: inf_commute fun_eq_iff)
```
```  1015 qed
```
```  1016
```
```  1017 lemma sup_Sup_fold_sup:
```
```  1018   assumes "finite A"
```
```  1019   shows "sup (Sup A) B = fold sup B A"
```
```  1020 proof -
```
```  1021   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
```
```  1022   from `finite A` fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
```
```  1023     (simp_all add: sup_commute fun_eq_iff)
```
```  1024 qed
```
```  1025
```
```  1026 lemma Inf_fold_inf:
```
```  1027   assumes "finite A"
```
```  1028   shows "Inf A = fold inf top A"
```
```  1029   using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
```
```  1030
```
```  1031 lemma Sup_fold_sup:
```
```  1032   assumes "finite A"
```
```  1033   shows "Sup A = fold sup bot A"
```
```  1034   using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
```
```  1035
```
```  1036 lemma inf_INF_fold_inf:
```
```  1037   assumes "finite A"
```
```  1038   shows "inf B (INFI A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold")
```
```  1039 proof (rule sym)
```
```  1040   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
```
```  1041   interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
```
```  1042   from `finite A` show "?fold = ?inf"
```
```  1043     by (induct A arbitrary: B)
```
```  1044       (simp_all add: inf_left_commute)
```
```  1045 qed
```
```  1046
```
```  1047 lemma sup_SUP_fold_sup:
```
```  1048   assumes "finite A"
```
```  1049   shows "sup B (SUPR A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold")
```
```  1050 proof (rule sym)
```
```  1051   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
```
```  1052   interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
```
```  1053   from `finite A` show "?fold = ?sup"
```
```  1054     by (induct A arbitrary: B)
```
```  1055       (simp_all add: sup_left_commute)
```
```  1056 qed
```
```  1057
```
```  1058 lemma INF_fold_inf:
```
```  1059   assumes "finite A"
```
```  1060   shows "INFI A f = fold (inf \<circ> f) top A"
```
```  1061   using assms inf_INF_fold_inf [of A top] by simp
```
```  1062
```
```  1063 lemma SUP_fold_sup:
```
```  1064   assumes "finite A"
```
```  1065   shows "SUPR A f = fold (sup \<circ> f) bot A"
```
```  1066   using assms sup_SUP_fold_sup [of A bot] by simp
```
```  1067
```
```  1068 end
```
```  1069
```
```  1070
```
```  1071 subsection {* Locales as mini-packages for fold operations *}
```
```  1072
```
```  1073 subsubsection {* The natural case *}
```
```  1074
```
```  1075 locale folding =
```
```  1076   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
```
```  1077   fixes z :: "'b"
```
```  1078   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
```
```  1079 begin
```
```  1080
```
```  1081 interpretation fold?: comp_fun_commute f
```
```  1082   by default (insert comp_fun_commute, simp add: fun_eq_iff)
```
```  1083
```
```  1084 definition F :: "'a set \<Rightarrow> 'b"
```
```  1085 where
```
```  1086   eq_fold: "F A = fold f z A"
```
```  1087
```
```  1088 lemma empty [simp]:
```
```  1089   "F {} = z"
```
```  1090   by (simp add: eq_fold)
```
```  1091
```
```  1092 lemma infinite [simp]:
```
```  1093   "\<not> finite A \<Longrightarrow> F A = z"
```
```  1094   by (simp add: eq_fold)
```
```  1095
```
```  1096 lemma insert [simp]:
```
```  1097   assumes "finite A" and "x \<notin> A"
```
```  1098   shows "F (insert x A) = f x (F A)"
```
```  1099 proof -
```
```  1100   from fold_insert assms
```
```  1101   have "fold f z (insert x A) = f x (fold f z A)" by simp
```
```  1102   with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
```
```  1103 qed
```
```  1104
```
```  1105 lemma remove:
```
```  1106   assumes "finite A" and "x \<in> A"
```
```  1107   shows "F A = f x (F (A - {x}))"
```
```  1108 proof -
```
```  1109   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
```
```  1110     by (auto dest: mk_disjoint_insert)
```
```  1111   moreover from `finite A` A have "finite B" by simp
```
```  1112   ultimately show ?thesis by simp
```
```  1113 qed
```
```  1114
```
```  1115 lemma insert_remove:
```
```  1116   assumes "finite A"
```
```  1117   shows "F (insert x A) = f x (F (A - {x}))"
```
```  1118   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
```
```  1119
```
```  1120 end
```
```  1121
```
```  1122
```
```  1123 subsubsection {* With idempotency *}
```
```  1124
```
```  1125 locale folding_idem = folding +
```
```  1126   assumes comp_fun_idem: "f x \<circ> f x = f x"
```
```  1127 begin
```
```  1128
```
```  1129 declare insert [simp del]
```
```  1130
```
```  1131 interpretation fold?: comp_fun_idem f
```
```  1132   by default (insert comp_fun_commute comp_fun_idem, simp add: fun_eq_iff)
```
```  1133
```
```  1134 lemma insert_idem [simp]:
```
```  1135   assumes "finite A"
```
```  1136   shows "F (insert x A) = f x (F A)"
```
```  1137 proof -
```
```  1138   from fold_insert_idem assms
```
```  1139   have "fold f z (insert x A) = f x (fold f z A)" by simp
```
```  1140   with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
```
```  1141 qed
```
```  1142
```
```  1143 end
```
```  1144
```
```  1145
```
```  1146 subsection {* Finite cardinality *}
```
```  1147
```
```  1148 text {*
```
```  1149   The traditional definition
```
```  1150   @{prop "card A \<equiv> LEAST n. EX f. A = {f i | i. i < n}"}
```
```  1151   is ugly to work with.
```
```  1152   But now that we have @{const fold} things are easy:
```
```  1153 *}
```
```  1154
```
```  1155 definition card :: "'a set \<Rightarrow> nat" where
```
```  1156   "card = folding.F (\<lambda>_. Suc) 0"
```
```  1157
```
```  1158 interpretation card!: folding "\<lambda>_. Suc" 0
```
```  1159 where
```
```  1160   "folding.F (\<lambda>_. Suc) 0 = card"
```
```  1161 proof -
```
```  1162   show "folding (\<lambda>_. Suc)" by default rule
```
```  1163   then interpret card!: folding "\<lambda>_. Suc" 0 .
```
```  1164   from card_def show "folding.F (\<lambda>_. Suc) 0 = card" by rule
```
```  1165 qed
```
```  1166
```
```  1167 lemma card_infinite:
```
```  1168   "\<not> finite A \<Longrightarrow> card A = 0"
```
```  1169   by (fact card.infinite)
```
```  1170
```
```  1171 lemma card_empty:
```
```  1172   "card {} = 0"
```
```  1173   by (fact card.empty)
```
```  1174
```
```  1175 lemma card_insert_disjoint:
```
```  1176   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> card (insert x A) = Suc (card A)"
```
```  1177   by (fact card.insert)
```
```  1178
```
```  1179 lemma card_insert_if:
```
```  1180   "finite A \<Longrightarrow> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
```
```  1181   by auto (simp add: card.insert_remove card.remove)
```
```  1182
```
```  1183 lemma card_ge_0_finite:
```
```  1184   "card A > 0 \<Longrightarrow> finite A"
```
```  1185   by (rule ccontr) simp
```
```  1186
```
```  1187 lemma card_0_eq [simp]:
```
```  1188   "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
```
```  1189   by (auto dest: mk_disjoint_insert)
```
```  1190
```
```  1191 lemma finite_UNIV_card_ge_0:
```
```  1192   "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
```
```  1193   by (rule ccontr) simp
```
```  1194
```
```  1195 lemma card_eq_0_iff:
```
```  1196   "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
```
```  1197   by auto
```
```  1198
```
```  1199 lemma card_gt_0_iff:
```
```  1200   "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
```
```  1201   by (simp add: neq0_conv [symmetric] card_eq_0_iff)
```
```  1202
```
```  1203 lemma card_Suc_Diff1:
```
```  1204   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> Suc (card (A - {x})) = card A"
```
```  1205 apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
```
```  1206 apply(simp del:insert_Diff_single)
```
```  1207 done
```
```  1208
```
```  1209 lemma card_Diff_singleton:
```
```  1210   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) = card A - 1"
```
```  1211   by (simp add: card_Suc_Diff1 [symmetric])
```
```  1212
```
```  1213 lemma card_Diff_singleton_if:
```
```  1214   "finite A \<Longrightarrow> card (A - {x}) = (if x \<in> A then card A - 1 else card A)"
```
```  1215   by (simp add: card_Diff_singleton)
```
```  1216
```
```  1217 lemma card_Diff_insert[simp]:
```
```  1218   assumes "finite A" and "a \<in> A" and "a \<notin> B"
```
```  1219   shows "card (A - insert a B) = card (A - B) - 1"
```
```  1220 proof -
```
```  1221   have "A - insert a B = (A - B) - {a}" using assms by blast
```
```  1222   then show ?thesis using assms by(simp add: card_Diff_singleton)
```
```  1223 qed
```
```  1224
```
```  1225 lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
```
```  1226   by (fact card.insert_remove)
```
```  1227
```
```  1228 lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
```
```  1229 by (simp add: card_insert_if)
```
```  1230
```
```  1231 lemma card_Collect_less_nat[simp]: "card{i::nat. i < n} = n"
```
```  1232 by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
```
```  1233
```
```  1234 lemma card_Collect_le_nat[simp]: "card{i::nat. i <= n} = Suc n"
```
```  1235 using card_Collect_less_nat[of "Suc n"] by(simp add: less_Suc_eq_le)
```
```  1236
```
```  1237 lemma card_mono:
```
```  1238   assumes "finite B" and "A \<subseteq> B"
```
```  1239   shows "card A \<le> card B"
```
```  1240 proof -
```
```  1241   from assms have "finite A" by (auto intro: finite_subset)
```
```  1242   then show ?thesis using assms proof (induct A arbitrary: B)
```
```  1243     case empty then show ?case by simp
```
```  1244   next
```
```  1245     case (insert x A)
```
```  1246     then have "x \<in> B" by simp
```
```  1247     from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
```
```  1248     with insert.hyps have "card A \<le> card (B - {x})" by auto
```
```  1249     with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
```
```  1250   qed
```
```  1251 qed
```
```  1252
```
```  1253 lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
```
```  1254 apply (induct rule: finite_induct)
```
```  1255 apply simp
```
```  1256 apply clarify
```
```  1257 apply (subgoal_tac "finite A & A - {x} <= F")
```
```  1258  prefer 2 apply (blast intro: finite_subset, atomize)
```
```  1259 apply (drule_tac x = "A - {x}" in spec)
```
```  1260 apply (simp add: card_Diff_singleton_if split add: split_if_asm)
```
```  1261 apply (case_tac "card A", auto)
```
```  1262 done
```
```  1263
```
```  1264 lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
```
```  1265 apply (simp add: psubset_eq linorder_not_le [symmetric])
```
```  1266 apply (blast dest: card_seteq)
```
```  1267 done
```
```  1268
```
```  1269 lemma card_Un_Int:
```
```  1270   assumes "finite A" and "finite B"
```
```  1271   shows "card A + card B = card (A \<union> B) + card (A \<inter> B)"
```
```  1272 using assms proof (induct A)
```
```  1273   case empty then show ?case by simp
```
```  1274 next
```
```  1275  case (insert x A) then show ?case
```
```  1276     by (auto simp add: insert_absorb Int_insert_left)
```
```  1277 qed
```
```  1278
```
```  1279 lemma card_Un_disjoint:
```
```  1280   assumes "finite A" and "finite B"
```
```  1281   assumes "A \<inter> B = {}"
```
```  1282   shows "card (A \<union> B) = card A + card B"
```
```  1283 using assms card_Un_Int [of A B] by simp
```
```  1284
```
```  1285 lemma card_Diff_subset:
```
```  1286   assumes "finite B" and "B \<subseteq> A"
```
```  1287   shows "card (A - B) = card A - card B"
```
```  1288 proof (cases "finite A")
```
```  1289   case False with assms show ?thesis by simp
```
```  1290 next
```
```  1291   case True with assms show ?thesis by (induct B arbitrary: A) simp_all
```
```  1292 qed
```
```  1293
```
```  1294 lemma card_Diff_subset_Int:
```
```  1295   assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
```
```  1296 proof -
```
```  1297   have "A - B = A - A \<inter> B" by auto
```
```  1298   thus ?thesis
```
```  1299     by (simp add: card_Diff_subset AB)
```
```  1300 qed
```
```  1301
```
```  1302 lemma diff_card_le_card_Diff:
```
```  1303 assumes "finite B" shows "card A - card B \<le> card(A - B)"
```
```  1304 proof-
```
```  1305   have "card A - card B \<le> card A - card (A \<inter> B)"
```
```  1306     using card_mono[OF assms Int_lower2, of A] by arith
```
```  1307   also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
```
```  1308   finally show ?thesis .
```
```  1309 qed
```
```  1310
```
```  1311 lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
```
```  1312 apply (rule Suc_less_SucD)
```
```  1313 apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
```
```  1314 done
```
```  1315
```
```  1316 lemma card_Diff2_less:
```
```  1317   "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
```
```  1318 apply (case_tac "x = y")
```
```  1319  apply (simp add: card_Diff1_less del:card_Diff_insert)
```
```  1320 apply (rule less_trans)
```
```  1321  prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
```
```  1322 done
```
```  1323
```
```  1324 lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
```
```  1325 apply (case_tac "x : A")
```
```  1326  apply (simp_all add: card_Diff1_less less_imp_le)
```
```  1327 done
```
```  1328
```
```  1329 lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
```
```  1330 by (erule psubsetI, blast)
```
```  1331
```
```  1332 lemma card_le_inj:
```
```  1333   assumes fA: "finite A"
```
```  1334     and fB: "finite B"
```
```  1335     and c: "card A \<le> card B"
```
```  1336   shows "\<exists>f. f ` A \<subseteq> B \<and> inj_on f A"
```
```  1337   using fA fB c
```
```  1338 proof (induct arbitrary: B rule: finite_induct)
```
```  1339   case empty
```
```  1340   then show ?case by simp
```
```  1341 next
```
```  1342   case (insert x s t)
```
```  1343   then show ?case
```
```  1344   proof (induct rule: finite_induct[OF "insert.prems"(1)])
```
```  1345     case 1
```
```  1346     then show ?case by simp
```
```  1347   next
```
```  1348     case (2 y t)
```
```  1349     from "2.prems"(1,2,5) "2.hyps"(1,2) have cst: "card s \<le> card t"
```
```  1350       by simp
```
```  1351     from "2.prems"(3) [OF "2.hyps"(1) cst]
```
```  1352     obtain f where "f ` s \<subseteq> t" "inj_on f s"
```
```  1353       by blast
```
```  1354     with "2.prems"(2) "2.hyps"(2) show ?case
```
```  1355       apply -
```
```  1356       apply (rule exI[where x = "\<lambda>z. if z = x then y else f z"])
```
```  1357       apply (auto simp add: inj_on_def)
```
```  1358       done
```
```  1359   qed
```
```  1360 qed
```
```  1361
```
```  1362 lemma card_subset_eq:
```
```  1363   assumes fB: "finite B"
```
```  1364     and AB: "A \<subseteq> B"
```
```  1365     and c: "card A = card B"
```
```  1366   shows "A = B"
```
```  1367 proof -
```
```  1368   from fB AB have fA: "finite A"
```
```  1369     by (auto intro: finite_subset)
```
```  1370   from fA fB have fBA: "finite (B - A)"
```
```  1371     by auto
```
```  1372   have e: "A \<inter> (B - A) = {}"
```
```  1373     by blast
```
```  1374   have eq: "A \<union> (B - A) = B"
```
```  1375     using AB by blast
```
```  1376   from card_Un_disjoint[OF fA fBA e, unfolded eq c] have "card (B - A) = 0"
```
```  1377     by arith
```
```  1378   then have "B - A = {}"
```
```  1379     unfolding card_eq_0_iff using fA fB by simp
```
```  1380   with AB show "A = B"
```
```  1381     by blast
```
```  1382 qed
```
```  1383
```
```  1384 lemma insert_partition:
```
```  1385   "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
```
```  1386   \<Longrightarrow> x \<inter> \<Union> F = {}"
```
```  1387 by auto
```
```  1388
```
```  1389 lemma finite_psubset_induct[consumes 1, case_names psubset]:
```
```  1390   assumes fin: "finite A"
```
```  1391   and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A"
```
```  1392   shows "P A"
```
```  1393 using fin
```
```  1394 proof (induct A taking: card rule: measure_induct_rule)
```
```  1395   case (less A)
```
```  1396   have fin: "finite A" by fact
```
```  1397   have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
```
```  1398   { fix B
```
```  1399     assume asm: "B \<subset> A"
```
```  1400     from asm have "card B < card A" using psubset_card_mono fin by blast
```
```  1401     moreover
```
```  1402     from asm have "B \<subseteq> A" by auto
```
```  1403     then have "finite B" using fin finite_subset by blast
```
```  1404     ultimately
```
```  1405     have "P B" using ih by simp
```
```  1406   }
```
```  1407   with fin show "P A" using major by blast
```
```  1408 qed
```
```  1409
```
```  1410 lemma finite_induct_select[consumes 1, case_names empty select]:
```
```  1411   assumes "finite S"
```
```  1412   assumes "P {}"
```
```  1413   assumes select: "\<And>T. T \<subset> S \<Longrightarrow> P T \<Longrightarrow> \<exists>s\<in>S - T. P (insert s T)"
```
```  1414   shows "P S"
```
```  1415 proof -
```
```  1416   have "0 \<le> card S" by simp
```
```  1417   then have "\<exists>T \<subseteq> S. card T = card S \<and> P T"
```
```  1418   proof (induct rule: dec_induct)
```
```  1419     case base with `P {}` show ?case
```
```  1420       by (intro exI[of _ "{}"]) auto
```
```  1421   next
```
```  1422     case (step n)
```
```  1423     then obtain T where T: "T \<subseteq> S" "card T = n" "P T"
```
```  1424       by auto
```
```  1425     with `n < card S` have "T \<subset> S" "P T"
```
```  1426       by auto
```
```  1427     with select[of T] obtain s where "s \<in> S" "s \<notin> T" "P (insert s T)"
```
```  1428       by auto
```
```  1429     with step(2) T `finite S` show ?case
```
```  1430       by (intro exI[of _ "insert s T"]) (auto dest: finite_subset)
```
```  1431   qed
```
```  1432   with `finite S` show "P S"
```
```  1433     by (auto dest: card_subset_eq)
```
```  1434 qed
```
```  1435
```
```  1436 text{* main cardinality theorem *}
```
```  1437 lemma card_partition [rule_format]:
```
```  1438   "finite C ==>
```
```  1439      finite (\<Union> C) -->
```
```  1440      (\<forall>c\<in>C. card c = k) -->
```
```  1441      (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
```
```  1442      k * card(C) = card (\<Union> C)"
```
```  1443 apply (erule finite_induct, simp)
```
```  1444 apply (simp add: card_Un_disjoint insert_partition
```
```  1445        finite_subset [of _ "\<Union> (insert x F)"])
```
```  1446 done
```
```  1447
```
```  1448 lemma card_eq_UNIV_imp_eq_UNIV:
```
```  1449   assumes fin: "finite (UNIV :: 'a set)"
```
```  1450   and card: "card A = card (UNIV :: 'a set)"
```
```  1451   shows "A = (UNIV :: 'a set)"
```
```  1452 proof
```
```  1453   show "A \<subseteq> UNIV" by simp
```
```  1454   show "UNIV \<subseteq> A"
```
```  1455   proof
```
```  1456     fix x
```
```  1457     show "x \<in> A"
```
```  1458     proof (rule ccontr)
```
```  1459       assume "x \<notin> A"
```
```  1460       then have "A \<subset> UNIV" by auto
```
```  1461       with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
```
```  1462       with card show False by simp
```
```  1463     qed
```
```  1464   qed
```
```  1465 qed
```
```  1466
```
```  1467 text{*The form of a finite set of given cardinality*}
```
```  1468
```
```  1469 lemma card_eq_SucD:
```
```  1470 assumes "card A = Suc k"
```
```  1471 shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
```
```  1472 proof -
```
```  1473   have fin: "finite A" using assms by (auto intro: ccontr)
```
```  1474   moreover have "card A \<noteq> 0" using assms by auto
```
```  1475   ultimately obtain b where b: "b \<in> A" by auto
```
```  1476   show ?thesis
```
```  1477   proof (intro exI conjI)
```
```  1478     show "A = insert b (A-{b})" using b by blast
```
```  1479     show "b \<notin> A - {b}" by blast
```
```  1480     show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
```
```  1481       using assms b fin by(fastforce dest:mk_disjoint_insert)+
```
```  1482   qed
```
```  1483 qed
```
```  1484
```
```  1485 lemma card_Suc_eq:
```
```  1486   "(card A = Suc k) =
```
```  1487    (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
```
```  1488  apply(auto elim!: card_eq_SucD)
```
```  1489  apply(subst card.insert)
```
```  1490  apply(auto simp add: intro:ccontr)
```
```  1491  done
```
```  1492
```
```  1493 lemma card_le_Suc_iff: "finite A \<Longrightarrow>
```
```  1494   Suc n \<le> card A = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
```
```  1495 by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
```
```  1496   dest: subset_singletonD split: nat.splits if_splits)
```
```  1497
```
```  1498 lemma finite_fun_UNIVD2:
```
```  1499   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
```
```  1500   shows "finite (UNIV :: 'b set)"
```
```  1501 proof -
```
```  1502   from fin have "\<And>arbitrary. finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
```
```  1503     by (rule finite_imageI)
```
```  1504   moreover have "\<And>arbitrary. UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
```
```  1505     by (rule UNIV_eq_I) auto
```
```  1506   ultimately show "finite (UNIV :: 'b set)" by simp
```
```  1507 qed
```
```  1508
```
```  1509 lemma card_UNIV_unit [simp]: "card (UNIV :: unit set) = 1"
```
```  1510   unfolding UNIV_unit by simp
```
```  1511
```
```  1512
```
```  1513 subsubsection {* Cardinality of image *}
```
```  1514
```
```  1515 lemma card_image_le: "finite A ==> card (f ` A) \<le> card A"
```
```  1516   by (induct rule: finite_induct) (simp_all add: le_SucI card_insert_if)
```
```  1517
```
```  1518 lemma card_image:
```
```  1519   assumes "inj_on f A"
```
```  1520   shows "card (f ` A) = card A"
```
```  1521 proof (cases "finite A")
```
```  1522   case True then show ?thesis using assms by (induct A) simp_all
```
```  1523 next
```
```  1524   case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
```
```  1525   with False show ?thesis by simp
```
```  1526 qed
```
```  1527
```
```  1528 lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
```
```  1529 by(auto simp: card_image bij_betw_def)
```
```  1530
```
```  1531 lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
```
```  1532 by (simp add: card_seteq card_image)
```
```  1533
```
```  1534 lemma eq_card_imp_inj_on:
```
```  1535   assumes "finite A" "card(f ` A) = card A" shows "inj_on f A"
```
```  1536 using assms
```
```  1537 proof (induct rule:finite_induct)
```
```  1538   case empty show ?case by simp
```
```  1539 next
```
```  1540   case (insert x A)
```
```  1541   then show ?case using card_image_le [of A f]
```
```  1542     by (simp add: card_insert_if split: if_splits)
```
```  1543 qed
```
```  1544
```
```  1545 lemma inj_on_iff_eq_card: "finite A \<Longrightarrow> inj_on f A \<longleftrightarrow> card(f ` A) = card A"
```
```  1546   by (blast intro: card_image eq_card_imp_inj_on)
```
```  1547
```
```  1548 lemma card_inj_on_le:
```
```  1549   assumes "inj_on f A" "f ` A \<subseteq> B" "finite B" shows "card A \<le> card B"
```
```  1550 proof -
```
```  1551   have "finite A" using assms
```
```  1552     by (blast intro: finite_imageD dest: finite_subset)
```
```  1553   then show ?thesis using assms
```
```  1554    by (force intro: card_mono simp: card_image [symmetric])
```
```  1555 qed
```
```  1556
```
```  1557 lemma card_bij_eq:
```
```  1558   "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
```
```  1559      finite A; finite B |] ==> card A = card B"
```
```  1560 by (auto intro: le_antisym card_inj_on_le)
```
```  1561
```
```  1562 lemma bij_betw_finite:
```
```  1563   assumes "bij_betw f A B"
```
```  1564   shows "finite A \<longleftrightarrow> finite B"
```
```  1565 using assms unfolding bij_betw_def
```
```  1566 using finite_imageD[of f A] by auto
```
```  1567
```
```  1568 lemma inj_on_finite:
```
```  1569 assumes "inj_on f A" "f ` A \<le> B" "finite B"
```
```  1570 shows "finite A"
```
```  1571 using assms finite_imageD finite_subset by blast
```
```  1572
```
```  1573
```
```  1574 subsubsection {* Pigeonhole Principles *}
```
```  1575
```
```  1576 lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
```
```  1577 by (auto dest: card_image less_irrefl_nat)
```
```  1578
```
```  1579 lemma pigeonhole_infinite:
```
```  1580 assumes  "~ finite A" and "finite(f`A)"
```
```  1581 shows "EX a0:A. ~finite{a:A. f a = f a0}"
```
```  1582 proof -
```
```  1583   have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
```
```  1584   proof(induct "f`A" arbitrary: A rule: finite_induct)
```
```  1585     case empty thus ?case by simp
```
```  1586   next
```
```  1587     case (insert b F)
```
```  1588     show ?case
```
```  1589     proof cases
```
```  1590       assume "finite{a:A. f a = b}"
```
```  1591       hence "~ finite(A - {a:A. f a = b})" using `\<not> finite A` by simp
```
```  1592       also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
```
```  1593       finally have "~ finite({a:A. f a \<noteq> b})" .
```
```  1594       from insert(3)[OF _ this]
```
```  1595       show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
```
```  1596     next
```
```  1597       assume 1: "~finite{a:A. f a = b}"
```
```  1598       hence "{a \<in> A. f a = b} \<noteq> {}" by force
```
```  1599       thus ?thesis using 1 by blast
```
```  1600     qed
```
```  1601   qed
```
```  1602   from this[OF assms(2,1)] show ?thesis .
```
```  1603 qed
```
```  1604
```
```  1605 lemma pigeonhole_infinite_rel:
```
```  1606 assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
```
```  1607 shows "EX b:B. ~finite{a:A. R a b}"
```
```  1608 proof -
```
```  1609    let ?F = "%a. {b:B. R a b}"
```
```  1610    from finite_Pow_iff[THEN iffD2, OF `finite B`]
```
```  1611    have "finite(?F ` A)" by(blast intro: rev_finite_subset)
```
```  1612    from pigeonhole_infinite[where f = ?F, OF assms(1) this]
```
```  1613    obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
```
```  1614    obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast
```
```  1615    { assume "finite{a:A. R a b0}"
```
```  1616      then have "finite {a\<in>A. ?F a = ?F a0}"
```
```  1617        using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)
```
```  1618    }
```
```  1619    with 1 `b0 : B` show ?thesis by blast
```
```  1620 qed
```
```  1621
```
```  1622
```
```  1623 subsubsection {* Cardinality of sums *}
```
```  1624
```
```  1625 lemma card_Plus:
```
```  1626   assumes "finite A" and "finite B"
```
```  1627   shows "card (A <+> B) = card A + card B"
```
```  1628 proof -
```
```  1629   have "Inl`A \<inter> Inr`B = {}" by fast
```
```  1630   with assms show ?thesis
```
```  1631     unfolding Plus_def
```
```  1632     by (simp add: card_Un_disjoint card_image)
```
```  1633 qed
```
```  1634
```
```  1635 lemma card_Plus_conv_if:
```
```  1636   "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
```
```  1637   by (auto simp add: card_Plus)
```
```  1638
```
```  1639 text {* Relates to equivalence classes.  Based on a theorem of F. Kamm\"uller.  *}
```
```  1640
```
```  1641 lemma dvd_partition:
```
```  1642   assumes f: "finite (\<Union>C)" and "\<forall>c\<in>C. k dvd card c" "\<forall>c1\<in>C. \<forall>c2\<in>C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {}"
```
```  1643     shows "k dvd card (\<Union>C)"
```
```  1644 proof -
```
```  1645   have "finite C"
```
```  1646     by (rule finite_UnionD [OF f])
```
```  1647   then show ?thesis using assms
```
```  1648   proof (induct rule: finite_induct)
```
```  1649     case empty show ?case by simp
```
```  1650   next
```
```  1651     case (insert c C)
```
```  1652     then show ?case
```
```  1653       apply simp
```
```  1654       apply (subst card_Un_disjoint)
```
```  1655       apply (auto simp add: disjoint_eq_subset_Compl)
```
```  1656       done
```
```  1657   qed
```
```  1658 qed
```
```  1659
```
```  1660 subsubsection {* Relating injectivity and surjectivity *}
```
```  1661
```
```  1662 lemma finite_surj_inj: assumes "finite A" "A \<subseteq> f ` A" shows "inj_on f A"
```
```  1663 proof -
```
```  1664   have "f ` A = A"
```
```  1665     by (rule card_seteq [THEN sym]) (auto simp add: assms card_image_le)
```
```  1666   then show ?thesis using assms
```
```  1667     by (simp add: eq_card_imp_inj_on)
```
```  1668 qed
```
```  1669
```
```  1670 lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
```
```  1671 shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
```
```  1672 by (blast intro: finite_surj_inj subset_UNIV)
```
```  1673
```
```  1674 lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
```
```  1675 shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
```
```  1676 by(fastforce simp:surj_def dest!: endo_inj_surj)
```
```  1677
```
```  1678 corollary infinite_UNIV_nat [iff]:
```
```  1679   "\<not> finite (UNIV :: nat set)"
```
```  1680 proof
```
```  1681   assume "finite (UNIV :: nat set)"
```
```  1682   with finite_UNIV_inj_surj [of Suc]
```
```  1683   show False by simp (blast dest: Suc_neq_Zero surjD)
```
```  1684 qed
```
```  1685
```
```  1686 lemma infinite_UNIV_char_0:
```
```  1687   "\<not> finite (UNIV :: 'a::semiring_char_0 set)"
```
```  1688 proof
```
```  1689   assume "finite (UNIV :: 'a set)"
```
```  1690   with subset_UNIV have "finite (range of_nat :: 'a set)"
```
```  1691     by (rule finite_subset)
```
```  1692   moreover have "inj (of_nat :: nat \<Rightarrow> 'a)"
```
```  1693     by (simp add: inj_on_def)
```
```  1694   ultimately have "finite (UNIV :: nat set)"
```
```  1695     by (rule finite_imageD)
```
```  1696   then show False
```
```  1697     by simp
```
```  1698 qed
```
```  1699
```
```  1700 hide_const (open) Finite_Set.fold
```
```  1701
```
```  1702 end
```