src/HOL/Finite_Set.thy
author hoelzl
Thu Jan 31 11:31:27 2013 +0100 (2013-01-31)
changeset 50999 3de230ed0547
parent 49806 acb6fa98e310
child 51290 c48477e76de5
permissions -rw-r--r--
introduce order topology
     1 (*  Title:      HOL/Finite_Set.thy
     2     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
     3                 with contributions by Jeremy Avigad
     4 *)
     5 
     6 header {* Finite sets *}
     7 
     8 theory Finite_Set
     9 imports Option Power
    10 begin
    11 
    12 subsection {* Predicate for finite sets *}
    13 
    14 inductive finite :: "'a set \<Rightarrow> bool"
    15   where
    16     emptyI [simp, intro!]: "finite {}"
    17   | insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)"
    18 
    19 simproc_setup finite_Collect ("finite (Collect P)") = {* K Set_Comprehension_Pointfree.simproc *}
    20 
    21 lemma finite_induct [case_names empty insert, induct set: finite]:
    22   -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
    23   assumes "finite F"
    24   assumes "P {}"
    25     and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
    26   shows "P F"
    27 using `finite F`
    28 proof induct
    29   show "P {}" by fact
    30   fix x F assume F: "finite F" and P: "P F"
    31   show "P (insert x F)"
    32   proof cases
    33     assume "x \<in> F"
    34     hence "insert x F = F" by (rule insert_absorb)
    35     with P show ?thesis by (simp only:)
    36   next
    37     assume "x \<notin> F"
    38     from F this P show ?thesis by (rule insert)
    39   qed
    40 qed
    41 
    42 
    43 subsubsection {* Choice principles *}
    44 
    45 lemma ex_new_if_finite: -- "does not depend on def of finite at all"
    46   assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
    47   shows "\<exists>a::'a. a \<notin> A"
    48 proof -
    49   from assms have "A \<noteq> UNIV" by blast
    50   then show ?thesis by blast
    51 qed
    52 
    53 text {* A finite choice principle. Does not need the SOME choice operator. *}
    54 
    55 lemma finite_set_choice:
    56   "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
    57 proof (induct rule: finite_induct)
    58   case empty then show ?case by simp
    59 next
    60   case (insert a A)
    61   then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
    62   show ?case (is "EX f. ?P f")
    63   proof
    64     show "?P(%x. if x = a then b else f x)" using f ab by auto
    65   qed
    66 qed
    67 
    68 
    69 subsubsection {* Finite sets are the images of initial segments of natural numbers *}
    70 
    71 lemma finite_imp_nat_seg_image_inj_on:
    72   assumes "finite A" 
    73   shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
    74 using assms
    75 proof induct
    76   case empty
    77   show ?case
    78   proof
    79     show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp 
    80   qed
    81 next
    82   case (insert a A)
    83   have notinA: "a \<notin> A" by fact
    84   from insert.hyps obtain n f
    85     where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
    86   hence "insert a A = f(n:=a) ` {i. i < Suc n}"
    87         "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
    88     by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
    89   thus ?case by blast
    90 qed
    91 
    92 lemma nat_seg_image_imp_finite:
    93   "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
    94 proof (induct n arbitrary: A)
    95   case 0 thus ?case by simp
    96 next
    97   case (Suc n)
    98   let ?B = "f ` {i. i < n}"
    99   have finB: "finite ?B" by(rule Suc.hyps[OF refl])
   100   show ?case
   101   proof cases
   102     assume "\<exists>k<n. f n = f k"
   103     hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
   104     thus ?thesis using finB by simp
   105   next
   106     assume "\<not>(\<exists> k<n. f n = f k)"
   107     hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
   108     thus ?thesis using finB by simp
   109   qed
   110 qed
   111 
   112 lemma finite_conv_nat_seg_image:
   113   "finite A \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"
   114   by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
   115 
   116 lemma finite_imp_inj_to_nat_seg:
   117   assumes "finite A"
   118   shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> inj_on f A"
   119 proof -
   120   from finite_imp_nat_seg_image_inj_on[OF `finite A`]
   121   obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
   122     by (auto simp:bij_betw_def)
   123   let ?f = "the_inv_into {i. i<n} f"
   124   have "inj_on ?f A & ?f ` A = {i. i<n}"
   125     by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
   126   thus ?thesis by blast
   127 qed
   128 
   129 lemma finite_Collect_less_nat [iff]:
   130   "finite {n::nat. n < k}"
   131   by (fastforce simp: finite_conv_nat_seg_image)
   132 
   133 lemma finite_Collect_le_nat [iff]:
   134   "finite {n::nat. n \<le> k}"
   135   by (simp add: le_eq_less_or_eq Collect_disj_eq)
   136 
   137 
   138 subsubsection {* Finiteness and common set operations *}
   139 
   140 lemma rev_finite_subset:
   141   "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
   142 proof (induct arbitrary: A rule: finite_induct)
   143   case empty
   144   then show ?case by simp
   145 next
   146   case (insert x F A)
   147   have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+
   148   show "finite A"
   149   proof cases
   150     assume x: "x \<in> A"
   151     with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
   152     with r have "finite (A - {x})" .
   153     hence "finite (insert x (A - {x}))" ..
   154     also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
   155     finally show ?thesis .
   156   next
   157     show "A \<subseteq> F ==> ?thesis" by fact
   158     assume "x \<notin> A"
   159     with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
   160   qed
   161 qed
   162 
   163 lemma finite_subset:
   164   "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
   165   by (rule rev_finite_subset)
   166 
   167 lemma finite_UnI:
   168   assumes "finite F" and "finite G"
   169   shows "finite (F \<union> G)"
   170   using assms by induct simp_all
   171 
   172 lemma finite_Un [iff]:
   173   "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
   174   by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])
   175 
   176 lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"
   177 proof -
   178   have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp
   179   then have "finite ({a} \<union> A) \<longleftrightarrow> finite A" by (simp only: finite_Un)
   180   then show ?thesis by simp
   181 qed
   182 
   183 lemma finite_Int [simp, intro]:
   184   "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
   185   by (blast intro: finite_subset)
   186 
   187 lemma finite_Collect_conjI [simp, intro]:
   188   "finite {x. P x} \<or> finite {x. Q x} \<Longrightarrow> finite {x. P x \<and> Q x}"
   189   by (simp add: Collect_conj_eq)
   190 
   191 lemma finite_Collect_disjI [simp]:
   192   "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"
   193   by (simp add: Collect_disj_eq)
   194 
   195 lemma finite_Diff [simp, intro]:
   196   "finite A \<Longrightarrow> finite (A - B)"
   197   by (rule finite_subset, rule Diff_subset)
   198 
   199 lemma finite_Diff2 [simp]:
   200   assumes "finite B"
   201   shows "finite (A - B) \<longleftrightarrow> finite A"
   202 proof -
   203   have "finite A \<longleftrightarrow> finite((A - B) \<union> (A \<inter> B))" by (simp add: Un_Diff_Int)
   204   also have "\<dots> \<longleftrightarrow> finite (A - B)" using `finite B` by simp
   205   finally show ?thesis ..
   206 qed
   207 
   208 lemma finite_Diff_insert [iff]:
   209   "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
   210 proof -
   211   have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
   212   moreover have "A - insert a B = A - B - {a}" by auto
   213   ultimately show ?thesis by simp
   214 qed
   215 
   216 lemma finite_compl[simp]:
   217   "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
   218   by (simp add: Compl_eq_Diff_UNIV)
   219 
   220 lemma finite_Collect_not[simp]:
   221   "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
   222   by (simp add: Collect_neg_eq)
   223 
   224 lemma finite_Union [simp, intro]:
   225   "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite(\<Union>A)"
   226   by (induct rule: finite_induct) simp_all
   227 
   228 lemma finite_UN_I [intro]:
   229   "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"
   230   by (induct rule: finite_induct) simp_all
   231 
   232 lemma finite_UN [simp]:
   233   "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
   234   by (blast intro: finite_subset)
   235 
   236 lemma finite_Inter [intro]:
   237   "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
   238   by (blast intro: Inter_lower finite_subset)
   239 
   240 lemma finite_INT [intro]:
   241   "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
   242   by (blast intro: INT_lower finite_subset)
   243 
   244 lemma finite_imageI [simp, intro]:
   245   "finite F \<Longrightarrow> finite (h ` F)"
   246   by (induct rule: finite_induct) simp_all
   247 
   248 lemma finite_image_set [simp]:
   249   "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
   250   by (simp add: image_Collect [symmetric])
   251 
   252 lemma finite_imageD:
   253   assumes "finite (f ` A)" and "inj_on f A"
   254   shows "finite A"
   255 using assms
   256 proof (induct "f ` A" arbitrary: A)
   257   case empty then show ?case by simp
   258 next
   259   case (insert x B)
   260   then have B_A: "insert x B = f ` A" by simp
   261   then obtain y where "x = f y" and "y \<in> A" by blast
   262   from B_A `x \<notin> B` have "B = f ` A - {x}" by blast
   263   with B_A `x \<notin> B` `x = f y` `inj_on f A` `y \<in> A` have "B = f ` (A - {y})" by (simp add: inj_on_image_set_diff)
   264   moreover from `inj_on f A` have "inj_on f (A - {y})" by (rule inj_on_diff)
   265   ultimately have "finite (A - {y})" by (rule insert.hyps)
   266   then show "finite A" by simp
   267 qed
   268 
   269 lemma finite_surj:
   270   "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
   271   by (erule finite_subset) (rule finite_imageI)
   272 
   273 lemma finite_range_imageI:
   274   "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
   275   by (drule finite_imageI) (simp add: range_composition)
   276 
   277 lemma finite_subset_image:
   278   assumes "finite B"
   279   shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
   280 using assms
   281 proof induct
   282   case empty then show ?case by simp
   283 next
   284   case insert then show ?case
   285     by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
   286        blast
   287 qed
   288 
   289 lemma finite_vimage_IntI:
   290   "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
   291   apply (induct rule: finite_induct)
   292    apply simp_all
   293   apply (subst vimage_insert)
   294   apply (simp add: finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2)
   295   done
   296 
   297 lemma finite_vimageI:
   298   "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
   299   using finite_vimage_IntI[of F h UNIV] by auto
   300 
   301 lemma finite_vimageD:
   302   assumes fin: "finite (h -` F)" and surj: "surj h"
   303   shows "finite F"
   304 proof -
   305   have "finite (h ` (h -` F))" using fin by (rule finite_imageI)
   306   also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)
   307   finally show "finite F" .
   308 qed
   309 
   310 lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
   311   unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
   312 
   313 lemma finite_Collect_bex [simp]:
   314   assumes "finite A"
   315   shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"
   316 proof -
   317   have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto
   318   with assms show ?thesis by simp
   319 qed
   320 
   321 lemma finite_Collect_bounded_ex [simp]:
   322   assumes "finite {y. P y}"
   323   shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
   324 proof -
   325   have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
   326   with assms show ?thesis by simp
   327 qed
   328 
   329 lemma finite_Plus:
   330   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
   331   by (simp add: Plus_def)
   332 
   333 lemma finite_PlusD: 
   334   fixes A :: "'a set" and B :: "'b set"
   335   assumes fin: "finite (A <+> B)"
   336   shows "finite A" "finite B"
   337 proof -
   338   have "Inl ` A \<subseteq> A <+> B" by auto
   339   then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
   340   then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
   341 next
   342   have "Inr ` B \<subseteq> A <+> B" by auto
   343   then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
   344   then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
   345 qed
   346 
   347 lemma finite_Plus_iff [simp]:
   348   "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
   349   by (auto intro: finite_PlusD finite_Plus)
   350 
   351 lemma finite_Plus_UNIV_iff [simp]:
   352   "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
   353   by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
   354 
   355 lemma finite_SigmaI [simp, intro]:
   356   "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
   357   by (unfold Sigma_def) blast
   358 
   359 lemma finite_cartesian_product:
   360   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
   361   by (rule finite_SigmaI)
   362 
   363 lemma finite_Prod_UNIV:
   364   "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
   365   by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
   366 
   367 lemma finite_cartesian_productD1:
   368   assumes "finite (A \<times> B)" and "B \<noteq> {}"
   369   shows "finite A"
   370 proof -
   371   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
   372     by (auto simp add: finite_conv_nat_seg_image)
   373   then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
   374   with `B \<noteq> {}` have "A = (fst \<circ> f) ` {i::nat. i < n}"
   375     by (simp add: image_compose)
   376   then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
   377   then show ?thesis
   378     by (auto simp add: finite_conv_nat_seg_image)
   379 qed
   380 
   381 lemma finite_cartesian_productD2:
   382   assumes "finite (A \<times> B)" and "A \<noteq> {}"
   383   shows "finite B"
   384 proof -
   385   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
   386     by (auto simp add: finite_conv_nat_seg_image)
   387   then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
   388   with `A \<noteq> {}` have "B = (snd \<circ> f) ` {i::nat. i < n}"
   389     by (simp add: image_compose)
   390   then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
   391   then show ?thesis
   392     by (auto simp add: finite_conv_nat_seg_image)
   393 qed
   394 
   395 lemma finite_prod: 
   396   "finite (UNIV :: ('a \<times> 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
   397 by(auto simp add: UNIV_Times_UNIV[symmetric] simp del: UNIV_Times_UNIV 
   398    dest: finite_cartesian_productD1 finite_cartesian_productD2)
   399 
   400 lemma finite_Pow_iff [iff]:
   401   "finite (Pow A) \<longleftrightarrow> finite A"
   402 proof
   403   assume "finite (Pow A)"
   404   then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
   405   then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
   406 next
   407   assume "finite A"
   408   then show "finite (Pow A)"
   409     by induct (simp_all add: Pow_insert)
   410 qed
   411 
   412 corollary finite_Collect_subsets [simp, intro]:
   413   "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
   414   by (simp add: Pow_def [symmetric])
   415 
   416 lemma finite_set: "finite (UNIV :: 'a set set) \<longleftrightarrow> finite (UNIV :: 'a set)"
   417 by(simp only: finite_Pow_iff Pow_UNIV[symmetric])
   418 
   419 lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
   420   by (blast intro: finite_subset [OF subset_Pow_Union])
   421 
   422 
   423 subsubsection {* Further induction rules on finite sets *}
   424 
   425 lemma finite_ne_induct [case_names singleton insert, consumes 2]:
   426   assumes "finite F" and "F \<noteq> {}"
   427   assumes "\<And>x. P {x}"
   428     and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
   429   shows "P F"
   430 using assms
   431 proof induct
   432   case empty then show ?case by simp
   433 next
   434   case (insert x F) then show ?case by cases auto
   435 qed
   436 
   437 lemma finite_subset_induct [consumes 2, case_names empty insert]:
   438   assumes "finite F" and "F \<subseteq> A"
   439   assumes empty: "P {}"
   440     and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
   441   shows "P F"
   442 using `finite F` `F \<subseteq> A`
   443 proof induct
   444   show "P {}" by fact
   445 next
   446   fix x F
   447   assume "finite F" and "x \<notin> F" and
   448     P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
   449   show "P (insert x F)"
   450   proof (rule insert)
   451     from i show "x \<in> A" by blast
   452     from i have "F \<subseteq> A" by blast
   453     with P show "P F" .
   454     show "finite F" by fact
   455     show "x \<notin> F" by fact
   456   qed
   457 qed
   458 
   459 lemma finite_empty_induct:
   460   assumes "finite A"
   461   assumes "P A"
   462     and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
   463   shows "P {}"
   464 proof -
   465   have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
   466   proof -
   467     fix B :: "'a set"
   468     assume "B \<subseteq> A"
   469     with `finite A` have "finite B" by (rule rev_finite_subset)
   470     from this `B \<subseteq> A` show "P (A - B)"
   471     proof induct
   472       case empty
   473       from `P A` show ?case by simp
   474     next
   475       case (insert b B)
   476       have "P (A - B - {b})"
   477       proof (rule remove)
   478         from `finite A` show "finite (A - B)" by induct auto
   479         from insert show "b \<in> A - B" by simp
   480         from insert show "P (A - B)" by simp
   481       qed
   482       also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
   483       finally show ?case .
   484     qed
   485   qed
   486   then have "P (A - A)" by blast
   487   then show ?thesis by simp
   488 qed
   489 
   490 
   491 subsection {* Class @{text finite}  *}
   492 
   493 class finite =
   494   assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
   495 begin
   496 
   497 lemma finite [simp]: "finite (A \<Colon> 'a set)"
   498   by (rule subset_UNIV finite_UNIV finite_subset)+
   499 
   500 lemma finite_code [code]: "finite (A \<Colon> 'a set) \<longleftrightarrow> True"
   501   by simp
   502 
   503 end
   504 
   505 instance prod :: (finite, finite) finite
   506   by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
   507 
   508 lemma inj_graph: "inj (%f. {(x, y). y = f x})"
   509   by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
   510 
   511 instance "fun" :: (finite, finite) finite
   512 proof
   513   show "finite (UNIV :: ('a => 'b) set)"
   514   proof (rule finite_imageD)
   515     let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
   516     have "range ?graph \<subseteq> Pow UNIV" by simp
   517     moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
   518       by (simp only: finite_Pow_iff finite)
   519     ultimately show "finite (range ?graph)"
   520       by (rule finite_subset)
   521     show "inj ?graph" by (rule inj_graph)
   522   qed
   523 qed
   524 
   525 instance bool :: finite
   526   by default (simp add: UNIV_bool)
   527 
   528 instance set :: (finite) finite
   529   by default (simp only: Pow_UNIV [symmetric] finite_Pow_iff finite)
   530 
   531 instance unit :: finite
   532   by default (simp add: UNIV_unit)
   533 
   534 instance sum :: (finite, finite) finite
   535   by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
   536 
   537 lemma finite_option_UNIV [simp]:
   538   "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
   539   by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
   540 
   541 instance option :: (finite) finite
   542   by default (simp add: UNIV_option_conv)
   543 
   544 
   545 subsection {* A basic fold functional for finite sets *}
   546 
   547 text {* The intended behaviour is
   548 @{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
   549 if @{text f} is ``left-commutative'':
   550 *}
   551 
   552 locale comp_fun_commute =
   553   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
   554   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
   555 begin
   556 
   557 lemma fun_left_comm: "f x (f y z) = f y (f x z)"
   558   using comp_fun_commute by (simp add: fun_eq_iff)
   559 
   560 end
   561 
   562 inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
   563 for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
   564   emptyI [intro]: "fold_graph f z {} z" |
   565   insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
   566       \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
   567 
   568 inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
   569 
   570 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
   571   "fold f z A = (THE y. fold_graph f z A y)"
   572 
   573 text{*A tempting alternative for the definiens is
   574 @{term "if finite A then THE y. fold_graph f z A y else e"}.
   575 It allows the removal of finiteness assumptions from the theorems
   576 @{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
   577 The proofs become ugly. It is not worth the effort. (???) *}
   578 
   579 lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
   580 by (induct rule: finite_induct) auto
   581 
   582 
   583 subsubsection{*From @{const fold_graph} to @{term fold}*}
   584 
   585 context comp_fun_commute
   586 begin
   587 
   588 lemma fold_graph_insertE_aux:
   589   "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'"
   590 proof (induct set: fold_graph)
   591   case (insertI x A y) show ?case
   592   proof (cases "x = a")
   593     assume "x = a" with insertI show ?case by auto
   594   next
   595     assume "x \<noteq> a"
   596     then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
   597       using insertI by auto
   598     have "f x y = f a (f x y')"
   599       unfolding y by (rule fun_left_comm)
   600     moreover have "fold_graph f z (insert x A - {a}) (f x y')"
   601       using y' and `x \<noteq> a` and `x \<notin> A`
   602       by (simp add: insert_Diff_if fold_graph.insertI)
   603     ultimately show ?case by fast
   604   qed
   605 qed simp
   606 
   607 lemma fold_graph_insertE:
   608   assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
   609   obtains y where "v = f x y" and "fold_graph f z A y"
   610 using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
   611 
   612 lemma fold_graph_determ:
   613   "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
   614 proof (induct arbitrary: y set: fold_graph)
   615   case (insertI x A y v)
   616   from `fold_graph f z (insert x A) v` and `x \<notin> A`
   617   obtain y' where "v = f x y'" and "fold_graph f z A y'"
   618     by (rule fold_graph_insertE)
   619   from `fold_graph f z A y'` have "y' = y" by (rule insertI)
   620   with `v = f x y'` show "v = f x y" by simp
   621 qed fast
   622 
   623 lemma fold_equality:
   624   "fold_graph f z A y \<Longrightarrow> fold f z A = y"
   625 by (unfold fold_def) (blast intro: fold_graph_determ)
   626 
   627 lemma fold_graph_fold:
   628   assumes "finite A"
   629   shows "fold_graph f z A (fold f z A)"
   630 proof -
   631   from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
   632   moreover note fold_graph_determ
   633   ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
   634   then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
   635   then show ?thesis by (unfold fold_def)
   636 qed
   637 
   638 text{* The base case for @{text fold}: *}
   639 
   640 lemma (in -) fold_empty [simp]: "fold f z {} = z"
   641 by (unfold fold_def) blast
   642 
   643 text{* The various recursion equations for @{const fold}: *}
   644 
   645 lemma fold_insert [simp]:
   646   assumes "finite A" and "x \<notin> A"
   647   shows "fold f z (insert x A) = f x (fold f z A)"
   648 proof (rule fold_equality)
   649   from `finite A` have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
   650   with `x \<notin> A`show "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)
   651 qed
   652 
   653 lemma fold_fun_comm:
   654   "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
   655 proof (induct rule: finite_induct)
   656   case empty then show ?case by simp
   657 next
   658   case (insert y A) then show ?case
   659     by (simp add: fun_left_comm[of x])
   660 qed
   661 
   662 lemma fold_insert2:
   663   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
   664 by (simp add: fold_fun_comm)
   665 
   666 lemma fold_rec:
   667   assumes "finite A" and "x \<in> A"
   668   shows "fold f z A = f x (fold f z (A - {x}))"
   669 proof -
   670   have A: "A = insert x (A - {x})" using `x \<in> A` by blast
   671   then have "fold f z A = fold f z (insert x (A - {x}))" by simp
   672   also have "\<dots> = f x (fold f z (A - {x}))"
   673     by (rule fold_insert) (simp add: `finite A`)+
   674   finally show ?thesis .
   675 qed
   676 
   677 lemma fold_insert_remove:
   678   assumes "finite A"
   679   shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
   680 proof -
   681   from `finite A` have "finite (insert x A)" by auto
   682   moreover have "x \<in> insert x A" by auto
   683   ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
   684     by (rule fold_rec)
   685   then show ?thesis by simp
   686 qed
   687 
   688 text{* Other properties of @{const fold}: *}
   689 
   690 lemma fold_image:
   691   assumes "finite A" and "inj_on g A"
   692   shows "fold f x (g ` A) = fold (f \<circ> g) x A"
   693 using assms
   694 proof induction
   695   case (insert a F)
   696     interpret comp_fun_commute "\<lambda>x. f (g x)" by default (simp add: comp_fun_commute)
   697     from insert show ?case by auto
   698 qed (simp)
   699 
   700 end
   701 
   702 lemma fold_cong:
   703   assumes "comp_fun_commute f" "comp_fun_commute g"
   704   assumes "finite A" and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
   705     and "A = B" and "s = t"
   706   shows "Finite_Set.fold f s A = Finite_Set.fold g t B"
   707 proof -
   708   have "Finite_Set.fold f s A = Finite_Set.fold g s A"  
   709   using `finite A` cong proof (induct A)
   710     case empty then show ?case by simp
   711   next
   712     case (insert x A)
   713     interpret f: comp_fun_commute f by (fact `comp_fun_commute f`)
   714     interpret g: comp_fun_commute g by (fact `comp_fun_commute g`)
   715     from insert show ?case by simp
   716   qed
   717   with assms show ?thesis by simp
   718 qed
   719 
   720 
   721 text{* A simplified version for idempotent functions: *}
   722 
   723 locale comp_fun_idem = comp_fun_commute +
   724   assumes comp_fun_idem: "f x o f x = f x"
   725 begin
   726 
   727 lemma fun_left_idem: "f x (f x z) = f x z"
   728   using comp_fun_idem by (simp add: fun_eq_iff)
   729 
   730 lemma fold_insert_idem:
   731   assumes fin: "finite A"
   732   shows "fold f z (insert x A) = f x (fold f z A)"
   733 proof cases
   734   assume "x \<in> A"
   735   then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
   736   then show ?thesis using assms by (simp add:fun_left_idem)
   737 next
   738   assume "x \<notin> A" then show ?thesis using assms by simp
   739 qed
   740 
   741 declare fold_insert[simp del] fold_insert_idem[simp]
   742 
   743 lemma fold_insert_idem2:
   744   "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
   745 by(simp add:fold_fun_comm)
   746 
   747 end
   748 
   749 
   750 subsubsection {* Liftings to @{text comp_fun_commute} etc. *}
   751 
   752 lemma (in comp_fun_commute) comp_comp_fun_commute:
   753   "comp_fun_commute (f \<circ> g)"
   754 proof
   755 qed (simp_all add: comp_fun_commute)
   756 
   757 lemma (in comp_fun_idem) comp_comp_fun_idem:
   758   "comp_fun_idem (f \<circ> g)"
   759   by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
   760     (simp_all add: comp_fun_idem)
   761 
   762 lemma (in comp_fun_commute) comp_fun_commute_funpow:
   763   "comp_fun_commute (\<lambda>x. f x ^^ g x)"
   764 proof
   765   fix y x
   766   show "f y ^^ g y \<circ> f x ^^ g x = f x ^^ g x \<circ> f y ^^ g y"
   767   proof (cases "x = y")
   768     case True then show ?thesis by simp
   769   next
   770     case False show ?thesis
   771     proof (induct "g x" arbitrary: g)
   772       case 0 then show ?case by simp
   773     next
   774       case (Suc n g)
   775       have hyp1: "f y ^^ g y \<circ> f x = f x \<circ> f y ^^ g y"
   776       proof (induct "g y" arbitrary: g)
   777         case 0 then show ?case by simp
   778       next
   779         case (Suc n g)
   780         def h \<equiv> "\<lambda>z. g z - 1"
   781         with Suc have "n = h y" by simp
   782         with Suc have hyp: "f y ^^ h y \<circ> f x = f x \<circ> f y ^^ h y"
   783           by auto
   784         from Suc h_def have "g y = Suc (h y)" by simp
   785         then show ?case by (simp add: comp_assoc hyp)
   786           (simp add: o_assoc comp_fun_commute)
   787       qed
   788       def h \<equiv> "\<lambda>z. if z = x then g x - 1 else g z"
   789       with Suc have "n = h x" by simp
   790       with Suc have "f y ^^ h y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ h y"
   791         by auto
   792       with False h_def have hyp2: "f y ^^ g y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ g y" by simp
   793       from Suc h_def have "g x = Suc (h x)" by simp
   794       then show ?case by (simp del: funpow.simps add: funpow_Suc_right o_assoc hyp2)
   795         (simp add: comp_assoc hyp1)
   796     qed
   797   qed
   798 qed
   799 
   800 
   801 subsubsection {* Expressing set operations via @{const fold} *}
   802 
   803 lemma comp_fun_idem_insert:
   804   "comp_fun_idem insert"
   805 proof
   806 qed auto
   807 
   808 lemma comp_fun_idem_remove:
   809   "comp_fun_idem Set.remove"
   810 proof
   811 qed auto
   812 
   813 lemma (in semilattice_inf) comp_fun_idem_inf:
   814   "comp_fun_idem inf"
   815 proof
   816 qed (auto simp add: inf_left_commute)
   817 
   818 lemma (in semilattice_sup) comp_fun_idem_sup:
   819   "comp_fun_idem sup"
   820 proof
   821 qed (auto simp add: sup_left_commute)
   822 
   823 lemma union_fold_insert:
   824   assumes "finite A"
   825   shows "A \<union> B = fold insert B A"
   826 proof -
   827   interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
   828   from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
   829 qed
   830 
   831 lemma minus_fold_remove:
   832   assumes "finite A"
   833   shows "B - A = fold Set.remove B A"
   834 proof -
   835   interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove)
   836   from `finite A` have "fold Set.remove B A = B - A" by (induct A arbitrary: B) auto
   837   then show ?thesis ..
   838 qed
   839 
   840 lemma comp_fun_commute_filter_fold: "comp_fun_commute (\<lambda>x A'. if P x then Set.insert x A' else A')"
   841 proof - 
   842   interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
   843   show ?thesis by default (auto simp: fun_eq_iff)
   844 qed
   845 
   846 lemma Set_filter_fold:
   847   assumes "finite A"
   848   shows "Set.filter P A = fold (\<lambda>x A'. if P x then Set.insert x A' else A') {} A"
   849 using assms
   850 by (induct A) 
   851   (auto simp add: Set.filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold])
   852 
   853 lemma inter_Set_filter:     
   854   assumes "finite B"
   855   shows "A \<inter> B = Set.filter (\<lambda>x. x \<in> A) B"
   856 using assms 
   857 by (induct B) (auto simp: Set.filter_def)
   858 
   859 lemma image_fold_insert:
   860   assumes "finite A"
   861   shows "image f A = fold (\<lambda>k A. Set.insert (f k) A) {} A"
   862 using assms
   863 proof -
   864   interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A" by default auto
   865   show ?thesis using assms by (induct A) auto
   866 qed
   867 
   868 lemma Ball_fold:
   869   assumes "finite A"
   870   shows "Ball A P = fold (\<lambda>k s. s \<and> P k) True A"
   871 using assms
   872 proof -
   873   interpret comp_fun_commute "\<lambda>k s. s \<and> P k" by default auto
   874   show ?thesis using assms by (induct A) auto
   875 qed
   876 
   877 lemma Bex_fold:
   878   assumes "finite A"
   879   shows "Bex A P = fold (\<lambda>k s. s \<or> P k) False A"
   880 using assms
   881 proof -
   882   interpret comp_fun_commute "\<lambda>k s. s \<or> P k" by default auto
   883   show ?thesis using assms by (induct A) auto
   884 qed
   885 
   886 lemma comp_fun_commute_Pow_fold: 
   887   "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)" 
   888   by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast
   889 
   890 lemma Pow_fold:
   891   assumes "finite A"
   892   shows "Pow A = fold (\<lambda>x A. A \<union> Set.insert x ` A) {{}} A"
   893 using assms
   894 proof -
   895   interpret comp_fun_commute "\<lambda>x A. A \<union> Set.insert x ` A" by (rule comp_fun_commute_Pow_fold)
   896   show ?thesis using assms by (induct A) (auto simp: Pow_insert)
   897 qed
   898 
   899 lemma fold_union_pair:
   900   assumes "finite B"
   901   shows "(\<Union>y\<in>B. {(x, y)}) \<union> A = fold (\<lambda>y. Set.insert (x, y)) A B"
   902 proof -
   903   interpret comp_fun_commute "\<lambda>y. Set.insert (x, y)" by default auto
   904   show ?thesis using assms  by (induct B arbitrary: A) simp_all
   905 qed
   906 
   907 lemma comp_fun_commute_product_fold: 
   908   assumes "finite B"
   909   shows "comp_fun_commute (\<lambda>x A. fold (\<lambda>y. Set.insert (x, y)) A B)" 
   910 by default (auto simp: fold_union_pair[symmetric] assms)
   911 
   912 lemma product_fold:
   913   assumes "finite A"
   914   assumes "finite B"
   915   shows "A \<times> B = fold (\<lambda>x A. fold (\<lambda>y. Set.insert (x, y)) A B) {} A"
   916 using assms unfolding Sigma_def 
   917 by (induct A) 
   918   (simp_all add: comp_fun_commute.fold_insert[OF comp_fun_commute_product_fold] fold_union_pair)
   919 
   920 
   921 context complete_lattice
   922 begin
   923 
   924 lemma inf_Inf_fold_inf:
   925   assumes "finite A"
   926   shows "inf B (Inf A) = fold inf B A"
   927 proof -
   928   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
   929   from `finite A` show ?thesis by (induct A arbitrary: B)
   930     (simp_all add: inf_commute fold_fun_comm)
   931 qed
   932 
   933 lemma sup_Sup_fold_sup:
   934   assumes "finite A"
   935   shows "sup B (Sup A) = fold sup B A"
   936 proof -
   937   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
   938   from `finite A` show ?thesis by (induct A arbitrary: B)
   939     (simp_all add: sup_commute fold_fun_comm)
   940 qed
   941 
   942 lemma Inf_fold_inf:
   943   assumes "finite A"
   944   shows "Inf A = fold inf top A"
   945   using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
   946 
   947 lemma Sup_fold_sup:
   948   assumes "finite A"
   949   shows "Sup A = fold sup bot A"
   950   using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
   951 
   952 lemma inf_INF_fold_inf:
   953   assumes "finite A"
   954   shows "inf B (INFI A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold") 
   955 proof (rule sym)
   956   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
   957   interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
   958   from `finite A` show "?fold = ?inf"
   959     by (induct A arbitrary: B)
   960       (simp_all add: INF_def inf_left_commute)
   961 qed
   962 
   963 lemma sup_SUP_fold_sup:
   964   assumes "finite A"
   965   shows "sup B (SUPR A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold") 
   966 proof (rule sym)
   967   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
   968   interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
   969   from `finite A` show "?fold = ?sup"
   970     by (induct A arbitrary: B)
   971       (simp_all add: SUP_def sup_left_commute)
   972 qed
   973 
   974 lemma INF_fold_inf:
   975   assumes "finite A"
   976   shows "INFI A f = fold (inf \<circ> f) top A"
   977   using assms inf_INF_fold_inf [of A top] by simp
   978 
   979 lemma SUP_fold_sup:
   980   assumes "finite A"
   981   shows "SUPR A f = fold (sup \<circ> f) bot A"
   982   using assms sup_SUP_fold_sup [of A bot] by simp
   983 
   984 end
   985 
   986 
   987 subsection {* The derived combinator @{text fold_image} *}
   988 
   989 definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
   990   where "fold_image f g = fold (\<lambda>x y. f (g x) y)"
   991 
   992 lemma fold_image_empty[simp]: "fold_image f g z {} = z"
   993   by (simp add:fold_image_def)
   994 
   995 context ab_semigroup_mult
   996 begin
   997 
   998 lemma fold_image_insert[simp]:
   999   assumes "finite A" and "a \<notin> A"
  1000   shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
  1001 proof -
  1002   interpret comp_fun_commute "%x y. (g x) * y"
  1003     by default (simp add: fun_eq_iff mult_ac)
  1004   from assms show ?thesis by (simp add: fold_image_def)
  1005 qed
  1006 
  1007 lemma fold_image_reindex:
  1008   assumes "finite A"
  1009   shows "inj_on h A \<Longrightarrow> fold_image times g z (h ` A) = fold_image times (g \<circ> h) z A"
  1010   using assms by induct auto
  1011 
  1012 lemma fold_image_cong:
  1013   assumes "finite A" and g_h: "\<And>x. x\<in>A \<Longrightarrow> g x = h x"
  1014   shows "fold_image times g z A = fold_image times h z A"
  1015 proof -
  1016   from `finite A`
  1017   have "\<And>C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C"
  1018   proof (induct arbitrary: C)
  1019     case empty then show ?case by simp
  1020   next
  1021     case (insert x F) then show ?case apply -
  1022     apply (simp add: subset_insert_iff, clarify)
  1023     apply (subgoal_tac "finite C")
  1024       prefer 2 apply (blast dest: finite_subset [rotated])
  1025     apply (subgoal_tac "C = insert x (C - {x})")
  1026       prefer 2 apply blast
  1027     apply (erule ssubst)
  1028     apply (simp add: Ball_def del: insert_Diff_single)
  1029     done
  1030   qed
  1031   with g_h show ?thesis by simp
  1032 qed
  1033 
  1034 end
  1035 
  1036 context comm_monoid_mult
  1037 begin
  1038 
  1039 lemma fold_image_1:
  1040   "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
  1041   apply (induct rule: finite_induct)
  1042   apply simp by auto
  1043 
  1044 lemma fold_image_Un_Int:
  1045   "finite A ==> finite B ==>
  1046     fold_image times g 1 A * fold_image times g 1 B =
  1047     fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
  1048   apply (induct rule: finite_induct)
  1049 by (induct set: finite) 
  1050    (auto simp add: mult_ac insert_absorb Int_insert_left)
  1051 
  1052 lemma fold_image_Un_one:
  1053   assumes fS: "finite S" and fT: "finite T"
  1054   and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
  1055   shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
  1056 proof-
  1057   have "fold_image op * f 1 (S \<inter> T) = 1" 
  1058     apply (rule fold_image_1)
  1059     using fS fT I0 by auto 
  1060   with fold_image_Un_Int[OF fS fT] show ?thesis by simp
  1061 qed
  1062 
  1063 corollary fold_Un_disjoint:
  1064   "finite A ==> finite B ==> A Int B = {} ==>
  1065    fold_image times g 1 (A Un B) =
  1066    fold_image times g 1 A * fold_image times g 1 B"
  1067 by (simp add: fold_image_Un_Int)
  1068 
  1069 lemma fold_image_UN_disjoint:
  1070   "\<lbrakk> finite I; ALL i:I. finite (A i);
  1071      ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
  1072    \<Longrightarrow> fold_image times g 1 (UNION I A) =
  1073        fold_image times (%i. fold_image times g 1 (A i)) 1 I"
  1074 apply (induct rule: finite_induct)
  1075 apply simp
  1076 apply atomize
  1077 apply (subgoal_tac "ALL i:F. x \<noteq> i")
  1078  prefer 2 apply blast
  1079 apply (subgoal_tac "A x Int UNION F A = {}")
  1080  prefer 2 apply blast
  1081 apply (simp add: fold_Un_disjoint)
  1082 done
  1083 
  1084 lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
  1085   fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A =
  1086   fold_image times (split g) 1 (SIGMA x:A. B x)"
  1087 apply (subst Sigma_def)
  1088 apply (subst fold_image_UN_disjoint, assumption, simp)
  1089  apply blast
  1090 apply (erule fold_image_cong)
  1091 apply (subst fold_image_UN_disjoint, simp, simp)
  1092  apply blast
  1093 apply simp
  1094 done
  1095 
  1096 lemma fold_image_distrib: "finite A \<Longrightarrow>
  1097    fold_image times (%x. g x * h x) 1 A =
  1098    fold_image times g 1 A *  fold_image times h 1 A"
  1099 by (erule finite_induct) (simp_all add: mult_ac)
  1100 
  1101 lemma fold_image_related: 
  1102   assumes Re: "R e e" 
  1103   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
  1104   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
  1105   shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
  1106   using fS by (rule finite_subset_induct) (insert assms, auto)
  1107 
  1108 lemma  fold_image_eq_general:
  1109   assumes fS: "finite S"
  1110   and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y" 
  1111   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
  1112   shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
  1113 proof-
  1114   from h f12 have hS: "h ` S = S'" by auto
  1115   {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
  1116     from f12 h H  have "x = y" by auto }
  1117   hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
  1118   from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
  1119   from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
  1120   also have "\<dots> = fold_image (op *) (f2 o h) e S" 
  1121     using fold_image_reindex[OF fS hinj, of f2 e] .
  1122   also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
  1123     by blast
  1124   finally show ?thesis ..
  1125 qed
  1126 
  1127 lemma fold_image_eq_general_inverses:
  1128   assumes fS: "finite S" 
  1129   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
  1130   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
  1131   shows "fold_image (op *) f e S = fold_image (op *) g e T"
  1132   (* metis solves it, but not yet available here *)
  1133   apply (rule fold_image_eq_general[OF fS, of T h g f e])
  1134   apply (rule ballI)
  1135   apply (frule kh)
  1136   apply (rule ex1I[])
  1137   apply blast
  1138   apply clarsimp
  1139   apply (drule hk) apply simp
  1140   apply (rule sym)
  1141   apply (erule conjunct1[OF conjunct2[OF hk]])
  1142   apply (rule ballI)
  1143   apply (drule  hk)
  1144   apply blast
  1145   done
  1146 
  1147 end
  1148 
  1149 
  1150 subsection {* A fold functional for non-empty sets *}
  1151 
  1152 text{* Does not require start value. *}
  1153 
  1154 inductive
  1155   fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
  1156   for f :: "'a => 'a => 'a"
  1157 where
  1158   fold1Set_insertI [intro]:
  1159    "\<lbrakk> fold_graph f a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
  1160 
  1161 definition fold1 :: "('a => 'a => 'a) => 'a set => 'a" where
  1162   "fold1 f A == THE x. fold1Set f A x"
  1163 
  1164 lemma fold1Set_nonempty:
  1165   "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
  1166 by(erule fold1Set.cases, simp_all)
  1167 
  1168 inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
  1169 
  1170 inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
  1171 
  1172 
  1173 lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
  1174 by (blast elim: fold_graph.cases)
  1175 
  1176 lemma fold1_singleton [simp]: "fold1 f {a} = a"
  1177 by (unfold fold1_def) blast
  1178 
  1179 lemma finite_nonempty_imp_fold1Set:
  1180   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
  1181 apply (induct A rule: finite_induct)
  1182 apply (auto dest: finite_imp_fold_graph [of _ f])
  1183 done
  1184 
  1185 text{*First, some lemmas about @{const fold_graph}.*}
  1186 
  1187 context ab_semigroup_mult
  1188 begin
  1189 
  1190 lemma comp_fun_commute: "comp_fun_commute (op *)"
  1191   by default (simp add: fun_eq_iff mult_ac)
  1192 
  1193 lemma fold_graph_insert_swap:
  1194 assumes fold: "fold_graph times (b::'a) A y" and "b \<notin> A"
  1195 shows "fold_graph times z (insert b A) (z * y)"
  1196 proof -
  1197   interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
  1198 from assms show ?thesis
  1199 proof (induct rule: fold_graph.induct)
  1200   case emptyI show ?case by (subst mult_commute [of z b], fast)
  1201 next
  1202   case (insertI x A y)
  1203     have "fold_graph times z (insert x (insert b A)) (x * (z * y))"
  1204       using insertI by force  --{*how does @{term id} get unfolded?*}
  1205     thus ?case by (simp add: insert_commute mult_ac)
  1206 qed
  1207 qed
  1208 
  1209 lemma fold_graph_permute_diff:
  1210 assumes fold: "fold_graph times b A x"
  1211 shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> fold_graph times a (insert b (A-{a})) x"
  1212 using fold
  1213 proof (induct rule: fold_graph.induct)
  1214   case emptyI thus ?case by simp
  1215 next
  1216   case (insertI x A y)
  1217   have "a = x \<or> a \<in> A" using insertI by simp
  1218   thus ?case
  1219   proof
  1220     assume "a = x"
  1221     with insertI show ?thesis
  1222       by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap)
  1223   next
  1224     assume ainA: "a \<in> A"
  1225     hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)"
  1226       using insertI by force
  1227     moreover
  1228     have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
  1229       using ainA insertI by blast
  1230     ultimately show ?thesis by simp
  1231   qed
  1232 qed
  1233 
  1234 lemma fold1_eq_fold:
  1235 assumes "finite A" "a \<notin> A" shows "fold1 times (insert a A) = fold times a A"
  1236 proof -
  1237   interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
  1238   from assms show ?thesis
  1239 apply (simp add: fold1_def fold_def)
  1240 apply (rule the_equality)
  1241 apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times])
  1242 apply (rule sym, clarify)
  1243 apply (case_tac "Aa=A")
  1244  apply (best intro: fold_graph_determ)
  1245 apply (subgoal_tac "fold_graph times a A x")
  1246  apply (best intro: fold_graph_determ)
  1247 apply (subgoal_tac "insert aa (Aa - {a}) = A")
  1248  prefer 2 apply (blast elim: equalityE)
  1249 apply (auto dest: fold_graph_permute_diff [where a=a])
  1250 done
  1251 qed
  1252 
  1253 lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
  1254 apply safe
  1255  apply simp
  1256  apply (drule_tac x=x in spec)
  1257  apply (drule_tac x="A-{x}" in spec, auto)
  1258 done
  1259 
  1260 lemma fold1_insert:
  1261   assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
  1262   shows "fold1 times (insert x A) = x * fold1 times A"
  1263 proof -
  1264   interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
  1265   from nonempty obtain a A' where "A = insert a A' & a ~: A'"
  1266     by (auto simp add: nonempty_iff)
  1267   with A show ?thesis
  1268     by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
  1269 qed
  1270 
  1271 end
  1272 
  1273 context ab_semigroup_idem_mult
  1274 begin
  1275 
  1276 lemma comp_fun_idem: "comp_fun_idem (op *)"
  1277   by default (simp_all add: fun_eq_iff mult_left_commute)
  1278 
  1279 lemma fold1_insert_idem [simp]:
  1280   assumes nonempty: "A \<noteq> {}" and A: "finite A" 
  1281   shows "fold1 times (insert x A) = x * fold1 times A"
  1282 proof -
  1283   interpret comp_fun_idem "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a"
  1284     by (rule comp_fun_idem)
  1285   from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
  1286     by (auto simp add: nonempty_iff)
  1287   show ?thesis
  1288   proof cases
  1289     assume a: "a = x"
  1290     show ?thesis
  1291     proof cases
  1292       assume "A' = {}"
  1293       with A' a show ?thesis by simp
  1294     next
  1295       assume "A' \<noteq> {}"
  1296       with A A' a show ?thesis
  1297         by (simp add: fold1_insert mult_assoc [symmetric])
  1298     qed
  1299   next
  1300     assume "a \<noteq> x"
  1301     with A A' show ?thesis
  1302       by (simp add: insert_commute fold1_eq_fold)
  1303   qed
  1304 qed
  1305 
  1306 lemma hom_fold1_commute:
  1307 assumes hom: "!!x y. h (x * y) = h x * h y"
  1308 and N: "finite N" "N \<noteq> {}" shows "h (fold1 times N) = fold1 times (h ` N)"
  1309 using N
  1310 proof (induct rule: finite_ne_induct)
  1311   case singleton thus ?case by simp
  1312 next
  1313   case (insert n N)
  1314   then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp
  1315   also have "\<dots> = h n * h (fold1 times N)" by(rule hom)
  1316   also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert)
  1317   also have "times (h n) \<dots> = fold1 times (insert (h n) (h ` N))"
  1318     using insert by(simp)
  1319   also have "insert (h n) (h ` N) = h ` insert n N" by simp
  1320   finally show ?case .
  1321 qed
  1322 
  1323 lemma fold1_eq_fold_idem:
  1324   assumes "finite A"
  1325   shows "fold1 times (insert a A) = fold times a A"
  1326 proof (cases "a \<in> A")
  1327   case False
  1328   with assms show ?thesis by (simp add: fold1_eq_fold)
  1329 next
  1330   interpret comp_fun_idem times by (fact comp_fun_idem)
  1331   case True then obtain b B
  1332     where A: "A = insert a B" and "a \<notin> B" by (rule set_insert)
  1333   with assms have "finite B" by auto
  1334   then have "fold times a (insert a B) = fold times (a * a) B"
  1335     using `a \<notin> B` by (rule fold_insert2)
  1336   then show ?thesis
  1337     using `a \<notin> B` `finite B` by (simp add: fold1_eq_fold A)
  1338 qed
  1339 
  1340 end
  1341 
  1342 
  1343 text{* Now the recursion rules for definitions: *}
  1344 
  1345 lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
  1346 by simp
  1347 
  1348 lemma (in ab_semigroup_mult) fold1_insert_def:
  1349   "\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
  1350 by (simp add:fold1_insert)
  1351 
  1352 lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:
  1353   "\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
  1354 by simp
  1355 
  1356 subsubsection{* Determinacy for @{term fold1Set} *}
  1357 
  1358 (*Not actually used!!*)
  1359 (*
  1360 context ab_semigroup_mult
  1361 begin
  1362 
  1363 lemma fold_graph_permute:
  1364   "[|fold_graph times id b (insert a A) x; a \<notin> A; b \<notin> A|]
  1365    ==> fold_graph times id a (insert b A) x"
  1366 apply (cases "a=b") 
  1367 apply (auto dest: fold_graph_permute_diff) 
  1368 done
  1369 
  1370 lemma fold1Set_determ:
  1371   "fold1Set times A x ==> fold1Set times A y ==> y = x"
  1372 proof (clarify elim!: fold1Set.cases)
  1373   fix A x B y a b
  1374   assume Ax: "fold_graph times id a A x"
  1375   assume By: "fold_graph times id b B y"
  1376   assume anotA:  "a \<notin> A"
  1377   assume bnotB:  "b \<notin> B"
  1378   assume eq: "insert a A = insert b B"
  1379   show "y=x"
  1380   proof cases
  1381     assume same: "a=b"
  1382     hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
  1383     thus ?thesis using Ax By same by (blast intro: fold_graph_determ)
  1384   next
  1385     assume diff: "a\<noteq>b"
  1386     let ?D = "B - {a}"
  1387     have B: "B = insert a ?D" and A: "A = insert b ?D"
  1388      and aB: "a \<in> B" and bA: "b \<in> A"
  1389       using eq anotA bnotB diff by (blast elim!:equalityE)+
  1390     with aB bnotB By
  1391     have "fold_graph times id a (insert b ?D) y" 
  1392       by (auto intro: fold_graph_permute simp add: insert_absorb)
  1393     moreover
  1394     have "fold_graph times id a (insert b ?D) x"
  1395       by (simp add: A [symmetric] Ax) 
  1396     ultimately show ?thesis by (blast intro: fold_graph_determ) 
  1397   qed
  1398 qed
  1399 
  1400 lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y"
  1401   by (unfold fold1_def) (blast intro: fold1Set_determ)
  1402 
  1403 end
  1404 *)
  1405 
  1406 declare
  1407   empty_fold_graphE [rule del]  fold_graph.intros [rule del]
  1408   empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
  1409   -- {* No more proofs involve these relations. *}
  1410 
  1411 subsubsection {* Lemmas about @{text fold1} *}
  1412 
  1413 context ab_semigroup_mult
  1414 begin
  1415 
  1416 lemma fold1_Un:
  1417 assumes A: "finite A" "A \<noteq> {}"
  1418 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
  1419        fold1 times (A Un B) = fold1 times A * fold1 times B"
  1420 using A by (induct rule: finite_ne_induct)
  1421   (simp_all add: fold1_insert mult_assoc)
  1422 
  1423 lemma fold1_in:
  1424   assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x,y}"
  1425   shows "fold1 times A \<in> A"
  1426 using A
  1427 proof (induct rule:finite_ne_induct)
  1428   case singleton thus ?case by simp
  1429 next
  1430   case insert thus ?case using elem by (force simp add:fold1_insert)
  1431 qed
  1432 
  1433 end
  1434 
  1435 lemma (in ab_semigroup_idem_mult) fold1_Un2:
  1436 assumes A: "finite A" "A \<noteq> {}"
  1437 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
  1438        fold1 times (A Un B) = fold1 times A * fold1 times B"
  1439 using A
  1440 proof(induct rule:finite_ne_induct)
  1441   case singleton thus ?case by simp
  1442 next
  1443   case insert thus ?case by (simp add: mult_assoc)
  1444 qed
  1445 
  1446 
  1447 subsection {* Locales as mini-packages for fold operations *}
  1448 
  1449 subsubsection {* The natural case *}
  1450 
  1451 locale folding =
  1452   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
  1453   fixes F :: "'a set \<Rightarrow> 'b \<Rightarrow> 'b"
  1454   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
  1455   assumes eq_fold: "finite A \<Longrightarrow> F A s = fold f s A"
  1456 begin
  1457 
  1458 lemma empty [simp]:
  1459   "F {} = id"
  1460   by (simp add: eq_fold fun_eq_iff)
  1461 
  1462 lemma insert [simp]:
  1463   assumes "finite A" and "x \<notin> A"
  1464   shows "F (insert x A) = F A \<circ> f x"
  1465 proof -
  1466   interpret comp_fun_commute f
  1467     by default (insert comp_fun_commute, simp add: fun_eq_iff)
  1468   from fold_insert2 assms
  1469   have "\<And>s. fold f s (insert x A) = fold f (f x s) A" .
  1470   with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
  1471 qed
  1472 
  1473 lemma remove:
  1474   assumes "finite A" and "x \<in> A"
  1475   shows "F A = F (A - {x}) \<circ> f x"
  1476 proof -
  1477   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
  1478     by (auto dest: mk_disjoint_insert)
  1479   moreover from `finite A` this have "finite B" by simp
  1480   ultimately show ?thesis by simp
  1481 qed
  1482 
  1483 lemma insert_remove:
  1484   assumes "finite A"
  1485   shows "F (insert x A) = F (A - {x}) \<circ> f x"
  1486   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
  1487 
  1488 lemma commute_left_comp:
  1489   "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
  1490   by (simp add: o_assoc comp_fun_commute)
  1491 
  1492 lemma comp_fun_commute':
  1493   assumes "finite A"
  1494   shows "f x \<circ> F A = F A \<circ> f x"
  1495   using assms by (induct A)
  1496     (simp, simp del: o_apply add: o_assoc, simp del: o_apply add: comp_assoc comp_fun_commute)
  1497 
  1498 lemma commute_left_comp':
  1499   assumes "finite A"
  1500   shows "f x \<circ> (F A \<circ> g) = F A \<circ> (f x \<circ> g)"
  1501   using assms by (simp add: o_assoc comp_fun_commute')
  1502 
  1503 lemma comp_fun_commute'':
  1504   assumes "finite A" and "finite B"
  1505   shows "F B \<circ> F A = F A \<circ> F B"
  1506   using assms by (induct A)
  1507     (simp_all add: o_assoc, simp add: comp_assoc comp_fun_commute')
  1508 
  1509 lemma commute_left_comp'':
  1510   assumes "finite A" and "finite B"
  1511   shows "F B \<circ> (F A \<circ> g) = F A \<circ> (F B \<circ> g)"
  1512   using assms by (simp add: o_assoc comp_fun_commute'')
  1513 
  1514 lemmas comp_fun_commutes = comp_assoc comp_fun_commute commute_left_comp
  1515   comp_fun_commute' commute_left_comp' comp_fun_commute'' commute_left_comp''
  1516 
  1517 lemma union_inter:
  1518   assumes "finite A" and "finite B"
  1519   shows "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B"
  1520   using assms by (induct A)
  1521     (simp_all del: o_apply add: insert_absorb Int_insert_left comp_fun_commutes,
  1522       simp add: o_assoc)
  1523 
  1524 lemma union:
  1525   assumes "finite A" and "finite B"
  1526   and "A \<inter> B = {}"
  1527   shows "F (A \<union> B) = F A \<circ> F B"
  1528 proof -
  1529   from union_inter `finite A` `finite B` have "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B" .
  1530   with `A \<inter> B = {}` show ?thesis by simp
  1531 qed
  1532 
  1533 end
  1534 
  1535 
  1536 subsubsection {* The natural case with idempotency *}
  1537 
  1538 locale folding_idem = folding +
  1539   assumes idem_comp: "f x \<circ> f x = f x"
  1540 begin
  1541 
  1542 lemma idem_left_comp:
  1543   "f x \<circ> (f x \<circ> g) = f x \<circ> g"
  1544   by (simp add: o_assoc idem_comp)
  1545 
  1546 lemma in_comp_idem:
  1547   assumes "finite A" and "x \<in> A"
  1548   shows "F A \<circ> f x = F A"
  1549 using assms by (induct A)
  1550   (auto simp add: comp_fun_commutes idem_comp, simp add: commute_left_comp' [symmetric] comp_fun_commute')
  1551 
  1552 lemma subset_comp_idem:
  1553   assumes "finite A" and "B \<subseteq> A"
  1554   shows "F A \<circ> F B = F A"
  1555 proof -
  1556   from assms have "finite B" by (blast dest: finite_subset)
  1557   then show ?thesis using `B \<subseteq> A` by (induct B)
  1558     (simp_all add: o_assoc in_comp_idem `finite A`)
  1559 qed
  1560 
  1561 declare insert [simp del]
  1562 
  1563 lemma insert_idem [simp]:
  1564   assumes "finite A"
  1565   shows "F (insert x A) = F A \<circ> f x"
  1566   using assms by (cases "x \<in> A") (simp_all add: insert in_comp_idem insert_absorb)
  1567 
  1568 lemma union_idem:
  1569   assumes "finite A" and "finite B"
  1570   shows "F (A \<union> B) = F A \<circ> F B"
  1571 proof -
  1572   from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
  1573   then have "F (A \<union> B) \<circ> F (A \<inter> B) = F (A \<union> B)" by (rule subset_comp_idem)
  1574   with assms show ?thesis by (simp add: union_inter)
  1575 qed
  1576 
  1577 end
  1578 
  1579 
  1580 subsubsection {* The image case with fixed function *}
  1581 
  1582 no_notation times (infixl "*" 70)
  1583 no_notation Groups.one ("1")
  1584 
  1585 locale folding_image_simple = comm_monoid +
  1586   fixes g :: "('b \<Rightarrow> 'a)"
  1587   fixes F :: "'b set \<Rightarrow> 'a"
  1588   assumes eq_fold_g: "finite A \<Longrightarrow> F A = fold_image f g 1 A"
  1589 begin
  1590 
  1591 lemma empty [simp]:
  1592   "F {} = 1"
  1593   by (simp add: eq_fold_g)
  1594 
  1595 lemma insert [simp]:
  1596   assumes "finite A" and "x \<notin> A"
  1597   shows "F (insert x A) = g x * F A"
  1598 proof -
  1599   interpret comp_fun_commute "%x y. (g x) * y"
  1600     by default (simp add: ac_simps fun_eq_iff)
  1601   from assms have "fold_image (op *) g 1 (insert x A) = g x * fold_image (op *) g 1 A"
  1602     by (simp add: fold_image_def)
  1603   with `finite A` show ?thesis by (simp add: eq_fold_g)
  1604 qed
  1605 
  1606 lemma remove:
  1607   assumes "finite A" and "x \<in> A"
  1608   shows "F A = g x * F (A - {x})"
  1609 proof -
  1610   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
  1611     by (auto dest: mk_disjoint_insert)
  1612   moreover from `finite A` this have "finite B" by simp
  1613   ultimately show ?thesis by simp
  1614 qed
  1615 
  1616 lemma insert_remove:
  1617   assumes "finite A"
  1618   shows "F (insert x A) = g x * F (A - {x})"
  1619   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
  1620 
  1621 lemma neutral:
  1622   assumes "finite A" and "\<forall>x\<in>A. g x = 1"
  1623   shows "F A = 1"
  1624   using assms by (induct A) simp_all
  1625 
  1626 lemma union_inter:
  1627   assumes "finite A" and "finite B"
  1628   shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
  1629 using assms proof (induct A)
  1630   case empty then show ?case by simp
  1631 next
  1632   case (insert x A) then show ?case
  1633     by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
  1634 qed
  1635 
  1636 corollary union_inter_neutral:
  1637   assumes "finite A" and "finite B"
  1638   and I0: "\<forall>x \<in> A\<inter>B. g x = 1"
  1639   shows "F (A \<union> B) = F A * F B"
  1640   using assms by (simp add: union_inter [symmetric] neutral)
  1641 
  1642 corollary union_disjoint:
  1643   assumes "finite A" and "finite B"
  1644   assumes "A \<inter> B = {}"
  1645   shows "F (A \<union> B) = F A * F B"
  1646   using assms by (simp add: union_inter_neutral)
  1647 
  1648 end
  1649 
  1650 
  1651 subsubsection {* The image case with flexible function *}
  1652 
  1653 locale folding_image = comm_monoid +
  1654   fixes F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
  1655   assumes eq_fold: "\<And>g. finite A \<Longrightarrow> F g A = fold_image f g 1 A"
  1656 
  1657 sublocale folding_image < folding_image_simple "op *" 1 g "F g" proof
  1658 qed (fact eq_fold)
  1659 
  1660 context folding_image
  1661 begin
  1662 
  1663 lemma reindex: (* FIXME polymorhism *)
  1664   assumes "finite A" and "inj_on h A"
  1665   shows "F g (h ` A) = F (g \<circ> h) A"
  1666   using assms by (induct A) auto
  1667 
  1668 lemma cong:
  1669   assumes "finite A" and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"
  1670   shows "F g A = F h A"
  1671 proof -
  1672   from assms have "ALL C. C <= A --> (ALL x:C. g x = h x) --> F g C = F h C"
  1673   apply - apply (erule finite_induct) apply simp
  1674   apply (simp add: subset_insert_iff, clarify)
  1675   apply (subgoal_tac "finite C")
  1676   prefer 2 apply (blast dest: finite_subset [rotated])
  1677   apply (subgoal_tac "C = insert x (C - {x})")
  1678   prefer 2 apply blast
  1679   apply (erule ssubst)
  1680   apply (drule spec)
  1681   apply (erule (1) notE impE)
  1682   apply (simp add: Ball_def del: insert_Diff_single)
  1683   done
  1684   with assms show ?thesis by simp
  1685 qed
  1686 
  1687 lemma UNION_disjoint:
  1688   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
  1689   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
  1690   shows "F g (UNION I A) = F (F g \<circ> A) I"
  1691 apply (insert assms)
  1692 apply (induct rule: finite_induct)
  1693 apply simp
  1694 apply atomize
  1695 apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
  1696  prefer 2 apply blast
  1697 apply (subgoal_tac "A x Int UNION Fa A = {}")
  1698  prefer 2 apply blast
  1699 apply (simp add: union_disjoint)
  1700 done
  1701 
  1702 lemma distrib:
  1703   assumes "finite A"
  1704   shows "F (\<lambda>x. g x * h x) A = F g A * F h A"
  1705   using assms by (rule finite_induct) (simp_all add: assoc commute left_commute)
  1706 
  1707 lemma related: 
  1708   assumes Re: "R 1 1" 
  1709   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
  1710   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
  1711   shows "R (F h S) (F g S)"
  1712   using fS by (rule finite_subset_induct) (insert assms, auto)
  1713 
  1714 lemma eq_general:
  1715   assumes fS: "finite S"
  1716   and h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y" 
  1717   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
  1718   shows "F f1 S = F f2 S'"
  1719 proof-
  1720   from h f12 have hS: "h ` S = S'" by blast
  1721   {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
  1722     from f12 h H  have "x = y" by auto }
  1723   hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
  1724   from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
  1725   from hS have "F f2 S' = F f2 (h ` S)" by simp
  1726   also have "\<dots> = F (f2 o h) S" using reindex [OF fS hinj, of f2] .
  1727   also have "\<dots> = F f1 S " using th cong [OF fS, of "f2 o h" f1]
  1728     by blast
  1729   finally show ?thesis ..
  1730 qed
  1731 
  1732 lemma eq_general_inverses:
  1733   assumes fS: "finite S" 
  1734   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
  1735   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"
  1736   shows "F j S = F g T"
  1737   (* metis solves it, but not yet available here *)
  1738   apply (rule eq_general [OF fS, of T h g j])
  1739   apply (rule ballI)
  1740   apply (frule kh)
  1741   apply (rule ex1I[])
  1742   apply blast
  1743   apply clarsimp
  1744   apply (drule hk) apply simp
  1745   apply (rule sym)
  1746   apply (erule conjunct1[OF conjunct2[OF hk]])
  1747   apply (rule ballI)
  1748   apply (drule hk)
  1749   apply blast
  1750   done
  1751 
  1752 end
  1753 
  1754 
  1755 subsubsection {* The image case with fixed function and idempotency *}
  1756 
  1757 locale folding_image_simple_idem = folding_image_simple +
  1758   assumes idem: "x * x = x"
  1759 
  1760 sublocale folding_image_simple_idem < semilattice: semilattice proof
  1761 qed (fact idem)
  1762 
  1763 context folding_image_simple_idem
  1764 begin
  1765 
  1766 lemma in_idem:
  1767   assumes "finite A" and "x \<in> A"
  1768   shows "g x * F A = F A"
  1769   using assms by (induct A) (auto simp add: left_commute)
  1770 
  1771 lemma subset_idem:
  1772   assumes "finite A" and "B \<subseteq> A"
  1773   shows "F B * F A = F A"
  1774 proof -
  1775   from assms have "finite B" by (blast dest: finite_subset)
  1776   then show ?thesis using `B \<subseteq> A` by (induct B)
  1777     (auto simp add: assoc in_idem `finite A`)
  1778 qed
  1779 
  1780 declare insert [simp del]
  1781 
  1782 lemma insert_idem [simp]:
  1783   assumes "finite A"
  1784   shows "F (insert x A) = g x * F A"
  1785   using assms by (cases "x \<in> A") (simp_all add: insert in_idem insert_absorb)
  1786 
  1787 lemma union_idem:
  1788   assumes "finite A" and "finite B"
  1789   shows "F (A \<union> B) = F A * F B"
  1790 proof -
  1791   from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
  1792   then have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (rule subset_idem)
  1793   with assms show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
  1794 qed
  1795 
  1796 end
  1797 
  1798 
  1799 subsubsection {* The image case with flexible function and idempotency *}
  1800 
  1801 locale folding_image_idem = folding_image +
  1802   assumes idem: "x * x = x"
  1803 
  1804 sublocale folding_image_idem < folding_image_simple_idem "op *" 1 g "F g" proof
  1805 qed (fact idem)
  1806 
  1807 
  1808 subsubsection {* The neutral-less case *}
  1809 
  1810 locale folding_one = abel_semigroup +
  1811   fixes F :: "'a set \<Rightarrow> 'a"
  1812   assumes eq_fold: "finite A \<Longrightarrow> F A = fold1 f A"
  1813 begin
  1814 
  1815 lemma singleton [simp]:
  1816   "F {x} = x"
  1817   by (simp add: eq_fold)
  1818 
  1819 lemma eq_fold':
  1820   assumes "finite A" and "x \<notin> A"
  1821   shows "F (insert x A) = fold (op *) x A"
  1822 proof -
  1823   interpret ab_semigroup_mult "op *" by default (simp_all add: ac_simps)
  1824   from assms show ?thesis by (simp add: eq_fold fold1_eq_fold)
  1825 qed
  1826 
  1827 lemma insert [simp]:
  1828   assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
  1829   shows "F (insert x A) = x * F A"
  1830 proof -
  1831   from `A \<noteq> {}` obtain b where "b \<in> A" by blast
  1832   then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
  1833   with `finite A` have "finite B" by simp
  1834   interpret fold: folding "op *" "\<lambda>a b. fold (op *) b a" proof
  1835   qed (simp_all add: fun_eq_iff ac_simps)
  1836   from `finite B` fold.comp_fun_commute' [of B x]
  1837     have "op * x \<circ> (\<lambda>b. fold op * b B) = (\<lambda>b. fold op * b B) \<circ> op * x" by simp
  1838   then have A: "x * fold op * b B = fold op * (b * x) B" by (simp add: fun_eq_iff commute)
  1839   from `finite B` * fold.insert [of B b]
  1840     have "(\<lambda>x. fold op * x (insert b B)) = (\<lambda>x. fold op * x B) \<circ> op * b" by simp
  1841   then have B: "fold op * x (insert b B) = fold op * (b * x) B" by (simp add: fun_eq_iff)
  1842   from A B assms * show ?thesis by (simp add: eq_fold' del: fold.insert)
  1843 qed
  1844 
  1845 lemma remove:
  1846   assumes "finite A" and "x \<in> A"
  1847   shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
  1848 proof -
  1849   from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
  1850   with assms show ?thesis by simp
  1851 qed
  1852 
  1853 lemma insert_remove:
  1854   assumes "finite A"
  1855   shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
  1856   using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
  1857 
  1858 lemma union_disjoint:
  1859   assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}" and "A \<inter> B = {}"
  1860   shows "F (A \<union> B) = F A * F B"
  1861   using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
  1862 
  1863 lemma union_inter:
  1864   assumes "finite A" and "finite B" and "A \<inter> B \<noteq> {}"
  1865   shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
  1866 proof -
  1867   from assms have "A \<noteq> {}" and "B \<noteq> {}" by auto
  1868   from `finite A` `A \<noteq> {}` `A \<inter> B \<noteq> {}` show ?thesis proof (induct A rule: finite_ne_induct)
  1869     case (singleton x) then show ?case by (simp add: insert_absorb ac_simps)
  1870   next
  1871     case (insert x A) show ?case proof (cases "x \<in> B")
  1872       case True then have "B \<noteq> {}" by auto
  1873       with insert True `finite B` show ?thesis by (cases "A \<inter> B = {}")
  1874         (simp_all add: insert_absorb ac_simps union_disjoint)
  1875     next
  1876       case False with insert have "F (A \<union> B) * F (A \<inter> B) = F A * F B" by simp
  1877       moreover from False `finite B` insert have "finite (A \<union> B)" "x \<notin> A \<union> B" "A \<union> B \<noteq> {}"
  1878         by auto
  1879       ultimately show ?thesis using False `finite A` `x \<notin> A` `A \<noteq> {}` by (simp add: assoc)
  1880     qed
  1881   qed
  1882 qed
  1883 
  1884 lemma closed:
  1885   assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
  1886   shows "F A \<in> A"
  1887 using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
  1888   case singleton then show ?case by simp
  1889 next
  1890   case insert with elem show ?case by force
  1891 qed
  1892 
  1893 end
  1894 
  1895 
  1896 subsubsection {* The neutral-less case with idempotency *}
  1897 
  1898 locale folding_one_idem = folding_one +
  1899   assumes idem: "x * x = x"
  1900 
  1901 sublocale folding_one_idem < semilattice: semilattice proof
  1902 qed (fact idem)
  1903 
  1904 context folding_one_idem
  1905 begin
  1906 
  1907 lemma in_idem:
  1908   assumes "finite A" and "x \<in> A"
  1909   shows "x * F A = F A"
  1910 proof -
  1911   from assms have "A \<noteq> {}" by auto
  1912   with `finite A` show ?thesis using `x \<in> A` by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
  1913 qed
  1914 
  1915 lemma subset_idem:
  1916   assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
  1917   shows "F B * F A = F A"
  1918 proof -
  1919   from assms have "finite B" by (blast dest: finite_subset)
  1920   then show ?thesis using `B \<noteq> {}` `B \<subseteq> A` by (induct B rule: finite_ne_induct)
  1921     (simp_all add: assoc in_idem `finite A`)
  1922 qed
  1923 
  1924 lemma eq_fold_idem':
  1925   assumes "finite A"
  1926   shows "F (insert a A) = fold (op *) a A"
  1927 proof -
  1928   interpret ab_semigroup_idem_mult "op *" by default (simp_all add: ac_simps)
  1929   from assms show ?thesis by (simp add: eq_fold fold1_eq_fold_idem)
  1930 qed
  1931 
  1932 lemma insert_idem [simp]:
  1933   assumes "finite A" and "A \<noteq> {}"
  1934   shows "F (insert x A) = x * F A"
  1935 proof (cases "x \<in> A")
  1936   case False from `finite A` `x \<notin> A` `A \<noteq> {}` show ?thesis by (rule insert)
  1937 next
  1938   case True
  1939   from `finite A` `A \<noteq> {}` show ?thesis by (simp add: in_idem insert_absorb True)
  1940 qed
  1941   
  1942 lemma union_idem:
  1943   assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
  1944   shows "F (A \<union> B) = F A * F B"
  1945 proof (cases "A \<inter> B = {}")
  1946   case True with assms show ?thesis by (simp add: union_disjoint)
  1947 next
  1948   case False
  1949   from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
  1950   with False have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (auto intro: subset_idem)
  1951   with assms False show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
  1952 qed
  1953 
  1954 lemma hom_commute:
  1955   assumes hom: "\<And>x y. h (x * y) = h x * h y"
  1956   and N: "finite N" "N \<noteq> {}" shows "h (F N) = F (h ` N)"
  1957 using N proof (induct rule: finite_ne_induct)
  1958   case singleton thus ?case by simp
  1959 next
  1960   case (insert n N)
  1961   then have "h (F (insert n N)) = h (n * F N)" by simp
  1962   also have "\<dots> = h n * h (F N)" by (rule hom)
  1963   also have "h (F N) = F (h ` N)" by(rule insert)
  1964   also have "h n * \<dots> = F (insert (h n) (h ` N))"
  1965     using insert by(simp)
  1966   also have "insert (h n) (h ` N) = h ` insert n N" by simp
  1967   finally show ?case .
  1968 qed
  1969 
  1970 end
  1971 
  1972 notation times (infixl "*" 70)
  1973 notation Groups.one ("1")
  1974 
  1975 
  1976 subsection {* Finite cardinality *}
  1977 
  1978 text {* This definition, although traditional, is ugly to work with:
  1979 @{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
  1980 But now that we have @{text fold_image} things are easy:
  1981 *}
  1982 
  1983 definition card :: "'a set \<Rightarrow> nat" where
  1984   "card A = (if finite A then fold_image (op +) (\<lambda>x. 1) 0 A else 0)"
  1985 
  1986 interpretation card: folding_image_simple "op +" 0 "\<lambda>x. 1" card proof
  1987 qed (simp add: card_def)
  1988 
  1989 lemma card_infinite [simp]:
  1990   "\<not> finite A \<Longrightarrow> card A = 0"
  1991   by (simp add: card_def)
  1992 
  1993 lemma card_empty:
  1994   "card {} = 0"
  1995   by (fact card.empty)
  1996 
  1997 lemma card_insert_disjoint:
  1998   "finite A ==> x \<notin> A ==> card (insert x A) = Suc (card A)"
  1999   by simp
  2000 
  2001 lemma card_insert_if:
  2002   "finite A ==> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
  2003   by auto (simp add: card.insert_remove card.remove)
  2004 
  2005 lemma card_ge_0_finite:
  2006   "card A > 0 \<Longrightarrow> finite A"
  2007   by (rule ccontr) simp
  2008 
  2009 lemma card_0_eq [simp, no_atp]:
  2010   "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
  2011   by (auto dest: mk_disjoint_insert)
  2012 
  2013 lemma finite_UNIV_card_ge_0:
  2014   "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
  2015   by (rule ccontr) simp
  2016 
  2017 lemma card_eq_0_iff:
  2018   "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
  2019   by auto
  2020 
  2021 lemma card_gt_0_iff:
  2022   "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
  2023   by (simp add: neq0_conv [symmetric] card_eq_0_iff) 
  2024 
  2025 lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
  2026 apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
  2027 apply(simp del:insert_Diff_single)
  2028 done
  2029 
  2030 lemma card_Diff_singleton:
  2031   "finite A ==> x: A ==> card (A - {x}) = card A - 1"
  2032 by (simp add: card_Suc_Diff1 [symmetric])
  2033 
  2034 lemma card_Diff_singleton_if:
  2035   "finite A ==> card (A - {x}) = (if x : A then card A - 1 else card A)"
  2036 by (simp add: card_Diff_singleton)
  2037 
  2038 lemma card_Diff_insert[simp]:
  2039 assumes "finite A" and "a:A" and "a ~: B"
  2040 shows "card(A - insert a B) = card(A - B) - 1"
  2041 proof -
  2042   have "A - insert a B = (A - B) - {a}" using assms by blast
  2043   then show ?thesis using assms by(simp add:card_Diff_singleton)
  2044 qed
  2045 
  2046 lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
  2047 by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert)
  2048 
  2049 lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
  2050 by (simp add: card_insert_if)
  2051 
  2052 lemma card_Collect_less_nat[simp]: "card{i::nat. i < n} = n"
  2053 by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
  2054 
  2055 lemma card_Collect_le_nat[simp]: "card{i::nat. i <= n} = Suc n"
  2056 using card_Collect_less_nat[of "Suc n"] by(simp add: less_Suc_eq_le)
  2057 
  2058 lemma card_mono:
  2059   assumes "finite B" and "A \<subseteq> B"
  2060   shows "card A \<le> card B"
  2061 proof -
  2062   from assms have "finite A" by (auto intro: finite_subset)
  2063   then show ?thesis using assms proof (induct A arbitrary: B)
  2064     case empty then show ?case by simp
  2065   next
  2066     case (insert x A)
  2067     then have "x \<in> B" by simp
  2068     from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
  2069     with insert.hyps have "card A \<le> card (B - {x})" by auto
  2070     with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
  2071   qed
  2072 qed
  2073 
  2074 lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
  2075 apply (induct rule: finite_induct)
  2076 apply simp
  2077 apply clarify
  2078 apply (subgoal_tac "finite A & A - {x} <= F")
  2079  prefer 2 apply (blast intro: finite_subset, atomize)
  2080 apply (drule_tac x = "A - {x}" in spec)
  2081 apply (simp add: card_Diff_singleton_if split add: split_if_asm)
  2082 apply (case_tac "card A", auto)
  2083 done
  2084 
  2085 lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
  2086 apply (simp add: psubset_eq linorder_not_le [symmetric])
  2087 apply (blast dest: card_seteq)
  2088 done
  2089 
  2090 lemma card_Un_Int: "finite A ==> finite B
  2091     ==> card A + card B = card (A Un B) + card (A Int B)"
  2092   by (fact card.union_inter [symmetric])
  2093 
  2094 lemma card_Un_disjoint: "finite A ==> finite B
  2095     ==> A Int B = {} ==> card (A Un B) = card A + card B"
  2096   by (fact card.union_disjoint)
  2097 
  2098 lemma card_Diff_subset:
  2099   assumes "finite B" and "B \<subseteq> A"
  2100   shows "card (A - B) = card A - card B"
  2101 proof (cases "finite A")
  2102   case False with assms show ?thesis by simp
  2103 next
  2104   case True with assms show ?thesis by (induct B arbitrary: A) simp_all
  2105 qed
  2106 
  2107 lemma card_Diff_subset_Int:
  2108   assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
  2109 proof -
  2110   have "A - B = A - A \<inter> B" by auto
  2111   thus ?thesis
  2112     by (simp add: card_Diff_subset AB) 
  2113 qed
  2114 
  2115 lemma diff_card_le_card_Diff:
  2116 assumes "finite B" shows "card A - card B \<le> card(A - B)"
  2117 proof-
  2118   have "card A - card B \<le> card A - card (A \<inter> B)"
  2119     using card_mono[OF assms Int_lower2, of A] by arith
  2120   also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
  2121   finally show ?thesis .
  2122 qed
  2123 
  2124 lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
  2125 apply (rule Suc_less_SucD)
  2126 apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
  2127 done
  2128 
  2129 lemma card_Diff2_less:
  2130   "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
  2131 apply (case_tac "x = y")
  2132  apply (simp add: card_Diff1_less del:card_Diff_insert)
  2133 apply (rule less_trans)
  2134  prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
  2135 done
  2136 
  2137 lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
  2138 apply (case_tac "x : A")
  2139  apply (simp_all add: card_Diff1_less less_imp_le)
  2140 done
  2141 
  2142 lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
  2143 by (erule psubsetI, blast)
  2144 
  2145 lemma insert_partition:
  2146   "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
  2147   \<Longrightarrow> x \<inter> \<Union> F = {}"
  2148 by auto
  2149 
  2150 lemma finite_psubset_induct[consumes 1, case_names psubset]:
  2151   assumes fin: "finite A" 
  2152   and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A" 
  2153   shows "P A"
  2154 using fin
  2155 proof (induct A taking: card rule: measure_induct_rule)
  2156   case (less A)
  2157   have fin: "finite A" by fact
  2158   have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
  2159   { fix B 
  2160     assume asm: "B \<subset> A"
  2161     from asm have "card B < card A" using psubset_card_mono fin by blast
  2162     moreover
  2163     from asm have "B \<subseteq> A" by auto
  2164     then have "finite B" using fin finite_subset by blast
  2165     ultimately 
  2166     have "P B" using ih by simp
  2167   }
  2168   with fin show "P A" using major by blast
  2169 qed
  2170 
  2171 text{* main cardinality theorem *}
  2172 lemma card_partition [rule_format]:
  2173   "finite C ==>
  2174      finite (\<Union> C) -->
  2175      (\<forall>c\<in>C. card c = k) -->
  2176      (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
  2177      k * card(C) = card (\<Union> C)"
  2178 apply (erule finite_induct, simp)
  2179 apply (simp add: card_Un_disjoint insert_partition 
  2180        finite_subset [of _ "\<Union> (insert x F)"])
  2181 done
  2182 
  2183 lemma card_eq_UNIV_imp_eq_UNIV:
  2184   assumes fin: "finite (UNIV :: 'a set)"
  2185   and card: "card A = card (UNIV :: 'a set)"
  2186   shows "A = (UNIV :: 'a set)"
  2187 proof
  2188   show "A \<subseteq> UNIV" by simp
  2189   show "UNIV \<subseteq> A"
  2190   proof
  2191     fix x
  2192     show "x \<in> A"
  2193     proof (rule ccontr)
  2194       assume "x \<notin> A"
  2195       then have "A \<subset> UNIV" by auto
  2196       with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
  2197       with card show False by simp
  2198     qed
  2199   qed
  2200 qed
  2201 
  2202 text{*The form of a finite set of given cardinality*}
  2203 
  2204 lemma card_eq_SucD:
  2205 assumes "card A = Suc k"
  2206 shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
  2207 proof -
  2208   have fin: "finite A" using assms by (auto intro: ccontr)
  2209   moreover have "card A \<noteq> 0" using assms by auto
  2210   ultimately obtain b where b: "b \<in> A" by auto
  2211   show ?thesis
  2212   proof (intro exI conjI)
  2213     show "A = insert b (A-{b})" using b by blast
  2214     show "b \<notin> A - {b}" by blast
  2215     show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
  2216       using assms b fin by(fastforce dest:mk_disjoint_insert)+
  2217   qed
  2218 qed
  2219 
  2220 lemma card_Suc_eq:
  2221   "(card A = Suc k) =
  2222    (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
  2223 apply(rule iffI)
  2224  apply(erule card_eq_SucD)
  2225 apply(auto)
  2226 apply(subst card_insert)
  2227  apply(auto intro:ccontr)
  2228 done
  2229 
  2230 lemma card_le_Suc_iff: "finite A \<Longrightarrow>
  2231   Suc n \<le> card A = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
  2232 by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
  2233   dest: subset_singletonD split: nat.splits if_splits)
  2234 
  2235 lemma finite_fun_UNIVD2:
  2236   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
  2237   shows "finite (UNIV :: 'b set)"
  2238 proof -
  2239   from fin have "\<And>arbitrary. finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
  2240     by (rule finite_imageI)
  2241   moreover have "\<And>arbitrary. UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
  2242     by (rule UNIV_eq_I) auto
  2243   ultimately show "finite (UNIV :: 'b set)" by simp
  2244 qed
  2245 
  2246 lemma card_UNIV_unit [simp]: "card (UNIV :: unit set) = 1"
  2247   unfolding UNIV_unit by simp
  2248 
  2249 lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
  2250   unfolding UNIV_bool by simp
  2251 
  2252 
  2253 subsubsection {* Cardinality of image *}
  2254 
  2255 lemma card_image_le: "finite A ==> card (f ` A) <= card A"
  2256 apply (induct rule: finite_induct)
  2257  apply simp
  2258 apply (simp add: le_SucI card_insert_if)
  2259 done
  2260 
  2261 lemma card_image:
  2262   assumes "inj_on f A"
  2263   shows "card (f ` A) = card A"
  2264 proof (cases "finite A")
  2265   case True then show ?thesis using assms by (induct A) simp_all
  2266 next
  2267   case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
  2268   with False show ?thesis by simp
  2269 qed
  2270 
  2271 lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
  2272 by(auto simp: card_image bij_betw_def)
  2273 
  2274 lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
  2275 by (simp add: card_seteq card_image)
  2276 
  2277 lemma eq_card_imp_inj_on:
  2278   "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
  2279 apply (induct rule:finite_induct)
  2280 apply simp
  2281 apply(frule card_image_le[where f = f])
  2282 apply(simp add:card_insert_if split:if_splits)
  2283 done
  2284 
  2285 lemma inj_on_iff_eq_card:
  2286   "finite A ==> inj_on f A = (card(f ` A) = card A)"
  2287 by(blast intro: card_image eq_card_imp_inj_on)
  2288 
  2289 
  2290 lemma card_inj_on_le:
  2291   "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
  2292 apply (subgoal_tac "finite A") 
  2293  apply (force intro: card_mono simp add: card_image [symmetric])
  2294 apply (blast intro: finite_imageD dest: finite_subset) 
  2295 done
  2296 
  2297 lemma card_bij_eq:
  2298   "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
  2299      finite A; finite B |] ==> card A = card B"
  2300 by (auto intro: le_antisym card_inj_on_le)
  2301 
  2302 lemma bij_betw_finite:
  2303   assumes "bij_betw f A B"
  2304   shows "finite A \<longleftrightarrow> finite B"
  2305 using assms unfolding bij_betw_def
  2306 using finite_imageD[of f A] by auto
  2307 
  2308 
  2309 subsubsection {* Pigeonhole Principles *}
  2310 
  2311 lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
  2312 by (auto dest: card_image less_irrefl_nat)
  2313 
  2314 lemma pigeonhole_infinite:
  2315 assumes  "~ finite A" and "finite(f`A)"
  2316 shows "EX a0:A. ~finite{a:A. f a = f a0}"
  2317 proof -
  2318   have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
  2319   proof(induct "f`A" arbitrary: A rule: finite_induct)
  2320     case empty thus ?case by simp
  2321   next
  2322     case (insert b F)
  2323     show ?case
  2324     proof cases
  2325       assume "finite{a:A. f a = b}"
  2326       hence "~ finite(A - {a:A. f a = b})" using `\<not> finite A` by simp
  2327       also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
  2328       finally have "~ finite({a:A. f a \<noteq> b})" .
  2329       from insert(3)[OF _ this]
  2330       show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
  2331     next
  2332       assume 1: "~finite{a:A. f a = b}"
  2333       hence "{a \<in> A. f a = b} \<noteq> {}" by force
  2334       thus ?thesis using 1 by blast
  2335     qed
  2336   qed
  2337   from this[OF assms(2,1)] show ?thesis .
  2338 qed
  2339 
  2340 lemma pigeonhole_infinite_rel:
  2341 assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
  2342 shows "EX b:B. ~finite{a:A. R a b}"
  2343 proof -
  2344    let ?F = "%a. {b:B. R a b}"
  2345    from finite_Pow_iff[THEN iffD2, OF `finite B`]
  2346    have "finite(?F ` A)" by(blast intro: rev_finite_subset)
  2347    from pigeonhole_infinite[where f = ?F, OF assms(1) this]
  2348    obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
  2349    obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast
  2350    { assume "finite{a:A. R a b0}"
  2351      then have "finite {a\<in>A. ?F a = ?F a0}"
  2352        using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)
  2353    }
  2354    with 1 `b0 : B` show ?thesis by blast
  2355 qed
  2356 
  2357 
  2358 subsubsection {* Cardinality of sums *}
  2359 
  2360 lemma card_Plus:
  2361   assumes "finite A" and "finite B"
  2362   shows "card (A <+> B) = card A + card B"
  2363 proof -
  2364   have "Inl`A \<inter> Inr`B = {}" by fast
  2365   with assms show ?thesis
  2366     unfolding Plus_def
  2367     by (simp add: card_Un_disjoint card_image)
  2368 qed
  2369 
  2370 lemma card_Plus_conv_if:
  2371   "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
  2372   by (auto simp add: card_Plus)
  2373 
  2374 
  2375 subsubsection {* Cardinality of the Powerset *}
  2376 
  2377 lemma card_Pow: "finite A ==> card (Pow A) = 2 ^ card A"
  2378 apply (induct rule: finite_induct)
  2379  apply (simp_all add: Pow_insert)
  2380 apply (subst card_Un_disjoint, blast)
  2381   apply (blast, blast)
  2382 apply (subgoal_tac "inj_on (insert x) (Pow F)")
  2383  apply (subst mult_2)
  2384  apply (simp add: card_image Pow_insert)
  2385 apply (unfold inj_on_def)
  2386 apply (blast elim!: equalityE)
  2387 done
  2388 
  2389 text {* Relates to equivalence classes.  Based on a theorem of F. Kamm\"uller.  *}
  2390 
  2391 lemma dvd_partition:
  2392   "finite (Union C) ==>
  2393     ALL c : C. k dvd card c ==>
  2394     (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
  2395   k dvd card (Union C)"
  2396 apply (frule finite_UnionD)
  2397 apply (rotate_tac -1)
  2398 apply (induct rule: finite_induct)
  2399 apply simp_all
  2400 apply clarify
  2401 apply (subst card_Un_disjoint)
  2402    apply (auto simp add: disjoint_eq_subset_Compl)
  2403 done
  2404 
  2405 
  2406 subsubsection {* Relating injectivity and surjectivity *}
  2407 
  2408 lemma finite_surj_inj: "finite A \<Longrightarrow> A \<subseteq> f ` A \<Longrightarrow> inj_on f A"
  2409 apply(rule eq_card_imp_inj_on, assumption)
  2410 apply(frule finite_imageI)
  2411 apply(drule (1) card_seteq)
  2412  apply(erule card_image_le)
  2413 apply simp
  2414 done
  2415 
  2416 lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
  2417 shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
  2418 by (blast intro: finite_surj_inj subset_UNIV)
  2419 
  2420 lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
  2421 shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
  2422 by(fastforce simp:surj_def dest!: endo_inj_surj)
  2423 
  2424 corollary infinite_UNIV_nat[iff]: "~finite(UNIV::nat set)"
  2425 proof
  2426   assume "finite(UNIV::nat set)"
  2427   with finite_UNIV_inj_surj[of Suc]
  2428   show False by simp (blast dest: Suc_neq_Zero surjD)
  2429 qed
  2430 
  2431 (* Often leads to bogus ATP proofs because of reduced type information, hence no_atp *)
  2432 lemma infinite_UNIV_char_0[no_atp]:
  2433   "\<not> finite (UNIV::'a::semiring_char_0 set)"
  2434 proof
  2435   assume "finite (UNIV::'a set)"
  2436   with subset_UNIV have "finite (range of_nat::'a set)"
  2437     by (rule finite_subset)
  2438   moreover have "inj (of_nat::nat \<Rightarrow> 'a)"
  2439     by (simp add: inj_on_def)
  2440   ultimately have "finite (UNIV::nat set)"
  2441     by (rule finite_imageD)
  2442   then show "False"
  2443     by simp
  2444 qed
  2445 
  2446 hide_const (open) Finite_Set.fold
  2447 
  2448 end
  2449