src/HOL/Big_Operators.thy
 author wenzelm Tue Sep 03 01:12:40 2013 +0200 (2013-09-03) changeset 53374 a14d2a854c02 parent 53174 71a2702da5e0 child 54147 97a8ff4e4ac9 permissions -rw-r--r--
tuned proofs -- clarified flow of facts wrt. calculation;
```     1 (*  Title:      HOL/Big_Operators.thy
```
```     2     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
```
```     3                 with contributions by Jeremy Avigad
```
```     4 *)
```
```     5
```
```     6 header {* Big operators and finite (non-empty) sets *}
```
```     7
```
```     8 theory Big_Operators
```
```     9 imports Finite_Set Option Metis
```
```    10 begin
```
```    11
```
```    12 subsection {* Generic monoid operation over a set *}
```
```    13
```
```    14 no_notation times (infixl "*" 70)
```
```    15 no_notation Groups.one ("1")
```
```    16
```
```    17 locale comm_monoid_set = comm_monoid
```
```    18 begin
```
```    19
```
```    20 interpretation comp_fun_commute f
```
```    21   by default (simp add: fun_eq_iff left_commute)
```
```    22
```
```    23 interpretation comp_fun_commute "f \<circ> g"
```
```    24   by (rule comp_comp_fun_commute)
```
```    25
```
```    26 definition F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
```
```    27 where
```
```    28   eq_fold: "F g A = Finite_Set.fold (f \<circ> g) 1 A"
```
```    29
```
```    30 lemma infinite [simp]:
```
```    31   "\<not> finite A \<Longrightarrow> F g A = 1"
```
```    32   by (simp add: eq_fold)
```
```    33
```
```    34 lemma empty [simp]:
```
```    35   "F g {} = 1"
```
```    36   by (simp add: eq_fold)
```
```    37
```
```    38 lemma insert [simp]:
```
```    39   assumes "finite A" and "x \<notin> A"
```
```    40   shows "F g (insert x A) = g x * F g A"
```
```    41   using assms by (simp add: eq_fold)
```
```    42
```
```    43 lemma remove:
```
```    44   assumes "finite A" and "x \<in> A"
```
```    45   shows "F g A = g x * F g (A - {x})"
```
```    46 proof -
```
```    47   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
```
```    48     by (auto dest: mk_disjoint_insert)
```
```    49   moreover from `finite A` A have "finite B" by simp
```
```    50   ultimately show ?thesis by simp
```
```    51 qed
```
```    52
```
```    53 lemma insert_remove:
```
```    54   assumes "finite A"
```
```    55   shows "F g (insert x A) = g x * F g (A - {x})"
```
```    56   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
```
```    57
```
```    58 lemma neutral:
```
```    59   assumes "\<forall>x\<in>A. g x = 1"
```
```    60   shows "F g A = 1"
```
```    61   using assms by (induct A rule: infinite_finite_induct) simp_all
```
```    62
```
```    63 lemma neutral_const [simp]:
```
```    64   "F (\<lambda>_. 1) A = 1"
```
```    65   by (simp add: neutral)
```
```    66
```
```    67 lemma union_inter:
```
```    68   assumes "finite A" and "finite B"
```
```    69   shows "F g (A \<union> B) * F g (A \<inter> B) = F g A * F g B"
```
```    70   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
```
```    71 using assms proof (induct A)
```
```    72   case empty then show ?case by simp
```
```    73 next
```
```    74   case (insert x A) then show ?case
```
```    75     by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
```
```    76 qed
```
```    77
```
```    78 corollary union_inter_neutral:
```
```    79   assumes "finite A" and "finite B"
```
```    80   and I0: "\<forall>x \<in> A \<inter> B. g x = 1"
```
```    81   shows "F g (A \<union> B) = F g A * F g B"
```
```    82   using assms by (simp add: union_inter [symmetric] neutral)
```
```    83
```
```    84 corollary union_disjoint:
```
```    85   assumes "finite A" and "finite B"
```
```    86   assumes "A \<inter> B = {}"
```
```    87   shows "F g (A \<union> B) = F g A * F g B"
```
```    88   using assms by (simp add: union_inter_neutral)
```
```    89
```
```    90 lemma subset_diff:
```
```    91   "B \<subseteq> A \<Longrightarrow> finite A \<Longrightarrow> F g A = F g (A - B) * F g B"
```
```    92   by (metis Diff_partition union_disjoint Diff_disjoint finite_Un inf_commute sup_commute)
```
```    93
```
```    94 lemma reindex:
```
```    95   assumes "inj_on h A"
```
```    96   shows "F g (h ` A) = F (g \<circ> h) A"
```
```    97 proof (cases "finite A")
```
```    98   case True
```
```    99   with assms show ?thesis by (simp add: eq_fold fold_image comp_assoc)
```
```   100 next
```
```   101   case False with assms have "\<not> finite (h ` A)" by (blast dest: finite_imageD)
```
```   102   with False show ?thesis by simp
```
```   103 qed
```
```   104
```
```   105 lemma cong:
```
```   106   assumes "A = B"
```
```   107   assumes g_h: "\<And>x. x \<in> B \<Longrightarrow> g x = h x"
```
```   108   shows "F g A = F h B"
```
```   109 proof (cases "finite A")
```
```   110   case True
```
```   111   then have "\<And>C. C \<subseteq> A \<longrightarrow> (\<forall>x\<in>C. g x = h x) \<longrightarrow> F g C = F h C"
```
```   112   proof induct
```
```   113     case empty then show ?case by simp
```
```   114   next
```
```   115     case (insert x F) then show ?case apply -
```
```   116     apply (simp add: subset_insert_iff, clarify)
```
```   117     apply (subgoal_tac "finite C")
```
```   118       prefer 2 apply (blast dest: finite_subset [rotated])
```
```   119     apply (subgoal_tac "C = insert x (C - {x})")
```
```   120       prefer 2 apply blast
```
```   121     apply (erule ssubst)
```
```   122     apply (simp add: Ball_def del: insert_Diff_single)
```
```   123     done
```
```   124   qed
```
```   125   with `A = B` g_h show ?thesis by simp
```
```   126 next
```
```   127   case False
```
```   128   with `A = B` show ?thesis by simp
```
```   129 qed
```
```   130
```
```   131 lemma strong_cong [cong]:
```
```   132   assumes "A = B" "\<And>x. x \<in> B =simp=> g x = h x"
```
```   133   shows "F (\<lambda>x. g x) A = F (\<lambda>x. h x) B"
```
```   134   by (rule cong) (insert assms, simp_all add: simp_implies_def)
```
```   135
```
```   136 lemma UNION_disjoint:
```
```   137   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
```
```   138   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
```
```   139   shows "F g (UNION I A) = F (\<lambda>x. F g (A x)) I"
```
```   140 apply (insert assms)
```
```   141 apply (induct rule: finite_induct)
```
```   142 apply simp
```
```   143 apply atomize
```
```   144 apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
```
```   145  prefer 2 apply blast
```
```   146 apply (subgoal_tac "A x Int UNION Fa A = {}")
```
```   147  prefer 2 apply blast
```
```   148 apply (simp add: union_disjoint)
```
```   149 done
```
```   150
```
```   151 lemma Union_disjoint:
```
```   152   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
```
```   153   shows "F g (Union C) = F (F g) C"
```
```   154 proof cases
```
```   155   assume "finite C"
```
```   156   from UNION_disjoint [OF this assms]
```
```   157   show ?thesis
```
```   158     by (simp add: SUP_def)
```
```   159 qed (auto dest: finite_UnionD intro: infinite)
```
```   160
```
```   161 lemma distrib:
```
```   162   "F (\<lambda>x. g x * h x) A = F g A * F h A"
```
```   163   using assms by (induct A rule: infinite_finite_induct) (simp_all add: assoc commute left_commute)
```
```   164
```
```   165 lemma Sigma:
```
```   166   "finite A \<Longrightarrow> \<forall>x\<in>A. finite (B x) \<Longrightarrow> F (\<lambda>x. F (g x) (B x)) A = F (split g) (SIGMA x:A. B x)"
```
```   167 apply (subst Sigma_def)
```
```   168 apply (subst UNION_disjoint, assumption, simp)
```
```   169  apply blast
```
```   170 apply (rule cong)
```
```   171 apply rule
```
```   172 apply (simp add: fun_eq_iff)
```
```   173 apply (subst UNION_disjoint, simp, simp)
```
```   174  apply blast
```
```   175 apply (simp add: comp_def)
```
```   176 done
```
```   177
```
```   178 lemma related:
```
```   179   assumes Re: "R 1 1"
```
```   180   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
```
```   181   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
```
```   182   shows "R (F h S) (F g S)"
```
```   183   using fS by (rule finite_subset_induct) (insert assms, auto)
```
```   184
```
```   185 lemma eq_general:
```
```   186   assumes h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y"
```
```   187   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
```
```   188   shows "F f1 S = F f2 S'"
```
```   189 proof-
```
```   190   from h f12 have hS: "h ` S = S'" by blast
```
```   191   {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
```
```   192     from f12 h H  have "x = y" by auto }
```
```   193   hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
```
```   194   from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
```
```   195   from hS have "F f2 S' = F f2 (h ` S)" by simp
```
```   196   also have "\<dots> = F (f2 o h) S" using reindex [OF hinj, of f2] .
```
```   197   also have "\<dots> = F f1 S " using th cong [of _ _ "f2 o h" f1]
```
```   198     by blast
```
```   199   finally show ?thesis ..
```
```   200 qed
```
```   201
```
```   202 lemma eq_general_reverses:
```
```   203   assumes kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
```
```   204   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"
```
```   205   shows "F j S = F g T"
```
```   206   (* metis solves it, but not yet available here *)
```
```   207   apply (rule eq_general [of T S h g j])
```
```   208   apply (rule ballI)
```
```   209   apply (frule kh)
```
```   210   apply (rule ex1I[])
```
```   211   apply blast
```
```   212   apply clarsimp
```
```   213   apply (drule hk) apply simp
```
```   214   apply (rule sym)
```
```   215   apply (erule conjunct1[OF conjunct2[OF hk]])
```
```   216   apply (rule ballI)
```
```   217   apply (drule hk)
```
```   218   apply blast
```
```   219   done
```
```   220
```
```   221 lemma mono_neutral_cong_left:
```
```   222   assumes "finite T" and "S \<subseteq> T" and "\<forall>i \<in> T - S. h i = 1"
```
```   223   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g S = F h T"
```
```   224 proof-
```
```   225   have eq: "T = S \<union> (T - S)" using `S \<subseteq> T` by blast
```
```   226   have d: "S \<inter> (T - S) = {}" using `S \<subseteq> T` by blast
```
```   227   from `finite T` `S \<subseteq> T` have f: "finite S" "finite (T - S)"
```
```   228     by (auto intro: finite_subset)
```
```   229   show ?thesis using assms(4)
```
```   230     by (simp add: union_disjoint [OF f d, unfolded eq [symmetric]] neutral [OF assms(3)])
```
```   231 qed
```
```   232
```
```   233 lemma mono_neutral_cong_right:
```
```   234   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1; \<And>x. x \<in> S \<Longrightarrow> g x = h x \<rbrakk>
```
```   235    \<Longrightarrow> F g T = F h S"
```
```   236   by (auto intro!: mono_neutral_cong_left [symmetric])
```
```   237
```
```   238 lemma mono_neutral_left:
```
```   239   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g S = F g T"
```
```   240   by (blast intro: mono_neutral_cong_left)
```
```   241
```
```   242 lemma mono_neutral_right:
```
```   243   "\<lbrakk> finite T;  S \<subseteq> T;  \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g T = F g S"
```
```   244   by (blast intro!: mono_neutral_left [symmetric])
```
```   245
```
```   246 lemma delta:
```
```   247   assumes fS: "finite S"
```
```   248   shows "F (\<lambda>k. if k = a then b k else 1) S = (if a \<in> S then b a else 1)"
```
```   249 proof-
```
```   250   let ?f = "(\<lambda>k. if k=a then b k else 1)"
```
```   251   { assume a: "a \<notin> S"
```
```   252     hence "\<forall>k\<in>S. ?f k = 1" by simp
```
```   253     hence ?thesis  using a by simp }
```
```   254   moreover
```
```   255   { assume a: "a \<in> S"
```
```   256     let ?A = "S - {a}"
```
```   257     let ?B = "{a}"
```
```   258     have eq: "S = ?A \<union> ?B" using a by blast
```
```   259     have dj: "?A \<inter> ?B = {}" by simp
```
```   260     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```   261     have "F ?f S = F ?f ?A * F ?f ?B"
```
```   262       using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]]
```
```   263       by simp
```
```   264     then have ?thesis using a by simp }
```
```   265   ultimately show ?thesis by blast
```
```   266 qed
```
```   267
```
```   268 lemma delta':
```
```   269   assumes fS: "finite S"
```
```   270   shows "F (\<lambda>k. if a = k then b k else 1) S = (if a \<in> S then b a else 1)"
```
```   271   using delta [OF fS, of a b, symmetric] by (auto intro: cong)
```
```   272
```
```   273 lemma If_cases:
```
```   274   fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
```
```   275   assumes fA: "finite A"
```
```   276   shows "F (\<lambda>x. if P x then h x else g x) A =
```
```   277     F h (A \<inter> {x. P x}) * F g (A \<inter> - {x. P x})"
```
```   278 proof -
```
```   279   have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}"
```
```   280           "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}"
```
```   281     by blast+
```
```   282   from fA
```
```   283   have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
```
```   284   let ?g = "\<lambda>x. if P x then h x else g x"
```
```   285   from union_disjoint [OF f a(2), of ?g] a(1)
```
```   286   show ?thesis
```
```   287     by (subst (1 2) cong) simp_all
```
```   288 qed
```
```   289
```
```   290 lemma cartesian_product:
```
```   291    "F (\<lambda>x. F (g x) B) A = F (split g) (A <*> B)"
```
```   292 apply (rule sym)
```
```   293 apply (cases "finite A")
```
```   294  apply (cases "finite B")
```
```   295   apply (simp add: Sigma)
```
```   296  apply (cases "A={}", simp)
```
```   297  apply simp
```
```   298 apply (auto intro: infinite dest: finite_cartesian_productD2)
```
```   299 apply (cases "B = {}") apply (auto intro: infinite dest: finite_cartesian_productD1)
```
```   300 done
```
```   301
```
```   302 end
```
```   303
```
```   304 notation times (infixl "*" 70)
```
```   305 notation Groups.one ("1")
```
```   306
```
```   307
```
```   308 subsection {* Generalized summation over a set *}
```
```   309
```
```   310 context comm_monoid_add
```
```   311 begin
```
```   312
```
```   313 definition setsum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
```
```   314 where
```
```   315   "setsum = comm_monoid_set.F plus 0"
```
```   316
```
```   317 sublocale setsum!: comm_monoid_set plus 0
```
```   318 where
```
```   319   "comm_monoid_set.F plus 0 = setsum"
```
```   320 proof -
```
```   321   show "comm_monoid_set plus 0" ..
```
```   322   then interpret setsum!: comm_monoid_set plus 0 .
```
```   323   from setsum_def show "comm_monoid_set.F plus 0 = setsum" by rule
```
```   324 qed
```
```   325
```
```   326 abbreviation
```
```   327   Setsum ("\<Sum>_"  999) where
```
```   328   "\<Sum>A \<equiv> setsum (%x. x) A"
```
```   329
```
```   330 end
```
```   331
```
```   332 text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
```
```   333 written @{text"\<Sum>x\<in>A. e"}. *}
```
```   334
```
```   335 syntax
```
```   336   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
```
```   337 syntax (xsymbols)
```
```   338   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```   339 syntax (HTML output)
```
```   340   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```   341
```
```   342 translations -- {* Beware of argument permutation! *}
```
```   343   "SUM i:A. b" == "CONST setsum (%i. b) A"
```
```   344   "\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A"
```
```   345
```
```   346 text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
```
```   347  @{text"\<Sum>x|P. e"}. *}
```
```   348
```
```   349 syntax
```
```   350   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
```
```   351 syntax (xsymbols)
```
```   352   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   353 syntax (HTML output)
```
```   354   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   355
```
```   356 translations
```
```   357   "SUM x|P. t" => "CONST setsum (%x. t) {x. P}"
```
```   358   "\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}"
```
```   359
```
```   360 print_translation {*
```
```   361 let
```
```   362   fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) \$ Abs (y, Ty, P)] =
```
```   363         if x <> y then raise Match
```
```   364         else
```
```   365           let
```
```   366             val x' = Syntax_Trans.mark_bound_body (x, Tx);
```
```   367             val t' = subst_bound (x', t);
```
```   368             val P' = subst_bound (x', P);
```
```   369           in
```
```   370             Syntax.const @{syntax_const "_qsetsum"} \$ Syntax_Trans.mark_bound_abs (x, Tx) \$ P' \$ t'
```
```   371           end
```
```   372     | setsum_tr' _ = raise Match;
```
```   373 in [(@{const_syntax setsum}, K setsum_tr')] end
```
```   374 *}
```
```   375
```
```   376 text {* TODO These are candidates for generalization *}
```
```   377
```
```   378 context comm_monoid_add
```
```   379 begin
```
```   380
```
```   381 lemma setsum_reindex_id:
```
```   382   "inj_on f B ==> setsum f B = setsum id (f ` B)"
```
```   383   by (simp add: setsum.reindex)
```
```   384
```
```   385 lemma setsum_reindex_nonzero:
```
```   386   assumes fS: "finite S"
```
```   387   and nz: "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x = f y \<Longrightarrow> h (f x) = 0"
```
```   388   shows "setsum h (f ` S) = setsum (h \<circ> f) S"
```
```   389 using nz proof (induct rule: finite_induct [OF fS])
```
```   390   case 1 thus ?case by simp
```
```   391 next
```
```   392   case (2 x F)
```
```   393   { assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
```
```   394     then obtain y where y: "y \<in> F" "f x = f y" by auto
```
```   395     from "2.hyps" y have xy: "x \<noteq> y" by auto
```
```   396     from "2.prems" [of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
```
```   397     have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
```
```   398     also have "\<dots> = setsum (h o f) (insert x F)"
```
```   399       unfolding setsum.insert[OF `finite F` `x\<notin>F`]
```
```   400       using h0
```
```   401       apply (simp cong del: setsum.strong_cong)
```
```   402       apply (rule "2.hyps"(3))
```
```   403       apply (rule_tac y="y" in  "2.prems")
```
```   404       apply simp_all
```
```   405       done
```
```   406     finally have ?case . }
```
```   407   moreover
```
```   408   { assume fxF: "f x \<notin> f ` F"
```
```   409     have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)"
```
```   410       using fxF "2.hyps" by simp
```
```   411     also have "\<dots> = setsum (h o f) (insert x F)"
```
```   412       unfolding setsum.insert[OF `finite F` `x\<notin>F`]
```
```   413       apply (simp cong del: setsum.strong_cong)
```
```   414       apply (rule cong [OF refl [of "op + (h (f x))"]])
```
```   415       apply (rule "2.hyps"(3))
```
```   416       apply (rule_tac y="y" in  "2.prems")
```
```   417       apply simp_all
```
```   418       done
```
```   419     finally have ?case . }
```
```   420   ultimately show ?case by blast
```
```   421 qed
```
```   422
```
```   423 lemma setsum_cong2:
```
```   424   "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> setsum f A = setsum g A"
```
```   425   by (auto intro: setsum.cong)
```
```   426
```
```   427 lemma setsum_reindex_cong:
```
```   428    "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|]
```
```   429     ==> setsum h B = setsum g A"
```
```   430   by (simp add: setsum.reindex)
```
```   431
```
```   432 lemma setsum_restrict_set:
```
```   433   assumes fA: "finite A"
```
```   434   shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"
```
```   435 proof-
```
```   436   from fA have fab: "finite (A \<inter> B)" by auto
```
```   437   have aba: "A \<inter> B \<subseteq> A" by blast
```
```   438   let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"
```
```   439   from setsum.mono_neutral_left [OF fA aba, of ?g]
```
```   440   show ?thesis by simp
```
```   441 qed
```
```   442
```
```   443 lemma setsum_Union_disjoint:
```
```   444   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"
```
```   445   shows "setsum f (Union C) = setsum (setsum f) C"
```
```   446   using assms by (fact setsum.Union_disjoint)
```
```   447
```
```   448 lemma setsum_cartesian_product:
```
```   449   "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
```
```   450   by (fact setsum.cartesian_product)
```
```   451
```
```   452 lemma setsum_UNION_zero:
```
```   453   assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"
```
```   454   and f0: "\<And>T1 T2 x. T1\<in>S \<Longrightarrow> T2\<in>S \<Longrightarrow> T1 \<noteq> T2 \<Longrightarrow> x \<in> T1 \<Longrightarrow> x \<in> T2 \<Longrightarrow> f x = 0"
```
```   455   shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"
```
```   456   using fSS f0
```
```   457 proof(induct rule: finite_induct[OF fS])
```
```   458   case 1 thus ?case by simp
```
```   459 next
```
```   460   case (2 T F)
```
```   461   then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F"
```
```   462     and H: "setsum f (\<Union> F) = setsum (setsum f) F" by auto
```
```   463   from fTF have fUF: "finite (\<Union>F)" by auto
```
```   464   from "2.prems" TF fTF
```
```   465   show ?case
```
```   466     by (auto simp add: H [symmetric] intro: setsum.union_inter_neutral [OF fTF(1) fUF, of f])
```
```   467 qed
```
```   468
```
```   469 text {* Commuting outer and inner summation *}
```
```   470
```
```   471 lemma setsum_commute:
```
```   472   "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
```
```   473 proof (simp add: setsum_cartesian_product)
```
```   474   have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
```
```   475     (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
```
```   476     (is "?s = _")
```
```   477     apply (simp add: setsum.reindex [where h = "%(i, j). (j, i)"] swap_inj_on)
```
```   478     apply (simp add: split_def)
```
```   479     done
```
```   480   also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
```
```   481     (is "_ = ?t")
```
```   482     apply (simp add: swap_product)
```
```   483     done
```
```   484   finally show "?s = ?t" .
```
```   485 qed
```
```   486
```
```   487 lemma setsum_Plus:
```
```   488   fixes A :: "'a set" and B :: "'b set"
```
```   489   assumes fin: "finite A" "finite B"
```
```   490   shows "setsum f (A <+> B) = setsum (f \<circ> Inl) A + setsum (f \<circ> Inr) B"
```
```   491 proof -
```
```   492   have "A <+> B = Inl ` A \<union> Inr ` B" by auto
```
```   493   moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"
```
```   494     by auto
```
```   495   moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto
```
```   496   moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)
```
```   497   ultimately show ?thesis using fin by(simp add: setsum.union_disjoint setsum.reindex)
```
```   498 qed
```
```   499
```
```   500 end
```
```   501
```
```   502 text {* TODO These are legacy *}
```
```   503
```
```   504 lemma setsum_empty:
```
```   505   "setsum f {} = 0"
```
```   506   by (fact setsum.empty)
```
```   507
```
```   508 lemma setsum_insert:
```
```   509   "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
```
```   510   by (fact setsum.insert)
```
```   511
```
```   512 lemma setsum_infinite:
```
```   513   "~ finite A ==> setsum f A = 0"
```
```   514   by (fact setsum.infinite)
```
```   515
```
```   516 lemma setsum_reindex:
```
```   517   "inj_on f B \<Longrightarrow> setsum h (f ` B) = setsum (h \<circ> f) B"
```
```   518   by (fact setsum.reindex)
```
```   519
```
```   520 lemma setsum_cong:
```
```   521   "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
```
```   522   by (fact setsum.cong)
```
```   523
```
```   524 lemma strong_setsum_cong:
```
```   525   "A = B ==> (!!x. x:B =simp=> f x = g x)
```
```   526    ==> setsum (%x. f x) A = setsum (%x. g x) B"
```
```   527   by (fact setsum.strong_cong)
```
```   528
```
```   529 lemmas setsum_0 = setsum.neutral_const
```
```   530 lemmas setsum_0' = setsum.neutral
```
```   531
```
```   532 lemma setsum_Un_Int: "finite A ==> finite B ==>
```
```   533   setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
```
```   534   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
```
```   535   by (fact setsum.union_inter)
```
```   536
```
```   537 lemma setsum_Un_disjoint: "finite A ==> finite B
```
```   538   ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
```
```   539   by (fact setsum.union_disjoint)
```
```   540
```
```   541 lemma setsum_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
```
```   542     setsum f A = setsum f (A - B) + setsum f B"
```
```   543   by (fact setsum.subset_diff)
```
```   544
```
```   545 lemma setsum_mono_zero_left:
```
```   546   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 0 \<rbrakk> \<Longrightarrow> setsum f S = setsum f T"
```
```   547   by (fact setsum.mono_neutral_left)
```
```   548
```
```   549 lemmas setsum_mono_zero_right = setsum.mono_neutral_right
```
```   550
```
```   551 lemma setsum_mono_zero_cong_left:
```
```   552   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 0; \<And>x. x \<in> S \<Longrightarrow> f x = g x \<rbrakk>
```
```   553   \<Longrightarrow> setsum f S = setsum g T"
```
```   554   by (fact setsum.mono_neutral_cong_left)
```
```   555
```
```   556 lemmas setsum_mono_zero_cong_right = setsum.mono_neutral_cong_right
```
```   557
```
```   558 lemma setsum_delta: "finite S \<Longrightarrow>
```
```   559   setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
```
```   560   by (fact setsum.delta)
```
```   561
```
```   562 lemma setsum_delta': "finite S \<Longrightarrow>
```
```   563   setsum (\<lambda>k. if a = k then b k else 0) S = (if a\<in> S then b a else 0)"
```
```   564   by (fact setsum.delta')
```
```   565
```
```   566 lemma setsum_cases:
```
```   567   assumes "finite A"
```
```   568   shows "setsum (\<lambda>x. if P x then f x else g x) A =
```
```   569          setsum f (A \<inter> {x. P x}) + setsum g (A \<inter> - {x. P x})"
```
```   570   using assms by (fact setsum.If_cases)
```
```   571
```
```   572 (*But we can't get rid of finite I. If infinite, although the rhs is 0,
```
```   573   the lhs need not be, since UNION I A could still be finite.*)
```
```   574 lemma setsum_UN_disjoint:
```
```   575   assumes "finite I" and "ALL i:I. finite (A i)"
```
```   576     and "ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}"
```
```   577   shows "setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
```
```   578   using assms by (fact setsum.UNION_disjoint)
```
```   579
```
```   580 (*But we can't get rid of finite A. If infinite, although the lhs is 0,
```
```   581   the rhs need not be, since SIGMA A B could still be finite.*)
```
```   582 lemma setsum_Sigma:
```
```   583   assumes "finite A" and  "ALL x:A. finite (B x)"
```
```   584   shows "(\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"
```
```   585   using assms by (fact setsum.Sigma)
```
```   586
```
```   587 lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
```
```   588   by (fact setsum.distrib)
```
```   589
```
```   590 lemma setsum_Un_zero:
```
```   591   "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 0 \<rbrakk> \<Longrightarrow>
```
```   592   setsum f (S \<union> T) = setsum f S + setsum f T"
```
```   593   by (fact setsum.union_inter_neutral)
```
```   594
```
```   595 lemma setsum_eq_general_reverses:
```
```   596   assumes fS: "finite S" and fT: "finite T"
```
```   597   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
```
```   598   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"
```
```   599   shows "setsum f S = setsum g T"
```
```   600   using kh hk by (fact setsum.eq_general_reverses)
```
```   601
```
```   602
```
```   603 subsubsection {* Properties in more restricted classes of structures *}
```
```   604
```
```   605 lemma setsum_Un: "finite A ==> finite B ==>
```
```   606   (setsum f (A Un B) :: 'a :: ab_group_add) =
```
```   607    setsum f A + setsum f B - setsum f (A Int B)"
```
```   608 by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
```
```   609
```
```   610 lemma setsum_Un2:
```
```   611   assumes "finite (A \<union> B)"
```
```   612   shows "setsum f (A \<union> B) = setsum f (A - B) + setsum f (B - A) + setsum f (A \<inter> B)"
```
```   613 proof -
```
```   614   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
```
```   615     by auto
```
```   616   with assms show ?thesis by simp (subst setsum_Un_disjoint, auto)+
```
```   617 qed
```
```   618
```
```   619 lemma setsum_diff1: "finite A \<Longrightarrow>
```
```   620   (setsum f (A - {a}) :: ('a::ab_group_add)) =
```
```   621   (if a:A then setsum f A - f a else setsum f A)"
```
```   622 by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```   623
```
```   624 lemma setsum_diff:
```
```   625   assumes le: "finite A" "B \<subseteq> A"
```
```   626   shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
```
```   627 proof -
```
```   628   from le have finiteB: "finite B" using finite_subset by auto
```
```   629   show ?thesis using finiteB le
```
```   630   proof induct
```
```   631     case empty
```
```   632     thus ?case by auto
```
```   633   next
```
```   634     case (insert x F)
```
```   635     thus ?case using le finiteB
```
```   636       by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
```
```   637   qed
```
```   638 qed
```
```   639
```
```   640 lemma setsum_mono:
```
```   641   assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))"
```
```   642   shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
```
```   643 proof (cases "finite K")
```
```   644   case True
```
```   645   thus ?thesis using le
```
```   646   proof induct
```
```   647     case empty
```
```   648     thus ?case by simp
```
```   649   next
```
```   650     case insert
```
```   651     thus ?case using add_mono by fastforce
```
```   652   qed
```
```   653 next
```
```   654   case False then show ?thesis by simp
```
```   655 qed
```
```   656
```
```   657 lemma setsum_strict_mono:
```
```   658   fixes f :: "'a \<Rightarrow> 'b::{ordered_cancel_ab_semigroup_add,comm_monoid_add}"
```
```   659   assumes "finite A"  "A \<noteq> {}"
```
```   660     and "!!x. x:A \<Longrightarrow> f x < g x"
```
```   661   shows "setsum f A < setsum g A"
```
```   662   using assms
```
```   663 proof (induct rule: finite_ne_induct)
```
```   664   case singleton thus ?case by simp
```
```   665 next
```
```   666   case insert thus ?case by (auto simp: add_strict_mono)
```
```   667 qed
```
```   668
```
```   669 lemma setsum_strict_mono_ex1:
```
```   670 fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_cancel_ab_semigroup_add}"
```
```   671 assumes "finite A" and "ALL x:A. f x \<le> g x" and "EX a:A. f a < g a"
```
```   672 shows "setsum f A < setsum g A"
```
```   673 proof-
```
```   674   from assms(3) obtain a where a: "a:A" "f a < g a" by blast
```
```   675   have "setsum f A = setsum f ((A-{a}) \<union> {a})"
```
```   676     by(simp add:insert_absorb[OF `a:A`])
```
```   677   also have "\<dots> = setsum f (A-{a}) + setsum f {a}"
```
```   678     using `finite A` by(subst setsum_Un_disjoint) auto
```
```   679   also have "setsum f (A-{a}) \<le> setsum g (A-{a})"
```
```   680     by(rule setsum_mono)(simp add: assms(2))
```
```   681   also have "setsum f {a} < setsum g {a}" using a by simp
```
```   682   also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"
```
```   683     using `finite A` by(subst setsum_Un_disjoint[symmetric]) auto
```
```   684   also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF `a:A`])
```
```   685   finally show ?thesis by (metis add_right_mono add_strict_left_mono)
```
```   686 qed
```
```   687
```
```   688 lemma setsum_negf:
```
```   689   "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
```
```   690 proof (cases "finite A")
```
```   691   case True thus ?thesis by (induct set: finite) auto
```
```   692 next
```
```   693   case False thus ?thesis by simp
```
```   694 qed
```
```   695
```
```   696 lemma setsum_subtractf:
```
```   697   "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
```
```   698     setsum f A - setsum g A"
```
```   699 proof (cases "finite A")
```
```   700   case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
```
```   701 next
```
```   702   case False thus ?thesis by simp
```
```   703 qed
```
```   704
```
```   705 lemma setsum_nonneg:
```
```   706   assumes nn: "\<forall>x\<in>A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
```
```   707   shows "0 \<le> setsum f A"
```
```   708 proof (cases "finite A")
```
```   709   case True thus ?thesis using nn
```
```   710   proof induct
```
```   711     case empty then show ?case by simp
```
```   712   next
```
```   713     case (insert x F)
```
```   714     then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
```
```   715     with insert show ?case by simp
```
```   716   qed
```
```   717 next
```
```   718   case False thus ?thesis by simp
```
```   719 qed
```
```   720
```
```   721 lemma setsum_nonpos:
```
```   722   assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})"
```
```   723   shows "setsum f A \<le> 0"
```
```   724 proof (cases "finite A")
```
```   725   case True thus ?thesis using np
```
```   726   proof induct
```
```   727     case empty then show ?case by simp
```
```   728   next
```
```   729     case (insert x F)
```
```   730     then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
```
```   731     with insert show ?case by simp
```
```   732   qed
```
```   733 next
```
```   734   case False thus ?thesis by simp
```
```   735 qed
```
```   736
```
```   737 lemma setsum_nonneg_leq_bound:
```
```   738   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
```
```   739   assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"
```
```   740   shows "f i \<le> B"
```
```   741 proof -
```
```   742   have "0 \<le> (\<Sum> i \<in> s - {i}. f i)" and "0 \<le> f i"
```
```   743     using assms by (auto intro!: setsum_nonneg)
```
```   744   moreover
```
```   745   have "(\<Sum> i \<in> s - {i}. f i) + f i = B"
```
```   746     using assms by (simp add: setsum_diff1)
```
```   747   ultimately show ?thesis by auto
```
```   748 qed
```
```   749
```
```   750 lemma setsum_nonneg_0:
```
```   751   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
```
```   752   assumes "finite s" and pos: "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"
```
```   753   and "(\<Sum> i \<in> s. f i) = 0" and i: "i \<in> s"
```
```   754   shows "f i = 0"
```
```   755   using setsum_nonneg_leq_bound[OF assms] pos[OF i] by auto
```
```   756
```
```   757 lemma setsum_mono2:
```
```   758 fixes f :: "'a \<Rightarrow> 'b :: ordered_comm_monoid_add"
```
```   759 assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
```
```   760 shows "setsum f A \<le> setsum f B"
```
```   761 proof -
```
```   762   have "setsum f A \<le> setsum f A + setsum f (B-A)"
```
```   763     by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
```
```   764   also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
```
```   765     by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
```
```   766   also have "A \<union> (B-A) = B" using sub by blast
```
```   767   finally show ?thesis .
```
```   768 qed
```
```   769
```
```   770 lemma setsum_mono3: "finite B ==> A <= B ==>
```
```   771     ALL x: B - A.
```
```   772       0 <= ((f x)::'a::{comm_monoid_add,ordered_ab_semigroup_add}) ==>
```
```   773         setsum f A <= setsum f B"
```
```   774   apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
```
```   775   apply (erule ssubst)
```
```   776   apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
```
```   777   apply simp
```
```   778   apply (rule add_left_mono)
```
```   779   apply (erule setsum_nonneg)
```
```   780   apply (subst setsum_Un_disjoint [THEN sym])
```
```   781   apply (erule finite_subset, assumption)
```
```   782   apply (rule finite_subset)
```
```   783   prefer 2
```
```   784   apply assumption
```
```   785   apply (auto simp add: sup_absorb2)
```
```   786 done
```
```   787
```
```   788 lemma setsum_right_distrib:
```
```   789   fixes f :: "'a => ('b::semiring_0)"
```
```   790   shows "r * setsum f A = setsum (%n. r * f n) A"
```
```   791 proof (cases "finite A")
```
```   792   case True
```
```   793   thus ?thesis
```
```   794   proof induct
```
```   795     case empty thus ?case by simp
```
```   796   next
```
```   797     case (insert x A) thus ?case by (simp add: distrib_left)
```
```   798   qed
```
```   799 next
```
```   800   case False thus ?thesis by simp
```
```   801 qed
```
```   802
```
```   803 lemma setsum_left_distrib:
```
```   804   "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
```
```   805 proof (cases "finite A")
```
```   806   case True
```
```   807   then show ?thesis
```
```   808   proof induct
```
```   809     case empty thus ?case by simp
```
```   810   next
```
```   811     case (insert x A) thus ?case by (simp add: distrib_right)
```
```   812   qed
```
```   813 next
```
```   814   case False thus ?thesis by simp
```
```   815 qed
```
```   816
```
```   817 lemma setsum_divide_distrib:
```
```   818   "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
```
```   819 proof (cases "finite A")
```
```   820   case True
```
```   821   then show ?thesis
```
```   822   proof induct
```
```   823     case empty thus ?case by simp
```
```   824   next
```
```   825     case (insert x A) thus ?case by (simp add: add_divide_distrib)
```
```   826   qed
```
```   827 next
```
```   828   case False thus ?thesis by simp
```
```   829 qed
```
```   830
```
```   831 lemma setsum_abs[iff]:
```
```   832   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   833   shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
```
```   834 proof (cases "finite A")
```
```   835   case True
```
```   836   thus ?thesis
```
```   837   proof induct
```
```   838     case empty thus ?case by simp
```
```   839   next
```
```   840     case (insert x A)
```
```   841     thus ?case by (auto intro: abs_triangle_ineq order_trans)
```
```   842   qed
```
```   843 next
```
```   844   case False thus ?thesis by simp
```
```   845 qed
```
```   846
```
```   847 lemma setsum_abs_ge_zero[iff]:
```
```   848   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   849   shows "0 \<le> setsum (%i. abs(f i)) A"
```
```   850 proof (cases "finite A")
```
```   851   case True
```
```   852   thus ?thesis
```
```   853   proof induct
```
```   854     case empty thus ?case by simp
```
```   855   next
```
```   856     case (insert x A) thus ?case by auto
```
```   857   qed
```
```   858 next
```
```   859   case False thus ?thesis by simp
```
```   860 qed
```
```   861
```
```   862 lemma abs_setsum_abs[simp]:
```
```   863   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   864   shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
```
```   865 proof (cases "finite A")
```
```   866   case True
```
```   867   thus ?thesis
```
```   868   proof induct
```
```   869     case empty thus ?case by simp
```
```   870   next
```
```   871     case (insert a A)
```
```   872     hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
```
```   873     also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
```
```   874     also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
```
```   875       by (simp del: abs_of_nonneg)
```
```   876     also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
```
```   877     finally show ?case .
```
```   878   qed
```
```   879 next
```
```   880   case False thus ?thesis by simp
```
```   881 qed
```
```   882
```
```   883 lemma setsum_diff1'[rule_format]:
```
```   884   "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
```
```   885 apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"])
```
```   886 apply (auto simp add: insert_Diff_if add_ac)
```
```   887 done
```
```   888
```
```   889 lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
```
```   890   shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
```
```   891 unfolding setsum_diff1'[OF assms] by auto
```
```   892
```
```   893 lemma setsum_product:
```
```   894   fixes f :: "'a => ('b::semiring_0)"
```
```   895   shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
```
```   896   by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)
```
```   897
```
```   898 lemma setsum_mult_setsum_if_inj:
```
```   899 fixes f :: "'a => ('b::semiring_0)"
```
```   900 shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
```
```   901   setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
```
```   902 by(auto simp: setsum_product setsum_cartesian_product
```
```   903         intro!:  setsum_reindex_cong[symmetric])
```
```   904
```
```   905 lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
```
```   906 apply (case_tac "finite A")
```
```   907  prefer 2 apply simp
```
```   908 apply (erule rev_mp)
```
```   909 apply (erule finite_induct, auto)
```
```   910 done
```
```   911
```
```   912 lemma setsum_eq_0_iff [simp]:
```
```   913   "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
```
```   914   by (induct set: finite) auto
```
```   915
```
```   916 lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
```
```   917   setsum f A = Suc 0 \<longleftrightarrow> (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
```
```   918 apply(erule finite_induct)
```
```   919 apply (auto simp add:add_is_1)
```
```   920 done
```
```   921
```
```   922 lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
```
```   923
```
```   924 lemma setsum_Un_nat: "finite A ==> finite B ==>
```
```   925   (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
```
```   926   -- {* For the natural numbers, we have subtraction. *}
```
```   927 by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
```
```   928
```
```   929 lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
```
```   930   (if a:A then setsum f A - f a else setsum f A)"
```
```   931 apply (case_tac "finite A")
```
```   932  prefer 2 apply simp
```
```   933 apply (erule finite_induct)
```
```   934  apply (auto simp add: insert_Diff_if)
```
```   935 apply (drule_tac a = a in mk_disjoint_insert, auto)
```
```   936 done
```
```   937
```
```   938 lemma setsum_diff_nat:
```
```   939 assumes "finite B" and "B \<subseteq> A"
```
```   940 shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
```
```   941 using assms
```
```   942 proof induct
```
```   943   show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
```
```   944 next
```
```   945   fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
```
```   946     and xFinA: "insert x F \<subseteq> A"
```
```   947     and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
```
```   948   from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
```
```   949   from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
```
```   950     by (simp add: setsum_diff1_nat)
```
```   951   from xFinA have "F \<subseteq> A" by simp
```
```   952   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
```
```   953   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
```
```   954     by simp
```
```   955   from xnotinF have "A - insert x F = (A - F) - {x}" by auto
```
```   956   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
```
```   957     by simp
```
```   958   from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
```
```   959   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
```
```   960     by simp
```
```   961   thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
```
```   962 qed
```
```   963
```
```   964 lemma setsum_comp_morphism:
```
```   965   assumes "h 0 = 0" and "\<And>x y. h (x + y) = h x + h y"
```
```   966   shows "setsum (h \<circ> g) A = h (setsum g A)"
```
```   967 proof (cases "finite A")
```
```   968   case False then show ?thesis by (simp add: assms)
```
```   969 next
```
```   970   case True then show ?thesis by (induct A) (simp_all add: assms)
```
```   971 qed
```
```   972
```
```   973
```
```   974 subsubsection {* Cardinality as special case of @{const setsum} *}
```
```   975
```
```   976 lemma card_eq_setsum:
```
```   977   "card A = setsum (\<lambda>x. 1) A"
```
```   978 proof -
```
```   979   have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)"
```
```   980     by (simp add: fun_eq_iff)
```
```   981   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) = Finite_Set.fold (\<lambda>_. Suc)"
```
```   982     by (rule arg_cong)
```
```   983   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
```
```   984     by (blast intro: fun_cong)
```
```   985   then show ?thesis by (simp add: card.eq_fold setsum.eq_fold)
```
```   986 qed
```
```   987
```
```   988 lemma setsum_constant [simp]:
```
```   989   "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
```
```   990 apply (cases "finite A")
```
```   991 apply (erule finite_induct)
```
```   992 apply (auto simp add: algebra_simps)
```
```   993 done
```
```   994
```
```   995 lemma setsum_bounded:
```
```   996   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"
```
```   997   shows "setsum f A \<le> of_nat (card A) * K"
```
```   998 proof (cases "finite A")
```
```   999   case True
```
```  1000   thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
```
```  1001 next
```
```  1002   case False thus ?thesis by simp
```
```  1003 qed
```
```  1004
```
```  1005 lemma card_UN_disjoint:
```
```  1006   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
```
```  1007     and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
```
```  1008   shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
```
```  1009 proof -
```
```  1010   have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)" by simp
```
```  1011   with assms show ?thesis by (simp add: card_eq_setsum setsum_UN_disjoint del: setsum_constant)
```
```  1012 qed
```
```  1013
```
```  1014 lemma card_Union_disjoint:
```
```  1015   "finite C ==> (ALL A:C. finite A) ==>
```
```  1016    (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {})
```
```  1017    ==> card (Union C) = setsum card C"
```
```  1018 apply (frule card_UN_disjoint [of C id])
```
```  1019 apply (simp_all add: SUP_def id_def)
```
```  1020 done
```
```  1021
```
```  1022
```
```  1023 subsubsection {* Cardinality of products *}
```
```  1024
```
```  1025 lemma card_SigmaI [simp]:
```
```  1026   "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
```
```  1027   \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
```
```  1028 by(simp add: card_eq_setsum setsum_Sigma del:setsum_constant)
```
```  1029
```
```  1030 (*
```
```  1031 lemma SigmaI_insert: "y \<notin> A ==>
```
```  1032   (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
```
```  1033   by auto
```
```  1034 *)
```
```  1035
```
```  1036 lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
```
```  1037   by (cases "finite A \<and> finite B")
```
```  1038     (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1039
```
```  1040 lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
```
```  1041 by (simp add: card_cartesian_product)
```
```  1042
```
```  1043
```
```  1044 subsection {* Generalized product over a set *}
```
```  1045
```
```  1046 context comm_monoid_mult
```
```  1047 begin
```
```  1048
```
```  1049 definition setprod :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
```
```  1050 where
```
```  1051   "setprod = comm_monoid_set.F times 1"
```
```  1052
```
```  1053 sublocale setprod!: comm_monoid_set times 1
```
```  1054 where
```
```  1055   "comm_monoid_set.F times 1 = setprod"
```
```  1056 proof -
```
```  1057   show "comm_monoid_set times 1" ..
```
```  1058   then interpret setprod!: comm_monoid_set times 1 .
```
```  1059   from setprod_def show "comm_monoid_set.F times 1 = setprod" by rule
```
```  1060 qed
```
```  1061
```
```  1062 abbreviation
```
```  1063   Setprod ("\<Prod>_"  999) where
```
```  1064   "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"
```
```  1065
```
```  1066 end
```
```  1067
```
```  1068 syntax
```
```  1069   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
```
```  1070 syntax (xsymbols)
```
```  1071   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```  1072 syntax (HTML output)
```
```  1073   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```  1074
```
```  1075 translations -- {* Beware of argument permutation! *}
```
```  1076   "PROD i:A. b" == "CONST setprod (%i. b) A"
```
```  1077   "\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A"
```
```  1078
```
```  1079 text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
```
```  1080  @{text"\<Prod>x|P. e"}. *}
```
```  1081
```
```  1082 syntax
```
```  1083   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
```
```  1084 syntax (xsymbols)
```
```  1085   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```  1086 syntax (HTML output)
```
```  1087   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```  1088
```
```  1089 translations
```
```  1090   "PROD x|P. t" => "CONST setprod (%x. t) {x. P}"
```
```  1091   "\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"
```
```  1092
```
```  1093 text {* TODO These are candidates for generalization *}
```
```  1094
```
```  1095 context comm_monoid_mult
```
```  1096 begin
```
```  1097
```
```  1098 lemma setprod_reindex_id:
```
```  1099   "inj_on f B ==> setprod f B = setprod id (f ` B)"
```
```  1100   by (auto simp add: setprod.reindex)
```
```  1101
```
```  1102 lemma setprod_reindex_cong:
```
```  1103   "inj_on f A ==> B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
```
```  1104   by (frule setprod.reindex, simp)
```
```  1105
```
```  1106 lemma strong_setprod_reindex_cong:
```
```  1107   assumes i: "inj_on f A"
```
```  1108   and B: "B = f ` A" and eq: "\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x"
```
```  1109   shows "setprod h B = setprod g A"
```
```  1110 proof-
```
```  1111   have "setprod h B = setprod (h o f) A"
```
```  1112     by (simp add: B setprod.reindex [OF i, of h])
```
```  1113   then show ?thesis apply simp
```
```  1114     apply (rule setprod.cong)
```
```  1115     apply simp
```
```  1116     by (simp add: eq)
```
```  1117 qed
```
```  1118
```
```  1119 lemma setprod_Union_disjoint:
```
```  1120   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"
```
```  1121   shows "setprod f (Union C) = setprod (setprod f) C"
```
```  1122   using assms by (fact setprod.Union_disjoint)
```
```  1123
```
```  1124 text{*Here we can eliminate the finiteness assumptions, by cases.*}
```
```  1125 lemma setprod_cartesian_product:
```
```  1126   "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
```
```  1127   by (fact setprod.cartesian_product)
```
```  1128
```
```  1129 lemma setprod_Un2:
```
```  1130   assumes "finite (A \<union> B)"
```
```  1131   shows "setprod f (A \<union> B) = setprod f (A - B) * setprod f (B - A) * setprod f (A \<inter> B)"
```
```  1132 proof -
```
```  1133   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
```
```  1134     by auto
```
```  1135   with assms show ?thesis by simp (subst setprod.union_disjoint, auto)+
```
```  1136 qed
```
```  1137
```
```  1138 end
```
```  1139
```
```  1140 text {* TODO These are legacy *}
```
```  1141
```
```  1142 lemma setprod_empty: "setprod f {} = 1"
```
```  1143   by (fact setprod.empty)
```
```  1144
```
```  1145 lemma setprod_insert: "[| finite A; a \<notin> A |] ==>
```
```  1146     setprod f (insert a A) = f a * setprod f A"
```
```  1147   by (fact setprod.insert)
```
```  1148
```
```  1149 lemma setprod_infinite: "~ finite A ==> setprod f A = 1"
```
```  1150   by (fact setprod.infinite)
```
```  1151
```
```  1152 lemma setprod_reindex:
```
```  1153   "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
```
```  1154   by (fact setprod.reindex)
```
```  1155
```
```  1156 lemma setprod_cong:
```
```  1157   "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
```
```  1158   by (fact setprod.cong)
```
```  1159
```
```  1160 lemma strong_setprod_cong:
```
```  1161   "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
```
```  1162   by (fact setprod.strong_cong)
```
```  1163
```
```  1164 lemma setprod_Un_one:
```
```  1165   "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 1 \<rbrakk>
```
```  1166   \<Longrightarrow> setprod f (S \<union> T) = setprod f S  * setprod f T"
```
```  1167   by (fact setprod.union_inter_neutral)
```
```  1168
```
```  1169 lemmas setprod_1 = setprod.neutral_const
```
```  1170 lemmas setprod_1' = setprod.neutral
```
```  1171
```
```  1172 lemma setprod_Un_Int: "finite A ==> finite B
```
```  1173     ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
```
```  1174   by (fact setprod.union_inter)
```
```  1175
```
```  1176 lemma setprod_Un_disjoint: "finite A ==> finite B
```
```  1177   ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
```
```  1178   by (fact setprod.union_disjoint)
```
```  1179
```
```  1180 lemma setprod_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
```
```  1181     setprod f A = setprod f (A - B) * setprod f B"
```
```  1182   by (fact setprod.subset_diff)
```
```  1183
```
```  1184 lemma setprod_mono_one_left:
```
```  1185   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 1 \<rbrakk> \<Longrightarrow> setprod f S = setprod f T"
```
```  1186   by (fact setprod.mono_neutral_left)
```
```  1187
```
```  1188 lemmas setprod_mono_one_right = setprod.mono_neutral_right
```
```  1189
```
```  1190 lemma setprod_mono_one_cong_left:
```
```  1191   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1; \<And>x. x \<in> S \<Longrightarrow> f x = g x \<rbrakk>
```
```  1192   \<Longrightarrow> setprod f S = setprod g T"
```
```  1193   by (fact setprod.mono_neutral_cong_left)
```
```  1194
```
```  1195 lemmas setprod_mono_one_cong_right = setprod.mono_neutral_cong_right
```
```  1196
```
```  1197 lemma setprod_delta: "finite S \<Longrightarrow>
```
```  1198   setprod (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"
```
```  1199   by (fact setprod.delta)
```
```  1200
```
```  1201 lemma setprod_delta': "finite S \<Longrightarrow>
```
```  1202   setprod (\<lambda>k. if a = k then b k else 1) S = (if a\<in> S then b a else 1)"
```
```  1203   by (fact setprod.delta')
```
```  1204
```
```  1205 lemma setprod_UN_disjoint:
```
```  1206     "finite I ==> (ALL i:I. finite (A i)) ==>
```
```  1207         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
```
```  1208       setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
```
```  1209   by (fact setprod.UNION_disjoint)
```
```  1210
```
```  1211 lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```  1212     (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
```
```  1213     (\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
```
```  1214   by (fact setprod.Sigma)
```
```  1215
```
```  1216 lemma setprod_timesf: "setprod (\<lambda>x. f x * g x) A = setprod f A * setprod g A"
```
```  1217   by (fact setprod.distrib)
```
```  1218
```
```  1219
```
```  1220 subsubsection {* Properties in more restricted classes of structures *}
```
```  1221
```
```  1222 lemma setprod_zero:
```
```  1223      "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0"
```
```  1224 apply (induct set: finite, force, clarsimp)
```
```  1225 apply (erule disjE, auto)
```
```  1226 done
```
```  1227
```
```  1228 lemma setprod_zero_iff[simp]: "finite A ==>
```
```  1229   (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) =
```
```  1230   (EX x: A. f x = 0)"
```
```  1231 by (erule finite_induct, auto simp:no_zero_divisors)
```
```  1232
```
```  1233 lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
```
```  1234   (setprod f (A Un B) :: 'a ::{field})
```
```  1235    = setprod f A * setprod f B / setprod f (A Int B)"
```
```  1236 by (subst setprod_Un_Int [symmetric], auto)
```
```  1237
```
```  1238 lemma setprod_nonneg [rule_format]:
```
```  1239    "(ALL x: A. (0::'a::linordered_semidom) \<le> f x) --> 0 \<le> setprod f A"
```
```  1240 by (cases "finite A", induct set: finite, simp_all add: mult_nonneg_nonneg)
```
```  1241
```
```  1242 lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::linordered_semidom) < f x)
```
```  1243   --> 0 < setprod f A"
```
```  1244 by (cases "finite A", induct set: finite, simp_all add: mult_pos_pos)
```
```  1245
```
```  1246 lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
```
```  1247   (setprod f (A - {a}) :: 'a :: {field}) =
```
```  1248   (if a:A then setprod f A / f a else setprod f A)"
```
```  1249   by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```  1250
```
```  1251 lemma setprod_inversef:
```
```  1252   fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
```
```  1253   shows "finite A ==> setprod (inverse \<circ> f) A = inverse (setprod f A)"
```
```  1254 by (erule finite_induct) auto
```
```  1255
```
```  1256 lemma setprod_dividef:
```
```  1257   fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
```
```  1258   shows "finite A
```
```  1259     ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
```
```  1260 apply (subgoal_tac
```
```  1261          "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
```
```  1262 apply (erule ssubst)
```
```  1263 apply (subst divide_inverse)
```
```  1264 apply (subst setprod_timesf)
```
```  1265 apply (subst setprod_inversef, assumption+, rule refl)
```
```  1266 apply (rule setprod_cong, rule refl)
```
```  1267 apply (subst divide_inverse, auto)
```
```  1268 done
```
```  1269
```
```  1270 lemma setprod_dvd_setprod [rule_format]:
```
```  1271     "(ALL x : A. f x dvd g x) \<longrightarrow> setprod f A dvd setprod g A"
```
```  1272   apply (cases "finite A")
```
```  1273   apply (induct set: finite)
```
```  1274   apply (auto simp add: dvd_def)
```
```  1275   apply (rule_tac x = "k * ka" in exI)
```
```  1276   apply (simp add: algebra_simps)
```
```  1277 done
```
```  1278
```
```  1279 lemma setprod_dvd_setprod_subset:
```
```  1280   "finite B \<Longrightarrow> A <= B \<Longrightarrow> setprod f A dvd setprod f B"
```
```  1281   apply (subgoal_tac "setprod f B = setprod f A * setprod f (B - A)")
```
```  1282   apply (unfold dvd_def, blast)
```
```  1283   apply (subst setprod_Un_disjoint [symmetric])
```
```  1284   apply (auto elim: finite_subset intro: setprod_cong)
```
```  1285 done
```
```  1286
```
```  1287 lemma setprod_dvd_setprod_subset2:
```
```  1288   "finite B \<Longrightarrow> A <= B \<Longrightarrow> ALL x : A. (f x::'a::comm_semiring_1) dvd g x \<Longrightarrow>
```
```  1289       setprod f A dvd setprod g B"
```
```  1290   apply (rule dvd_trans)
```
```  1291   apply (rule setprod_dvd_setprod, erule (1) bspec)
```
```  1292   apply (erule (1) setprod_dvd_setprod_subset)
```
```  1293 done
```
```  1294
```
```  1295 lemma dvd_setprod: "finite A \<Longrightarrow> i:A \<Longrightarrow>
```
```  1296     (f i ::'a::comm_semiring_1) dvd setprod f A"
```
```  1297 by (induct set: finite) (auto intro: dvd_mult)
```
```  1298
```
```  1299 lemma dvd_setsum [rule_format]: "(ALL i : A. d dvd f i) \<longrightarrow>
```
```  1300     (d::'a::comm_semiring_1) dvd (SUM x : A. f x)"
```
```  1301   apply (cases "finite A")
```
```  1302   apply (induct set: finite)
```
```  1303   apply auto
```
```  1304 done
```
```  1305
```
```  1306 lemma setprod_mono:
```
```  1307   fixes f :: "'a \<Rightarrow> 'b\<Colon>linordered_semidom"
```
```  1308   assumes "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i"
```
```  1309   shows "setprod f A \<le> setprod g A"
```
```  1310 proof (cases "finite A")
```
```  1311   case True
```
```  1312   hence ?thesis "setprod f A \<ge> 0" using subset_refl[of A]
```
```  1313   proof (induct A rule: finite_subset_induct)
```
```  1314     case (insert a F)
```
```  1315     thus "setprod f (insert a F) \<le> setprod g (insert a F)" "0 \<le> setprod f (insert a F)"
```
```  1316       unfolding setprod_insert[OF insert(1,3)]
```
```  1317       using assms[rule_format,OF insert(2)] insert
```
```  1318       by (auto intro: mult_mono mult_nonneg_nonneg)
```
```  1319   qed auto
```
```  1320   thus ?thesis by simp
```
```  1321 qed auto
```
```  1322
```
```  1323 lemma abs_setprod:
```
```  1324   fixes f :: "'a \<Rightarrow> 'b\<Colon>{linordered_field,abs}"
```
```  1325   shows "abs (setprod f A) = setprod (\<lambda>x. abs (f x)) A"
```
```  1326 proof (cases "finite A")
```
```  1327   case True thus ?thesis
```
```  1328     by induct (auto simp add: field_simps abs_mult)
```
```  1329 qed auto
```
```  1330
```
```  1331 lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
```
```  1332 apply (erule finite_induct)
```
```  1333 apply auto
```
```  1334 done
```
```  1335
```
```  1336 lemma setprod_gen_delta:
```
```  1337   assumes fS: "finite S"
```
```  1338   shows "setprod (\<lambda>k. if k=a then b k else c) S = (if a \<in> S then (b a ::'a::comm_monoid_mult) * c^ (card S - 1) else c^ card S)"
```
```  1339 proof-
```
```  1340   let ?f = "(\<lambda>k. if k=a then b k else c)"
```
```  1341   {assume a: "a \<notin> S"
```
```  1342     hence "\<forall> k\<in> S. ?f k = c" by simp
```
```  1343     hence ?thesis  using a setprod_constant[OF fS, of c] by simp }
```
```  1344   moreover
```
```  1345   {assume a: "a \<in> S"
```
```  1346     let ?A = "S - {a}"
```
```  1347     let ?B = "{a}"
```
```  1348     have eq: "S = ?A \<union> ?B" using a by blast
```
```  1349     have dj: "?A \<inter> ?B = {}" by simp
```
```  1350     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```  1351     have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
```
```  1352       apply (rule setprod_cong) by auto
```
```  1353     have cA: "card ?A = card S - 1" using fS a by auto
```
```  1354     have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
```
```  1355     have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
```
```  1356       using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
```
```  1357       by simp
```
```  1358     then have ?thesis using a cA
```
```  1359       by (simp add: fA1 field_simps cong add: setprod_cong cong del: if_weak_cong)}
```
```  1360   ultimately show ?thesis by blast
```
```  1361 qed
```
```  1362
```
```  1363 lemma setprod_eq_1_iff [simp]:
```
```  1364   "finite F ==> setprod f F = 1 \<longleftrightarrow> (ALL a:F. f a = (1::nat))"
```
```  1365   by (induct set: finite) auto
```
```  1366
```
```  1367 lemma setprod_pos_nat:
```
```  1368   "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0"
```
```  1369 using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
```
```  1370
```
```  1371 lemma setprod_pos_nat_iff[simp]:
```
```  1372   "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))"
```
```  1373 using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
```
```  1374
```
```  1375
```
```  1376 subsection {* Generic lattice operations over a set *}
```
```  1377
```
```  1378 no_notation times (infixl "*" 70)
```
```  1379 no_notation Groups.one ("1")
```
```  1380
```
```  1381
```
```  1382 subsubsection {* Without neutral element *}
```
```  1383
```
```  1384 locale semilattice_set = semilattice
```
```  1385 begin
```
```  1386
```
```  1387 interpretation comp_fun_idem f
```
```  1388   by default (simp_all add: fun_eq_iff left_commute)
```
```  1389
```
```  1390 definition F :: "'a set \<Rightarrow> 'a"
```
```  1391 where
```
```  1392   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)"
```
```  1393
```
```  1394 lemma eq_fold:
```
```  1395   assumes "finite A"
```
```  1396   shows "F (insert x A) = Finite_Set.fold f x A"
```
```  1397 proof (rule sym)
```
```  1398   let ?f = "\<lambda>x y. Some (case y of None \<Rightarrow> x | Some z \<Rightarrow> f x z)"
```
```  1399   interpret comp_fun_idem "?f"
```
```  1400     by default (simp_all add: fun_eq_iff commute left_commute split: option.split)
```
```  1401   from assms show "Finite_Set.fold f x A = F (insert x A)"
```
```  1402   proof induct
```
```  1403     case empty then show ?case by (simp add: eq_fold')
```
```  1404   next
```
```  1405     case (insert y B) then show ?case by (simp add: insert_commute [of x] eq_fold')
```
```  1406   qed
```
```  1407 qed
```
```  1408
```
```  1409 lemma singleton [simp]:
```
```  1410   "F {x} = x"
```
```  1411   by (simp add: eq_fold)
```
```  1412
```
```  1413 lemma insert_not_elem:
```
```  1414   assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
```
```  1415   shows "F (insert x A) = x * F A"
```
```  1416 proof -
```
```  1417   from `A \<noteq> {}` obtain b where "b \<in> A" by blast
```
```  1418   then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
```
```  1419   with `finite A` and `x \<notin> A`
```
```  1420     have "finite (insert x B)" and "b \<notin> insert x B" by auto
```
```  1421   then have "F (insert b (insert x B)) = x * F (insert b B)"
```
```  1422     by (simp add: eq_fold)
```
```  1423   then show ?thesis by (simp add: * insert_commute)
```
```  1424 qed
```
```  1425
```
```  1426 lemma in_idem:
```
```  1427   assumes "finite A" and "x \<in> A"
```
```  1428   shows "x * F A = F A"
```
```  1429 proof -
```
```  1430   from assms have "A \<noteq> {}" by auto
```
```  1431   with `finite A` show ?thesis using `x \<in> A`
```
```  1432     by (induct A rule: finite_ne_induct) (auto simp add: ac_simps insert_not_elem)
```
```  1433 qed
```
```  1434
```
```  1435 lemma insert [simp]:
```
```  1436   assumes "finite A" and "A \<noteq> {}"
```
```  1437   shows "F (insert x A) = x * F A"
```
```  1438   using assms by (cases "x \<in> A") (simp_all add: insert_absorb in_idem insert_not_elem)
```
```  1439
```
```  1440 lemma union:
```
```  1441   assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
```
```  1442   shows "F (A \<union> B) = F A * F B"
```
```  1443   using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
```
```  1444
```
```  1445 lemma remove:
```
```  1446   assumes "finite A" and "x \<in> A"
```
```  1447   shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
```
```  1448 proof -
```
```  1449   from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
```
```  1450   with assms show ?thesis by simp
```
```  1451 qed
```
```  1452
```
```  1453 lemma insert_remove:
```
```  1454   assumes "finite A"
```
```  1455   shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
```
```  1456   using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
```
```  1457
```
```  1458 lemma subset:
```
```  1459   assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
```
```  1460   shows "F B * F A = F A"
```
```  1461 proof -
```
```  1462   from assms have "A \<noteq> {}" and "finite B" by (auto dest: finite_subset)
```
```  1463   with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
```
```  1464 qed
```
```  1465
```
```  1466 lemma closed:
```
```  1467   assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
```
```  1468   shows "F A \<in> A"
```
```  1469 using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
```
```  1470   case singleton then show ?case by simp
```
```  1471 next
```
```  1472   case insert with elem show ?case by force
```
```  1473 qed
```
```  1474
```
```  1475 lemma hom_commute:
```
```  1476   assumes hom: "\<And>x y. h (x * y) = h x * h y"
```
```  1477   and N: "finite N" "N \<noteq> {}"
```
```  1478   shows "h (F N) = F (h ` N)"
```
```  1479 using N proof (induct rule: finite_ne_induct)
```
```  1480   case singleton thus ?case by simp
```
```  1481 next
```
```  1482   case (insert n N)
```
```  1483   then have "h (F (insert n N)) = h (n * F N)" by simp
```
```  1484   also have "\<dots> = h n * h (F N)" by (rule hom)
```
```  1485   also have "h (F N) = F (h ` N)" by (rule insert)
```
```  1486   also have "h n * \<dots> = F (insert (h n) (h ` N))"
```
```  1487     using insert by simp
```
```  1488   also have "insert (h n) (h ` N) = h ` insert n N" by simp
```
```  1489   finally show ?case .
```
```  1490 qed
```
```  1491
```
```  1492 end
```
```  1493
```
```  1494 locale semilattice_order_set = semilattice_order + semilattice_set
```
```  1495 begin
```
```  1496
```
```  1497 lemma bounded_iff:
```
```  1498   assumes "finite A" and "A \<noteq> {}"
```
```  1499   shows "x \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> a)"
```
```  1500   using assms by (induct rule: finite_ne_induct) (simp_all add: bounded_iff)
```
```  1501
```
```  1502 lemma boundedI:
```
```  1503   assumes "finite A"
```
```  1504   assumes "A \<noteq> {}"
```
```  1505   assumes "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
```
```  1506   shows "x \<preceq> F A"
```
```  1507   using assms by (simp add: bounded_iff)
```
```  1508
```
```  1509 lemma boundedE:
```
```  1510   assumes "finite A" and "A \<noteq> {}" and "x \<preceq> F A"
```
```  1511   obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
```
```  1512   using assms by (simp add: bounded_iff)
```
```  1513
```
```  1514 lemma coboundedI:
```
```  1515   assumes "finite A"
```
```  1516     and "a \<in> A"
```
```  1517   shows "F A \<preceq> a"
```
```  1518 proof -
```
```  1519   from assms have "A \<noteq> {}" by auto
```
```  1520   from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
```
```  1521   proof (induct rule: finite_ne_induct)
```
```  1522     case singleton thus ?case by (simp add: refl)
```
```  1523   next
```
```  1524     case (insert x B)
```
```  1525     from insert have "a = x \<or> a \<in> B" by simp
```
```  1526     then show ?case using insert by (auto intro: coboundedI2)
```
```  1527   qed
```
```  1528 qed
```
```  1529
```
```  1530 lemma antimono:
```
```  1531   assumes "A \<subseteq> B" and "A \<noteq> {}" and "finite B"
```
```  1532   shows "F B \<preceq> F A"
```
```  1533 proof (cases "A = B")
```
```  1534   case True then show ?thesis by (simp add: refl)
```
```  1535 next
```
```  1536   case False
```
```  1537   have B: "B = A \<union> (B - A)" using `A \<subseteq> B` by blast
```
```  1538   then have "F B = F (A \<union> (B - A))" by simp
```
```  1539   also have "\<dots> = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
```
```  1540   also have "\<dots> \<preceq> F A" by simp
```
```  1541   finally show ?thesis .
```
```  1542 qed
```
```  1543
```
```  1544 end
```
```  1545
```
```  1546
```
```  1547 subsubsection {* With neutral element *}
```
```  1548
```
```  1549 locale semilattice_neutr_set = semilattice_neutr
```
```  1550 begin
```
```  1551
```
```  1552 interpretation comp_fun_idem f
```
```  1553   by default (simp_all add: fun_eq_iff left_commute)
```
```  1554
```
```  1555 definition F :: "'a set \<Rightarrow> 'a"
```
```  1556 where
```
```  1557   eq_fold: "F A = Finite_Set.fold f 1 A"
```
```  1558
```
```  1559 lemma infinite [simp]:
```
```  1560   "\<not> finite A \<Longrightarrow> F A = 1"
```
```  1561   by (simp add: eq_fold)
```
```  1562
```
```  1563 lemma empty [simp]:
```
```  1564   "F {} = 1"
```
```  1565   by (simp add: eq_fold)
```
```  1566
```
```  1567 lemma insert [simp]:
```
```  1568   assumes "finite A"
```
```  1569   shows "F (insert x A) = x * F A"
```
```  1570   using assms by (simp add: eq_fold)
```
```  1571
```
```  1572 lemma in_idem:
```
```  1573   assumes "finite A" and "x \<in> A"
```
```  1574   shows "x * F A = F A"
```
```  1575 proof -
```
```  1576   from assms have "A \<noteq> {}" by auto
```
```  1577   with `finite A` show ?thesis using `x \<in> A`
```
```  1578     by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
```
```  1579 qed
```
```  1580
```
```  1581 lemma union:
```
```  1582   assumes "finite A" and "finite B"
```
```  1583   shows "F (A \<union> B) = F A * F B"
```
```  1584   using assms by (induct A) (simp_all add: ac_simps)
```
```  1585
```
```  1586 lemma remove:
```
```  1587   assumes "finite A" and "x \<in> A"
```
```  1588   shows "F A = x * F (A - {x})"
```
```  1589 proof -
```
```  1590   from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
```
```  1591   with assms show ?thesis by simp
```
```  1592 qed
```
```  1593
```
```  1594 lemma insert_remove:
```
```  1595   assumes "finite A"
```
```  1596   shows "F (insert x A) = x * F (A - {x})"
```
```  1597   using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
```
```  1598
```
```  1599 lemma subset:
```
```  1600   assumes "finite A" and "B \<subseteq> A"
```
```  1601   shows "F B * F A = F A"
```
```  1602 proof -
```
```  1603   from assms have "finite B" by (auto dest: finite_subset)
```
```  1604   with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
```
```  1605 qed
```
```  1606
```
```  1607 lemma closed:
```
```  1608   assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
```
```  1609   shows "F A \<in> A"
```
```  1610 using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
```
```  1611   case singleton then show ?case by simp
```
```  1612 next
```
```  1613   case insert with elem show ?case by force
```
```  1614 qed
```
```  1615
```
```  1616 end
```
```  1617
```
```  1618 locale semilattice_order_neutr_set = semilattice_neutr_order + semilattice_neutr_set
```
```  1619 begin
```
```  1620
```
```  1621 lemma bounded_iff:
```
```  1622   assumes "finite A"
```
```  1623   shows "x \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> a)"
```
```  1624   using assms by (induct A) (simp_all add: bounded_iff)
```
```  1625
```
```  1626 lemma boundedI:
```
```  1627   assumes "finite A"
```
```  1628   assumes "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
```
```  1629   shows "x \<preceq> F A"
```
```  1630   using assms by (simp add: bounded_iff)
```
```  1631
```
```  1632 lemma boundedE:
```
```  1633   assumes "finite A" and "x \<preceq> F A"
```
```  1634   obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
```
```  1635   using assms by (simp add: bounded_iff)
```
```  1636
```
```  1637 lemma coboundedI:
```
```  1638   assumes "finite A"
```
```  1639     and "a \<in> A"
```
```  1640   shows "F A \<preceq> a"
```
```  1641 proof -
```
```  1642   from assms have "A \<noteq> {}" by auto
```
```  1643   from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
```
```  1644   proof (induct rule: finite_ne_induct)
```
```  1645     case singleton thus ?case by (simp add: refl)
```
```  1646   next
```
```  1647     case (insert x B)
```
```  1648     from insert have "a = x \<or> a \<in> B" by simp
```
```  1649     then show ?case using insert by (auto intro: coboundedI2)
```
```  1650   qed
```
```  1651 qed
```
```  1652
```
```  1653 lemma antimono:
```
```  1654   assumes "A \<subseteq> B" and "finite B"
```
```  1655   shows "F B \<preceq> F A"
```
```  1656 proof (cases "A = B")
```
```  1657   case True then show ?thesis by (simp add: refl)
```
```  1658 next
```
```  1659   case False
```
```  1660   have B: "B = A \<union> (B - A)" using `A \<subseteq> B` by blast
```
```  1661   then have "F B = F (A \<union> (B - A))" by simp
```
```  1662   also have "\<dots> = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
```
```  1663   also have "\<dots> \<preceq> F A" by simp
```
```  1664   finally show ?thesis .
```
```  1665 qed
```
```  1666
```
```  1667 end
```
```  1668
```
```  1669 notation times (infixl "*" 70)
```
```  1670 notation Groups.one ("1")
```
```  1671
```
```  1672
```
```  1673 subsection {* Lattice operations on finite sets *}
```
```  1674
```
```  1675 text {*
```
```  1676   For historic reasons, there is the sublocale dependency from @{class distrib_lattice}
```
```  1677   to @{class linorder}.  This is badly designed: both should depend on a common abstract
```
```  1678   distributive lattice rather than having this non-subclass dependecy between two
```
```  1679   classes.  But for the moment we have to live with it.  This forces us to setup
```
```  1680   this sublocale dependency simultaneously with the lattice operations on finite
```
```  1681   sets, to avoid garbage.
```
```  1682 *}
```
```  1683
```
```  1684 definition (in semilattice_inf) Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>\<^sub>f\<^sub>i\<^sub>n_"  900)
```
```  1685 where
```
```  1686   "Inf_fin = semilattice_set.F inf"
```
```  1687
```
```  1688 definition (in semilattice_sup) Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>\<^sub>f\<^sub>i\<^sub>n_"  900)
```
```  1689 where
```
```  1690   "Sup_fin = semilattice_set.F sup"
```
```  1691
```
```  1692 context linorder
```
```  1693 begin
```
```  1694
```
```  1695 definition Min :: "'a set \<Rightarrow> 'a"
```
```  1696 where
```
```  1697   "Min = semilattice_set.F min"
```
```  1698
```
```  1699 definition Max :: "'a set \<Rightarrow> 'a"
```
```  1700 where
```
```  1701   "Max = semilattice_set.F max"
```
```  1702
```
```  1703 sublocale Min!: semilattice_order_set min less_eq less
```
```  1704   + Max!: semilattice_order_set max greater_eq greater
```
```  1705 where
```
```  1706   "semilattice_set.F min = Min"
```
```  1707   and "semilattice_set.F max = Max"
```
```  1708 proof -
```
```  1709   show "semilattice_order_set min less_eq less" by default (auto simp add: min_def)
```
```  1710   then interpret Min!: semilattice_order_set min less_eq less .
```
```  1711   show "semilattice_order_set max greater_eq greater" by default (auto simp add: max_def)
```
```  1712   then interpret Max!: semilattice_order_set max greater_eq greater .
```
```  1713   from Min_def show "semilattice_set.F min = Min" by rule
```
```  1714   from Max_def show "semilattice_set.F max = Max" by rule
```
```  1715 qed
```
```  1716
```
```  1717
```
```  1718 text {* An aside: @{const min}/@{const max} on linear orders as special case of @{const inf}/@{const sup} *}
```
```  1719
```
```  1720 sublocale min_max!: distrib_lattice min less_eq less max
```
```  1721 where
```
```  1722   "semilattice_inf.Inf_fin min = Min"
```
```  1723   and "semilattice_sup.Sup_fin max = Max"
```
```  1724 proof -
```
```  1725   show "class.distrib_lattice min less_eq less max"
```
```  1726   proof
```
```  1727     fix x y z
```
```  1728     show "max x (min y z) = min (max x y) (max x z)"
```
```  1729       by (auto simp add: min_def max_def)
```
```  1730   qed (auto simp add: min_def max_def not_le less_imp_le)
```
```  1731   then interpret min_max!: distrib_lattice min less_eq less max .
```
```  1732   show "semilattice_inf.Inf_fin min = Min"
```
```  1733     by (simp only: min_max.Inf_fin_def Min_def)
```
```  1734   show "semilattice_sup.Sup_fin max = Max"
```
```  1735     by (simp only: min_max.Sup_fin_def Max_def)
```
```  1736 qed
```
```  1737
```
```  1738 lemmas le_maxI1 = min_max.sup_ge1
```
```  1739 lemmas le_maxI2 = min_max.sup_ge2
```
```  1740
```
```  1741 lemmas min_ac = min_max.inf_assoc min_max.inf_commute
```
```  1742   min.left_commute
```
```  1743
```
```  1744 lemmas max_ac = min_max.sup_assoc min_max.sup_commute
```
```  1745   max.left_commute
```
```  1746
```
```  1747 end
```
```  1748
```
```  1749
```
```  1750 text {* Lattice operations proper *}
```
```  1751
```
```  1752 sublocale semilattice_inf < Inf_fin!: semilattice_order_set inf less_eq less
```
```  1753 where
```
```  1754   "semilattice_set.F inf = Inf_fin"
```
```  1755 proof -
```
```  1756   show "semilattice_order_set inf less_eq less" ..
```
```  1757   then interpret Inf_fin!: semilattice_order_set inf less_eq less .
```
```  1758   from Inf_fin_def show "semilattice_set.F inf = Inf_fin" by rule
```
```  1759 qed
```
```  1760
```
```  1761 sublocale semilattice_sup < Sup_fin!: semilattice_order_set sup greater_eq greater
```
```  1762 where
```
```  1763   "semilattice_set.F sup = Sup_fin"
```
```  1764 proof -
```
```  1765   show "semilattice_order_set sup greater_eq greater" ..
```
```  1766   then interpret Sup_fin!: semilattice_order_set sup greater_eq greater .
```
```  1767   from Sup_fin_def show "semilattice_set.F sup = Sup_fin" by rule
```
```  1768 qed
```
```  1769
```
```  1770
```
```  1771 text {* An aside again: @{const Min}/@{const Max} on linear orders as special case of @{const Inf_fin}/@{const Sup_fin} *}
```
```  1772
```
```  1773 lemma Inf_fin_Min:
```
```  1774   "Inf_fin = (Min :: 'a::{semilattice_inf, linorder} set \<Rightarrow> 'a)"
```
```  1775   by (simp add: Inf_fin_def Min_def inf_min)
```
```  1776
```
```  1777 lemma Sup_fin_Max:
```
```  1778   "Sup_fin = (Max :: 'a::{semilattice_sup, linorder} set \<Rightarrow> 'a)"
```
```  1779   by (simp add: Sup_fin_def Max_def sup_max)
```
```  1780
```
```  1781
```
```  1782
```
```  1783 subsection {* Infimum and Supremum over non-empty sets *}
```
```  1784
```
```  1785 text {*
```
```  1786   After this non-regular bootstrap, things continue canonically.
```
```  1787 *}
```
```  1788
```
```  1789 context lattice
```
```  1790 begin
```
```  1791
```
```  1792 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"
```
```  1793 apply(subgoal_tac "EX a. a:A")
```
```  1794 prefer 2 apply blast
```
```  1795 apply(erule exE)
```
```  1796 apply(rule order_trans)
```
```  1797 apply(erule (1) Inf_fin.coboundedI)
```
```  1798 apply(erule (1) Sup_fin.coboundedI)
```
```  1799 done
```
```  1800
```
```  1801 lemma sup_Inf_absorb [simp]:
```
```  1802   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA) = a"
```
```  1803 apply(subst sup_commute)
```
```  1804 apply(simp add: sup_absorb2 Inf_fin.coboundedI)
```
```  1805 done
```
```  1806
```
```  1807 lemma inf_Sup_absorb [simp]:
```
```  1808   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA) = a"
```
```  1809 by (simp add: inf_absorb1 Sup_fin.coboundedI)
```
```  1810
```
```  1811 end
```
```  1812
```
```  1813 context distrib_lattice
```
```  1814 begin
```
```  1815
```
```  1816 lemma sup_Inf1_distrib:
```
```  1817   assumes "finite A"
```
```  1818     and "A \<noteq> {}"
```
```  1819   shows "sup x (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA) = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n{sup x a|a. a \<in> A}"
```
```  1820 using assms by (simp add: image_def Inf_fin.hom_commute [where h="sup x", OF sup_inf_distrib1])
```
```  1821   (rule arg_cong [where f="Inf_fin"], blast)
```
```  1822
```
```  1823 lemma sup_Inf2_distrib:
```
```  1824   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
```
```  1825   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}"
```
```  1826 using A proof (induct rule: finite_ne_induct)
```
```  1827   case singleton then show ?case
```
```  1828     by (simp add: sup_Inf1_distrib [OF B])
```
```  1829 next
```
```  1830   case (insert x A)
```
```  1831   have finB: "finite {sup x b |b. b \<in> B}"
```
```  1832     by (rule finite_surj [where f = "sup x", OF B(1)], auto)
```
```  1833   have finAB: "finite {sup a b |a b. a \<in> A \<and> b \<in> B}"
```
```  1834   proof -
```
```  1835     have "{sup a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {sup a b})"
```
```  1836       by blast
```
```  1837     thus ?thesis by(simp add: insert(1) B(1))
```
```  1838   qed
```
```  1839   have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
```
```  1840   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)"
```
```  1841     using insert by simp
```
```  1842   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)
```
```  1843   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})"
```
```  1844     using insert by(simp add:sup_Inf1_distrib[OF B])
```
```  1845   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})"
```
```  1846     (is "_ = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n?M")
```
```  1847     using B insert
```
```  1848     by (simp add: Inf_fin.union [OF finB _ finAB ne])
```
```  1849   also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"
```
```  1850     by blast
```
```  1851   finally show ?case .
```
```  1852 qed
```
```  1853
```
```  1854 lemma inf_Sup1_distrib:
```
```  1855   assumes "finite A" and "A \<noteq> {}"
```
```  1856   shows "inf x (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA) = \<Squnion>\<^sub>f\<^sub>i\<^sub>n{inf x a|a. a \<in> A}"
```
```  1857 using assms by (simp add: image_def Sup_fin.hom_commute [where h="inf x", OF inf_sup_distrib1])
```
```  1858   (rule arg_cong [where f="Sup_fin"], blast)
```
```  1859
```
```  1860 lemma inf_Sup2_distrib:
```
```  1861   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
```
```  1862   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}"
```
```  1863 using A proof (induct rule: finite_ne_induct)
```
```  1864   case singleton thus ?case
```
```  1865     by(simp add: inf_Sup1_distrib [OF B])
```
```  1866 next
```
```  1867   case (insert x A)
```
```  1868   have finB: "finite {inf x b |b. b \<in> B}"
```
```  1869     by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto)
```
```  1870   have finAB: "finite {inf a b |a b. a \<in> A \<and> b \<in> B}"
```
```  1871   proof -
```
```  1872     have "{inf a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {inf a b})"
```
```  1873       by blast
```
```  1874     thus ?thesis by(simp add: insert(1) B(1))
```
```  1875   qed
```
```  1876   have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
```
```  1877   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)"
```
```  1878     using insert by simp
```
```  1879   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)
```
```  1880   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})"
```
```  1881     using insert by(simp add:inf_Sup1_distrib[OF B])
```
```  1882   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})"
```
```  1883     (is "_ = \<Squnion>\<^sub>f\<^sub>i\<^sub>n?M")
```
```  1884     using B insert
```
```  1885     by (simp add: Sup_fin.union [OF finB _ finAB ne])
```
```  1886   also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"
```
```  1887     by blast
```
```  1888   finally show ?case .
```
```  1889 qed
```
```  1890
```
```  1891 end
```
```  1892
```
```  1893 context complete_lattice
```
```  1894 begin
```
```  1895
```
```  1896 lemma Inf_fin_Inf:
```
```  1897   assumes "finite A" and "A \<noteq> {}"
```
```  1898   shows "\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA = Inf A"
```
```  1899 proof -
```
```  1900   from assms obtain b B where "A = insert b B" and "finite B" by auto
```
```  1901   then show ?thesis
```
```  1902     by (simp add: Inf_fin.eq_fold inf_Inf_fold_inf inf.commute [of b])
```
```  1903 qed
```
```  1904
```
```  1905 lemma Sup_fin_Sup:
```
```  1906   assumes "finite A" and "A \<noteq> {}"
```
```  1907   shows "\<Squnion>\<^sub>f\<^sub>i\<^sub>nA = Sup A"
```
```  1908 proof -
```
```  1909   from assms obtain b B where "A = insert b B" and "finite B" by auto
```
```  1910   then show ?thesis
```
```  1911     by (simp add: Sup_fin.eq_fold sup_Sup_fold_sup sup.commute [of b])
```
```  1912 qed
```
```  1913
```
```  1914 end
```
```  1915
```
```  1916
```
```  1917 subsection {* Minimum and Maximum over non-empty sets *}
```
```  1918
```
```  1919 context linorder
```
```  1920 begin
```
```  1921
```
```  1922 lemma dual_min:
```
```  1923   "ord.min greater_eq = max"
```
```  1924   by (auto simp add: ord.min_def max_def fun_eq_iff)
```
```  1925
```
```  1926 lemma dual_max:
```
```  1927   "ord.max greater_eq = min"
```
```  1928   by (auto simp add: ord.max_def min_def fun_eq_iff)
```
```  1929
```
```  1930 lemma dual_Min:
```
```  1931   "linorder.Min greater_eq = Max"
```
```  1932 proof -
```
```  1933   interpret dual!: linorder greater_eq greater by (fact dual_linorder)
```
```  1934   show ?thesis by (simp add: dual.Min_def dual_min Max_def)
```
```  1935 qed
```
```  1936
```
```  1937 lemma dual_Max:
```
```  1938   "linorder.Max greater_eq = Min"
```
```  1939 proof -
```
```  1940   interpret dual!: linorder greater_eq greater by (fact dual_linorder)
```
```  1941   show ?thesis by (simp add: dual.Max_def dual_max Min_def)
```
```  1942 qed
```
```  1943
```
```  1944 lemmas Min_singleton = Min.singleton
```
```  1945 lemmas Max_singleton = Max.singleton
```
```  1946 lemmas Min_insert = Min.insert
```
```  1947 lemmas Max_insert = Max.insert
```
```  1948 lemmas Min_Un = Min.union
```
```  1949 lemmas Max_Un = Max.union
```
```  1950 lemmas hom_Min_commute = Min.hom_commute
```
```  1951 lemmas hom_Max_commute = Max.hom_commute
```
```  1952
```
```  1953 lemma Min_in [simp]:
```
```  1954   assumes "finite A" and "A \<noteq> {}"
```
```  1955   shows "Min A \<in> A"
```
```  1956   using assms by (auto simp add: min_def Min.closed)
```
```  1957
```
```  1958 lemma Max_in [simp]:
```
```  1959   assumes "finite A" and "A \<noteq> {}"
```
```  1960   shows "Max A \<in> A"
```
```  1961   using assms by (auto simp add: max_def Max.closed)
```
```  1962
```
```  1963 lemma Min_le [simp]:
```
```  1964   assumes "finite A" and "x \<in> A"
```
```  1965   shows "Min A \<le> x"
```
```  1966   using assms by (fact Min.coboundedI)
```
```  1967
```
```  1968 lemma Max_ge [simp]:
```
```  1969   assumes "finite A" and "x \<in> A"
```
```  1970   shows "x \<le> Max A"
```
```  1971   using assms by (fact Max.coboundedI)
```
```  1972
```
```  1973 lemma Min_eqI:
```
```  1974   assumes "finite A"
```
```  1975   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<ge> x"
```
```  1976     and "x \<in> A"
```
```  1977   shows "Min A = x"
```
```  1978 proof (rule antisym)
```
```  1979   from `x \<in> A` have "A \<noteq> {}" by auto
```
```  1980   with assms show "Min A \<ge> x" by simp
```
```  1981 next
```
```  1982   from assms show "x \<ge> Min A" by simp
```
```  1983 qed
```
```  1984
```
```  1985 lemma Max_eqI:
```
```  1986   assumes "finite A"
```
```  1987   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
```
```  1988     and "x \<in> A"
```
```  1989   shows "Max A = x"
```
```  1990 proof (rule antisym)
```
```  1991   from `x \<in> A` have "A \<noteq> {}" by auto
```
```  1992   with assms show "Max A \<le> x" by simp
```
```  1993 next
```
```  1994   from assms show "x \<le> Max A" by simp
```
```  1995 qed
```
```  1996
```
```  1997 context
```
```  1998   fixes A :: "'a set"
```
```  1999   assumes fin_nonempty: "finite A" "A \<noteq> {}"
```
```  2000 begin
```
```  2001
```
```  2002 lemma Min_ge_iff [simp, no_atp]:
```
```  2003   "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
```
```  2004   using fin_nonempty by (fact Min.bounded_iff)
```
```  2005
```
```  2006 lemma Max_le_iff [simp, no_atp]:
```
```  2007   "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
```
```  2008   using fin_nonempty by (fact Max.bounded_iff)
```
```  2009
```
```  2010 lemma Min_gr_iff [simp, no_atp]:
```
```  2011   "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
```
```  2012   using fin_nonempty  by (induct rule: finite_ne_induct) simp_all
```
```  2013
```
```  2014 lemma Max_less_iff [simp, no_atp]:
```
```  2015   "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
```
```  2016   using fin_nonempty by (induct rule: finite_ne_induct) simp_all
```
```  2017
```
```  2018 lemma Min_le_iff [no_atp]:
```
```  2019   "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
```
```  2020   using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_le_iff_disj)
```
```  2021
```
```  2022 lemma Max_ge_iff [no_atp]:
```
```  2023   "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
```
```  2024   using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: le_max_iff_disj)
```
```  2025
```
```  2026 lemma Min_less_iff [no_atp]:
```
```  2027   "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
```
```  2028   using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_less_iff_disj)
```
```  2029
```
```  2030 lemma Max_gr_iff [no_atp]:
```
```  2031   "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
```
```  2032   using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: less_max_iff_disj)
```
```  2033
```
```  2034 end
```
```  2035
```
```  2036 lemma Min_antimono:
```
```  2037   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
```
```  2038   shows "Min N \<le> Min M"
```
```  2039   using assms by (fact Min.antimono)
```
```  2040
```
```  2041 lemma Max_mono:
```
```  2042   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
```
```  2043   shows "Max M \<le> Max N"
```
```  2044   using assms by (fact Max.antimono)
```
```  2045
```
```  2046 lemma mono_Min_commute:
```
```  2047   assumes "mono f"
```
```  2048   assumes "finite A" and "A \<noteq> {}"
```
```  2049   shows "f (Min A) = Min (f ` A)"
```
```  2050 proof (rule linorder_class.Min_eqI [symmetric])
```
```  2051   from `finite A` show "finite (f ` A)" by simp
```
```  2052   from assms show "f (Min A) \<in> f ` A" by simp
```
```  2053   fix x
```
```  2054   assume "x \<in> f ` A"
```
```  2055   then obtain y where "y \<in> A" and "x = f y" ..
```
```  2056   with assms have "Min A \<le> y" by auto
```
```  2057   with `mono f` have "f (Min A) \<le> f y" by (rule monoE)
```
```  2058   with `x = f y` show "f (Min A) \<le> x" by simp
```
```  2059 qed
```
```  2060
```
```  2061 lemma mono_Max_commute:
```
```  2062   assumes "mono f"
```
```  2063   assumes "finite A" and "A \<noteq> {}"
```
```  2064   shows "f (Max A) = Max (f ` A)"
```
```  2065 proof (rule linorder_class.Max_eqI [symmetric])
```
```  2066   from `finite A` show "finite (f ` A)" by simp
```
```  2067   from assms show "f (Max A) \<in> f ` A" by simp
```
```  2068   fix x
```
```  2069   assume "x \<in> f ` A"
```
```  2070   then obtain y where "y \<in> A" and "x = f y" ..
```
```  2071   with assms have "y \<le> Max A" by auto
```
```  2072   with `mono f` have "f y \<le> f (Max A)" by (rule monoE)
```
```  2073   with `x = f y` show "x \<le> f (Max A)" by simp
```
```  2074 qed
```
```  2075
```
```  2076 lemma finite_linorder_max_induct [consumes 1, case_names empty insert]:
```
```  2077   assumes fin: "finite A"
```
```  2078   and empty: "P {}"
```
```  2079   and insert: "\<And>b A. finite A \<Longrightarrow> \<forall>a\<in>A. a < b \<Longrightarrow> P A \<Longrightarrow> P (insert b A)"
```
```  2080   shows "P A"
```
```  2081 using fin empty insert
```
```  2082 proof (induct rule: finite_psubset_induct)
```
```  2083   case (psubset A)
```
```  2084   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
```
```  2085   have fin: "finite A" by fact
```
```  2086   have empty: "P {}" by fact
```
```  2087   have step: "\<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. a < b; P A\<rbrakk> \<Longrightarrow> P (insert b A)" by fact
```
```  2088   show "P A"
```
```  2089   proof (cases "A = {}")
```
```  2090     assume "A = {}"
```
```  2091     then show "P A" using `P {}` by simp
```
```  2092   next
```
```  2093     let ?B = "A - {Max A}"
```
```  2094     let ?A = "insert (Max A) ?B"
```
```  2095     have "finite ?B" using `finite A` by simp
```
```  2096     assume "A \<noteq> {}"
```
```  2097     with `finite A` have "Max A : A" by auto
```
```  2098     then have A: "?A = A" using insert_Diff_single insert_absorb by auto
```
```  2099     then have "P ?B" using `P {}` step IH [of ?B] by blast
```
```  2100     moreover
```
```  2101     have "\<forall>a\<in>?B. a < Max A" using Max_ge [OF `finite A`] by fastforce
```
```  2102     ultimately show "P A" using A insert_Diff_single step [OF `finite ?B`] by fastforce
```
```  2103   qed
```
```  2104 qed
```
```  2105
```
```  2106 lemma finite_linorder_min_induct [consumes 1, case_names empty insert]:
```
```  2107   "\<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"
```
```  2108   by (rule linorder.finite_linorder_max_induct [OF dual_linorder])
```
```  2109
```
```  2110 lemma Least_Min:
```
```  2111   assumes "finite {a. P a}" and "\<exists>a. P a"
```
```  2112   shows "(LEAST a. P a) = Min {a. P a}"
```
```  2113 proof -
```
```  2114   { fix A :: "'a set"
```
```  2115     assume A: "finite A" "A \<noteq> {}"
```
```  2116     have "(LEAST a. a \<in> A) = Min A"
```
```  2117     using A proof (induct A rule: finite_ne_induct)
```
```  2118       case singleton show ?case by (rule Least_equality) simp_all
```
```  2119     next
```
```  2120       case (insert a A)
```
```  2121       have "(LEAST b. b = a \<or> b \<in> A) = min a (LEAST a. a \<in> A)"
```
```  2122         by (auto intro!: Least_equality simp add: min_def not_le Min_le_iff insert.hyps dest!: less_imp_le)
```
```  2123       with insert show ?case by simp
```
```  2124     qed
```
```  2125   } from this [of "{a. P a}"] assms show ?thesis by simp
```
```  2126 qed
```
```  2127
```
```  2128 end
```
```  2129
```
```  2130 context linordered_ab_semigroup_add
```
```  2131 begin
```
```  2132
```
```  2133 lemma add_Min_commute:
```
```  2134   fixes k
```
```  2135   assumes "finite N" and "N \<noteq> {}"
```
```  2136   shows "k + Min N = Min {k + m | m. m \<in> N}"
```
```  2137 proof -
```
```  2138   have "\<And>x y. k + min x y = min (k + x) (k + y)"
```
```  2139     by (simp add: min_def not_le)
```
```  2140       (blast intro: antisym less_imp_le add_left_mono)
```
```  2141   with assms show ?thesis
```
```  2142     using hom_Min_commute [of "plus k" N]
```
```  2143     by simp (blast intro: arg_cong [where f = Min])
```
```  2144 qed
```
```  2145
```
```  2146 lemma add_Max_commute:
```
```  2147   fixes k
```
```  2148   assumes "finite N" and "N \<noteq> {}"
```
```  2149   shows "k + Max N = Max {k + m | m. m \<in> N}"
```
```  2150 proof -
```
```  2151   have "\<And>x y. k + max x y = max (k + x) (k + y)"
```
```  2152     by (simp add: max_def not_le)
```
```  2153       (blast intro: antisym less_imp_le add_left_mono)
```
```  2154   with assms show ?thesis
```
```  2155     using hom_Max_commute [of "plus k" N]
```
```  2156     by simp (blast intro: arg_cong [where f = Max])
```
```  2157 qed
```
```  2158
```
```  2159 end
```
```  2160
```
```  2161 context linordered_ab_group_add
```
```  2162 begin
```
```  2163
```
```  2164 lemma minus_Max_eq_Min [simp]:
```
```  2165   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Max S = Min (uminus ` S)"
```
```  2166   by (induct S rule: finite_ne_induct) (simp_all add: minus_max_eq_min)
```
```  2167
```
```  2168 lemma minus_Min_eq_Max [simp]:
```
```  2169   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Min S = Max (uminus ` S)"
```
```  2170   by (induct S rule: finite_ne_induct) (simp_all add: minus_min_eq_max)
```
```  2171
```
```  2172 end
```
```  2173
```
```  2174 context complete_linorder
```
```  2175 begin
```
```  2176
```
```  2177 lemma Min_Inf:
```
```  2178   assumes "finite A" and "A \<noteq> {}"
```
```  2179   shows "Min A = Inf A"
```
```  2180 proof -
```
```  2181   from assms obtain b B where "A = insert b B" and "finite B" by auto
```
```  2182   then show ?thesis
```
```  2183     by (simp add: Min.eq_fold complete_linorder_inf_min [symmetric] inf_Inf_fold_inf inf.commute [of b])
```
```  2184 qed
```
```  2185
```
```  2186 lemma Max_Sup:
```
```  2187   assumes "finite A" and "A \<noteq> {}"
```
```  2188   shows "Max A = Sup A"
```
```  2189 proof -
```
```  2190   from assms obtain b B where "A = insert b B" and "finite B" by auto
```
```  2191   then show ?thesis
```
```  2192     by (simp add: Max.eq_fold complete_linorder_sup_max [symmetric] sup_Sup_fold_sup sup.commute [of b])
```
```  2193 qed
```
```  2194
```
```  2195 end
```
```  2196
```
```  2197 end
```