src/HOL/Lattices_Big.thy
 author haftmann Sun Dec 15 15:10:14 2013 +0100 (2013-12-15) changeset 54744 1e7f2d296e19 child 54745 46e441e61ff5 permissions -rw-r--r--
more algebraic terminology for theories about big operators
```     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 header {* Big infimum (minimum) and supremum (maximum) over finite (non-empty) sets *}
```
```     7
```
```     8 theory Lattices_Big
```
```     9 imports Finite_Set
```
```    10 begin
```
```    11
```
```    12 subsection {* Generic lattice operations over a set *}
```
```    13
```
```    14 no_notation times (infixl "*" 70)
```
```    15 no_notation Groups.one ("1")
```
```    16
```
```    17
```
```    18 subsubsection {* Without neutral element *}
```
```    19
```
```    20 locale semilattice_set = semilattice
```
```    21 begin
```
```    22
```
```    23 interpretation comp_fun_idem f
```
```    24   by default (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 default (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 `A \<noteq> {}` 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 `finite A` and `x \<notin> A`
```
```    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 `finite A` show ?thesis using `x \<in> A`
```
```    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 `finite A` `A \<noteq> {}` 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 end
```
```   129
```
```   130 locale semilattice_order_set = semilattice_order + semilattice_set
```
```   131 begin
```
```   132
```
```   133 lemma bounded_iff:
```
```   134   assumes "finite A" and "A \<noteq> {}"
```
```   135   shows "x \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> a)"
```
```   136   using assms by (induct rule: finite_ne_induct) (simp_all add: bounded_iff)
```
```   137
```
```   138 lemma boundedI:
```
```   139   assumes "finite A"
```
```   140   assumes "A \<noteq> {}"
```
```   141   assumes "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
```
```   142   shows "x \<preceq> F A"
```
```   143   using assms by (simp add: bounded_iff)
```
```   144
```
```   145 lemma boundedE:
```
```   146   assumes "finite A" and "A \<noteq> {}" and "x \<preceq> F A"
```
```   147   obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
```
```   148   using assms by (simp add: bounded_iff)
```
```   149
```
```   150 lemma coboundedI:
```
```   151   assumes "finite A"
```
```   152     and "a \<in> A"
```
```   153   shows "F A \<preceq> a"
```
```   154 proof -
```
```   155   from assms have "A \<noteq> {}" by auto
```
```   156   from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
```
```   157   proof (induct rule: finite_ne_induct)
```
```   158     case singleton thus ?case by (simp add: refl)
```
```   159   next
```
```   160     case (insert x B)
```
```   161     from insert have "a = x \<or> a \<in> B" by simp
```
```   162     then show ?case using insert by (auto intro: coboundedI2)
```
```   163   qed
```
```   164 qed
```
```   165
```
```   166 lemma antimono:
```
```   167   assumes "A \<subseteq> B" and "A \<noteq> {}" and "finite B"
```
```   168   shows "F B \<preceq> F A"
```
```   169 proof (cases "A = B")
```
```   170   case True then show ?thesis by (simp add: refl)
```
```   171 next
```
```   172   case False
```
```   173   have B: "B = A \<union> (B - A)" using `A \<subseteq> B` by blast
```
```   174   then have "F B = F (A \<union> (B - A))" by simp
```
```   175   also have "\<dots> = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
```
```   176   also have "\<dots> \<preceq> F A" by simp
```
```   177   finally show ?thesis .
```
```   178 qed
```
```   179
```
```   180 end
```
```   181
```
```   182
```
```   183 subsubsection {* With neutral element *}
```
```   184
```
```   185 locale semilattice_neutr_set = semilattice_neutr
```
```   186 begin
```
```   187
```
```   188 interpretation comp_fun_idem f
```
```   189   by default (simp_all add: fun_eq_iff left_commute)
```
```   190
```
```   191 definition F :: "'a set \<Rightarrow> 'a"
```
```   192 where
```
```   193   eq_fold: "F A = Finite_Set.fold f 1 A"
```
```   194
```
```   195 lemma infinite [simp]:
```
```   196   "\<not> finite A \<Longrightarrow> F A = 1"
```
```   197   by (simp add: eq_fold)
```
```   198
```
```   199 lemma empty [simp]:
```
```   200   "F {} = 1"
```
```   201   by (simp add: eq_fold)
```
```   202
```
```   203 lemma insert [simp]:
```
```   204   assumes "finite A"
```
```   205   shows "F (insert x A) = x * F A"
```
```   206   using assms by (simp add: eq_fold)
```
```   207
```
```   208 lemma in_idem:
```
```   209   assumes "finite A" and "x \<in> A"
```
```   210   shows "x * F A = F A"
```
```   211 proof -
```
```   212   from assms have "A \<noteq> {}" by auto
```
```   213   with `finite A` show ?thesis using `x \<in> A`
```
```   214     by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
```
```   215 qed
```
```   216
```
```   217 lemma union:
```
```   218   assumes "finite A" and "finite B"
```
```   219   shows "F (A \<union> B) = F A * F B"
```
```   220   using assms by (induct A) (simp_all add: ac_simps)
```
```   221
```
```   222 lemma remove:
```
```   223   assumes "finite A" and "x \<in> A"
```
```   224   shows "F A = x * F (A - {x})"
```
```   225 proof -
```
```   226   from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
```
```   227   with assms show ?thesis by simp
```
```   228 qed
```
```   229
```
```   230 lemma insert_remove:
```
```   231   assumes "finite A"
```
```   232   shows "F (insert x A) = x * F (A - {x})"
```
```   233   using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
```
```   234
```
```   235 lemma subset:
```
```   236   assumes "finite A" and "B \<subseteq> A"
```
```   237   shows "F B * F A = F A"
```
```   238 proof -
```
```   239   from assms have "finite B" by (auto dest: finite_subset)
```
```   240   with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
```
```   241 qed
```
```   242
```
```   243 lemma closed:
```
```   244   assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
```
```   245   shows "F A \<in> A"
```
```   246 using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
```
```   247   case singleton then show ?case by simp
```
```   248 next
```
```   249   case insert with elem show ?case by force
```
```   250 qed
```
```   251
```
```   252 end
```
```   253
```
```   254 locale semilattice_order_neutr_set = semilattice_neutr_order + semilattice_neutr_set
```
```   255 begin
```
```   256
```
```   257 lemma bounded_iff:
```
```   258   assumes "finite A"
```
```   259   shows "x \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> a)"
```
```   260   using assms by (induct A) (simp_all add: bounded_iff)
```
```   261
```
```   262 lemma boundedI:
```
```   263   assumes "finite A"
```
```   264   assumes "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
```
```   265   shows "x \<preceq> F A"
```
```   266   using assms by (simp add: bounded_iff)
```
```   267
```
```   268 lemma boundedE:
```
```   269   assumes "finite A" and "x \<preceq> F A"
```
```   270   obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
```
```   271   using assms by (simp add: bounded_iff)
```
```   272
```
```   273 lemma coboundedI:
```
```   274   assumes "finite A"
```
```   275     and "a \<in> A"
```
```   276   shows "F A \<preceq> a"
```
```   277 proof -
```
```   278   from assms have "A \<noteq> {}" by auto
```
```   279   from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
```
```   280   proof (induct rule: finite_ne_induct)
```
```   281     case singleton thus ?case by (simp add: refl)
```
```   282   next
```
```   283     case (insert x B)
```
```   284     from insert have "a = x \<or> a \<in> B" by simp
```
```   285     then show ?case using insert by (auto intro: coboundedI2)
```
```   286   qed
```
```   287 qed
```
```   288
```
```   289 lemma antimono:
```
```   290   assumes "A \<subseteq> B" and "finite B"
```
```   291   shows "F B \<preceq> F A"
```
```   292 proof (cases "A = B")
```
```   293   case True then show ?thesis by (simp add: refl)
```
```   294 next
```
```   295   case False
```
```   296   have B: "B = A \<union> (B - A)" using `A \<subseteq> B` by blast
```
```   297   then have "F B = F (A \<union> (B - A))" by simp
```
```   298   also have "\<dots> = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
```
```   299   also have "\<dots> \<preceq> F A" by simp
```
```   300   finally show ?thesis .
```
```   301 qed
```
```   302
```
```   303 end
```
```   304
```
```   305 notation times (infixl "*" 70)
```
```   306 notation Groups.one ("1")
```
```   307
```
```   308
```
```   309 subsection {* Lattice operations on finite sets *}
```
```   310
```
```   311 text {*
```
```   312   For historic reasons, there is the sublocale dependency from @{class distrib_lattice}
```
```   313   to @{class linorder}.  This is badly designed: both should depend on a common abstract
```
```   314   distributive lattice rather than having this non-subclass dependecy between two
```
```   315   classes.  But for the moment we have to live with it.  This forces us to setup
```
```   316   this sublocale dependency simultaneously with the lattice operations on finite
```
```   317   sets, to avoid garbage.
```
```   318 *}
```
```   319
```
```   320 definition (in semilattice_inf) Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>\<^sub>f\<^sub>i\<^sub>n_"  900)
```
```   321 where
```
```   322   "Inf_fin = semilattice_set.F inf"
```
```   323
```
```   324 definition (in semilattice_sup) Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>\<^sub>f\<^sub>i\<^sub>n_"  900)
```
```   325 where
```
```   326   "Sup_fin = semilattice_set.F sup"
```
```   327
```
```   328 context linorder
```
```   329 begin
```
```   330
```
```   331 definition Min :: "'a set \<Rightarrow> 'a"
```
```   332 where
```
```   333   "Min = semilattice_set.F min"
```
```   334
```
```   335 definition Max :: "'a set \<Rightarrow> 'a"
```
```   336 where
```
```   337   "Max = semilattice_set.F max"
```
```   338
```
```   339 sublocale Min!: semilattice_order_set min less_eq less
```
```   340   + Max!: semilattice_order_set max greater_eq greater
```
```   341 where
```
```   342   "semilattice_set.F min = Min"
```
```   343   and "semilattice_set.F max = Max"
```
```   344 proof -
```
```   345   show "semilattice_order_set min less_eq less" by default (auto simp add: min_def)
```
```   346   then interpret Min!: semilattice_order_set min less_eq less .
```
```   347   show "semilattice_order_set max greater_eq greater" by default (auto simp add: max_def)
```
```   348   then interpret Max!: semilattice_order_set max greater_eq greater .
```
```   349   from Min_def show "semilattice_set.F min = Min" by rule
```
```   350   from Max_def show "semilattice_set.F max = Max" by rule
```
```   351 qed
```
```   352
```
```   353
```
```   354 text {* An aside: @{const min}/@{const max} on linear orders as special case of @{const inf}/@{const sup} *}
```
```   355
```
```   356 sublocale min_max!: distrib_lattice min less_eq less max
```
```   357 where
```
```   358   "semilattice_inf.Inf_fin min = Min"
```
```   359   and "semilattice_sup.Sup_fin max = Max"
```
```   360 proof -
```
```   361   show "class.distrib_lattice min less_eq less max"
```
```   362   proof
```
```   363     fix x y z
```
```   364     show "max x (min y z) = min (max x y) (max x z)"
```
```   365       by (auto simp add: min_def max_def)
```
```   366   qed (auto simp add: min_def max_def not_le less_imp_le)
```
```   367   then interpret min_max!: distrib_lattice min less_eq less max .
```
```   368   show "semilattice_inf.Inf_fin min = Min"
```
```   369     by (simp only: min_max.Inf_fin_def Min_def)
```
```   370   show "semilattice_sup.Sup_fin max = Max"
```
```   371     by (simp only: min_max.Sup_fin_def Max_def)
```
```   372 qed
```
```   373
```
```   374 lemmas le_maxI1 = min_max.sup_ge1
```
```   375 lemmas le_maxI2 = min_max.sup_ge2
```
```   376
```
```   377 lemmas min_ac = min_max.inf_assoc min_max.inf_commute
```
```   378   min.left_commute
```
```   379
```
```   380 lemmas max_ac = min_max.sup_assoc min_max.sup_commute
```
```   381   max.left_commute
```
```   382
```
```   383 end
```
```   384
```
```   385
```
```   386 text {* Lattice operations proper *}
```
```   387
```
```   388 sublocale semilattice_inf < Inf_fin!: semilattice_order_set inf less_eq less
```
```   389 where
```
```   390   "semilattice_set.F inf = Inf_fin"
```
```   391 proof -
```
```   392   show "semilattice_order_set inf less_eq less" ..
```
```   393   then interpret Inf_fin!: semilattice_order_set inf less_eq less .
```
```   394   from Inf_fin_def show "semilattice_set.F inf = Inf_fin" by rule
```
```   395 qed
```
```   396
```
```   397 sublocale semilattice_sup < Sup_fin!: semilattice_order_set sup greater_eq greater
```
```   398 where
```
```   399   "semilattice_set.F sup = Sup_fin"
```
```   400 proof -
```
```   401   show "semilattice_order_set sup greater_eq greater" ..
```
```   402   then interpret Sup_fin!: semilattice_order_set sup greater_eq greater .
```
```   403   from Sup_fin_def show "semilattice_set.F sup = Sup_fin" by rule
```
```   404 qed
```
```   405
```
```   406
```
```   407 text {* An aside again: @{const Min}/@{const Max} on linear orders as special case of @{const Inf_fin}/@{const Sup_fin} *}
```
```   408
```
```   409 lemma Inf_fin_Min:
```
```   410   "Inf_fin = (Min :: 'a::{semilattice_inf, linorder} set \<Rightarrow> 'a)"
```
```   411   by (simp add: Inf_fin_def Min_def inf_min)
```
```   412
```
```   413 lemma Sup_fin_Max:
```
```   414   "Sup_fin = (Max :: 'a::{semilattice_sup, linorder} set \<Rightarrow> 'a)"
```
```   415   by (simp add: Sup_fin_def Max_def sup_max)
```
```   416
```
```   417
```
```   418
```
```   419 subsection {* Infimum and Supremum over non-empty sets *}
```
```   420
```
```   421 text {*
```
```   422   After this non-regular bootstrap, things continue canonically.
```
```   423 *}
```
```   424
```
```   425 context lattice
```
```   426 begin
```
```   427
```
```   428 lemma Inf_le_Sup [simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>\<^sub>f\<^sub>i\<^sub>nA \<le> \<Squnion>\<^sub>f\<^sub>i\<^sub>nA"
```
```   429 apply(subgoal_tac "EX a. a:A")
```
```   430 prefer 2 apply blast
```
```   431 apply(erule exE)
```
```   432 apply(rule order_trans)
```
```   433 apply(erule (1) Inf_fin.coboundedI)
```
```   434 apply(erule (1) Sup_fin.coboundedI)
```
```   435 done
```
```   436
```
```   437 lemma sup_Inf_absorb [simp]:
```
```   438   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA) = a"
```
```   439 apply(subst sup_commute)
```
```   440 apply(simp add: sup_absorb2 Inf_fin.coboundedI)
```
```   441 done
```
```   442
```
```   443 lemma inf_Sup_absorb [simp]:
```
```   444   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA) = a"
```
```   445 by (simp add: inf_absorb1 Sup_fin.coboundedI)
```
```   446
```
```   447 end
```
```   448
```
```   449 context distrib_lattice
```
```   450 begin
```
```   451
```
```   452 lemma sup_Inf1_distrib:
```
```   453   assumes "finite A"
```
```   454     and "A \<noteq> {}"
```
```   455   shows "sup x (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA) = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n{sup x a|a. a \<in> A}"
```
```   456 using assms by (simp add: image_def Inf_fin.hom_commute [where h="sup x", OF sup_inf_distrib1])
```
```   457   (rule arg_cong [where f="Inf_fin"], blast)
```
```   458
```
```   459 lemma sup_Inf2_distrib:
```
```   460   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
```
```   461   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}"
```
```   462 using A proof (induct rule: finite_ne_induct)
```
```   463   case singleton then show ?case
```
```   464     by (simp add: sup_Inf1_distrib [OF B])
```
```   465 next
```
```   466   case (insert x A)
```
```   467   have finB: "finite {sup x b |b. b \<in> B}"
```
```   468     by (rule finite_surj [where f = "sup x", OF B(1)], auto)
```
```   469   have finAB: "finite {sup a b |a b. a \<in> A \<and> b \<in> B}"
```
```   470   proof -
```
```   471     have "{sup a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {sup a b})"
```
```   472       by blast
```
```   473     thus ?thesis by(simp add: insert(1) B(1))
```
```   474   qed
```
```   475   have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
```
```   476   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)"
```
```   477     using insert by simp
```
```   478   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)
```
```   479   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})"
```
```   480     using insert by(simp add:sup_Inf1_distrib[OF B])
```
```   481   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})"
```
```   482     (is "_ = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n?M")
```
```   483     using B insert
```
```   484     by (simp add: Inf_fin.union [OF finB _ finAB ne])
```
```   485   also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"
```
```   486     by blast
```
```   487   finally show ?case .
```
```   488 qed
```
```   489
```
```   490 lemma inf_Sup1_distrib:
```
```   491   assumes "finite A" and "A \<noteq> {}"
```
```   492   shows "inf x (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA) = \<Squnion>\<^sub>f\<^sub>i\<^sub>n{inf x a|a. a \<in> A}"
```
```   493 using assms by (simp add: image_def Sup_fin.hom_commute [where h="inf x", OF inf_sup_distrib1])
```
```   494   (rule arg_cong [where f="Sup_fin"], blast)
```
```   495
```
```   496 lemma inf_Sup2_distrib:
```
```   497   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
```
```   498   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}"
```
```   499 using A proof (induct rule: finite_ne_induct)
```
```   500   case singleton thus ?case
```
```   501     by(simp add: inf_Sup1_distrib [OF B])
```
```   502 next
```
```   503   case (insert x A)
```
```   504   have finB: "finite {inf x b |b. b \<in> B}"
```
```   505     by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto)
```
```   506   have finAB: "finite {inf a b |a b. a \<in> A \<and> b \<in> B}"
```
```   507   proof -
```
```   508     have "{inf a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {inf a b})"
```
```   509       by blast
```
```   510     thus ?thesis by(simp add: insert(1) B(1))
```
```   511   qed
```
```   512   have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
```
```   513   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)"
```
```   514     using insert by simp
```
```   515   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)
```
```   516   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})"
```
```   517     using insert by(simp add:inf_Sup1_distrib[OF B])
```
```   518   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})"
```
```   519     (is "_ = \<Squnion>\<^sub>f\<^sub>i\<^sub>n?M")
```
```   520     using B insert
```
```   521     by (simp add: Sup_fin.union [OF finB _ finAB ne])
```
```   522   also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"
```
```   523     by blast
```
```   524   finally show ?case .
```
```   525 qed
```
```   526
```
```   527 end
```
```   528
```
```   529 context complete_lattice
```
```   530 begin
```
```   531
```
```   532 lemma Inf_fin_Inf:
```
```   533   assumes "finite A" and "A \<noteq> {}"
```
```   534   shows "\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA = Inf A"
```
```   535 proof -
```
```   536   from assms obtain b B where "A = insert b B" and "finite B" by auto
```
```   537   then show ?thesis
```
```   538     by (simp add: Inf_fin.eq_fold inf_Inf_fold_inf inf.commute [of b])
```
```   539 qed
```
```   540
```
```   541 lemma Sup_fin_Sup:
```
```   542   assumes "finite A" and "A \<noteq> {}"
```
```   543   shows "\<Squnion>\<^sub>f\<^sub>i\<^sub>nA = Sup A"
```
```   544 proof -
```
```   545   from assms obtain b B where "A = insert b B" and "finite B" by auto
```
```   546   then show ?thesis
```
```   547     by (simp add: Sup_fin.eq_fold sup_Sup_fold_sup sup.commute [of b])
```
```   548 qed
```
```   549
```
```   550 end
```
```   551
```
```   552
```
```   553 subsection {* Minimum and Maximum over non-empty sets *}
```
```   554
```
```   555 context linorder
```
```   556 begin
```
```   557
```
```   558 lemma dual_min:
```
```   559   "ord.min greater_eq = max"
```
```   560   by (auto simp add: ord.min_def max_def fun_eq_iff)
```
```   561
```
```   562 lemma dual_max:
```
```   563   "ord.max greater_eq = min"
```
```   564   by (auto simp add: ord.max_def min_def fun_eq_iff)
```
```   565
```
```   566 lemma dual_Min:
```
```   567   "linorder.Min greater_eq = Max"
```
```   568 proof -
```
```   569   interpret dual!: linorder greater_eq greater by (fact dual_linorder)
```
```   570   show ?thesis by (simp add: dual.Min_def dual_min Max_def)
```
```   571 qed
```
```   572
```
```   573 lemma dual_Max:
```
```   574   "linorder.Max greater_eq = Min"
```
```   575 proof -
```
```   576   interpret dual!: linorder greater_eq greater by (fact dual_linorder)
```
```   577   show ?thesis by (simp add: dual.Max_def dual_max Min_def)
```
```   578 qed
```
```   579
```
```   580 lemmas Min_singleton = Min.singleton
```
```   581 lemmas Max_singleton = Max.singleton
```
```   582 lemmas Min_insert = Min.insert
```
```   583 lemmas Max_insert = Max.insert
```
```   584 lemmas Min_Un = Min.union
```
```   585 lemmas Max_Un = Max.union
```
```   586 lemmas hom_Min_commute = Min.hom_commute
```
```   587 lemmas hom_Max_commute = Max.hom_commute
```
```   588
```
```   589 lemma Min_in [simp]:
```
```   590   assumes "finite A" and "A \<noteq> {}"
```
```   591   shows "Min A \<in> A"
```
```   592   using assms by (auto simp add: min_def Min.closed)
```
```   593
```
```   594 lemma Max_in [simp]:
```
```   595   assumes "finite A" and "A \<noteq> {}"
```
```   596   shows "Max A \<in> A"
```
```   597   using assms by (auto simp add: max_def Max.closed)
```
```   598
```
```   599 lemma Min_le [simp]:
```
```   600   assumes "finite A" and "x \<in> A"
```
```   601   shows "Min A \<le> x"
```
```   602   using assms by (fact Min.coboundedI)
```
```   603
```
```   604 lemma Max_ge [simp]:
```
```   605   assumes "finite A" and "x \<in> A"
```
```   606   shows "x \<le> Max A"
```
```   607   using assms by (fact Max.coboundedI)
```
```   608
```
```   609 lemma Min_eqI:
```
```   610   assumes "finite A"
```
```   611   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<ge> x"
```
```   612     and "x \<in> A"
```
```   613   shows "Min A = x"
```
```   614 proof (rule antisym)
```
```   615   from `x \<in> A` have "A \<noteq> {}" by auto
```
```   616   with assms show "Min A \<ge> x" by simp
```
```   617 next
```
```   618   from assms show "x \<ge> Min A" by simp
```
```   619 qed
```
```   620
```
```   621 lemma Max_eqI:
```
```   622   assumes "finite A"
```
```   623   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
```
```   624     and "x \<in> A"
```
```   625   shows "Max A = x"
```
```   626 proof (rule antisym)
```
```   627   from `x \<in> A` have "A \<noteq> {}" by auto
```
```   628   with assms show "Max A \<le> x" by simp
```
```   629 next
```
```   630   from assms show "x \<le> Max A" by simp
```
```   631 qed
```
```   632
```
```   633 context
```
```   634   fixes A :: "'a set"
```
```   635   assumes fin_nonempty: "finite A" "A \<noteq> {}"
```
```   636 begin
```
```   637
```
```   638 lemma Min_ge_iff [simp]:
```
```   639   "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
```
```   640   using fin_nonempty by (fact Min.bounded_iff)
```
```   641
```
```   642 lemma Max_le_iff [simp]:
```
```   643   "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
```
```   644   using fin_nonempty by (fact Max.bounded_iff)
```
```   645
```
```   646 lemma Min_gr_iff [simp]:
```
```   647   "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
```
```   648   using fin_nonempty  by (induct rule: finite_ne_induct) simp_all
```
```   649
```
```   650 lemma Max_less_iff [simp]:
```
```   651   "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
```
```   652   using fin_nonempty by (induct rule: finite_ne_induct) simp_all
```
```   653
```
```   654 lemma Min_le_iff:
```
```   655   "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
```
```   656   using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_le_iff_disj)
```
```   657
```
```   658 lemma Max_ge_iff:
```
```   659   "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
```
```   660   using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: le_max_iff_disj)
```
```   661
```
```   662 lemma Min_less_iff:
```
```   663   "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
```
```   664   using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_less_iff_disj)
```
```   665
```
```   666 lemma Max_gr_iff:
```
```   667   "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
```
```   668   using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: less_max_iff_disj)
```
```   669
```
```   670 end
```
```   671
```
```   672 lemma Min_antimono:
```
```   673   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
```
```   674   shows "Min N \<le> Min M"
```
```   675   using assms by (fact Min.antimono)
```
```   676
```
```   677 lemma Max_mono:
```
```   678   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
```
```   679   shows "Max M \<le> Max N"
```
```   680   using assms by (fact Max.antimono)
```
```   681
```
```   682 lemma mono_Min_commute:
```
```   683   assumes "mono f"
```
```   684   assumes "finite A" and "A \<noteq> {}"
```
```   685   shows "f (Min A) = Min (f ` A)"
```
```   686 proof (rule linorder_class.Min_eqI [symmetric])
```
```   687   from `finite A` show "finite (f ` A)" by simp
```
```   688   from assms show "f (Min A) \<in> f ` A" by simp
```
```   689   fix x
```
```   690   assume "x \<in> f ` A"
```
```   691   then obtain y where "y \<in> A" and "x = f y" ..
```
```   692   with assms have "Min A \<le> y" by auto
```
```   693   with `mono f` have "f (Min A) \<le> f y" by (rule monoE)
```
```   694   with `x = f y` show "f (Min A) \<le> x" by simp
```
```   695 qed
```
```   696
```
```   697 lemma mono_Max_commute:
```
```   698   assumes "mono f"
```
```   699   assumes "finite A" and "A \<noteq> {}"
```
```   700   shows "f (Max A) = Max (f ` A)"
```
```   701 proof (rule linorder_class.Max_eqI [symmetric])
```
```   702   from `finite A` show "finite (f ` A)" by simp
```
```   703   from assms show "f (Max A) \<in> f ` A" by simp
```
```   704   fix x
```
```   705   assume "x \<in> f ` A"
```
```   706   then obtain y where "y \<in> A" and "x = f y" ..
```
```   707   with assms have "y \<le> Max A" by auto
```
```   708   with `mono f` have "f y \<le> f (Max A)" by (rule monoE)
```
```   709   with `x = f y` show "x \<le> f (Max A)" by simp
```
```   710 qed
```
```   711
```
```   712 lemma finite_linorder_max_induct [consumes 1, case_names empty insert]:
```
```   713   assumes fin: "finite A"
```
```   714   and empty: "P {}"
```
```   715   and insert: "\<And>b A. finite A \<Longrightarrow> \<forall>a\<in>A. a < b \<Longrightarrow> P A \<Longrightarrow> P (insert b A)"
```
```   716   shows "P A"
```
```   717 using fin empty insert
```
```   718 proof (induct rule: finite_psubset_induct)
```
```   719   case (psubset A)
```
```   720   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
```
```   721   have fin: "finite A" by fact
```
```   722   have empty: "P {}" by fact
```
```   723   have step: "\<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. a < b; P A\<rbrakk> \<Longrightarrow> P (insert b A)" by fact
```
```   724   show "P A"
```
```   725   proof (cases "A = {}")
```
```   726     assume "A = {}"
```
```   727     then show "P A" using `P {}` by simp
```
```   728   next
```
```   729     let ?B = "A - {Max A}"
```
```   730     let ?A = "insert (Max A) ?B"
```
```   731     have "finite ?B" using `finite A` by simp
```
```   732     assume "A \<noteq> {}"
```
```   733     with `finite A` have "Max A : A" by auto
```
```   734     then have A: "?A = A" using insert_Diff_single insert_absorb by auto
```
```   735     then have "P ?B" using `P {}` step IH [of ?B] by blast
```
```   736     moreover
```
```   737     have "\<forall>a\<in>?B. a < Max A" using Max_ge [OF `finite A`] by fastforce
```
```   738     ultimately show "P A" using A insert_Diff_single step [OF `finite ?B`] by fastforce
```
```   739   qed
```
```   740 qed
```
```   741
```
```   742 lemma finite_linorder_min_induct [consumes 1, case_names empty insert]:
```
```   743   "\<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"
```
```   744   by (rule linorder.finite_linorder_max_induct [OF dual_linorder])
```
```   745
```
```   746 lemma Least_Min:
```
```   747   assumes "finite {a. P a}" and "\<exists>a. P a"
```
```   748   shows "(LEAST a. P a) = Min {a. P a}"
```
```   749 proof -
```
```   750   { fix A :: "'a set"
```
```   751     assume A: "finite A" "A \<noteq> {}"
```
```   752     have "(LEAST a. a \<in> A) = Min A"
```
```   753     using A proof (induct A rule: finite_ne_induct)
```
```   754       case singleton show ?case by (rule Least_equality) simp_all
```
```   755     next
```
```   756       case (insert a A)
```
```   757       have "(LEAST b. b = a \<or> b \<in> A) = min a (LEAST a. a \<in> A)"
```
```   758         by (auto intro!: Least_equality simp add: min_def not_le Min_le_iff insert.hyps dest!: less_imp_le)
```
```   759       with insert show ?case by simp
```
```   760     qed
```
```   761   } from this [of "{a. P a}"] assms show ?thesis by simp
```
```   762 qed
```
```   763
```
```   764 end
```
```   765
```
```   766 context linordered_ab_semigroup_add
```
```   767 begin
```
```   768
```
```   769 lemma add_Min_commute:
```
```   770   fixes k
```
```   771   assumes "finite N" and "N \<noteq> {}"
```
```   772   shows "k + Min N = Min {k + m | m. m \<in> N}"
```
```   773 proof -
```
```   774   have "\<And>x y. k + min x y = min (k + x) (k + y)"
```
```   775     by (simp add: min_def not_le)
```
```   776       (blast intro: antisym less_imp_le add_left_mono)
```
```   777   with assms show ?thesis
```
```   778     using hom_Min_commute [of "plus k" N]
```
```   779     by simp (blast intro: arg_cong [where f = Min])
```
```   780 qed
```
```   781
```
```   782 lemma add_Max_commute:
```
```   783   fixes k
```
```   784   assumes "finite N" and "N \<noteq> {}"
```
```   785   shows "k + Max N = Max {k + m | m. m \<in> N}"
```
```   786 proof -
```
```   787   have "\<And>x y. k + max x y = max (k + x) (k + y)"
```
```   788     by (simp add: max_def not_le)
```
```   789       (blast intro: antisym less_imp_le add_left_mono)
```
```   790   with assms show ?thesis
```
```   791     using hom_Max_commute [of "plus k" N]
```
```   792     by simp (blast intro: arg_cong [where f = Max])
```
```   793 qed
```
```   794
```
```   795 end
```
```   796
```
```   797 context linordered_ab_group_add
```
```   798 begin
```
```   799
```
```   800 lemma minus_Max_eq_Min [simp]:
```
```   801   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Max S = Min (uminus ` S)"
```
```   802   by (induct S rule: finite_ne_induct) (simp_all add: minus_max_eq_min)
```
```   803
```
```   804 lemma minus_Min_eq_Max [simp]:
```
```   805   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Min S = Max (uminus ` S)"
```
```   806   by (induct S rule: finite_ne_induct) (simp_all add: minus_min_eq_max)
```
```   807
```
```   808 end
```
```   809
```
```   810 context complete_linorder
```
```   811 begin
```
```   812
```
```   813 lemma Min_Inf:
```
```   814   assumes "finite A" and "A \<noteq> {}"
```
```   815   shows "Min A = Inf A"
```
```   816 proof -
```
```   817   from assms obtain b B where "A = insert b B" and "finite B" by auto
```
```   818   then show ?thesis
```
```   819     by (simp add: Min.eq_fold complete_linorder_inf_min [symmetric] inf_Inf_fold_inf inf.commute [of b])
```
```   820 qed
```
```   821
```
```   822 lemma Max_Sup:
```
```   823   assumes "finite A" and "A \<noteq> {}"
```
```   824   shows "Max A = Sup A"
```
```   825 proof -
```
```   826   from assms obtain b B where "A = insert b B" and "finite B" by auto
```
```   827   then show ?thesis
```
```   828     by (simp add: Max.eq_fold complete_linorder_sup_max [symmetric] sup_Sup_fold_sup sup.commute [of b])
```
```   829 qed
```
```   830
```
```   831 end
```
```   832
```
```   833 end
```