src/HOL/Lattices_Big.thy
author paulson <lp15@cam.ac.uk>
Mon Feb 22 14:37:56 2016 +0000 (2016-02-22)
changeset 62379 340738057c8c
parent 61776 57bb7da5c867
child 63290 9ac558ab0906
permissions -rw-r--r--
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
     1 (*  Title:      HOL/Lattices_Big.thy
     2     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
     3                 with contributions by Jeremy Avigad
     4 *)
     5 
     6 section \<open>Big infimum (minimum) and supremum (maximum) over finite (non-empty) sets\<close>
     7 
     8 theory Lattices_Big
     9 imports Finite_Set Option
    10 begin
    11 
    12 subsection \<open>Generic lattice operations over a set\<close>
    13 
    14 no_notation times (infixl "*" 70)
    15 no_notation Groups.one ("1")
    16 
    17 
    18 subsubsection \<open>Without neutral element\<close>
    19 
    20 locale semilattice_set = semilattice
    21 begin
    22 
    23 interpretation comp_fun_idem f
    24   by standard (simp_all add: fun_eq_iff left_commute)
    25 
    26 definition F :: "'a set \<Rightarrow> 'a"
    27 where
    28   eq_fold': "F A = the (Finite_Set.fold (\<lambda>x y. Some (case y of None \<Rightarrow> x | Some z \<Rightarrow> f x z)) None A)"
    29 
    30 lemma eq_fold:
    31   assumes "finite A"
    32   shows "F (insert x A) = Finite_Set.fold f x A"
    33 proof (rule sym)
    34   let ?f = "\<lambda>x y. Some (case y of None \<Rightarrow> x | Some z \<Rightarrow> f x z)"
    35   interpret comp_fun_idem "?f"
    36     by standard (simp_all add: fun_eq_iff commute left_commute split: option.split)
    37   from assms show "Finite_Set.fold f x A = F (insert x A)"
    38   proof induct
    39     case empty then show ?case by (simp add: eq_fold')
    40   next
    41     case (insert y B) then show ?case by (simp add: insert_commute [of x] eq_fold')
    42   qed
    43 qed
    44 
    45 lemma singleton [simp]:
    46   "F {x} = x"
    47   by (simp add: eq_fold)
    48 
    49 lemma insert_not_elem:
    50   assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
    51   shows "F (insert x A) = x * F A"
    52 proof -
    53   from \<open>A \<noteq> {}\<close> obtain b where "b \<in> A" by blast
    54   then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
    55   with \<open>finite A\<close> and \<open>x \<notin> A\<close>
    56     have "finite (insert x B)" and "b \<notin> insert x B" by auto
    57   then have "F (insert b (insert x B)) = x * F (insert b B)"
    58     by (simp add: eq_fold)
    59   then show ?thesis by (simp add: * insert_commute)
    60 qed
    61 
    62 lemma in_idem:
    63   assumes "finite A" and "x \<in> A"
    64   shows "x * F A = F A"
    65 proof -
    66   from assms have "A \<noteq> {}" by auto
    67   with \<open>finite A\<close> show ?thesis using \<open>x \<in> A\<close>
    68     by (induct A rule: finite_ne_induct) (auto simp add: ac_simps insert_not_elem)
    69 qed
    70 
    71 lemma insert [simp]:
    72   assumes "finite A" and "A \<noteq> {}"
    73   shows "F (insert x A) = x * F A"
    74   using assms by (cases "x \<in> A") (simp_all add: insert_absorb in_idem insert_not_elem)
    75 
    76 lemma union:
    77   assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
    78   shows "F (A \<union> B) = F A * F B"
    79   using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
    80 
    81 lemma remove:
    82   assumes "finite A" and "x \<in> A"
    83   shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
    84 proof -
    85   from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
    86   with assms show ?thesis by simp
    87 qed
    88 
    89 lemma insert_remove:
    90   assumes "finite A"
    91   shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
    92   using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
    93 
    94 lemma subset:
    95   assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
    96   shows "F B * F A = F A"
    97 proof -
    98   from assms have "A \<noteq> {}" and "finite B" by (auto dest: finite_subset)
    99   with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
   100 qed
   101 
   102 lemma closed:
   103   assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
   104   shows "F A \<in> A"
   105 using \<open>finite A\<close> \<open>A \<noteq> {}\<close> proof (induct rule: finite_ne_induct)
   106   case singleton then show ?case by simp
   107 next
   108   case insert with elem show ?case by force
   109 qed
   110 
   111 lemma hom_commute:
   112   assumes hom: "\<And>x y. h (x * y) = h x * h y"
   113   and N: "finite N" "N \<noteq> {}"
   114   shows "h (F N) = F (h ` N)"
   115 using N proof (induct rule: finite_ne_induct)
   116   case singleton thus ?case by simp
   117 next
   118   case (insert n N)
   119   then have "h (F (insert n N)) = h (n * F N)" by simp
   120   also have "\<dots> = h n * h (F N)" by (rule hom)
   121   also have "h (F N) = F (h ` N)" by (rule insert)
   122   also have "h n * \<dots> = F (insert (h n) (h ` N))"
   123     using insert by simp
   124   also have "insert (h n) (h ` N) = h ` insert n N" by simp
   125   finally show ?case .
   126 qed
   127 
   128 lemma infinite: "\<not> finite A \<Longrightarrow> F A = the None"
   129   unfolding eq_fold' by (cases "finite (UNIV::'a set)") (auto intro: finite_subset fold_infinite)
   130 
   131 end
   132 
   133 locale semilattice_order_set = binary?: semilattice_order + semilattice_set
   134 begin
   135 
   136 lemma bounded_iff:
   137   assumes "finite A" and "A \<noteq> {}"
   138   shows "x \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> a)"
   139   using assms by (induct rule: finite_ne_induct) (simp_all add: bounded_iff)
   140 
   141 lemma boundedI:
   142   assumes "finite A"
   143   assumes "A \<noteq> {}"
   144   assumes "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
   145   shows "x \<preceq> F A"
   146   using assms by (simp add: bounded_iff)
   147 
   148 lemma boundedE:
   149   assumes "finite A" and "A \<noteq> {}" and "x \<preceq> F A"
   150   obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
   151   using assms by (simp add: bounded_iff)
   152 
   153 lemma coboundedI:
   154   assumes "finite A"
   155     and "a \<in> A"
   156   shows "F A \<preceq> a"
   157 proof -
   158   from assms have "A \<noteq> {}" by auto
   159   from \<open>finite A\<close> \<open>A \<noteq> {}\<close> \<open>a \<in> A\<close> show ?thesis
   160   proof (induct rule: finite_ne_induct)
   161     case singleton thus ?case by (simp add: refl)
   162   next
   163     case (insert x B)
   164     from insert have "a = x \<or> a \<in> B" by simp
   165     then show ?case using insert by (auto intro: coboundedI2)
   166   qed
   167 qed
   168 
   169 lemma antimono:
   170   assumes "A \<subseteq> B" and "A \<noteq> {}" and "finite B"
   171   shows "F B \<preceq> F A"
   172 proof (cases "A = B")
   173   case True then show ?thesis by (simp add: refl)
   174 next
   175   case False
   176   have B: "B = A \<union> (B - A)" using \<open>A \<subseteq> B\<close> by blast
   177   then have "F B = F (A \<union> (B - A))" by simp
   178   also have "\<dots> = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
   179   also have "\<dots> \<preceq> F A" by simp
   180   finally show ?thesis .
   181 qed
   182 
   183 end
   184 
   185 
   186 subsubsection \<open>With neutral element\<close>
   187 
   188 locale semilattice_neutr_set = semilattice_neutr
   189 begin
   190 
   191 interpretation comp_fun_idem f
   192   by standard (simp_all add: fun_eq_iff left_commute)
   193 
   194 definition F :: "'a set \<Rightarrow> 'a"
   195 where
   196   eq_fold: "F A = Finite_Set.fold f 1 A"
   197 
   198 lemma infinite [simp]:
   199   "\<not> finite A \<Longrightarrow> F A = 1"
   200   by (simp add: eq_fold)
   201 
   202 lemma empty [simp]:
   203   "F {} = 1"
   204   by (simp add: eq_fold)
   205 
   206 lemma insert [simp]:
   207   assumes "finite A"
   208   shows "F (insert x A) = x * F A"
   209   using assms by (simp add: eq_fold)
   210 
   211 lemma in_idem:
   212   assumes "finite A" and "x \<in> A"
   213   shows "x * F A = F A"
   214 proof -
   215   from assms have "A \<noteq> {}" by auto
   216   with \<open>finite A\<close> show ?thesis using \<open>x \<in> A\<close>
   217     by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
   218 qed
   219 
   220 lemma union:
   221   assumes "finite A" and "finite B"
   222   shows "F (A \<union> B) = F A * F B"
   223   using assms by (induct A) (simp_all add: ac_simps)
   224 
   225 lemma remove:
   226   assumes "finite A" and "x \<in> A"
   227   shows "F A = x * F (A - {x})"
   228 proof -
   229   from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
   230   with assms show ?thesis by simp
   231 qed
   232 
   233 lemma insert_remove:
   234   assumes "finite A"
   235   shows "F (insert x A) = x * F (A - {x})"
   236   using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
   237 
   238 lemma subset:
   239   assumes "finite A" and "B \<subseteq> A"
   240   shows "F B * F A = F A"
   241 proof -
   242   from assms have "finite B" by (auto dest: finite_subset)
   243   with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
   244 qed
   245 
   246 lemma closed:
   247   assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
   248   shows "F A \<in> A"
   249 using \<open>finite A\<close> \<open>A \<noteq> {}\<close> proof (induct rule: finite_ne_induct)
   250   case singleton then show ?case by simp
   251 next
   252   case insert with elem show ?case by force
   253 qed
   254 
   255 end
   256 
   257 locale semilattice_order_neutr_set = binary?: semilattice_neutr_order + semilattice_neutr_set
   258 begin
   259 
   260 lemma bounded_iff:
   261   assumes "finite A"
   262   shows "x \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> a)"
   263   using assms by (induct A) (simp_all add: bounded_iff)
   264 
   265 lemma boundedI:
   266   assumes "finite A"
   267   assumes "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
   268   shows "x \<preceq> F A"
   269   using assms by (simp add: bounded_iff)
   270 
   271 lemma boundedE:
   272   assumes "finite A" and "x \<preceq> F A"
   273   obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
   274   using assms by (simp add: bounded_iff)
   275 
   276 lemma coboundedI:
   277   assumes "finite A"
   278     and "a \<in> A"
   279   shows "F A \<preceq> a"
   280 proof -
   281   from assms have "A \<noteq> {}" by auto
   282   from \<open>finite A\<close> \<open>A \<noteq> {}\<close> \<open>a \<in> A\<close> show ?thesis
   283   proof (induct rule: finite_ne_induct)
   284     case singleton thus ?case by (simp add: refl)
   285   next
   286     case (insert x B)
   287     from insert have "a = x \<or> a \<in> B" by simp
   288     then show ?case using insert by (auto intro: coboundedI2)
   289   qed
   290 qed
   291 
   292 lemma antimono:
   293   assumes "A \<subseteq> B" and "finite B"
   294   shows "F B \<preceq> F A"
   295 proof (cases "A = B")
   296   case True then show ?thesis by (simp add: refl)
   297 next
   298   case False
   299   have B: "B = A \<union> (B - A)" using \<open>A \<subseteq> B\<close> by blast
   300   then have "F B = F (A \<union> (B - A))" by simp
   301   also have "\<dots> = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
   302   also have "\<dots> \<preceq> F A" by simp
   303   finally show ?thesis .
   304 qed
   305 
   306 end
   307 
   308 notation times (infixl "*" 70)
   309 notation Groups.one ("1")
   310 
   311 
   312 subsection \<open>Lattice operations on finite sets\<close>
   313 
   314 context semilattice_inf
   315 begin
   316 
   317 sublocale Inf_fin: semilattice_order_set inf less_eq less
   318 defines
   319   Inf_fin ("\<Sqinter>\<^sub>f\<^sub>i\<^sub>n_" [900] 900) = Inf_fin.F ..
   320 
   321 end
   322 
   323 context semilattice_sup
   324 begin
   325 
   326 sublocale Sup_fin: semilattice_order_set sup greater_eq greater
   327 defines
   328   Sup_fin ("\<Squnion>\<^sub>f\<^sub>i\<^sub>n_" [900] 900) = Sup_fin.F ..
   329 
   330 end
   331 
   332 
   333 subsection \<open>Infimum and Supremum over non-empty sets\<close>
   334 
   335 context lattice
   336 begin
   337 
   338 lemma Inf_fin_le_Sup_fin [simp]: 
   339   assumes "finite A" and "A \<noteq> {}"
   340   shows "\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA \<le> \<Squnion>\<^sub>f\<^sub>i\<^sub>nA"
   341 proof -
   342   from \<open>A \<noteq> {}\<close> obtain a where "a \<in> A" by blast
   343   with \<open>finite A\<close> have "\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA \<le> a" by (rule Inf_fin.coboundedI)
   344   moreover from \<open>finite A\<close> \<open>a \<in> A\<close> have "a \<le> \<Squnion>\<^sub>f\<^sub>i\<^sub>nA" by (rule Sup_fin.coboundedI)
   345   ultimately show ?thesis by (rule order_trans)
   346 qed
   347 
   348 lemma sup_Inf_absorb [simp]:
   349   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> \<Sqinter>\<^sub>f\<^sub>i\<^sub>nA \<squnion> a = a"
   350   by (rule sup_absorb2) (rule Inf_fin.coboundedI)
   351 
   352 lemma inf_Sup_absorb [simp]:
   353   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> a \<sqinter> \<Squnion>\<^sub>f\<^sub>i\<^sub>nA = a"
   354   by (rule inf_absorb1) (rule Sup_fin.coboundedI)
   355 
   356 end
   357 
   358 context distrib_lattice
   359 begin
   360 
   361 lemma sup_Inf1_distrib:
   362   assumes "finite A"
   363     and "A \<noteq> {}"
   364   shows "sup x (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA) = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n{sup x a|a. a \<in> A}"
   365 using assms by (simp add: image_def Inf_fin.hom_commute [where h="sup x", OF sup_inf_distrib1])
   366   (rule arg_cong [where f="Inf_fin"], blast)
   367 
   368 lemma sup_Inf2_distrib:
   369   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
   370   shows "sup (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA) (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nB) = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n{sup a b|a b. a \<in> A \<and> b \<in> B}"
   371 using A proof (induct rule: finite_ne_induct)
   372   case singleton then show ?case
   373     by (simp add: sup_Inf1_distrib [OF B])
   374 next
   375   case (insert x A)
   376   have finB: "finite {sup x b |b. b \<in> B}"
   377     by (rule finite_surj [where f = "sup x", OF B(1)], auto)
   378   have finAB: "finite {sup a b |a b. a \<in> A \<and> b \<in> B}"
   379   proof -
   380     have "{sup a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {sup a b})"
   381       by blast
   382     thus ?thesis by(simp add: insert(1) B(1))
   383   qed
   384   have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
   385   have "sup (\<Sqinter>\<^sub>f\<^sub>i\<^sub>n(insert x A)) (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nB) = sup (inf x (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA)) (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nB)"
   386     using insert by simp
   387   also have "\<dots> = inf (sup x (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nB)) (sup (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA) (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nB))" by(rule sup_inf_distrib2)
   388   also have "\<dots> = inf (\<Sqinter>\<^sub>f\<^sub>i\<^sub>n{sup x b|b. b \<in> B}) (\<Sqinter>\<^sub>f\<^sub>i\<^sub>n{sup a b|a b. a \<in> A \<and> b \<in> B})"
   389     using insert by(simp add:sup_Inf1_distrib[OF B])
   390   also have "\<dots> = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n({sup x b |b. b \<in> B} \<union> {sup a b |a b. a \<in> A \<and> b \<in> B})"
   391     (is "_ = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n?M")
   392     using B insert
   393     by (simp add: Inf_fin.union [OF finB _ finAB ne])
   394   also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"
   395     by blast
   396   finally show ?case .
   397 qed
   398 
   399 lemma inf_Sup1_distrib:
   400   assumes "finite A" and "A \<noteq> {}"
   401   shows "inf x (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA) = \<Squnion>\<^sub>f\<^sub>i\<^sub>n{inf x a|a. a \<in> A}"
   402 using assms by (simp add: image_def Sup_fin.hom_commute [where h="inf x", OF inf_sup_distrib1])
   403   (rule arg_cong [where f="Sup_fin"], blast)
   404 
   405 lemma inf_Sup2_distrib:
   406   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
   407   shows "inf (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA) (\<Squnion>\<^sub>f\<^sub>i\<^sub>nB) = \<Squnion>\<^sub>f\<^sub>i\<^sub>n{inf a b|a b. a \<in> A \<and> b \<in> B}"
   408 using A proof (induct rule: finite_ne_induct)
   409   case singleton thus ?case
   410     by(simp add: inf_Sup1_distrib [OF B])
   411 next
   412   case (insert x A)
   413   have finB: "finite {inf x b |b. b \<in> B}"
   414     by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto)
   415   have finAB: "finite {inf a b |a b. a \<in> A \<and> b \<in> B}"
   416   proof -
   417     have "{inf a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {inf a b})"
   418       by blast
   419     thus ?thesis by(simp add: insert(1) B(1))
   420   qed
   421   have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
   422   have "inf (\<Squnion>\<^sub>f\<^sub>i\<^sub>n(insert x A)) (\<Squnion>\<^sub>f\<^sub>i\<^sub>nB) = inf (sup x (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA)) (\<Squnion>\<^sub>f\<^sub>i\<^sub>nB)"
   423     using insert by simp
   424   also have "\<dots> = sup (inf x (\<Squnion>\<^sub>f\<^sub>i\<^sub>nB)) (inf (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA) (\<Squnion>\<^sub>f\<^sub>i\<^sub>nB))" by(rule inf_sup_distrib2)
   425   also have "\<dots> = sup (\<Squnion>\<^sub>f\<^sub>i\<^sub>n{inf x b|b. b \<in> B}) (\<Squnion>\<^sub>f\<^sub>i\<^sub>n{inf a b|a b. a \<in> A \<and> b \<in> B})"
   426     using insert by(simp add:inf_Sup1_distrib[OF B])
   427   also have "\<dots> = \<Squnion>\<^sub>f\<^sub>i\<^sub>n({inf x b |b. b \<in> B} \<union> {inf a b |a b. a \<in> A \<and> b \<in> B})"
   428     (is "_ = \<Squnion>\<^sub>f\<^sub>i\<^sub>n?M")
   429     using B insert
   430     by (simp add: Sup_fin.union [OF finB _ finAB ne])
   431   also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"
   432     by blast
   433   finally show ?case .
   434 qed
   435 
   436 end
   437 
   438 context complete_lattice
   439 begin
   440 
   441 lemma Inf_fin_Inf:
   442   assumes "finite A" and "A \<noteq> {}"
   443   shows "\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA = \<Sqinter>A"
   444 proof -
   445   from assms obtain b B where "A = insert b B" and "finite B" by auto
   446   then show ?thesis
   447     by (simp add: Inf_fin.eq_fold inf_Inf_fold_inf inf.commute [of b])
   448 qed
   449 
   450 lemma Sup_fin_Sup:
   451   assumes "finite A" and "A \<noteq> {}"
   452   shows "\<Squnion>\<^sub>f\<^sub>i\<^sub>nA = \<Squnion>A"
   453 proof -
   454   from assms obtain b B where "A = insert b B" and "finite B" by auto
   455   then show ?thesis
   456     by (simp add: Sup_fin.eq_fold sup_Sup_fold_sup sup.commute [of b])
   457 qed
   458 
   459 end
   460 
   461 
   462 subsection \<open>Minimum and Maximum over non-empty sets\<close>
   463 
   464 context linorder
   465 begin
   466 
   467 sublocale Min: semilattice_order_set min less_eq less
   468   + Max: semilattice_order_set max greater_eq greater
   469 defines
   470   Min = Min.F and Max = Max.F ..
   471 
   472 end
   473 
   474 text \<open>An aside: @{const Min}/@{const Max} on linear orders as special case of @{const Inf_fin}/@{const Sup_fin}\<close>
   475 
   476 lemma Inf_fin_Min:
   477   "Inf_fin = (Min :: 'a::{semilattice_inf, linorder} set \<Rightarrow> 'a)"
   478   by (simp add: Inf_fin_def Min_def inf_min)
   479 
   480 lemma Sup_fin_Max:
   481   "Sup_fin = (Max :: 'a::{semilattice_sup, linorder} set \<Rightarrow> 'a)"
   482   by (simp add: Sup_fin_def Max_def sup_max)
   483 
   484 context linorder
   485 begin
   486 
   487 lemma dual_min:
   488   "ord.min greater_eq = max"
   489   by (auto simp add: ord.min_def max_def fun_eq_iff)
   490 
   491 lemma dual_max:
   492   "ord.max greater_eq = min"
   493   by (auto simp add: ord.max_def min_def fun_eq_iff)
   494 
   495 lemma dual_Min:
   496   "linorder.Min greater_eq = Max"
   497 proof -
   498   interpret dual: linorder greater_eq greater by (fact dual_linorder)
   499   show ?thesis by (simp add: dual.Min_def dual_min Max_def)
   500 qed
   501 
   502 lemma dual_Max:
   503   "linorder.Max greater_eq = Min"
   504 proof -
   505   interpret dual: linorder greater_eq greater by (fact dual_linorder)
   506   show ?thesis by (simp add: dual.Max_def dual_max Min_def)
   507 qed
   508 
   509 lemmas Min_singleton = Min.singleton
   510 lemmas Max_singleton = Max.singleton
   511 lemmas Min_insert = Min.insert
   512 lemmas Max_insert = Max.insert
   513 lemmas Min_Un = Min.union
   514 lemmas Max_Un = Max.union
   515 lemmas hom_Min_commute = Min.hom_commute
   516 lemmas hom_Max_commute = Max.hom_commute
   517 
   518 lemma Min_in [simp]:
   519   assumes "finite A" and "A \<noteq> {}"
   520   shows "Min A \<in> A"
   521   using assms by (auto simp add: min_def Min.closed)
   522 
   523 lemma Max_in [simp]:
   524   assumes "finite A" and "A \<noteq> {}"
   525   shows "Max A \<in> A"
   526   using assms by (auto simp add: max_def Max.closed)
   527 
   528 lemma Min_insert2:
   529   assumes "finite A" and min: "\<And>b. b \<in> A \<Longrightarrow> a \<le> b"
   530   shows "Min (insert a A) = a"
   531 proof (cases "A = {}")
   532   case True then show ?thesis by simp
   533 next
   534   case False with \<open>finite A\<close> have "Min (insert a A) = min a (Min A)"
   535     by simp
   536   moreover from \<open>finite A\<close> \<open>A \<noteq> {}\<close> min have "a \<le> Min A" by simp
   537   ultimately show ?thesis by (simp add: min.absorb1)
   538 qed
   539 
   540 lemma Max_insert2:
   541   assumes "finite A" and max: "\<And>b. b \<in> A \<Longrightarrow> b \<le> a"
   542   shows "Max (insert a A) = a"
   543 proof (cases "A = {}")
   544   case True then show ?thesis by simp
   545 next
   546   case False with \<open>finite A\<close> have "Max (insert a A) = max a (Max A)"
   547     by simp
   548   moreover from \<open>finite A\<close> \<open>A \<noteq> {}\<close> max have "Max A \<le> a" by simp
   549   ultimately show ?thesis by (simp add: max.absorb1)
   550 qed
   551 
   552 lemma Min_le [simp]:
   553   assumes "finite A" and "x \<in> A"
   554   shows "Min A \<le> x"
   555   using assms by (fact Min.coboundedI)
   556 
   557 lemma Max_ge [simp]:
   558   assumes "finite A" and "x \<in> A"
   559   shows "x \<le> Max A"
   560   using assms by (fact Max.coboundedI)
   561 
   562 lemma Min_eqI:
   563   assumes "finite A"
   564   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<ge> x"
   565     and "x \<in> A"
   566   shows "Min A = x"
   567 proof (rule antisym)
   568   from \<open>x \<in> A\<close> have "A \<noteq> {}" by auto
   569   with assms show "Min A \<ge> x" by simp
   570 next
   571   from assms show "x \<ge> Min A" by simp
   572 qed
   573 
   574 lemma Max_eqI:
   575   assumes "finite A"
   576   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
   577     and "x \<in> A"
   578   shows "Max A = x"
   579 proof (rule antisym)
   580   from \<open>x \<in> A\<close> have "A \<noteq> {}" by auto
   581   with assms show "Max A \<le> x" by simp
   582 next
   583   from assms show "x \<le> Max A" by simp
   584 qed
   585 
   586 context
   587   fixes A :: "'a set"
   588   assumes fin_nonempty: "finite A" "A \<noteq> {}"
   589 begin
   590 
   591 lemma Min_ge_iff [simp]:
   592   "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
   593   using fin_nonempty by (fact Min.bounded_iff)
   594 
   595 lemma Max_le_iff [simp]:
   596   "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
   597   using fin_nonempty by (fact Max.bounded_iff)
   598 
   599 lemma Min_gr_iff [simp]:
   600   "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
   601   using fin_nonempty  by (induct rule: finite_ne_induct) simp_all
   602 
   603 lemma Max_less_iff [simp]:
   604   "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
   605   using fin_nonempty by (induct rule: finite_ne_induct) simp_all
   606 
   607 lemma Min_le_iff:
   608   "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
   609   using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_le_iff_disj)
   610 
   611 lemma Max_ge_iff:
   612   "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
   613   using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: le_max_iff_disj)
   614 
   615 lemma Min_less_iff:
   616   "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
   617   using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_less_iff_disj)
   618 
   619 lemma Max_gr_iff:
   620   "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
   621   using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: less_max_iff_disj)
   622 
   623 end
   624 
   625 lemma Max_eq_if:
   626   assumes "finite A"  "finite B"  "\<forall>a\<in>A. \<exists>b\<in>B. a \<le> b"  "\<forall>b\<in>B. \<exists>a\<in>A. b \<le> a"
   627   shows "Max A = Max B"
   628 proof cases
   629   assume "A = {}" thus ?thesis using assms by simp
   630 next
   631   assume "A \<noteq> {}" thus ?thesis using assms
   632     by(blast intro: antisym Max_in Max_ge_iff[THEN iffD2])
   633 qed
   634 
   635 lemma Min_antimono:
   636   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
   637   shows "Min N \<le> Min M"
   638   using assms by (fact Min.antimono)
   639 
   640 lemma Max_mono:
   641   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
   642   shows "Max M \<le> Max N"
   643   using assms by (fact Max.antimono)
   644 
   645 end
   646 
   647 context linorder  (* FIXME *)
   648 begin
   649 
   650 lemma mono_Min_commute:
   651   assumes "mono f"
   652   assumes "finite A" and "A \<noteq> {}"
   653   shows "f (Min A) = Min (f ` A)"
   654 proof (rule linorder_class.Min_eqI [symmetric])
   655   from \<open>finite A\<close> show "finite (f ` A)" by simp
   656   from assms show "f (Min A) \<in> f ` A" by simp
   657   fix x
   658   assume "x \<in> f ` A"
   659   then obtain y where "y \<in> A" and "x = f y" ..
   660   with assms have "Min A \<le> y" by auto
   661   with \<open>mono f\<close> have "f (Min A) \<le> f y" by (rule monoE)
   662   with \<open>x = f y\<close> show "f (Min A) \<le> x" by simp
   663 qed
   664 
   665 lemma mono_Max_commute:
   666   assumes "mono f"
   667   assumes "finite A" and "A \<noteq> {}"
   668   shows "f (Max A) = Max (f ` A)"
   669 proof (rule linorder_class.Max_eqI [symmetric])
   670   from \<open>finite A\<close> show "finite (f ` A)" by simp
   671   from assms show "f (Max A) \<in> f ` A" by simp
   672   fix x
   673   assume "x \<in> f ` A"
   674   then obtain y where "y \<in> A" and "x = f y" ..
   675   with assms have "y \<le> Max A" by auto
   676   with \<open>mono f\<close> have "f y \<le> f (Max A)" by (rule monoE)
   677   with \<open>x = f y\<close> show "x \<le> f (Max A)" by simp
   678 qed
   679 
   680 lemma finite_linorder_max_induct [consumes 1, case_names empty insert]:
   681   assumes fin: "finite A"
   682   and empty: "P {}" 
   683   and insert: "\<And>b A. finite A \<Longrightarrow> \<forall>a\<in>A. a < b \<Longrightarrow> P A \<Longrightarrow> P (insert b A)"
   684   shows "P A"
   685 using fin empty insert
   686 proof (induct rule: finite_psubset_induct)
   687   case (psubset A)
   688   have IH: "\<And>B. \<lbrakk>B < A; P {}; (\<And>A b. \<lbrakk>finite A; \<forall>a\<in>A. a<b; P A\<rbrakk> \<Longrightarrow> P (insert b A))\<rbrakk> \<Longrightarrow> P B" by fact 
   689   have fin: "finite A" by fact 
   690   have empty: "P {}" by fact
   691   have step: "\<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. a < b; P A\<rbrakk> \<Longrightarrow> P (insert b A)" by fact
   692   show "P A"
   693   proof (cases "A = {}")
   694     assume "A = {}" 
   695     then show "P A" using \<open>P {}\<close> by simp
   696   next
   697     let ?B = "A - {Max A}" 
   698     let ?A = "insert (Max A) ?B"
   699     have "finite ?B" using \<open>finite A\<close> by simp
   700     assume "A \<noteq> {}"
   701     with \<open>finite A\<close> have "Max A : A" by auto
   702     then have A: "?A = A" using insert_Diff_single insert_absorb by auto
   703     then have "P ?B" using \<open>P {}\<close> step IH [of ?B] by blast
   704     moreover 
   705     have "\<forall>a\<in>?B. a < Max A" using Max_ge [OF \<open>finite A\<close>] by fastforce
   706     ultimately show "P A" using A insert_Diff_single step [OF \<open>finite ?B\<close>] by fastforce
   707   qed
   708 qed
   709 
   710 lemma finite_linorder_min_induct [consumes 1, case_names empty insert]:
   711   "\<lbrakk>finite A; P {}; \<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. b < a; P A\<rbrakk> \<Longrightarrow> P (insert b A)\<rbrakk> \<Longrightarrow> P A"
   712   by (rule linorder.finite_linorder_max_induct [OF dual_linorder])
   713 
   714 lemma Least_Min:
   715   assumes "finite {a. P a}" and "\<exists>a. P a"
   716   shows "(LEAST a. P a) = Min {a. P a}"
   717 proof -
   718   { fix A :: "'a set"
   719     assume A: "finite A" "A \<noteq> {}"
   720     have "(LEAST a. a \<in> A) = Min A"
   721     using A proof (induct A rule: finite_ne_induct)
   722       case singleton show ?case by (rule Least_equality) simp_all
   723     next
   724       case (insert a A)
   725       have "(LEAST b. b = a \<or> b \<in> A) = min a (LEAST a. a \<in> A)"
   726         by (auto intro!: Least_equality simp add: min_def not_le Min_le_iff insert.hyps dest!: less_imp_le)
   727       with insert show ?case by simp
   728     qed
   729   } from this [of "{a. P a}"] assms show ?thesis by simp
   730 qed
   731 
   732 lemma infinite_growing:
   733   assumes "X \<noteq> {}"
   734   assumes *: "\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>X. y > x"
   735   shows "\<not> finite X"
   736 proof
   737   assume "finite X"
   738   with \<open>X \<noteq> {}\<close> have "Max X \<in> X" "\<forall>x\<in>X. x \<le> Max X"
   739     by auto
   740   with *[of "Max X"] show False
   741     by auto
   742 qed
   743 
   744 end
   745 
   746 context linordered_ab_semigroup_add
   747 begin
   748 
   749 lemma add_Min_commute:
   750   fixes k
   751   assumes "finite N" and "N \<noteq> {}"
   752   shows "k + Min N = Min {k + m | m. m \<in> N}"
   753 proof -
   754   have "\<And>x y. k + min x y = min (k + x) (k + y)"
   755     by (simp add: min_def not_le)
   756       (blast intro: antisym less_imp_le add_left_mono)
   757   with assms show ?thesis
   758     using hom_Min_commute [of "plus k" N]
   759     by simp (blast intro: arg_cong [where f = Min])
   760 qed
   761 
   762 lemma add_Max_commute:
   763   fixes k
   764   assumes "finite N" and "N \<noteq> {}"
   765   shows "k + Max N = Max {k + m | m. m \<in> N}"
   766 proof -
   767   have "\<And>x y. k + max x y = max (k + x) (k + y)"
   768     by (simp add: max_def not_le)
   769       (blast intro: antisym less_imp_le add_left_mono)
   770   with assms show ?thesis
   771     using hom_Max_commute [of "plus k" N]
   772     by simp (blast intro: arg_cong [where f = Max])
   773 qed
   774 
   775 end
   776 
   777 context linordered_ab_group_add
   778 begin
   779 
   780 lemma minus_Max_eq_Min [simp]:
   781   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Max S = Min (uminus ` S)"
   782   by (induct S rule: finite_ne_induct) (simp_all add: minus_max_eq_min)
   783 
   784 lemma minus_Min_eq_Max [simp]:
   785   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Min S = Max (uminus ` S)"
   786   by (induct S rule: finite_ne_induct) (simp_all add: minus_min_eq_max)
   787 
   788 end
   789 
   790 context complete_linorder
   791 begin
   792 
   793 lemma Min_Inf:
   794   assumes "finite A" and "A \<noteq> {}"
   795   shows "Min A = Inf A"
   796 proof -
   797   from assms obtain b B where "A = insert b B" and "finite B" by auto
   798   then show ?thesis
   799     by (simp add: Min.eq_fold complete_linorder_inf_min [symmetric] inf_Inf_fold_inf inf.commute [of b])
   800 qed
   801 
   802 lemma Max_Sup:
   803   assumes "finite A" and "A \<noteq> {}"
   804   shows "Max A = Sup A"
   805 proof -
   806   from assms obtain b B where "A = insert b B" and "finite B" by auto
   807   then show ?thesis
   808     by (simp add: Max.eq_fold complete_linorder_sup_max [symmetric] sup_Sup_fold_sup sup.commute [of b])
   809 qed
   810 
   811 end
   812 
   813 end