src/HOL/Finite_Set.thy
 author haftmann Sun Jul 17 19:48:02 2011 +0200 (2011-07-17) changeset 43866 8a50dc70cbff parent 42875 d1aad0957eb2 child 43991 f4a7697011c5 permissions -rw-r--r--
moving UNIV = ... equations to their proper theories
```     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` proof induct
```
```    26   show "P {}" by fact
```
```    27   fix x F assume F: "finite F" and P: "P F"
```
```    28   show "P (insert x F)"
```
```    29   proof cases
```
```    30     assume "x \<in> F"
```
```    31     hence "insert x F = F" by (rule insert_absorb)
```
```    32     with P show ?thesis by (simp only:)
```
```    33   next
```
```    34     assume "x \<notin> F"
```
```    35     from F this P show ?thesis by (rule insert)
```
```    36   qed
```
```    37 qed
```
```    38
```
```    39
```
```    40 subsubsection {* Choice principles *}
```
```    41
```
```    42 lemma ex_new_if_finite: -- "does not depend on def of finite at all"
```
```    43   assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
```
```    44   shows "\<exists>a::'a. a \<notin> A"
```
```    45 proof -
```
```    46   from assms have "A \<noteq> UNIV" by blast
```
```    47   then show ?thesis by blast
```
```    48 qed
```
```    49
```
```    50 text {* A finite choice principle. Does not need the SOME choice operator. *}
```
```    51
```
```    52 lemma finite_set_choice:
```
```    53   "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
```
```    54 proof (induct rule: finite_induct)
```
```    55   case empty then show ?case by simp
```
```    56 next
```
```    57   case (insert a A)
```
```    58   then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
```
```    59   show ?case (is "EX f. ?P f")
```
```    60   proof
```
```    61     show "?P(%x. if x = a then b else f x)" using f ab by auto
```
```    62   qed
```
```    63 qed
```
```    64
```
```    65
```
```    66 subsubsection {* Finite sets are the images of initial segments of natural numbers *}
```
```    67
```
```    68 lemma finite_imp_nat_seg_image_inj_on:
```
```    69   assumes "finite A"
```
```    70   shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
```
```    71 using assms proof induct
```
```    72   case empty
```
```    73   show ?case
```
```    74   proof
```
```    75     show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp
```
```    76   qed
```
```    77 next
```
```    78   case (insert a A)
```
```    79   have notinA: "a \<notin> A" by fact
```
```    80   from insert.hyps obtain n f
```
```    81     where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
```
```    82   hence "insert a A = f(n:=a) ` {i. i < Suc n}"
```
```    83         "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
```
```    84     by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
```
```    85   thus ?case by blast
```
```    86 qed
```
```    87
```
```    88 lemma nat_seg_image_imp_finite:
```
```    89   "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
```
```    90 proof (induct n arbitrary: A)
```
```    91   case 0 thus ?case by simp
```
```    92 next
```
```    93   case (Suc n)
```
```    94   let ?B = "f ` {i. i < n}"
```
```    95   have finB: "finite ?B" by(rule Suc.hyps[OF refl])
```
```    96   show ?case
```
```    97   proof cases
```
```    98     assume "\<exists>k<n. f n = f k"
```
```    99     hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   100     thus ?thesis using finB by simp
```
```   101   next
```
```   102     assume "\<not>(\<exists> k<n. f n = f k)"
```
```   103     hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   104     thus ?thesis using finB by simp
```
```   105   qed
```
```   106 qed
```
```   107
```
```   108 lemma finite_conv_nat_seg_image:
```
```   109   "finite A \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"
```
```   110   by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
```
```   111
```
```   112 lemma finite_imp_inj_to_nat_seg:
```
```   113   assumes "finite A"
```
```   114   shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> inj_on f A"
```
```   115 proof -
```
```   116   from finite_imp_nat_seg_image_inj_on[OF `finite A`]
```
```   117   obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
```
```   118     by (auto simp:bij_betw_def)
```
```   119   let ?f = "the_inv_into {i. i<n} f"
```
```   120   have "inj_on ?f A & ?f ` A = {i. i<n}"
```
```   121     by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
```
```   122   thus ?thesis by blast
```
```   123 qed
```
```   124
```
```   125 lemma finite_Collect_less_nat [iff]:
```
```   126   "finite {n::nat. n < k}"
```
```   127   by (fastsimp simp: finite_conv_nat_seg_image)
```
```   128
```
```   129 lemma finite_Collect_le_nat [iff]:
```
```   130   "finite {n::nat. n \<le> k}"
```
```   131   by (simp add: le_eq_less_or_eq Collect_disj_eq)
```
```   132
```
```   133
```
```   134 subsubsection {* Finiteness and common set operations *}
```
```   135
```
```   136 lemma rev_finite_subset:
```
```   137   "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
```
```   138 proof (induct arbitrary: A rule: finite_induct)
```
```   139   case empty
```
```   140   then show ?case by simp
```
```   141 next
```
```   142   case (insert x F A)
```
```   143   have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+
```
```   144   show "finite A"
```
```   145   proof cases
```
```   146     assume x: "x \<in> A"
```
```   147     with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
```
```   148     with r have "finite (A - {x})" .
```
```   149     hence "finite (insert x (A - {x}))" ..
```
```   150     also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
```
```   151     finally show ?thesis .
```
```   152   next
```
```   153     show "A \<subseteq> F ==> ?thesis" by fact
```
```   154     assume "x \<notin> A"
```
```   155     with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
```
```   156   qed
```
```   157 qed
```
```   158
```
```   159 lemma finite_subset:
```
```   160   "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
```
```   161   by (rule rev_finite_subset)
```
```   162
```
```   163 lemma finite_UnI:
```
```   164   assumes "finite F" and "finite G"
```
```   165   shows "finite (F \<union> G)"
```
```   166   using assms by induct simp_all
```
```   167
```
```   168 lemma finite_Un [iff]:
```
```   169   "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
```
```   170   by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])
```
```   171
```
```   172 lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"
```
```   173 proof -
```
```   174   have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp
```
```   175   then have "finite ({a} \<union> A) \<longleftrightarrow> finite A" by (simp only: finite_Un)
```
```   176   then show ?thesis by simp
```
```   177 qed
```
```   178
```
```   179 lemma finite_Int [simp, intro]:
```
```   180   "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
```
```   181   by (blast intro: finite_subset)
```
```   182
```
```   183 lemma finite_Collect_conjI [simp, intro]:
```
```   184   "finite {x. P x} \<or> finite {x. Q x} \<Longrightarrow> finite {x. P x \<and> Q x}"
```
```   185   by (simp add: Collect_conj_eq)
```
```   186
```
```   187 lemma finite_Collect_disjI [simp]:
```
```   188   "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"
```
```   189   by (simp add: Collect_disj_eq)
```
```   190
```
```   191 lemma finite_Diff [simp, intro]:
```
```   192   "finite A \<Longrightarrow> finite (A - B)"
```
```   193   by (rule finite_subset, rule Diff_subset)
```
```   194
```
```   195 lemma finite_Diff2 [simp]:
```
```   196   assumes "finite B"
```
```   197   shows "finite (A - B) \<longleftrightarrow> finite A"
```
```   198 proof -
```
```   199   have "finite A \<longleftrightarrow> finite((A - B) \<union> (A \<inter> B))" by (simp add: Un_Diff_Int)
```
```   200   also have "\<dots> \<longleftrightarrow> finite (A - B)" using `finite B` by simp
```
```   201   finally show ?thesis ..
```
```   202 qed
```
```   203
```
```   204 lemma finite_Diff_insert [iff]:
```
```   205   "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
```
```   206 proof -
```
```   207   have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
```
```   208   moreover have "A - insert a B = A - B - {a}" by auto
```
```   209   ultimately show ?thesis by simp
```
```   210 qed
```
```   211
```
```   212 lemma finite_compl[simp]:
```
```   213   "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   214   by (simp add: Compl_eq_Diff_UNIV)
```
```   215
```
```   216 lemma finite_Collect_not[simp]:
```
```   217   "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   218   by (simp add: Collect_neg_eq)
```
```   219
```
```   220 lemma finite_Union [simp, intro]:
```
```   221   "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite(\<Union>A)"
```
```   222   by (induct rule: finite_induct) simp_all
```
```   223
```
```   224 lemma finite_UN_I [intro]:
```
```   225   "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"
```
```   226   by (induct rule: finite_induct) simp_all
```
```   227
```
```   228 lemma finite_UN [simp]:
```
```   229   "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
```
```   230   by (blast intro: finite_subset)
```
```   231
```
```   232 lemma finite_Inter [intro]:
```
```   233   "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
```
```   234   by (blast intro: Inter_lower finite_subset)
```
```   235
```
```   236 lemma finite_INT [intro]:
```
```   237   "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
```
```   238   by (blast intro: INT_lower finite_subset)
```
```   239
```
```   240 lemma finite_imageI [simp, intro]:
```
```   241   "finite F \<Longrightarrow> finite (h ` F)"
```
```   242   by (induct rule: finite_induct) simp_all
```
```   243
```
```   244 lemma finite_image_set [simp]:
```
```   245   "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
```
```   246   by (simp add: image_Collect [symmetric])
```
```   247
```
```   248 lemma finite_imageD:
```
```   249   assumes "finite (f ` A)" and "inj_on f A"
```
```   250   shows "finite A"
```
```   251 using assms proof (induct "f ` A" arbitrary: A)
```
```   252   case empty then show ?case by simp
```
```   253 next
```
```   254   case (insert x B)
```
```   255   then have B_A: "insert x B = f ` A" by simp
```
```   256   then obtain y where "x = f y" and "y \<in> A" by blast
```
```   257   from B_A `x \<notin> B` have "B = f ` A - {x}" by blast
```
```   258   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)
```
```   259   moreover from `inj_on f A` have "inj_on f (A - {y})" by (rule inj_on_diff)
```
```   260   ultimately have "finite (A - {y})" by (rule insert.hyps)
```
```   261   then show "finite A" by simp
```
```   262 qed
```
```   263
```
```   264 lemma finite_surj:
```
```   265   "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
```
```   266   by (erule finite_subset) (rule finite_imageI)
```
```   267
```
```   268 lemma finite_range_imageI:
```
```   269   "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
```
```   270   by (drule finite_imageI) (simp add: range_composition)
```
```   271
```
```   272 lemma finite_subset_image:
```
```   273   assumes "finite B"
```
```   274   shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
```
```   275 using assms proof induct
```
```   276   case empty then show ?case by simp
```
```   277 next
```
```   278   case insert then show ?case
```
```   279     by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
```
```   280        blast
```
```   281 qed
```
```   282
```
```   283 lemma finite_vimageI:
```
```   284   "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
```
```   285   apply (induct rule: finite_induct)
```
```   286    apply simp_all
```
```   287   apply (subst vimage_insert)
```
```   288   apply (simp add: finite_subset [OF inj_vimage_singleton])
```
```   289   done
```
```   290
```
```   291 lemma finite_vimageD:
```
```   292   assumes fin: "finite (h -` F)" and surj: "surj h"
```
```   293   shows "finite F"
```
```   294 proof -
```
```   295   have "finite (h ` (h -` F))" using fin by (rule finite_imageI)
```
```   296   also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)
```
```   297   finally show "finite F" .
```
```   298 qed
```
```   299
```
```   300 lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
```
```   301   unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
```
```   302
```
```   303 lemma finite_Collect_bex [simp]:
```
```   304   assumes "finite A"
```
```   305   shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"
```
```   306 proof -
```
```   307   have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto
```
```   308   with assms show ?thesis by simp
```
```   309 qed
```
```   310
```
```   311 lemma finite_Collect_bounded_ex [simp]:
```
```   312   assumes "finite {y. P y}"
```
```   313   shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
```
```   314 proof -
```
```   315   have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
```
```   316   with assms show ?thesis by simp
```
```   317 qed
```
```   318
```
```   319 lemma finite_Plus:
```
```   320   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
```
```   321   by (simp add: Plus_def)
```
```   322
```
```   323 lemma finite_PlusD:
```
```   324   fixes A :: "'a set" and B :: "'b set"
```
```   325   assumes fin: "finite (A <+> B)"
```
```   326   shows "finite A" "finite B"
```
```   327 proof -
```
```   328   have "Inl ` A \<subseteq> A <+> B" by auto
```
```   329   then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
```
```   330   then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
```
```   331 next
```
```   332   have "Inr ` B \<subseteq> A <+> B" by auto
```
```   333   then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
```
```   334   then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
```
```   335 qed
```
```   336
```
```   337 lemma finite_Plus_iff [simp]:
```
```   338   "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
```
```   339   by (auto intro: finite_PlusD finite_Plus)
```
```   340
```
```   341 lemma finite_Plus_UNIV_iff [simp]:
```
```   342   "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
```
```   343   by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
```
```   344
```
```   345 lemma finite_SigmaI [simp, intro]:
```
```   346   "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
```
```   347   by (unfold Sigma_def) blast
```
```   348
```
```   349 lemma finite_cartesian_product:
```
```   350   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
```
```   351   by (rule finite_SigmaI)
```
```   352
```
```   353 lemma finite_Prod_UNIV:
```
```   354   "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
```
```   355   by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
```
```   356
```
```   357 lemma finite_cartesian_productD1:
```
```   358   assumes "finite (A \<times> B)" and "B \<noteq> {}"
```
```   359   shows "finite A"
```
```   360 proof -
```
```   361   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
```
```   362     by (auto simp add: finite_conv_nat_seg_image)
```
```   363   then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
```
```   364   with `B \<noteq> {}` have "A = (fst \<circ> f) ` {i::nat. i < n}"
```
```   365     by (simp add: image_compose)
```
```   366   then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
```
```   367   then show ?thesis
```
```   368     by (auto simp add: finite_conv_nat_seg_image)
```
```   369 qed
```
```   370
```
```   371 lemma finite_cartesian_productD2:
```
```   372   assumes "finite (A \<times> B)" and "A \<noteq> {}"
```
```   373   shows "finite B"
```
```   374 proof -
```
```   375   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
```
```   376     by (auto simp add: finite_conv_nat_seg_image)
```
```   377   then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
```
```   378   with `A \<noteq> {}` have "B = (snd \<circ> f) ` {i::nat. i < n}"
```
```   379     by (simp add: image_compose)
```
```   380   then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
```
```   381   then show ?thesis
```
```   382     by (auto simp add: finite_conv_nat_seg_image)
```
```   383 qed
```
```   384
```
```   385 lemma finite_Pow_iff [iff]:
```
```   386   "finite (Pow A) \<longleftrightarrow> finite A"
```
```   387 proof
```
```   388   assume "finite (Pow A)"
```
```   389   then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
```
```   390   then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
```
```   391 next
```
```   392   assume "finite A"
```
```   393   then show "finite (Pow A)"
```
```   394     by induct (simp_all add: Pow_insert)
```
```   395 qed
```
```   396
```
```   397 corollary finite_Collect_subsets [simp, intro]:
```
```   398   "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
```
```   399   by (simp add: Pow_def [symmetric])
```
```   400
```
```   401 lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
```
```   402   by (blast intro: finite_subset [OF subset_Pow_Union])
```
```   403
```
```   404
```
```   405 subsubsection {* Further induction rules on finite sets *}
```
```   406
```
```   407 lemma finite_ne_induct [case_names singleton insert, consumes 2]:
```
```   408   assumes "finite F" and "F \<noteq> {}"
```
```   409   assumes "\<And>x. P {x}"
```
```   410     and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
```
```   411   shows "P F"
```
```   412 using assms proof induct
```
```   413   case empty then show ?case by simp
```
```   414 next
```
```   415   case (insert x F) then show ?case by cases auto
```
```   416 qed
```
```   417
```
```   418 lemma finite_subset_induct [consumes 2, case_names empty insert]:
```
```   419   assumes "finite F" and "F \<subseteq> A"
```
```   420   assumes empty: "P {}"
```
```   421     and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
```
```   422   shows "P F"
```
```   423 using `finite F` `F \<subseteq> A` proof induct
```
```   424   show "P {}" by fact
```
```   425 next
```
```   426   fix x F
```
```   427   assume "finite F" and "x \<notin> F" and
```
```   428     P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
```
```   429   show "P (insert x F)"
```
```   430   proof (rule insert)
```
```   431     from i show "x \<in> A" by blast
```
```   432     from i have "F \<subseteq> A" by blast
```
```   433     with P show "P F" .
```
```   434     show "finite F" by fact
```
```   435     show "x \<notin> F" by fact
```
```   436   qed
```
```   437 qed
```
```   438
```
```   439 lemma finite_empty_induct:
```
```   440   assumes "finite A"
```
```   441   assumes "P A"
```
```   442     and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
```
```   443   shows "P {}"
```
```   444 proof -
```
```   445   have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
```
```   446   proof -
```
```   447     fix B :: "'a set"
```
```   448     assume "B \<subseteq> A"
```
```   449     with `finite A` have "finite B" by (rule rev_finite_subset)
```
```   450     from this `B \<subseteq> A` show "P (A - B)"
```
```   451     proof induct
```
```   452       case empty
```
```   453       from `P A` show ?case by simp
```
```   454     next
```
```   455       case (insert b B)
```
```   456       have "P (A - B - {b})"
```
```   457       proof (rule remove)
```
```   458         from `finite A` show "finite (A - B)" by induct auto
```
```   459         from insert show "b \<in> A - B" by simp
```
```   460         from insert show "P (A - B)" by simp
```
```   461       qed
```
```   462       also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
```
```   463       finally show ?case .
```
```   464     qed
```
```   465   qed
```
```   466   then have "P (A - A)" by blast
```
```   467   then show ?thesis by simp
```
```   468 qed
```
```   469
```
```   470
```
```   471 subsection {* Class @{text finite}  *}
```
```   472
```
```   473 class finite =
```
```   474   assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
```
```   475 begin
```
```   476
```
```   477 lemma finite [simp]: "finite (A \<Colon> 'a set)"
```
```   478   by (rule subset_UNIV finite_UNIV finite_subset)+
```
```   479
```
```   480 lemma finite_code [code]: "finite (A \<Colon> 'a set) \<longleftrightarrow> True"
```
```   481   by simp
```
```   482
```
```   483 end
```
```   484
```
```   485 instance unit :: finite proof
```
```   486 qed (simp add: UNIV_unit)
```
```   487
```
```   488 instance bool :: finite proof
```
```   489 qed (simp add: UNIV_bool)
```
```   490
```
```   491 instance prod :: (finite, finite) finite proof
```
```   492 qed (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
```
```   493
```
```   494 lemma finite_option_UNIV [simp]:
```
```   495   "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
```
```   496   by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
```
```   497
```
```   498 instance option :: (finite) finite proof
```
```   499 qed (simp add: UNIV_option_conv)
```
```   500
```
```   501 lemma inj_graph: "inj (%f. {(x, y). y = f x})"
```
```   502   by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
```
```   503
```
```   504 instance "fun" :: (finite, finite) finite
```
```   505 proof
```
```   506   show "finite (UNIV :: ('a => 'b) set)"
```
```   507   proof (rule finite_imageD)
```
```   508     let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
```
```   509     have "range ?graph \<subseteq> Pow UNIV" by simp
```
```   510     moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
```
```   511       by (simp only: finite_Pow_iff finite)
```
```   512     ultimately show "finite (range ?graph)"
```
```   513       by (rule finite_subset)
```
```   514     show "inj ?graph" by (rule inj_graph)
```
```   515   qed
```
```   516 qed
```
```   517
```
```   518 instance sum :: (finite, finite) finite proof
```
```   519 qed (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
```
```   520
```
```   521
```
```   522 subsection {* A basic fold functional for finite sets *}
```
```   523
```
```   524 text {* The intended behaviour is
```
```   525 @{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
```
```   526 if @{text f} is ``left-commutative'':
```
```   527 *}
```
```   528
```
```   529 locale comp_fun_commute =
```
```   530   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
```
```   531   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
```
```   532 begin
```
```   533
```
```   534 lemma fun_left_comm: "f x (f y z) = f y (f x z)"
```
```   535   using comp_fun_commute by (simp add: fun_eq_iff)
```
```   536
```
```   537 end
```
```   538
```
```   539 inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
```
```   540 for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
```
```   541   emptyI [intro]: "fold_graph f z {} z" |
```
```   542   insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
```
```   543       \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
```
```   544
```
```   545 inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
```
```   546
```
```   547 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
```
```   548   "fold f z A = (THE y. fold_graph f z A y)"
```
```   549
```
```   550 text{*A tempting alternative for the definiens is
```
```   551 @{term "if finite A then THE y. fold_graph f z A y else e"}.
```
```   552 It allows the removal of finiteness assumptions from the theorems
```
```   553 @{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
```
```   554 The proofs become ugly. It is not worth the effort. (???) *}
```
```   555
```
```   556 lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
```
```   557 by (induct rule: finite_induct) auto
```
```   558
```
```   559
```
```   560 subsubsection{*From @{const fold_graph} to @{term fold}*}
```
```   561
```
```   562 context comp_fun_commute
```
```   563 begin
```
```   564
```
```   565 lemma fold_graph_insertE_aux:
```
```   566   "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'"
```
```   567 proof (induct set: fold_graph)
```
```   568   case (insertI x A y) show ?case
```
```   569   proof (cases "x = a")
```
```   570     assume "x = a" with insertI show ?case by auto
```
```   571   next
```
```   572     assume "x \<noteq> a"
```
```   573     then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
```
```   574       using insertI by auto
```
```   575     have "f x y = f a (f x y')"
```
```   576       unfolding y by (rule fun_left_comm)
```
```   577     moreover have "fold_graph f z (insert x A - {a}) (f x y')"
```
```   578       using y' and `x \<noteq> a` and `x \<notin> A`
```
```   579       by (simp add: insert_Diff_if fold_graph.insertI)
```
```   580     ultimately show ?case by fast
```
```   581   qed
```
```   582 qed simp
```
```   583
```
```   584 lemma fold_graph_insertE:
```
```   585   assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
```
```   586   obtains y where "v = f x y" and "fold_graph f z A y"
```
```   587 using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
```
```   588
```
```   589 lemma fold_graph_determ:
```
```   590   "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
```
```   591 proof (induct arbitrary: y set: fold_graph)
```
```   592   case (insertI x A y v)
```
```   593   from `fold_graph f z (insert x A) v` and `x \<notin> A`
```
```   594   obtain y' where "v = f x y'" and "fold_graph f z A y'"
```
```   595     by (rule fold_graph_insertE)
```
```   596   from `fold_graph f z A y'` have "y' = y" by (rule insertI)
```
```   597   with `v = f x y'` show "v = f x y" by simp
```
```   598 qed fast
```
```   599
```
```   600 lemma fold_equality:
```
```   601   "fold_graph f z A y \<Longrightarrow> fold f z A = y"
```
```   602 by (unfold fold_def) (blast intro: fold_graph_determ)
```
```   603
```
```   604 lemma fold_graph_fold:
```
```   605   assumes "finite A"
```
```   606   shows "fold_graph f z A (fold f z A)"
```
```   607 proof -
```
```   608   from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
```
```   609   moreover note fold_graph_determ
```
```   610   ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
```
```   611   then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
```
```   612   then show ?thesis by (unfold fold_def)
```
```   613 qed
```
```   614
```
```   615 text{* The base case for @{text fold}: *}
```
```   616
```
```   617 lemma (in -) fold_empty [simp]: "fold f z {} = z"
```
```   618 by (unfold fold_def) blast
```
```   619
```
```   620 text{* The various recursion equations for @{const fold}: *}
```
```   621
```
```   622 lemma fold_insert [simp]:
```
```   623   assumes "finite A" and "x \<notin> A"
```
```   624   shows "fold f z (insert x A) = f x (fold f z A)"
```
```   625 proof (rule fold_equality)
```
```   626   from `finite A` have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
```
```   627   with `x \<notin> A`show "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)
```
```   628 qed
```
```   629
```
```   630 lemma fold_fun_comm:
```
```   631   "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
```
```   632 proof (induct rule: finite_induct)
```
```   633   case empty then show ?case by simp
```
```   634 next
```
```   635   case (insert y A) then show ?case
```
```   636     by (simp add: fun_left_comm[of x])
```
```   637 qed
```
```   638
```
```   639 lemma fold_insert2:
```
```   640   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
```
```   641 by (simp add: fold_fun_comm)
```
```   642
```
```   643 lemma fold_rec:
```
```   644   assumes "finite A" and "x \<in> A"
```
```   645   shows "fold f z A = f x (fold f z (A - {x}))"
```
```   646 proof -
```
```   647   have A: "A = insert x (A - {x})" using `x \<in> A` by blast
```
```   648   then have "fold f z A = fold f z (insert x (A - {x}))" by simp
```
```   649   also have "\<dots> = f x (fold f z (A - {x}))"
```
```   650     by (rule fold_insert) (simp add: `finite A`)+
```
```   651   finally show ?thesis .
```
```   652 qed
```
```   653
```
```   654 lemma fold_insert_remove:
```
```   655   assumes "finite A"
```
```   656   shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
```
```   657 proof -
```
```   658   from `finite A` have "finite (insert x A)" by auto
```
```   659   moreover have "x \<in> insert x A" by auto
```
```   660   ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
```
```   661     by (rule fold_rec)
```
```   662   then show ?thesis by simp
```
```   663 qed
```
```   664
```
```   665 end
```
```   666
```
```   667 text{* A simplified version for idempotent functions: *}
```
```   668
```
```   669 locale comp_fun_idem = comp_fun_commute +
```
```   670   assumes comp_fun_idem: "f x o f x = f x"
```
```   671 begin
```
```   672
```
```   673 lemma fun_left_idem: "f x (f x z) = f x z"
```
```   674   using comp_fun_idem by (simp add: fun_eq_iff)
```
```   675
```
```   676 lemma fold_insert_idem:
```
```   677   assumes fin: "finite A"
```
```   678   shows "fold f z (insert x A) = f x (fold f z A)"
```
```   679 proof cases
```
```   680   assume "x \<in> A"
```
```   681   then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
```
```   682   then show ?thesis using assms by (simp add:fun_left_idem)
```
```   683 next
```
```   684   assume "x \<notin> A" then show ?thesis using assms by simp
```
```   685 qed
```
```   686
```
```   687 declare fold_insert[simp del] fold_insert_idem[simp]
```
```   688
```
```   689 lemma fold_insert_idem2:
```
```   690   "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
```
```   691 by(simp add:fold_fun_comm)
```
```   692
```
```   693 end
```
```   694
```
```   695
```
```   696 subsubsection {* Expressing set operations via @{const fold} *}
```
```   697
```
```   698 lemma (in comp_fun_commute) comp_comp_fun_commute:
```
```   699   "comp_fun_commute (f \<circ> g)"
```
```   700 proof
```
```   701 qed (simp_all add: comp_fun_commute)
```
```   702
```
```   703 lemma (in comp_fun_idem) comp_comp_fun_idem:
```
```   704   "comp_fun_idem (f \<circ> g)"
```
```   705   by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
```
```   706     (simp_all add: comp_fun_idem)
```
```   707
```
```   708 lemma comp_fun_idem_insert:
```
```   709   "comp_fun_idem insert"
```
```   710 proof
```
```   711 qed auto
```
```   712
```
```   713 lemma comp_fun_idem_remove:
```
```   714   "comp_fun_idem (\<lambda>x A. A - {x})"
```
```   715 proof
```
```   716 qed auto
```
```   717
```
```   718 lemma (in semilattice_inf) comp_fun_idem_inf:
```
```   719   "comp_fun_idem inf"
```
```   720 proof
```
```   721 qed (auto simp add: inf_left_commute)
```
```   722
```
```   723 lemma (in semilattice_sup) comp_fun_idem_sup:
```
```   724   "comp_fun_idem sup"
```
```   725 proof
```
```   726 qed (auto simp add: sup_left_commute)
```
```   727
```
```   728 lemma union_fold_insert:
```
```   729   assumes "finite A"
```
```   730   shows "A \<union> B = fold insert B A"
```
```   731 proof -
```
```   732   interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
```
```   733   from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
```
```   734 qed
```
```   735
```
```   736 lemma minus_fold_remove:
```
```   737   assumes "finite A"
```
```   738   shows "B - A = fold (\<lambda>x A. A - {x}) B A"
```
```   739 proof -
```
```   740   interpret comp_fun_idem "\<lambda>x A. A - {x}" by (fact comp_fun_idem_remove)
```
```   741   from `finite A` show ?thesis by (induct A arbitrary: B) auto
```
```   742 qed
```
```   743
```
```   744 context complete_lattice
```
```   745 begin
```
```   746
```
```   747 lemma inf_Inf_fold_inf:
```
```   748   assumes "finite A"
```
```   749   shows "inf B (Inf A) = fold inf B A"
```
```   750 proof -
```
```   751   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
```
```   752   from `finite A` show ?thesis by (induct A arbitrary: B)
```
```   753     (simp_all add: Inf_insert inf_commute fold_fun_comm)
```
```   754 qed
```
```   755
```
```   756 lemma sup_Sup_fold_sup:
```
```   757   assumes "finite A"
```
```   758   shows "sup B (Sup A) = fold sup B A"
```
```   759 proof -
```
```   760   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
```
```   761   from `finite A` show ?thesis by (induct A arbitrary: B)
```
```   762     (simp_all add: Sup_insert sup_commute fold_fun_comm)
```
```   763 qed
```
```   764
```
```   765 lemma Inf_fold_inf:
```
```   766   assumes "finite A"
```
```   767   shows "Inf A = fold inf top A"
```
```   768   using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
```
```   769
```
```   770 lemma Sup_fold_sup:
```
```   771   assumes "finite A"
```
```   772   shows "Sup A = fold sup bot A"
```
```   773   using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
```
```   774
```
```   775 lemma inf_INFI_fold_inf:
```
```   776   assumes "finite A"
```
```   777   shows "inf B (INFI A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold")
```
```   778 proof (rule sym)
```
```   779   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
```
```   780   interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
```
```   781   from `finite A` show "?fold = ?inf"
```
```   782     by (induct A arbitrary: B)
```
```   783       (simp_all add: INFI_def Inf_insert inf_left_commute)
```
```   784 qed
```
```   785
```
```   786 lemma sup_SUPR_fold_sup:
```
```   787   assumes "finite A"
```
```   788   shows "sup B (SUPR A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold")
```
```   789 proof (rule sym)
```
```   790   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
```
```   791   interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
```
```   792   from `finite A` show "?fold = ?sup"
```
```   793     by (induct A arbitrary: B)
```
```   794       (simp_all add: SUPR_def Sup_insert sup_left_commute)
```
```   795 qed
```
```   796
```
```   797 lemma INFI_fold_inf:
```
```   798   assumes "finite A"
```
```   799   shows "INFI A f = fold (inf \<circ> f) top A"
```
```   800   using assms inf_INFI_fold_inf [of A top] by simp
```
```   801
```
```   802 lemma SUPR_fold_sup:
```
```   803   assumes "finite A"
```
```   804   shows "SUPR A f = fold (sup \<circ> f) bot A"
```
```   805   using assms sup_SUPR_fold_sup [of A bot] by simp
```
```   806
```
```   807 end
```
```   808
```
```   809
```
```   810 subsection {* The derived combinator @{text fold_image} *}
```
```   811
```
```   812 definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
```
```   813   where "fold_image f g = fold (\<lambda>x y. f (g x) y)"
```
```   814
```
```   815 lemma fold_image_empty[simp]: "fold_image f g z {} = z"
```
```   816   by (simp add:fold_image_def)
```
```   817
```
```   818 context ab_semigroup_mult
```
```   819 begin
```
```   820
```
```   821 lemma fold_image_insert[simp]:
```
```   822   assumes "finite A" and "a \<notin> A"
```
```   823   shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
```
```   824 proof -
```
```   825   interpret comp_fun_commute "%x y. (g x) * y" proof
```
```   826   qed (simp add: fun_eq_iff mult_ac)
```
```   827   show ?thesis using assms by (simp add: fold_image_def)
```
```   828 qed
```
```   829
```
```   830 lemma fold_image_reindex:
```
```   831   assumes "finite A"
```
```   832   shows "inj_on h A \<Longrightarrow> fold_image times g z (h ` A) = fold_image times (g \<circ> h) z A"
```
```   833   using assms by induct auto
```
```   834
```
```   835 lemma fold_image_cong:
```
```   836   assumes "finite A" and g_h: "\<And>x. x\<in>A \<Longrightarrow> g x = h x"
```
```   837   shows "fold_image times g z A = fold_image times h z A"
```
```   838 proof -
```
```   839   from `finite A`
```
```   840   have "\<And>C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C"
```
```   841   proof (induct arbitrary: C)
```
```   842     case empty then show ?case by simp
```
```   843   next
```
```   844     case (insert x F) then show ?case apply -
```
```   845     apply (simp add: subset_insert_iff, clarify)
```
```   846     apply (subgoal_tac "finite C")
```
```   847       prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
```
```   848     apply (subgoal_tac "C = insert x (C - {x})")
```
```   849       prefer 2 apply blast
```
```   850     apply (erule ssubst)
```
```   851     apply (simp add: Ball_def del: insert_Diff_single)
```
```   852     done
```
```   853   qed
```
```   854   with g_h show ?thesis by simp
```
```   855 qed
```
```   856
```
```   857 end
```
```   858
```
```   859 context comm_monoid_mult
```
```   860 begin
```
```   861
```
```   862 lemma fold_image_1:
```
```   863   "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
```
```   864   apply (induct rule: finite_induct)
```
```   865   apply simp by auto
```
```   866
```
```   867 lemma fold_image_Un_Int:
```
```   868   "finite A ==> finite B ==>
```
```   869     fold_image times g 1 A * fold_image times g 1 B =
```
```   870     fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
```
```   871   apply (induct rule: finite_induct)
```
```   872 by (induct set: finite)
```
```   873    (auto simp add: mult_ac insert_absorb Int_insert_left)
```
```   874
```
```   875 lemma fold_image_Un_one:
```
```   876   assumes fS: "finite S" and fT: "finite T"
```
```   877   and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
```
```   878   shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
```
```   879 proof-
```
```   880   have "fold_image op * f 1 (S \<inter> T) = 1"
```
```   881     apply (rule fold_image_1)
```
```   882     using fS fT I0 by auto
```
```   883   with fold_image_Un_Int[OF fS fT] show ?thesis by simp
```
```   884 qed
```
```   885
```
```   886 corollary fold_Un_disjoint:
```
```   887   "finite A ==> finite B ==> A Int B = {} ==>
```
```   888    fold_image times g 1 (A Un B) =
```
```   889    fold_image times g 1 A * fold_image times g 1 B"
```
```   890 by (simp add: fold_image_Un_Int)
```
```   891
```
```   892 lemma fold_image_UN_disjoint:
```
```   893   "\<lbrakk> finite I; ALL i:I. finite (A i);
```
```   894      ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
```
```   895    \<Longrightarrow> fold_image times g 1 (UNION I A) =
```
```   896        fold_image times (%i. fold_image times g 1 (A i)) 1 I"
```
```   897 apply (induct rule: finite_induct)
```
```   898 apply simp
```
```   899 apply atomize
```
```   900 apply (subgoal_tac "ALL i:F. x \<noteq> i")
```
```   901  prefer 2 apply blast
```
```   902 apply (subgoal_tac "A x Int UNION F A = {}")
```
```   903  prefer 2 apply blast
```
```   904 apply (simp add: fold_Un_disjoint)
```
```   905 done
```
```   906
```
```   907 lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```   908   fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A =
```
```   909   fold_image times (split g) 1 (SIGMA x:A. B x)"
```
```   910 apply (subst Sigma_def)
```
```   911 apply (subst fold_image_UN_disjoint, assumption, simp)
```
```   912  apply blast
```
```   913 apply (erule fold_image_cong)
```
```   914 apply (subst fold_image_UN_disjoint, simp, simp)
```
```   915  apply blast
```
```   916 apply simp
```
```   917 done
```
```   918
```
```   919 lemma fold_image_distrib: "finite A \<Longrightarrow>
```
```   920    fold_image times (%x. g x * h x) 1 A =
```
```   921    fold_image times g 1 A *  fold_image times h 1 A"
```
```   922 by (erule finite_induct) (simp_all add: mult_ac)
```
```   923
```
```   924 lemma fold_image_related:
```
```   925   assumes Re: "R e e"
```
```   926   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
```
```   927   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
```
```   928   shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
```
```   929   using fS by (rule finite_subset_induct) (insert assms, auto)
```
```   930
```
```   931 lemma  fold_image_eq_general:
```
```   932   assumes fS: "finite S"
```
```   933   and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y"
```
```   934   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
```
```   935   shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
```
```   936 proof-
```
```   937   from h f12 have hS: "h ` S = S'" by auto
```
```   938   {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
```
```   939     from f12 h H  have "x = y" by auto }
```
```   940   hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
```
```   941   from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
```
```   942   from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
```
```   943   also have "\<dots> = fold_image (op *) (f2 o h) e S"
```
```   944     using fold_image_reindex[OF fS hinj, of f2 e] .
```
```   945   also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
```
```   946     by blast
```
```   947   finally show ?thesis ..
```
```   948 qed
```
```   949
```
```   950 lemma fold_image_eq_general_inverses:
```
```   951   assumes fS: "finite S"
```
```   952   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
```
```   953   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
```
```   954   shows "fold_image (op *) f e S = fold_image (op *) g e T"
```
```   955   (* metis solves it, but not yet available here *)
```
```   956   apply (rule fold_image_eq_general[OF fS, of T h g f e])
```
```   957   apply (rule ballI)
```
```   958   apply (frule kh)
```
```   959   apply (rule ex1I[])
```
```   960   apply blast
```
```   961   apply clarsimp
```
```   962   apply (drule hk) apply simp
```
```   963   apply (rule sym)
```
```   964   apply (erule conjunct1[OF conjunct2[OF hk]])
```
```   965   apply (rule ballI)
```
```   966   apply (drule  hk)
```
```   967   apply blast
```
```   968   done
```
```   969
```
```   970 end
```
```   971
```
```   972
```
```   973 subsection {* A fold functional for non-empty sets *}
```
```   974
```
```   975 text{* Does not require start value. *}
```
```   976
```
```   977 inductive
```
```   978   fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
```
```   979   for f :: "'a => 'a => 'a"
```
```   980 where
```
```   981   fold1Set_insertI [intro]:
```
```   982    "\<lbrakk> fold_graph f a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
```
```   983
```
```   984 definition fold1 :: "('a => 'a => 'a) => 'a set => 'a" where
```
```   985   "fold1 f A == THE x. fold1Set f A x"
```
```   986
```
```   987 lemma fold1Set_nonempty:
```
```   988   "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
```
```   989 by(erule fold1Set.cases, simp_all)
```
```   990
```
```   991 inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
```
```   992
```
```   993 inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
```
```   994
```
```   995
```
```   996 lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
```
```   997 by (blast elim: fold_graph.cases)
```
```   998
```
```   999 lemma fold1_singleton [simp]: "fold1 f {a} = a"
```
```  1000 by (unfold fold1_def) blast
```
```  1001
```
```  1002 lemma finite_nonempty_imp_fold1Set:
```
```  1003   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
```
```  1004 apply (induct A rule: finite_induct)
```
```  1005 apply (auto dest: finite_imp_fold_graph [of _ f])
```
```  1006 done
```
```  1007
```
```  1008 text{*First, some lemmas about @{const fold_graph}.*}
```
```  1009
```
```  1010 context ab_semigroup_mult
```
```  1011 begin
```
```  1012
```
```  1013 lemma comp_fun_commute: "comp_fun_commute (op *)" proof
```
```  1014 qed (simp add: fun_eq_iff mult_ac)
```
```  1015
```
```  1016 lemma fold_graph_insert_swap:
```
```  1017 assumes fold: "fold_graph times (b::'a) A y" and "b \<notin> A"
```
```  1018 shows "fold_graph times z (insert b A) (z * y)"
```
```  1019 proof -
```
```  1020   interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
```
```  1021 from assms show ?thesis
```
```  1022 proof (induct rule: fold_graph.induct)
```
```  1023   case emptyI show ?case by (subst mult_commute [of z b], fast)
```
```  1024 next
```
```  1025   case (insertI x A y)
```
```  1026     have "fold_graph times z (insert x (insert b A)) (x * (z * y))"
```
```  1027       using insertI by force  --{*how does @{term id} get unfolded?*}
```
```  1028     thus ?case by (simp add: insert_commute mult_ac)
```
```  1029 qed
```
```  1030 qed
```
```  1031
```
```  1032 lemma fold_graph_permute_diff:
```
```  1033 assumes fold: "fold_graph times b A x"
```
```  1034 shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> fold_graph times a (insert b (A-{a})) x"
```
```  1035 using fold
```
```  1036 proof (induct rule: fold_graph.induct)
```
```  1037   case emptyI thus ?case by simp
```
```  1038 next
```
```  1039   case (insertI x A y)
```
```  1040   have "a = x \<or> a \<in> A" using insertI by simp
```
```  1041   thus ?case
```
```  1042   proof
```
```  1043     assume "a = x"
```
```  1044     with insertI show ?thesis
```
```  1045       by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap)
```
```  1046   next
```
```  1047     assume ainA: "a \<in> A"
```
```  1048     hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)"
```
```  1049       using insertI by force
```
```  1050     moreover
```
```  1051     have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
```
```  1052       using ainA insertI by blast
```
```  1053     ultimately show ?thesis by simp
```
```  1054   qed
```
```  1055 qed
```
```  1056
```
```  1057 lemma fold1_eq_fold:
```
```  1058 assumes "finite A" "a \<notin> A" shows "fold1 times (insert a A) = fold times a A"
```
```  1059 proof -
```
```  1060   interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
```
```  1061   from assms show ?thesis
```
```  1062 apply (simp add: fold1_def fold_def)
```
```  1063 apply (rule the_equality)
```
```  1064 apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times])
```
```  1065 apply (rule sym, clarify)
```
```  1066 apply (case_tac "Aa=A")
```
```  1067  apply (best intro: fold_graph_determ)
```
```  1068 apply (subgoal_tac "fold_graph times a A x")
```
```  1069  apply (best intro: fold_graph_determ)
```
```  1070 apply (subgoal_tac "insert aa (Aa - {a}) = A")
```
```  1071  prefer 2 apply (blast elim: equalityE)
```
```  1072 apply (auto dest: fold_graph_permute_diff [where a=a])
```
```  1073 done
```
```  1074 qed
```
```  1075
```
```  1076 lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
```
```  1077 apply safe
```
```  1078  apply simp
```
```  1079  apply (drule_tac x=x in spec)
```
```  1080  apply (drule_tac x="A-{x}" in spec, auto)
```
```  1081 done
```
```  1082
```
```  1083 lemma fold1_insert:
```
```  1084   assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
```
```  1085   shows "fold1 times (insert x A) = x * fold1 times A"
```
```  1086 proof -
```
```  1087   interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
```
```  1088   from nonempty obtain a A' where "A = insert a A' & a ~: A'"
```
```  1089     by (auto simp add: nonempty_iff)
```
```  1090   with A show ?thesis
```
```  1091     by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
```
```  1092 qed
```
```  1093
```
```  1094 end
```
```  1095
```
```  1096 context ab_semigroup_idem_mult
```
```  1097 begin
```
```  1098
```
```  1099 lemma comp_fun_idem: "comp_fun_idem (op *)" proof
```
```  1100 qed (simp_all add: fun_eq_iff mult_left_commute)
```
```  1101
```
```  1102 lemma fold1_insert_idem [simp]:
```
```  1103   assumes nonempty: "A \<noteq> {}" and A: "finite A"
```
```  1104   shows "fold1 times (insert x A) = x * fold1 times A"
```
```  1105 proof -
```
```  1106   interpret comp_fun_idem "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```  1107     by (rule comp_fun_idem)
```
```  1108   from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
```
```  1109     by (auto simp add: nonempty_iff)
```
```  1110   show ?thesis
```
```  1111   proof cases
```
```  1112     assume a: "a = x"
```
```  1113     show ?thesis
```
```  1114     proof cases
```
```  1115       assume "A' = {}"
```
```  1116       with A' a show ?thesis by simp
```
```  1117     next
```
```  1118       assume "A' \<noteq> {}"
```
```  1119       with A A' a show ?thesis
```
```  1120         by (simp add: fold1_insert mult_assoc [symmetric])
```
```  1121     qed
```
```  1122   next
```
```  1123     assume "a \<noteq> x"
```
```  1124     with A A' show ?thesis
```
```  1125       by (simp add: insert_commute fold1_eq_fold)
```
```  1126   qed
```
```  1127 qed
```
```  1128
```
```  1129 lemma hom_fold1_commute:
```
```  1130 assumes hom: "!!x y. h (x * y) = h x * h y"
```
```  1131 and N: "finite N" "N \<noteq> {}" shows "h (fold1 times N) = fold1 times (h ` N)"
```
```  1132 using N proof (induct rule: finite_ne_induct)
```
```  1133   case singleton thus ?case by simp
```
```  1134 next
```
```  1135   case (insert n N)
```
```  1136   then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp
```
```  1137   also have "\<dots> = h n * h (fold1 times N)" by(rule hom)
```
```  1138   also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert)
```
```  1139   also have "times (h n) \<dots> = fold1 times (insert (h n) (h ` N))"
```
```  1140     using insert by(simp)
```
```  1141   also have "insert (h n) (h ` N) = h ` insert n N" by simp
```
```  1142   finally show ?case .
```
```  1143 qed
```
```  1144
```
```  1145 lemma fold1_eq_fold_idem:
```
```  1146   assumes "finite A"
```
```  1147   shows "fold1 times (insert a A) = fold times a A"
```
```  1148 proof (cases "a \<in> A")
```
```  1149   case False
```
```  1150   with assms show ?thesis by (simp add: fold1_eq_fold)
```
```  1151 next
```
```  1152   interpret comp_fun_idem times by (fact comp_fun_idem)
```
```  1153   case True then obtain b B
```
```  1154     where A: "A = insert a B" and "a \<notin> B" by (rule set_insert)
```
```  1155   with assms have "finite B" by auto
```
```  1156   then have "fold times a (insert a B) = fold times (a * a) B"
```
```  1157     using `a \<notin> B` by (rule fold_insert2)
```
```  1158   then show ?thesis
```
```  1159     using `a \<notin> B` `finite B` by (simp add: fold1_eq_fold A)
```
```  1160 qed
```
```  1161
```
```  1162 end
```
```  1163
```
```  1164
```
```  1165 text{* Now the recursion rules for definitions: *}
```
```  1166
```
```  1167 lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
```
```  1168 by simp
```
```  1169
```
```  1170 lemma (in ab_semigroup_mult) fold1_insert_def:
```
```  1171   "\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
```
```  1172 by (simp add:fold1_insert)
```
```  1173
```
```  1174 lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:
```
```  1175   "\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
```
```  1176 by simp
```
```  1177
```
```  1178 subsubsection{* Determinacy for @{term fold1Set} *}
```
```  1179
```
```  1180 (*Not actually used!!*)
```
```  1181 (*
```
```  1182 context ab_semigroup_mult
```
```  1183 begin
```
```  1184
```
```  1185 lemma fold_graph_permute:
```
```  1186   "[|fold_graph times id b (insert a A) x; a \<notin> A; b \<notin> A|]
```
```  1187    ==> fold_graph times id a (insert b A) x"
```
```  1188 apply (cases "a=b")
```
```  1189 apply (auto dest: fold_graph_permute_diff)
```
```  1190 done
```
```  1191
```
```  1192 lemma fold1Set_determ:
```
```  1193   "fold1Set times A x ==> fold1Set times A y ==> y = x"
```
```  1194 proof (clarify elim!: fold1Set.cases)
```
```  1195   fix A x B y a b
```
```  1196   assume Ax: "fold_graph times id a A x"
```
```  1197   assume By: "fold_graph times id b B y"
```
```  1198   assume anotA:  "a \<notin> A"
```
```  1199   assume bnotB:  "b \<notin> B"
```
```  1200   assume eq: "insert a A = insert b B"
```
```  1201   show "y=x"
```
```  1202   proof cases
```
```  1203     assume same: "a=b"
```
```  1204     hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
```
```  1205     thus ?thesis using Ax By same by (blast intro: fold_graph_determ)
```
```  1206   next
```
```  1207     assume diff: "a\<noteq>b"
```
```  1208     let ?D = "B - {a}"
```
```  1209     have B: "B = insert a ?D" and A: "A = insert b ?D"
```
```  1210      and aB: "a \<in> B" and bA: "b \<in> A"
```
```  1211       using eq anotA bnotB diff by (blast elim!:equalityE)+
```
```  1212     with aB bnotB By
```
```  1213     have "fold_graph times id a (insert b ?D) y"
```
```  1214       by (auto intro: fold_graph_permute simp add: insert_absorb)
```
```  1215     moreover
```
```  1216     have "fold_graph times id a (insert b ?D) x"
```
```  1217       by (simp add: A [symmetric] Ax)
```
```  1218     ultimately show ?thesis by (blast intro: fold_graph_determ)
```
```  1219   qed
```
```  1220 qed
```
```  1221
```
```  1222 lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y"
```
```  1223   by (unfold fold1_def) (blast intro: fold1Set_determ)
```
```  1224
```
```  1225 end
```
```  1226 *)
```
```  1227
```
```  1228 declare
```
```  1229   empty_fold_graphE [rule del]  fold_graph.intros [rule del]
```
```  1230   empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
```
```  1231   -- {* No more proofs involve these relations. *}
```
```  1232
```
```  1233 subsubsection {* Lemmas about @{text fold1} *}
```
```  1234
```
```  1235 context ab_semigroup_mult
```
```  1236 begin
```
```  1237
```
```  1238 lemma fold1_Un:
```
```  1239 assumes A: "finite A" "A \<noteq> {}"
```
```  1240 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
```
```  1241        fold1 times (A Un B) = fold1 times A * fold1 times B"
```
```  1242 using A by (induct rule: finite_ne_induct)
```
```  1243   (simp_all add: fold1_insert mult_assoc)
```
```  1244
```
```  1245 lemma fold1_in:
```
```  1246   assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x,y}"
```
```  1247   shows "fold1 times A \<in> A"
```
```  1248 using A
```
```  1249 proof (induct rule:finite_ne_induct)
```
```  1250   case singleton thus ?case by simp
```
```  1251 next
```
```  1252   case insert thus ?case using elem by (force simp add:fold1_insert)
```
```  1253 qed
```
```  1254
```
```  1255 end
```
```  1256
```
```  1257 lemma (in ab_semigroup_idem_mult) fold1_Un2:
```
```  1258 assumes A: "finite A" "A \<noteq> {}"
```
```  1259 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
```
```  1260        fold1 times (A Un B) = fold1 times A * fold1 times B"
```
```  1261 using A
```
```  1262 proof(induct rule:finite_ne_induct)
```
```  1263   case singleton thus ?case by simp
```
```  1264 next
```
```  1265   case insert thus ?case by (simp add: mult_assoc)
```
```  1266 qed
```
```  1267
```
```  1268
```
```  1269 subsection {* Locales as mini-packages for fold operations *}
```
```  1270
```
```  1271 subsubsection {* The natural case *}
```
```  1272
```
```  1273 locale folding =
```
```  1274   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
```
```  1275   fixes F :: "'a set \<Rightarrow> 'b \<Rightarrow> 'b"
```
```  1276   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
```
```  1277   assumes eq_fold: "finite A \<Longrightarrow> F A s = fold f s A"
```
```  1278 begin
```
```  1279
```
```  1280 lemma empty [simp]:
```
```  1281   "F {} = id"
```
```  1282   by (simp add: eq_fold fun_eq_iff)
```
```  1283
```
```  1284 lemma insert [simp]:
```
```  1285   assumes "finite A" and "x \<notin> A"
```
```  1286   shows "F (insert x A) = F A \<circ> f x"
```
```  1287 proof -
```
```  1288   interpret comp_fun_commute f proof
```
```  1289   qed (insert comp_fun_commute, simp add: fun_eq_iff)
```
```  1290   from fold_insert2 assms
```
```  1291   have "\<And>s. fold f s (insert x A) = fold f (f x s) A" .
```
```  1292   with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
```
```  1293 qed
```
```  1294
```
```  1295 lemma remove:
```
```  1296   assumes "finite A" and "x \<in> A"
```
```  1297   shows "F A = F (A - {x}) \<circ> f x"
```
```  1298 proof -
```
```  1299   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
```
```  1300     by (auto dest: mk_disjoint_insert)
```
```  1301   moreover from `finite A` this have "finite B" by simp
```
```  1302   ultimately show ?thesis by simp
```
```  1303 qed
```
```  1304
```
```  1305 lemma insert_remove:
```
```  1306   assumes "finite A"
```
```  1307   shows "F (insert x A) = F (A - {x}) \<circ> f x"
```
```  1308   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
```
```  1309
```
```  1310 lemma commute_left_comp:
```
```  1311   "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
```
```  1312   by (simp add: o_assoc comp_fun_commute)
```
```  1313
```
```  1314 lemma comp_fun_commute':
```
```  1315   assumes "finite A"
```
```  1316   shows "f x \<circ> F A = F A \<circ> f x"
```
```  1317   using assms by (induct A)
```
```  1318     (simp, simp del: o_apply add: o_assoc, simp del: o_apply add: o_assoc [symmetric] comp_fun_commute)
```
```  1319
```
```  1320 lemma commute_left_comp':
```
```  1321   assumes "finite A"
```
```  1322   shows "f x \<circ> (F A \<circ> g) = F A \<circ> (f x \<circ> g)"
```
```  1323   using assms by (simp add: o_assoc comp_fun_commute')
```
```  1324
```
```  1325 lemma comp_fun_commute'':
```
```  1326   assumes "finite A" and "finite B"
```
```  1327   shows "F B \<circ> F A = F A \<circ> F B"
```
```  1328   using assms by (induct A)
```
```  1329     (simp_all add: o_assoc, simp add: o_assoc [symmetric] comp_fun_commute')
```
```  1330
```
```  1331 lemma commute_left_comp'':
```
```  1332   assumes "finite A" and "finite B"
```
```  1333   shows "F B \<circ> (F A \<circ> g) = F A \<circ> (F B \<circ> g)"
```
```  1334   using assms by (simp add: o_assoc comp_fun_commute'')
```
```  1335
```
```  1336 lemmas comp_fun_commutes = o_assoc [symmetric] comp_fun_commute commute_left_comp
```
```  1337   comp_fun_commute' commute_left_comp' comp_fun_commute'' commute_left_comp''
```
```  1338
```
```  1339 lemma union_inter:
```
```  1340   assumes "finite A" and "finite B"
```
```  1341   shows "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B"
```
```  1342   using assms by (induct A)
```
```  1343     (simp_all del: o_apply add: insert_absorb Int_insert_left comp_fun_commutes,
```
```  1344       simp add: o_assoc)
```
```  1345
```
```  1346 lemma union:
```
```  1347   assumes "finite A" and "finite B"
```
```  1348   and "A \<inter> B = {}"
```
```  1349   shows "F (A \<union> B) = F A \<circ> F B"
```
```  1350 proof -
```
```  1351   from union_inter `finite A` `finite B` have "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B" .
```
```  1352   with `A \<inter> B = {}` show ?thesis by simp
```
```  1353 qed
```
```  1354
```
```  1355 end
```
```  1356
```
```  1357
```
```  1358 subsubsection {* The natural case with idempotency *}
```
```  1359
```
```  1360 locale folding_idem = folding +
```
```  1361   assumes idem_comp: "f x \<circ> f x = f x"
```
```  1362 begin
```
```  1363
```
```  1364 lemma idem_left_comp:
```
```  1365   "f x \<circ> (f x \<circ> g) = f x \<circ> g"
```
```  1366   by (simp add: o_assoc idem_comp)
```
```  1367
```
```  1368 lemma in_comp_idem:
```
```  1369   assumes "finite A" and "x \<in> A"
```
```  1370   shows "F A \<circ> f x = F A"
```
```  1371 using assms by (induct A)
```
```  1372   (auto simp add: comp_fun_commutes idem_comp, simp add: commute_left_comp' [symmetric] comp_fun_commute')
```
```  1373
```
```  1374 lemma subset_comp_idem:
```
```  1375   assumes "finite A" and "B \<subseteq> A"
```
```  1376   shows "F A \<circ> F B = F A"
```
```  1377 proof -
```
```  1378   from assms have "finite B" by (blast dest: finite_subset)
```
```  1379   then show ?thesis using `B \<subseteq> A` by (induct B)
```
```  1380     (simp_all add: o_assoc in_comp_idem `finite A`)
```
```  1381 qed
```
```  1382
```
```  1383 declare insert [simp del]
```
```  1384
```
```  1385 lemma insert_idem [simp]:
```
```  1386   assumes "finite A"
```
```  1387   shows "F (insert x A) = F A \<circ> f x"
```
```  1388   using assms by (cases "x \<in> A") (simp_all add: insert in_comp_idem insert_absorb)
```
```  1389
```
```  1390 lemma union_idem:
```
```  1391   assumes "finite A" and "finite B"
```
```  1392   shows "F (A \<union> B) = F A \<circ> F B"
```
```  1393 proof -
```
```  1394   from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
```
```  1395   then have "F (A \<union> B) \<circ> F (A \<inter> B) = F (A \<union> B)" by (rule subset_comp_idem)
```
```  1396   with assms show ?thesis by (simp add: union_inter)
```
```  1397 qed
```
```  1398
```
```  1399 end
```
```  1400
```
```  1401
```
```  1402 subsubsection {* The image case with fixed function *}
```
```  1403
```
```  1404 no_notation times (infixl "*" 70)
```
```  1405 no_notation Groups.one ("1")
```
```  1406
```
```  1407 locale folding_image_simple = comm_monoid +
```
```  1408   fixes g :: "('b \<Rightarrow> 'a)"
```
```  1409   fixes F :: "'b set \<Rightarrow> 'a"
```
```  1410   assumes eq_fold_g: "finite A \<Longrightarrow> F A = fold_image f g 1 A"
```
```  1411 begin
```
```  1412
```
```  1413 lemma empty [simp]:
```
```  1414   "F {} = 1"
```
```  1415   by (simp add: eq_fold_g)
```
```  1416
```
```  1417 lemma insert [simp]:
```
```  1418   assumes "finite A" and "x \<notin> A"
```
```  1419   shows "F (insert x A) = g x * F A"
```
```  1420 proof -
```
```  1421   interpret comp_fun_commute "%x y. (g x) * y" proof
```
```  1422   qed (simp add: ac_simps fun_eq_iff)
```
```  1423   with assms have "fold_image (op *) g 1 (insert x A) = g x * fold_image (op *) g 1 A"
```
```  1424     by (simp add: fold_image_def)
```
```  1425   with `finite A` show ?thesis by (simp add: eq_fold_g)
```
```  1426 qed
```
```  1427
```
```  1428 lemma remove:
```
```  1429   assumes "finite A" and "x \<in> A"
```
```  1430   shows "F A = g x * F (A - {x})"
```
```  1431 proof -
```
```  1432   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
```
```  1433     by (auto dest: mk_disjoint_insert)
```
```  1434   moreover from `finite A` this have "finite B" by simp
```
```  1435   ultimately show ?thesis by simp
```
```  1436 qed
```
```  1437
```
```  1438 lemma insert_remove:
```
```  1439   assumes "finite A"
```
```  1440   shows "F (insert x A) = g x * F (A - {x})"
```
```  1441   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
```
```  1442
```
```  1443 lemma neutral:
```
```  1444   assumes "finite A" and "\<forall>x\<in>A. g x = 1"
```
```  1445   shows "F A = 1"
```
```  1446   using assms by (induct A) simp_all
```
```  1447
```
```  1448 lemma union_inter:
```
```  1449   assumes "finite A" and "finite B"
```
```  1450   shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
```
```  1451 using assms proof (induct A)
```
```  1452   case empty then show ?case by simp
```
```  1453 next
```
```  1454   case (insert x A) then show ?case
```
```  1455     by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
```
```  1456 qed
```
```  1457
```
```  1458 corollary union_inter_neutral:
```
```  1459   assumes "finite A" and "finite B"
```
```  1460   and I0: "\<forall>x \<in> A\<inter>B. g x = 1"
```
```  1461   shows "F (A \<union> B) = F A * F B"
```
```  1462   using assms by (simp add: union_inter [symmetric] neutral)
```
```  1463
```
```  1464 corollary union_disjoint:
```
```  1465   assumes "finite A" and "finite B"
```
```  1466   assumes "A \<inter> B = {}"
```
```  1467   shows "F (A \<union> B) = F A * F B"
```
```  1468   using assms by (simp add: union_inter_neutral)
```
```  1469
```
```  1470 end
```
```  1471
```
```  1472
```
```  1473 subsubsection {* The image case with flexible function *}
```
```  1474
```
```  1475 locale folding_image = comm_monoid +
```
```  1476   fixes F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
```
```  1477   assumes eq_fold: "\<And>g. finite A \<Longrightarrow> F g A = fold_image f g 1 A"
```
```  1478
```
```  1479 sublocale folding_image < folding_image_simple "op *" 1 g "F g" proof
```
```  1480 qed (fact eq_fold)
```
```  1481
```
```  1482 context folding_image
```
```  1483 begin
```
```  1484
```
```  1485 lemma reindex: (* FIXME polymorhism *)
```
```  1486   assumes "finite A" and "inj_on h A"
```
```  1487   shows "F g (h ` A) = F (g \<circ> h) A"
```
```  1488   using assms by (induct A) auto
```
```  1489
```
```  1490 lemma cong:
```
```  1491   assumes "finite A" and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"
```
```  1492   shows "F g A = F h A"
```
```  1493 proof -
```
```  1494   from assms have "ALL C. C <= A --> (ALL x:C. g x = h x) --> F g C = F h C"
```
```  1495   apply - apply (erule finite_induct) apply simp
```
```  1496   apply (simp add: subset_insert_iff, clarify)
```
```  1497   apply (subgoal_tac "finite C")
```
```  1498   prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
```
```  1499   apply (subgoal_tac "C = insert x (C - {x})")
```
```  1500   prefer 2 apply blast
```
```  1501   apply (erule ssubst)
```
```  1502   apply (drule spec)
```
```  1503   apply (erule (1) notE impE)
```
```  1504   apply (simp add: Ball_def del: insert_Diff_single)
```
```  1505   done
```
```  1506   with assms show ?thesis by simp
```
```  1507 qed
```
```  1508
```
```  1509 lemma UNION_disjoint:
```
```  1510   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
```
```  1511   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
```
```  1512   shows "F g (UNION I A) = F (F g \<circ> A) I"
```
```  1513 apply (insert assms)
```
```  1514 apply (induct rule: finite_induct)
```
```  1515 apply simp
```
```  1516 apply atomize
```
```  1517 apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
```
```  1518  prefer 2 apply blast
```
```  1519 apply (subgoal_tac "A x Int UNION Fa A = {}")
```
```  1520  prefer 2 apply blast
```
```  1521 apply (simp add: union_disjoint)
```
```  1522 done
```
```  1523
```
```  1524 lemma distrib:
```
```  1525   assumes "finite A"
```
```  1526   shows "F (\<lambda>x. g x * h x) A = F g A * F h A"
```
```  1527   using assms by (rule finite_induct) (simp_all add: assoc commute left_commute)
```
```  1528
```
```  1529 lemma related:
```
```  1530   assumes Re: "R 1 1"
```
```  1531   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
```
```  1532   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
```
```  1533   shows "R (F h S) (F g S)"
```
```  1534   using fS by (rule finite_subset_induct) (insert assms, auto)
```
```  1535
```
```  1536 lemma eq_general:
```
```  1537   assumes fS: "finite S"
```
```  1538   and h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y"
```
```  1539   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
```
```  1540   shows "F f1 S = F f2 S'"
```
```  1541 proof-
```
```  1542   from h f12 have hS: "h ` S = S'" by blast
```
```  1543   {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
```
```  1544     from f12 h H  have "x = y" by auto }
```
```  1545   hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
```
```  1546   from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
```
```  1547   from hS have "F f2 S' = F f2 (h ` S)" by simp
```
```  1548   also have "\<dots> = F (f2 o h) S" using reindex [OF fS hinj, of f2] .
```
```  1549   also have "\<dots> = F f1 S " using th cong [OF fS, of "f2 o h" f1]
```
```  1550     by blast
```
```  1551   finally show ?thesis ..
```
```  1552 qed
```
```  1553
```
```  1554 lemma eq_general_inverses:
```
```  1555   assumes fS: "finite S"
```
```  1556   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
```
```  1557   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"
```
```  1558   shows "F j S = F g T"
```
```  1559   (* metis solves it, but not yet available here *)
```
```  1560   apply (rule eq_general [OF fS, of T h g j])
```
```  1561   apply (rule ballI)
```
```  1562   apply (frule kh)
```
```  1563   apply (rule ex1I[])
```
```  1564   apply blast
```
```  1565   apply clarsimp
```
```  1566   apply (drule hk) apply simp
```
```  1567   apply (rule sym)
```
```  1568   apply (erule conjunct1[OF conjunct2[OF hk]])
```
```  1569   apply (rule ballI)
```
```  1570   apply (drule hk)
```
```  1571   apply blast
```
```  1572   done
```
```  1573
```
```  1574 end
```
```  1575
```
```  1576
```
```  1577 subsubsection {* The image case with fixed function and idempotency *}
```
```  1578
```
```  1579 locale folding_image_simple_idem = folding_image_simple +
```
```  1580   assumes idem: "x * x = x"
```
```  1581
```
```  1582 sublocale folding_image_simple_idem < semilattice proof
```
```  1583 qed (fact idem)
```
```  1584
```
```  1585 context folding_image_simple_idem
```
```  1586 begin
```
```  1587
```
```  1588 lemma in_idem:
```
```  1589   assumes "finite A" and "x \<in> A"
```
```  1590   shows "g x * F A = F A"
```
```  1591   using assms by (induct A) (auto simp add: left_commute)
```
```  1592
```
```  1593 lemma subset_idem:
```
```  1594   assumes "finite A" and "B \<subseteq> A"
```
```  1595   shows "F B * F A = F A"
```
```  1596 proof -
```
```  1597   from assms have "finite B" by (blast dest: finite_subset)
```
```  1598   then show ?thesis using `B \<subseteq> A` by (induct B)
```
```  1599     (auto simp add: assoc in_idem `finite A`)
```
```  1600 qed
```
```  1601
```
```  1602 declare insert [simp del]
```
```  1603
```
```  1604 lemma insert_idem [simp]:
```
```  1605   assumes "finite A"
```
```  1606   shows "F (insert x A) = g x * F A"
```
```  1607   using assms by (cases "x \<in> A") (simp_all add: insert in_idem insert_absorb)
```
```  1608
```
```  1609 lemma union_idem:
```
```  1610   assumes "finite A" and "finite B"
```
```  1611   shows "F (A \<union> B) = F A * F B"
```
```  1612 proof -
```
```  1613   from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
```
```  1614   then have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (rule subset_idem)
```
```  1615   with assms show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
```
```  1616 qed
```
```  1617
```
```  1618 end
```
```  1619
```
```  1620
```
```  1621 subsubsection {* The image case with flexible function and idempotency *}
```
```  1622
```
```  1623 locale folding_image_idem = folding_image +
```
```  1624   assumes idem: "x * x = x"
```
```  1625
```
```  1626 sublocale folding_image_idem < folding_image_simple_idem "op *" 1 g "F g" proof
```
```  1627 qed (fact idem)
```
```  1628
```
```  1629
```
```  1630 subsubsection {* The neutral-less case *}
```
```  1631
```
```  1632 locale folding_one = abel_semigroup +
```
```  1633   fixes F :: "'a set \<Rightarrow> 'a"
```
```  1634   assumes eq_fold: "finite A \<Longrightarrow> F A = fold1 f A"
```
```  1635 begin
```
```  1636
```
```  1637 lemma singleton [simp]:
```
```  1638   "F {x} = x"
```
```  1639   by (simp add: eq_fold)
```
```  1640
```
```  1641 lemma eq_fold':
```
```  1642   assumes "finite A" and "x \<notin> A"
```
```  1643   shows "F (insert x A) = fold (op *) x A"
```
```  1644 proof -
```
```  1645   interpret ab_semigroup_mult "op *" proof qed (simp_all add: ac_simps)
```
```  1646   with assms show ?thesis by (simp add: eq_fold fold1_eq_fold)
```
```  1647 qed
```
```  1648
```
```  1649 lemma insert [simp]:
```
```  1650   assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
```
```  1651   shows "F (insert x A) = x * F A"
```
```  1652 proof -
```
```  1653   from `A \<noteq> {}` obtain b where "b \<in> A" by blast
```
```  1654   then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
```
```  1655   with `finite A` have "finite B" by simp
```
```  1656   interpret fold: folding "op *" "\<lambda>a b. fold (op *) b a" proof
```
```  1657   qed (simp_all add: fun_eq_iff ac_simps)
```
```  1658   thm fold.comp_fun_commute' [of B b, simplified fun_eq_iff, simplified]
```
```  1659   from `finite B` fold.comp_fun_commute' [of B x]
```
```  1660     have "op * x \<circ> (\<lambda>b. fold op * b B) = (\<lambda>b. fold op * b B) \<circ> op * x" by simp
```
```  1661   then have A: "x * fold op * b B = fold op * (b * x) B" by (simp add: fun_eq_iff commute)
```
```  1662   from `finite B` * fold.insert [of B b]
```
```  1663     have "(\<lambda>x. fold op * x (insert b B)) = (\<lambda>x. fold op * x B) \<circ> op * b" by simp
```
```  1664   then have B: "fold op * x (insert b B) = fold op * (b * x) B" by (simp add: fun_eq_iff)
```
```  1665   from A B assms * show ?thesis by (simp add: eq_fold' del: fold.insert)
```
```  1666 qed
```
```  1667
```
```  1668 lemma remove:
```
```  1669   assumes "finite A" and "x \<in> A"
```
```  1670   shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
```
```  1671 proof -
```
```  1672   from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
```
```  1673   with assms show ?thesis by simp
```
```  1674 qed
```
```  1675
```
```  1676 lemma insert_remove:
```
```  1677   assumes "finite A"
```
```  1678   shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
```
```  1679   using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
```
```  1680
```
```  1681 lemma union_disjoint:
```
```  1682   assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}" and "A \<inter> B = {}"
```
```  1683   shows "F (A \<union> B) = F A * F B"
```
```  1684   using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
```
```  1685
```
```  1686 lemma union_inter:
```
```  1687   assumes "finite A" and "finite B" and "A \<inter> B \<noteq> {}"
```
```  1688   shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
```
```  1689 proof -
```
```  1690   from assms have "A \<noteq> {}" and "B \<noteq> {}" by auto
```
```  1691   from `finite A` `A \<noteq> {}` `A \<inter> B \<noteq> {}` show ?thesis proof (induct A rule: finite_ne_induct)
```
```  1692     case (singleton x) then show ?case by (simp add: insert_absorb ac_simps)
```
```  1693   next
```
```  1694     case (insert x A) show ?case proof (cases "x \<in> B")
```
```  1695       case True then have "B \<noteq> {}" by auto
```
```  1696       with insert True `finite B` show ?thesis by (cases "A \<inter> B = {}")
```
```  1697         (simp_all add: insert_absorb ac_simps union_disjoint)
```
```  1698     next
```
```  1699       case False with insert have "F (A \<union> B) * F (A \<inter> B) = F A * F B" by simp
```
```  1700       moreover from False `finite B` insert have "finite (A \<union> B)" "x \<notin> A \<union> B" "A \<union> B \<noteq> {}"
```
```  1701         by auto
```
```  1702       ultimately show ?thesis using False `finite A` `x \<notin> A` `A \<noteq> {}` by (simp add: assoc)
```
```  1703     qed
```
```  1704   qed
```
```  1705 qed
```
```  1706
```
```  1707 lemma closed:
```
```  1708   assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
```
```  1709   shows "F A \<in> A"
```
```  1710 using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
```
```  1711   case singleton then show ?case by simp
```
```  1712 next
```
```  1713   case insert with elem show ?case by force
```
```  1714 qed
```
```  1715
```
```  1716 end
```
```  1717
```
```  1718
```
```  1719 subsubsection {* The neutral-less case with idempotency *}
```
```  1720
```
```  1721 locale folding_one_idem = folding_one +
```
```  1722   assumes idem: "x * x = x"
```
```  1723
```
```  1724 sublocale folding_one_idem < semilattice proof
```
```  1725 qed (fact idem)
```
```  1726
```
```  1727 context folding_one_idem
```
```  1728 begin
```
```  1729
```
```  1730 lemma in_idem:
```
```  1731   assumes "finite A" and "x \<in> A"
```
```  1732   shows "x * F A = F A"
```
```  1733 proof -
```
```  1734   from assms have "A \<noteq> {}" by auto
```
```  1735   with `finite A` show ?thesis using `x \<in> A` by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
```
```  1736 qed
```
```  1737
```
```  1738 lemma subset_idem:
```
```  1739   assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
```
```  1740   shows "F B * F A = F A"
```
```  1741 proof -
```
```  1742   from assms have "finite B" by (blast dest: finite_subset)
```
```  1743   then show ?thesis using `B \<noteq> {}` `B \<subseteq> A` by (induct B rule: finite_ne_induct)
```
```  1744     (simp_all add: assoc in_idem `finite A`)
```
```  1745 qed
```
```  1746
```
```  1747 lemma eq_fold_idem':
```
```  1748   assumes "finite A"
```
```  1749   shows "F (insert a A) = fold (op *) a A"
```
```  1750 proof -
```
```  1751   interpret ab_semigroup_idem_mult "op *" proof qed (simp_all add: ac_simps)
```
```  1752   with assms show ?thesis by (simp add: eq_fold fold1_eq_fold_idem)
```
```  1753 qed
```
```  1754
```
```  1755 lemma insert_idem [simp]:
```
```  1756   assumes "finite A" and "A \<noteq> {}"
```
```  1757   shows "F (insert x A) = x * F A"
```
```  1758 proof (cases "x \<in> A")
```
```  1759   case False from `finite A` `x \<notin> A` `A \<noteq> {}` show ?thesis by (rule insert)
```
```  1760 next
```
```  1761   case True
```
```  1762   from `finite A` `A \<noteq> {}` show ?thesis by (simp add: in_idem insert_absorb True)
```
```  1763 qed
```
```  1764
```
```  1765 lemma union_idem:
```
```  1766   assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
```
```  1767   shows "F (A \<union> B) = F A * F B"
```
```  1768 proof (cases "A \<inter> B = {}")
```
```  1769   case True with assms show ?thesis by (simp add: union_disjoint)
```
```  1770 next
```
```  1771   case False
```
```  1772   from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
```
```  1773   with False have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (auto intro: subset_idem)
```
```  1774   with assms False show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
```
```  1775 qed
```
```  1776
```
```  1777 lemma hom_commute:
```
```  1778   assumes hom: "\<And>x y. h (x * y) = h x * h y"
```
```  1779   and N: "finite N" "N \<noteq> {}" shows "h (F N) = F (h ` N)"
```
```  1780 using N proof (induct rule: finite_ne_induct)
```
```  1781   case singleton thus ?case by simp
```
```  1782 next
```
```  1783   case (insert n N)
```
```  1784   then have "h (F (insert n N)) = h (n * F N)" by simp
```
```  1785   also have "\<dots> = h n * h (F N)" by (rule hom)
```
```  1786   also have "h (F N) = F (h ` N)" by(rule insert)
```
```  1787   also have "h n * \<dots> = F (insert (h n) (h ` N))"
```
```  1788     using insert by(simp)
```
```  1789   also have "insert (h n) (h ` N) = h ` insert n N" by simp
```
```  1790   finally show ?case .
```
```  1791 qed
```
```  1792
```
```  1793 end
```
```  1794
```
```  1795 notation times (infixl "*" 70)
```
```  1796 notation Groups.one ("1")
```
```  1797
```
```  1798
```
```  1799 subsection {* Finite cardinality *}
```
```  1800
```
```  1801 text {* This definition, although traditional, is ugly to work with:
```
```  1802 @{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
```
```  1803 But now that we have @{text fold_image} things are easy:
```
```  1804 *}
```
```  1805
```
```  1806 definition card :: "'a set \<Rightarrow> nat" where
```
```  1807   "card A = (if finite A then fold_image (op +) (\<lambda>x. 1) 0 A else 0)"
```
```  1808
```
```  1809 interpretation card: folding_image_simple "op +" 0 "\<lambda>x. 1" card proof
```
```  1810 qed (simp add: card_def)
```
```  1811
```
```  1812 lemma card_infinite [simp]:
```
```  1813   "\<not> finite A \<Longrightarrow> card A = 0"
```
```  1814   by (simp add: card_def)
```
```  1815
```
```  1816 lemma card_empty:
```
```  1817   "card {} = 0"
```
```  1818   by (fact card.empty)
```
```  1819
```
```  1820 lemma card_insert_disjoint:
```
```  1821   "finite A ==> x \<notin> A ==> card (insert x A) = Suc (card A)"
```
```  1822   by simp
```
```  1823
```
```  1824 lemma card_insert_if:
```
```  1825   "finite A ==> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
```
```  1826   by auto (simp add: card.insert_remove card.remove)
```
```  1827
```
```  1828 lemma card_ge_0_finite:
```
```  1829   "card A > 0 \<Longrightarrow> finite A"
```
```  1830   by (rule ccontr) simp
```
```  1831
```
```  1832 lemma card_0_eq [simp, no_atp]:
```
```  1833   "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
```
```  1834   by (auto dest: mk_disjoint_insert)
```
```  1835
```
```  1836 lemma finite_UNIV_card_ge_0:
```
```  1837   "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
```
```  1838   by (rule ccontr) simp
```
```  1839
```
```  1840 lemma card_eq_0_iff:
```
```  1841   "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
```
```  1842   by auto
```
```  1843
```
```  1844 lemma card_gt_0_iff:
```
```  1845   "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
```
```  1846   by (simp add: neq0_conv [symmetric] card_eq_0_iff)
```
```  1847
```
```  1848 lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
```
```  1849 apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
```
```  1850 apply(simp del:insert_Diff_single)
```
```  1851 done
```
```  1852
```
```  1853 lemma card_Diff_singleton:
```
```  1854   "finite A ==> x: A ==> card (A - {x}) = card A - 1"
```
```  1855 by (simp add: card_Suc_Diff1 [symmetric])
```
```  1856
```
```  1857 lemma card_Diff_singleton_if:
```
```  1858   "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
```
```  1859 by (simp add: card_Diff_singleton)
```
```  1860
```
```  1861 lemma card_Diff_insert[simp]:
```
```  1862 assumes "finite A" and "a:A" and "a ~: B"
```
```  1863 shows "card(A - insert a B) = card(A - B) - 1"
```
```  1864 proof -
```
```  1865   have "A - insert a B = (A - B) - {a}" using assms by blast
```
```  1866   then show ?thesis using assms by(simp add:card_Diff_singleton)
```
```  1867 qed
```
```  1868
```
```  1869 lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
```
```  1870 by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert)
```
```  1871
```
```  1872 lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
```
```  1873 by (simp add: card_insert_if)
```
```  1874
```
```  1875 lemma card_Collect_less_nat[simp]: "card{i::nat. i < n} = n"
```
```  1876 by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
```
```  1877
```
```  1878 lemma card_Collect_le_nat[simp]: "card{i::nat. i <= n} = Suc n"
```
```  1879 using card_Collect_less_nat[of "Suc n"] by(simp add: less_Suc_eq_le)
```
```  1880
```
```  1881 lemma card_mono:
```
```  1882   assumes "finite B" and "A \<subseteq> B"
```
```  1883   shows "card A \<le> card B"
```
```  1884 proof -
```
```  1885   from assms have "finite A" by (auto intro: finite_subset)
```
```  1886   then show ?thesis using assms proof (induct A arbitrary: B)
```
```  1887     case empty then show ?case by simp
```
```  1888   next
```
```  1889     case (insert x A)
```
```  1890     then have "x \<in> B" by simp
```
```  1891     from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
```
```  1892     with insert.hyps have "card A \<le> card (B - {x})" by auto
```
```  1893     with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
```
```  1894   qed
```
```  1895 qed
```
```  1896
```
```  1897 lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
```
```  1898 apply (induct rule: finite_induct)
```
```  1899 apply simp
```
```  1900 apply clarify
```
```  1901 apply (subgoal_tac "finite A & A - {x} <= F")
```
```  1902  prefer 2 apply (blast intro: finite_subset, atomize)
```
```  1903 apply (drule_tac x = "A - {x}" in spec)
```
```  1904 apply (simp add: card_Diff_singleton_if split add: split_if_asm)
```
```  1905 apply (case_tac "card A", auto)
```
```  1906 done
```
```  1907
```
```  1908 lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
```
```  1909 apply (simp add: psubset_eq linorder_not_le [symmetric])
```
```  1910 apply (blast dest: card_seteq)
```
```  1911 done
```
```  1912
```
```  1913 lemma card_Un_Int: "finite A ==> finite B
```
```  1914     ==> card A + card B = card (A Un B) + card (A Int B)"
```
```  1915   by (fact card.union_inter [symmetric])
```
```  1916
```
```  1917 lemma card_Un_disjoint: "finite A ==> finite B
```
```  1918     ==> A Int B = {} ==> card (A Un B) = card A + card B"
```
```  1919   by (fact card.union_disjoint)
```
```  1920
```
```  1921 lemma card_Diff_subset:
```
```  1922   assumes "finite B" and "B \<subseteq> A"
```
```  1923   shows "card (A - B) = card A - card B"
```
```  1924 proof (cases "finite A")
```
```  1925   case False with assms show ?thesis by simp
```
```  1926 next
```
```  1927   case True with assms show ?thesis by (induct B arbitrary: A) simp_all
```
```  1928 qed
```
```  1929
```
```  1930 lemma card_Diff_subset_Int:
```
```  1931   assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
```
```  1932 proof -
```
```  1933   have "A - B = A - A \<inter> B" by auto
```
```  1934   thus ?thesis
```
```  1935     by (simp add: card_Diff_subset AB)
```
```  1936 qed
```
```  1937
```
```  1938 lemma diff_card_le_card_Diff:
```
```  1939 assumes "finite B" shows "card A - card B \<le> card(A - B)"
```
```  1940 proof-
```
```  1941   have "card A - card B \<le> card A - card (A \<inter> B)"
```
```  1942     using card_mono[OF assms Int_lower2, of A] by arith
```
```  1943   also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
```
```  1944   finally show ?thesis .
```
```  1945 qed
```
```  1946
```
```  1947 lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
```
```  1948 apply (rule Suc_less_SucD)
```
```  1949 apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
```
```  1950 done
```
```  1951
```
```  1952 lemma card_Diff2_less:
```
```  1953   "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
```
```  1954 apply (case_tac "x = y")
```
```  1955  apply (simp add: card_Diff1_less del:card_Diff_insert)
```
```  1956 apply (rule less_trans)
```
```  1957  prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
```
```  1958 done
```
```  1959
```
```  1960 lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
```
```  1961 apply (case_tac "x : A")
```
```  1962  apply (simp_all add: card_Diff1_less less_imp_le)
```
```  1963 done
```
```  1964
```
```  1965 lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
```
```  1966 by (erule psubsetI, blast)
```
```  1967
```
```  1968 lemma insert_partition:
```
```  1969   "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
```
```  1970   \<Longrightarrow> x \<inter> \<Union> F = {}"
```
```  1971 by auto
```
```  1972
```
```  1973 lemma finite_psubset_induct[consumes 1, case_names psubset]:
```
```  1974   assumes fin: "finite A"
```
```  1975   and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A"
```
```  1976   shows "P A"
```
```  1977 using fin
```
```  1978 proof (induct A taking: card rule: measure_induct_rule)
```
```  1979   case (less A)
```
```  1980   have fin: "finite A" by fact
```
```  1981   have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
```
```  1982   { fix B
```
```  1983     assume asm: "B \<subset> A"
```
```  1984     from asm have "card B < card A" using psubset_card_mono fin by blast
```
```  1985     moreover
```
```  1986     from asm have "B \<subseteq> A" by auto
```
```  1987     then have "finite B" using fin finite_subset by blast
```
```  1988     ultimately
```
```  1989     have "P B" using ih by simp
```
```  1990   }
```
```  1991   with fin show "P A" using major by blast
```
```  1992 qed
```
```  1993
```
```  1994 text{* main cardinality theorem *}
```
```  1995 lemma card_partition [rule_format]:
```
```  1996   "finite C ==>
```
```  1997      finite (\<Union> C) -->
```
```  1998      (\<forall>c\<in>C. card c = k) -->
```
```  1999      (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
```
```  2000      k * card(C) = card (\<Union> C)"
```
```  2001 apply (erule finite_induct, simp)
```
```  2002 apply (simp add: card_Un_disjoint insert_partition
```
```  2003        finite_subset [of _ "\<Union> (insert x F)"])
```
```  2004 done
```
```  2005
```
```  2006 lemma card_eq_UNIV_imp_eq_UNIV:
```
```  2007   assumes fin: "finite (UNIV :: 'a set)"
```
```  2008   and card: "card A = card (UNIV :: 'a set)"
```
```  2009   shows "A = (UNIV :: 'a set)"
```
```  2010 proof
```
```  2011   show "A \<subseteq> UNIV" by simp
```
```  2012   show "UNIV \<subseteq> A"
```
```  2013   proof
```
```  2014     fix x
```
```  2015     show "x \<in> A"
```
```  2016     proof (rule ccontr)
```
```  2017       assume "x \<notin> A"
```
```  2018       then have "A \<subset> UNIV" by auto
```
```  2019       with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
```
```  2020       with card show False by simp
```
```  2021     qed
```
```  2022   qed
```
```  2023 qed
```
```  2024
```
```  2025 text{*The form of a finite set of given cardinality*}
```
```  2026
```
```  2027 lemma card_eq_SucD:
```
```  2028 assumes "card A = Suc k"
```
```  2029 shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
```
```  2030 proof -
```
```  2031   have fin: "finite A" using assms by (auto intro: ccontr)
```
```  2032   moreover have "card A \<noteq> 0" using assms by auto
```
```  2033   ultimately obtain b where b: "b \<in> A" by auto
```
```  2034   show ?thesis
```
```  2035   proof (intro exI conjI)
```
```  2036     show "A = insert b (A-{b})" using b by blast
```
```  2037     show "b \<notin> A - {b}" by blast
```
```  2038     show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
```
```  2039       using assms b fin by(fastsimp dest:mk_disjoint_insert)+
```
```  2040   qed
```
```  2041 qed
```
```  2042
```
```  2043 lemma card_Suc_eq:
```
```  2044   "(card A = Suc k) =
```
```  2045    (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
```
```  2046 apply(rule iffI)
```
```  2047  apply(erule card_eq_SucD)
```
```  2048 apply(auto)
```
```  2049 apply(subst card_insert)
```
```  2050  apply(auto intro:ccontr)
```
```  2051 done
```
```  2052
```
```  2053 lemma finite_fun_UNIVD2:
```
```  2054   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
```
```  2055   shows "finite (UNIV :: 'b set)"
```
```  2056 proof -
```
```  2057   from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
```
```  2058     by(rule finite_imageI)
```
```  2059   moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
```
```  2060     by(rule UNIV_eq_I) auto
```
```  2061   ultimately show "finite (UNIV :: 'b set)" by simp
```
```  2062 qed
```
```  2063
```
```  2064 lemma card_UNIV_unit: "card (UNIV :: unit set) = 1"
```
```  2065   unfolding UNIV_unit by simp
```
```  2066
```
```  2067
```
```  2068 subsubsection {* Cardinality of image *}
```
```  2069
```
```  2070 lemma card_image_le: "finite A ==> card (f ` A) <= card A"
```
```  2071 apply (induct rule: finite_induct)
```
```  2072  apply simp
```
```  2073 apply (simp add: le_SucI card_insert_if)
```
```  2074 done
```
```  2075
```
```  2076 lemma card_image:
```
```  2077   assumes "inj_on f A"
```
```  2078   shows "card (f ` A) = card A"
```
```  2079 proof (cases "finite A")
```
```  2080   case True then show ?thesis using assms by (induct A) simp_all
```
```  2081 next
```
```  2082   case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
```
```  2083   with False show ?thesis by simp
```
```  2084 qed
```
```  2085
```
```  2086 lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
```
```  2087 by(auto simp: card_image bij_betw_def)
```
```  2088
```
```  2089 lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
```
```  2090 by (simp add: card_seteq card_image)
```
```  2091
```
```  2092 lemma eq_card_imp_inj_on:
```
```  2093   "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
```
```  2094 apply (induct rule:finite_induct)
```
```  2095 apply simp
```
```  2096 apply(frule card_image_le[where f = f])
```
```  2097 apply(simp add:card_insert_if split:if_splits)
```
```  2098 done
```
```  2099
```
```  2100 lemma inj_on_iff_eq_card:
```
```  2101   "finite A ==> inj_on f A = (card(f ` A) = card A)"
```
```  2102 by(blast intro: card_image eq_card_imp_inj_on)
```
```  2103
```
```  2104
```
```  2105 lemma card_inj_on_le:
```
```  2106   "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
```
```  2107 apply (subgoal_tac "finite A")
```
```  2108  apply (force intro: card_mono simp add: card_image [symmetric])
```
```  2109 apply (blast intro: finite_imageD dest: finite_subset)
```
```  2110 done
```
```  2111
```
```  2112 lemma card_bij_eq:
```
```  2113   "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
```
```  2114      finite A; finite B |] ==> card A = card B"
```
```  2115 by (auto intro: le_antisym card_inj_on_le)
```
```  2116
```
```  2117 lemma bij_betw_finite:
```
```  2118   assumes "bij_betw f A B"
```
```  2119   shows "finite A \<longleftrightarrow> finite B"
```
```  2120 using assms unfolding bij_betw_def
```
```  2121 using finite_imageD[of f A] by auto
```
```  2122
```
```  2123
```
```  2124 subsubsection {* Pigeonhole Principles *}
```
```  2125
```
```  2126 lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
```
```  2127 by (auto dest: card_image less_irrefl_nat)
```
```  2128
```
```  2129 lemma pigeonhole_infinite:
```
```  2130 assumes  "~ finite A" and "finite(f`A)"
```
```  2131 shows "EX a0:A. ~finite{a:A. f a = f a0}"
```
```  2132 proof -
```
```  2133   have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
```
```  2134   proof(induct "f`A" arbitrary: A rule: finite_induct)
```
```  2135     case empty thus ?case by simp
```
```  2136   next
```
```  2137     case (insert b F)
```
```  2138     show ?case
```
```  2139     proof cases
```
```  2140       assume "finite{a:A. f a = b}"
```
```  2141       hence "~ finite(A - {a:A. f a = b})" using `\<not> finite A` by simp
```
```  2142       also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
```
```  2143       finally have "~ finite({a:A. f a \<noteq> b})" .
```
```  2144       from insert(3)[OF _ this]
```
```  2145       show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
```
```  2146     next
```
```  2147       assume 1: "~finite{a:A. f a = b}"
```
```  2148       hence "{a \<in> A. f a = b} \<noteq> {}" by force
```
```  2149       thus ?thesis using 1 by blast
```
```  2150     qed
```
```  2151   qed
```
```  2152   from this[OF assms(2,1)] show ?thesis .
```
```  2153 qed
```
```  2154
```
```  2155 lemma pigeonhole_infinite_rel:
```
```  2156 assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
```
```  2157 shows "EX b:B. ~finite{a:A. R a b}"
```
```  2158 proof -
```
```  2159    let ?F = "%a. {b:B. R a b}"
```
```  2160    from finite_Pow_iff[THEN iffD2, OF `finite B`]
```
```  2161    have "finite(?F ` A)" by(blast intro: rev_finite_subset)
```
```  2162    from pigeonhole_infinite[where f = ?F, OF assms(1) this]
```
```  2163    obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
```
```  2164    obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast
```
```  2165    { assume "finite{a:A. R a b0}"
```
```  2166      then have "finite {a\<in>A. ?F a = ?F a0}"
```
```  2167        using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)
```
```  2168    }
```
```  2169    with 1 `b0 : B` show ?thesis by blast
```
```  2170 qed
```
```  2171
```
```  2172
```
```  2173 subsubsection {* Cardinality of sums *}
```
```  2174
```
```  2175 lemma card_Plus:
```
```  2176   assumes "finite A" and "finite B"
```
```  2177   shows "card (A <+> B) = card A + card B"
```
```  2178 proof -
```
```  2179   have "Inl`A \<inter> Inr`B = {}" by fast
```
```  2180   with assms show ?thesis
```
```  2181     unfolding Plus_def
```
```  2182     by (simp add: card_Un_disjoint card_image)
```
```  2183 qed
```
```  2184
```
```  2185 lemma card_Plus_conv_if:
```
```  2186   "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
```
```  2187   by (auto simp add: card_Plus)
```
```  2188
```
```  2189
```
```  2190 subsubsection {* Cardinality of the Powerset *}
```
```  2191
```
```  2192 lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
```
```  2193 apply (induct rule: finite_induct)
```
```  2194  apply (simp_all add: Pow_insert)
```
```  2195 apply (subst card_Un_disjoint, blast)
```
```  2196   apply (blast, blast)
```
```  2197 apply (subgoal_tac "inj_on (insert x) (Pow F)")
```
```  2198  apply (simp add: card_image Pow_insert)
```
```  2199 apply (unfold inj_on_def)
```
```  2200 apply (blast elim!: equalityE)
```
```  2201 done
```
```  2202
```
```  2203 text {* Relates to equivalence classes.  Based on a theorem of F. Kamm\"uller.  *}
```
```  2204
```
```  2205 lemma dvd_partition:
```
```  2206   "finite (Union C) ==>
```
```  2207     ALL c : C. k dvd card c ==>
```
```  2208     (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
```
```  2209   k dvd card (Union C)"
```
```  2210 apply (frule finite_UnionD)
```
```  2211 apply (rotate_tac -1)
```
```  2212 apply (induct rule: finite_induct)
```
```  2213 apply simp_all
```
```  2214 apply clarify
```
```  2215 apply (subst card_Un_disjoint)
```
```  2216    apply (auto simp add: disjoint_eq_subset_Compl)
```
```  2217 done
```
```  2218
```
```  2219
```
```  2220 subsubsection {* Relating injectivity and surjectivity *}
```
```  2221
```
```  2222 lemma finite_surj_inj: "finite A \<Longrightarrow> A \<subseteq> f ` A \<Longrightarrow> inj_on f A"
```
```  2223 apply(rule eq_card_imp_inj_on, assumption)
```
```  2224 apply(frule finite_imageI)
```
```  2225 apply(drule (1) card_seteq)
```
```  2226  apply(erule card_image_le)
```
```  2227 apply simp
```
```  2228 done
```
```  2229
```
```  2230 lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
```
```  2231 shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
```
```  2232 by (blast intro: finite_surj_inj subset_UNIV)
```
```  2233
```
```  2234 lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
```
```  2235 shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
```
```  2236 by(fastsimp simp:surj_def dest!: endo_inj_surj)
```
```  2237
```
```  2238 corollary infinite_UNIV_nat[iff]: "~finite(UNIV::nat set)"
```
```  2239 proof
```
```  2240   assume "finite(UNIV::nat set)"
```
```  2241   with finite_UNIV_inj_surj[of Suc]
```
```  2242   show False by simp (blast dest: Suc_neq_Zero surjD)
```
```  2243 qed
```
```  2244
```
```  2245 (* Often leads to bogus ATP proofs because of reduced type information, hence no_atp *)
```
```  2246 lemma infinite_UNIV_char_0[no_atp]:
```
```  2247   "\<not> finite (UNIV::'a::semiring_char_0 set)"
```
```  2248 proof
```
```  2249   assume "finite (UNIV::'a set)"
```
```  2250   with subset_UNIV have "finite (range of_nat::'a set)"
```
```  2251     by (rule finite_subset)
```
```  2252   moreover have "inj (of_nat::nat \<Rightarrow> 'a)"
```
```  2253     by (simp add: inj_on_def)
```
```  2254   ultimately have "finite (UNIV::nat set)"
```
```  2255     by (rule finite_imageD)
```
```  2256   then show "False"
```
```  2257     by simp
```
```  2258 qed
```
```  2259
```
```  2260 end
```