src/HOL/Big_Operators.thy
 author haftmann Tue Mar 26 21:53:56 2013 +0100 (2013-03-26) changeset 51546 2e26df807dc7 parent 51540 eea5c4ca4a0e child 51586 7c59fe17f495 permissions -rw-r--r--
more uniform style for interpretation and sublocale declarations
```     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 definition F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
```
```    21 where
```
```    22   eq_fold: "F g A = Finite_Set.fold (f \<circ> g) 1 A"
```
```    23
```
```    24 lemma infinite [simp]:
```
```    25   "\<not> finite A \<Longrightarrow> F g A = 1"
```
```    26   by (simp add: eq_fold)
```
```    27
```
```    28 lemma empty [simp]:
```
```    29   "F g {} = 1"
```
```    30   by (simp add: eq_fold)
```
```    31
```
```    32 lemma insert [simp]:
```
```    33   assumes "finite A" and "x \<notin> A"
```
```    34   shows "F g (insert x A) = g x * F g A"
```
```    35 proof -
```
```    36   interpret comp_fun_commute f
```
```    37     by default (simp add: fun_eq_iff left_commute)
```
```    38   interpret comp_fun_commute "f \<circ> g"
```
```    39     by (rule comp_comp_fun_commute)
```
```    40   from assms show ?thesis by (simp add: eq_fold)
```
```    41 qed
```
```    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` this 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 proof (cases "finite A")
```
```    62   case True from `finite A` assms show ?thesis by (induct A) simp_all
```
```    63 next
```
```    64   case False then show ?thesis by simp
```
```    65 qed
```
```    66
```
```    67 lemma neutral_const [simp]:
```
```    68   "F (\<lambda>_. 1) A = 1"
```
```    69   by (simp add: neutral)
```
```    70
```
```    71 lemma union_inter:
```
```    72   assumes "finite A" and "finite B"
```
```    73   shows "F g (A \<union> B) * F g (A \<inter> B) = F g A * F g B"
```
```    74   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
```
```    75 using assms proof (induct A)
```
```    76   case empty then show ?case by simp
```
```    77 next
```
```    78   case (insert x A) then show ?case
```
```    79     by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
```
```    80 qed
```
```    81
```
```    82 corollary union_inter_neutral:
```
```    83   assumes "finite A" and "finite B"
```
```    84   and I0: "\<forall>x \<in> A \<inter> B. g x = 1"
```
```    85   shows "F g (A \<union> B) = F g A * F g B"
```
```    86   using assms by (simp add: union_inter [symmetric] neutral)
```
```    87
```
```    88 corollary union_disjoint:
```
```    89   assumes "finite A" and "finite B"
```
```    90   assumes "A \<inter> B = {}"
```
```    91   shows "F g (A \<union> B) = F g A * F g B"
```
```    92   using assms by (simp add: union_inter_neutral)
```
```    93
```
```    94 lemma subset_diff:
```
```    95   "B \<subseteq> A \<Longrightarrow> finite A \<Longrightarrow> F g A = F g (A - B) * F g B"
```
```    96   by (metis Diff_partition union_disjoint Diff_disjoint finite_Un inf_commute sup_commute)
```
```    97
```
```    98 lemma reindex:
```
```    99   assumes "inj_on h A"
```
```   100   shows "F g (h ` A) = F (g \<circ> h) A"
```
```   101 proof (cases "finite A")
```
```   102   case True
```
```   103   interpret comp_fun_commute f
```
```   104     by default (simp add: fun_eq_iff left_commute)
```
```   105   interpret comp_fun_commute "f \<circ> g"
```
```   106     by (rule comp_comp_fun_commute)
```
```   107   from assms `finite A` show ?thesis by (simp add: eq_fold fold_image comp_assoc)
```
```   108 next
```
```   109   case False with assms have "\<not> finite (h ` A)" by (blast dest: finite_imageD)
```
```   110   with False show ?thesis by simp
```
```   111 qed
```
```   112
```
```   113 lemma cong:
```
```   114   assumes "A = B"
```
```   115   assumes g_h: "\<And>x. x \<in> B \<Longrightarrow> g x = h x"
```
```   116   shows "F g A = F h B"
```
```   117 proof (cases "finite A")
```
```   118   case True
```
```   119   then have "\<And>C. C \<subseteq> A \<longrightarrow> (\<forall>x\<in>C. g x = h x) \<longrightarrow> F g C = F h C"
```
```   120   proof induct
```
```   121     case empty then show ?case by simp
```
```   122   next
```
```   123     case (insert x F) then show ?case apply -
```
```   124     apply (simp add: subset_insert_iff, clarify)
```
```   125     apply (subgoal_tac "finite C")
```
```   126       prefer 2 apply (blast dest: finite_subset [rotated])
```
```   127     apply (subgoal_tac "C = insert x (C - {x})")
```
```   128       prefer 2 apply blast
```
```   129     apply (erule ssubst)
```
```   130     apply (simp add: Ball_def del: insert_Diff_single)
```
```   131     done
```
```   132   qed
```
```   133   with `A = B` g_h show ?thesis by simp
```
```   134 next
```
```   135   case False
```
```   136   with `A = B` show ?thesis by simp
```
```   137 qed
```
```   138
```
```   139 lemma strong_cong [cong]:
```
```   140   assumes "A = B" "\<And>x. x \<in> B =simp=> g x = h x"
```
```   141   shows "F (\<lambda>x. g x) A = F (\<lambda>x. h x) B"
```
```   142   by (rule cong) (insert assms, simp_all add: simp_implies_def)
```
```   143
```
```   144 lemma UNION_disjoint:
```
```   145   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
```
```   146   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
```
```   147   shows "F g (UNION I A) = F (\<lambda>x. F g (A x)) I"
```
```   148 apply (insert assms)
```
```   149 apply (induct rule: finite_induct)
```
```   150 apply simp
```
```   151 apply atomize
```
```   152 apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
```
```   153  prefer 2 apply blast
```
```   154 apply (subgoal_tac "A x Int UNION Fa A = {}")
```
```   155  prefer 2 apply blast
```
```   156 apply (simp add: union_disjoint)
```
```   157 done
```
```   158
```
```   159 lemma Union_disjoint:
```
```   160   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
```
```   161   shows "F g (Union C) = F (F g) C"
```
```   162 proof cases
```
```   163   assume "finite C"
```
```   164   from UNION_disjoint [OF this assms]
```
```   165   show ?thesis
```
```   166     by (simp add: SUP_def)
```
```   167 qed (auto dest: finite_UnionD intro: infinite)
```
```   168
```
```   169 lemma distrib:
```
```   170   "F (\<lambda>x. g x * h x) A = F g A * F h A"
```
```   171 proof (cases "finite A")
```
```   172   case False then show ?thesis by simp
```
```   173 next
```
```   174   case True then show ?thesis by (rule finite_induct) (simp_all add: assoc commute left_commute)
```
```   175 qed
```
```   176
```
```   177 lemma Sigma:
```
```   178   "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)"
```
```   179 apply (subst Sigma_def)
```
```   180 apply (subst UNION_disjoint, assumption, simp)
```
```   181  apply blast
```
```   182 apply (rule cong)
```
```   183 apply rule
```
```   184 apply (simp add: fun_eq_iff)
```
```   185 apply (subst UNION_disjoint, simp, simp)
```
```   186  apply blast
```
```   187 apply (simp add: comp_def)
```
```   188 done
```
```   189
```
```   190 lemma related:
```
```   191   assumes Re: "R 1 1"
```
```   192   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
```
```   193   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
```
```   194   shows "R (F h S) (F g S)"
```
```   195   using fS by (rule finite_subset_induct) (insert assms, auto)
```
```   196
```
```   197 lemma eq_general:
```
```   198   assumes h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y"
```
```   199   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
```
```   200   shows "F f1 S = F f2 S'"
```
```   201 proof-
```
```   202   from h f12 have hS: "h ` S = S'" by blast
```
```   203   {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
```
```   204     from f12 h H  have "x = y" by auto }
```
```   205   hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
```
```   206   from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
```
```   207   from hS have "F f2 S' = F f2 (h ` S)" by simp
```
```   208   also have "\<dots> = F (f2 o h) S" using reindex [OF hinj, of f2] .
```
```   209   also have "\<dots> = F f1 S " using th cong [of _ _ "f2 o h" f1]
```
```   210     by blast
```
```   211   finally show ?thesis ..
```
```   212 qed
```
```   213
```
```   214 lemma eq_general_reverses:
```
```   215   assumes kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
```
```   216   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"
```
```   217   shows "F j S = F g T"
```
```   218   (* metis solves it, but not yet available here *)
```
```   219   apply (rule eq_general [of T S h g j])
```
```   220   apply (rule ballI)
```
```   221   apply (frule kh)
```
```   222   apply (rule ex1I[])
```
```   223   apply blast
```
```   224   apply clarsimp
```
```   225   apply (drule hk) apply simp
```
```   226   apply (rule sym)
```
```   227   apply (erule conjunct1[OF conjunct2[OF hk]])
```
```   228   apply (rule ballI)
```
```   229   apply (drule hk)
```
```   230   apply blast
```
```   231   done
```
```   232
```
```   233 lemma mono_neutral_cong_left:
```
```   234   assumes "finite T" and "S \<subseteq> T" and "\<forall>i \<in> T - S. h i = 1"
```
```   235   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g S = F h T"
```
```   236 proof-
```
```   237   have eq: "T = S \<union> (T - S)" using `S \<subseteq> T` by blast
```
```   238   have d: "S \<inter> (T - S) = {}" using `S \<subseteq> T` by blast
```
```   239   from `finite T` `S \<subseteq> T` have f: "finite S" "finite (T - S)"
```
```   240     by (auto intro: finite_subset)
```
```   241   show ?thesis using assms(4)
```
```   242     by (simp add: union_disjoint [OF f d, unfolded eq [symmetric]] neutral [OF assms(3)])
```
```   243 qed
```
```   244
```
```   245 lemma mono_neutral_cong_right:
```
```   246   "\<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>
```
```   247    \<Longrightarrow> F g T = F h S"
```
```   248   by (auto intro!: mono_neutral_cong_left [symmetric])
```
```   249
```
```   250 lemma mono_neutral_left:
```
```   251   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g S = F g T"
```
```   252   by (blast intro: mono_neutral_cong_left)
```
```   253
```
```   254 lemma mono_neutral_right:
```
```   255   "\<lbrakk> finite T;  S \<subseteq> T;  \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g T = F g S"
```
```   256   by (blast intro!: mono_neutral_left [symmetric])
```
```   257
```
```   258 lemma delta:
```
```   259   assumes fS: "finite S"
```
```   260   shows "F (\<lambda>k. if k = a then b k else 1) S = (if a \<in> S then b a else 1)"
```
```   261 proof-
```
```   262   let ?f = "(\<lambda>k. if k=a then b k else 1)"
```
```   263   { assume a: "a \<notin> S"
```
```   264     hence "\<forall>k\<in>S. ?f k = 1" by simp
```
```   265     hence ?thesis  using a by simp }
```
```   266   moreover
```
```   267   { assume a: "a \<in> S"
```
```   268     let ?A = "S - {a}"
```
```   269     let ?B = "{a}"
```
```   270     have eq: "S = ?A \<union> ?B" using a by blast
```
```   271     have dj: "?A \<inter> ?B = {}" by simp
```
```   272     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```   273     have "F ?f S = F ?f ?A * F ?f ?B"
```
```   274       using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]]
```
```   275       by simp
```
```   276     then have ?thesis using a by simp }
```
```   277   ultimately show ?thesis by blast
```
```   278 qed
```
```   279
```
```   280 lemma delta':
```
```   281   assumes fS: "finite S"
```
```   282   shows "F (\<lambda>k. if a = k then b k else 1) S = (if a \<in> S then b a else 1)"
```
```   283   using delta [OF fS, of a b, symmetric] by (auto intro: cong)
```
```   284
```
```   285 lemma If_cases:
```
```   286   fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
```
```   287   assumes fA: "finite A"
```
```   288   shows "F (\<lambda>x. if P x then h x else g x) A =
```
```   289     F h (A \<inter> {x. P x}) * F g (A \<inter> - {x. P x})"
```
```   290 proof -
```
```   291   have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}"
```
```   292           "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}"
```
```   293     by blast+
```
```   294   from fA
```
```   295   have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
```
```   296   let ?g = "\<lambda>x. if P x then h x else g x"
```
```   297   from union_disjoint [OF f a(2), of ?g] a(1)
```
```   298   show ?thesis
```
```   299     by (subst (1 2) cong) simp_all
```
```   300 qed
```
```   301
```
```   302 lemma cartesian_product:
```
```   303    "F (\<lambda>x. F (g x) B) A = F (split g) (A <*> B)"
```
```   304 apply (rule sym)
```
```   305 apply (cases "finite A")
```
```   306  apply (cases "finite B")
```
```   307   apply (simp add: Sigma)
```
```   308  apply (cases "A={}", simp)
```
```   309  apply simp
```
```   310 apply (auto intro: infinite dest: finite_cartesian_productD2)
```
```   311 apply (cases "B = {}") apply (auto intro: infinite dest: finite_cartesian_productD1)
```
```   312 done
```
```   313
```
```   314 end
```
```   315
```
```   316 notation times (infixl "*" 70)
```
```   317 notation Groups.one ("1")
```
```   318
```
```   319
```
```   320 subsection {* Generalized summation over a set *}
```
```   321
```
```   322 definition (in comm_monoid_add) setsum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
```
```   323 where
```
```   324   "setsum = comm_monoid_set.F plus 0"
```
```   325
```
```   326 sublocale comm_monoid_add < setsum!: comm_monoid_set plus 0
```
```   327 where
```
```   328   "comm_monoid_set.F plus 0 = setsum"
```
```   329 proof -
```
```   330   show "comm_monoid_set plus 0" ..
```
```   331   then interpret setsum!: comm_monoid_set plus 0 .
```
```   332   from setsum_def show "comm_monoid_set.F plus 0 = setsum" by rule
```
```   333 qed
```
```   334
```
```   335 abbreviation
```
```   336   Setsum ("\<Sum>_" [1000] 999) where
```
```   337   "\<Sum>A \<equiv> setsum (%x. x) A"
```
```   338
```
```   339 text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
```
```   340 written @{text"\<Sum>x\<in>A. e"}. *}
```
```   341
```
```   342 syntax
```
```   343   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
```
```   344 syntax (xsymbols)
```
```   345   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```   346 syntax (HTML output)
```
```   347   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```   348
```
```   349 translations -- {* Beware of argument permutation! *}
```
```   350   "SUM i:A. b" == "CONST setsum (%i. b) A"
```
```   351   "\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A"
```
```   352
```
```   353 text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
```
```   354  @{text"\<Sum>x|P. e"}. *}
```
```   355
```
```   356 syntax
```
```   357   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
```
```   358 syntax (xsymbols)
```
```   359   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   360 syntax (HTML output)
```
```   361   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   362
```
```   363 translations
```
```   364   "SUM x|P. t" => "CONST setsum (%x. t) {x. P}"
```
```   365   "\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}"
```
```   366
```
```   367 print_translation {*
```
```   368 let
```
```   369   fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) \$ Abs (y, Ty, P)] =
```
```   370         if x <> y then raise Match
```
```   371         else
```
```   372           let
```
```   373             val x' = Syntax_Trans.mark_bound_body (x, Tx);
```
```   374             val t' = subst_bound (x', t);
```
```   375             val P' = subst_bound (x', P);
```
```   376           in
```
```   377             Syntax.const @{syntax_const "_qsetsum"} \$ Syntax_Trans.mark_bound_abs (x, Tx) \$ P' \$ t'
```
```   378           end
```
```   379     | setsum_tr' _ = raise Match;
```
```   380 in [(@{const_syntax setsum}, setsum_tr')] end
```
```   381 *}
```
```   382
```
```   383 text {* TODO These are candidates for generalization *}
```
```   384
```
```   385 context comm_monoid_add
```
```   386 begin
```
```   387
```
```   388 lemma setsum_reindex_id:
```
```   389   "inj_on f B ==> setsum f B = setsum id (f ` B)"
```
```   390   by (simp add: setsum.reindex)
```
```   391
```
```   392 lemma setsum_reindex_nonzero:
```
```   393   assumes fS: "finite S"
```
```   394   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"
```
```   395   shows "setsum h (f ` S) = setsum (h \<circ> f) S"
```
```   396 using nz proof (induct rule: finite_induct [OF fS])
```
```   397   case 1 thus ?case by simp
```
```   398 next
```
```   399   case (2 x F)
```
```   400   { assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
```
```   401     then obtain y where y: "y \<in> F" "f x = f y" by auto
```
```   402     from "2.hyps" y have xy: "x \<noteq> y" by auto
```
```   403     from "2.prems" [of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
```
```   404     have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
```
```   405     also have "\<dots> = setsum (h o f) (insert x F)"
```
```   406       unfolding setsum.insert[OF `finite F` `x\<notin>F`]
```
```   407       using h0
```
```   408       apply (simp cong del: setsum.strong_cong)
```
```   409       apply (rule "2.hyps"(3))
```
```   410       apply (rule_tac y="y" in  "2.prems")
```
```   411       apply simp_all
```
```   412       done
```
```   413     finally have ?case . }
```
```   414   moreover
```
```   415   { assume fxF: "f x \<notin> f ` F"
```
```   416     have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)"
```
```   417       using fxF "2.hyps" by simp
```
```   418     also have "\<dots> = setsum (h o f) (insert x F)"
```
```   419       unfolding setsum.insert[OF `finite F` `x\<notin>F`]
```
```   420       apply (simp cong del: setsum.strong_cong)
```
```   421       apply (rule cong [OF refl [of "op + (h (f x))"]])
```
```   422       apply (rule "2.hyps"(3))
```
```   423       apply (rule_tac y="y" in  "2.prems")
```
```   424       apply simp_all
```
```   425       done
```
```   426     finally have ?case . }
```
```   427   ultimately show ?case by blast
```
```   428 qed
```
```   429
```
```   430 lemma setsum_cong2:
```
```   431   "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> setsum f A = setsum g A"
```
```   432   by (auto intro: setsum.cong)
```
```   433
```
```   434 lemma setsum_reindex_cong:
```
```   435    "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|]
```
```   436     ==> setsum h B = setsum g A"
```
```   437   by (simp add: setsum.reindex)
```
```   438
```
```   439 lemma setsum_restrict_set:
```
```   440   assumes fA: "finite A"
```
```   441   shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"
```
```   442 proof-
```
```   443   from fA have fab: "finite (A \<inter> B)" by auto
```
```   444   have aba: "A \<inter> B \<subseteq> A" by blast
```
```   445   let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"
```
```   446   from setsum.mono_neutral_left [OF fA aba, of ?g]
```
```   447   show ?thesis by simp
```
```   448 qed
```
```   449
```
```   450 lemma setsum_Union_disjoint:
```
```   451   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"
```
```   452   shows "setsum f (Union C) = setsum (setsum f) C"
```
```   453   using assms by (fact setsum.Union_disjoint)
```
```   454
```
```   455 lemma setsum_cartesian_product:
```
```   456   "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
```
```   457   by (fact setsum.cartesian_product)
```
```   458
```
```   459 lemma setsum_UNION_zero:
```
```   460   assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"
```
```   461   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"
```
```   462   shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"
```
```   463   using fSS f0
```
```   464 proof(induct rule: finite_induct[OF fS])
```
```   465   case 1 thus ?case by simp
```
```   466 next
```
```   467   case (2 T F)
```
```   468   then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F"
```
```   469     and H: "setsum f (\<Union> F) = setsum (setsum f) F" by auto
```
```   470   from fTF have fUF: "finite (\<Union>F)" by auto
```
```   471   from "2.prems" TF fTF
```
```   472   show ?case
```
```   473     by (auto simp add: H [symmetric] intro: setsum.union_inter_neutral [OF fTF(1) fUF, of f])
```
```   474 qed
```
```   475
```
```   476 text {* Commuting outer and inner summation *}
```
```   477
```
```   478 lemma setsum_commute:
```
```   479   "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
```
```   480 proof (simp add: setsum_cartesian_product)
```
```   481   have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
```
```   482     (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
```
```   483     (is "?s = _")
```
```   484     apply (simp add: setsum.reindex [where h = "%(i, j). (j, i)"] swap_inj_on)
```
```   485     apply (simp add: split_def)
```
```   486     done
```
```   487   also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
```
```   488     (is "_ = ?t")
```
```   489     apply (simp add: swap_product)
```
```   490     done
```
```   491   finally show "?s = ?t" .
```
```   492 qed
```
```   493
```
```   494 lemma setsum_Plus:
```
```   495   fixes A :: "'a set" and B :: "'b set"
```
```   496   assumes fin: "finite A" "finite B"
```
```   497   shows "setsum f (A <+> B) = setsum (f \<circ> Inl) A + setsum (f \<circ> Inr) B"
```
```   498 proof -
```
```   499   have "A <+> B = Inl ` A \<union> Inr ` B" by auto
```
```   500   moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"
```
```   501     by auto
```
```   502   moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto
```
```   503   moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)
```
```   504   ultimately show ?thesis using fin by(simp add: setsum.union_disjoint setsum.reindex)
```
```   505 qed
```
```   506
```
```   507 end
```
```   508
```
```   509 text {* TODO These are legacy *}
```
```   510
```
```   511 lemma setsum_empty:
```
```   512   "setsum f {} = 0"
```
```   513   by (fact setsum.empty)
```
```   514
```
```   515 lemma setsum_insert:
```
```   516   "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
```
```   517   by (fact setsum.insert)
```
```   518
```
```   519 lemma setsum_infinite:
```
```   520   "~ finite A ==> setsum f A = 0"
```
```   521   by (fact setsum.infinite)
```
```   522
```
```   523 lemma setsum_reindex:
```
```   524   "inj_on f B \<Longrightarrow> setsum h (f ` B) = setsum (h \<circ> f) B"
```
```   525   by (fact setsum.reindex)
```
```   526
```
```   527 lemma setsum_cong:
```
```   528   "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
```
```   529   by (fact setsum.cong)
```
```   530
```
```   531 lemma strong_setsum_cong:
```
```   532   "A = B ==> (!!x. x:B =simp=> f x = g x)
```
```   533    ==> setsum (%x. f x) A = setsum (%x. g x) B"
```
```   534   by (fact setsum.strong_cong)
```
```   535
```
```   536 lemmas setsum_0 = setsum.neutral_const
```
```   537 lemmas setsum_0' = setsum.neutral
```
```   538
```
```   539 lemma setsum_Un_Int: "finite A ==> finite B ==>
```
```   540   setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
```
```   541   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
```
```   542   by (fact setsum.union_inter)
```
```   543
```
```   544 lemma setsum_Un_disjoint: "finite A ==> finite B
```
```   545   ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
```
```   546   by (fact setsum.union_disjoint)
```
```   547
```
```   548 lemma setsum_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
```
```   549     setsum f A = setsum f (A - B) + setsum f B"
```
```   550   by (fact setsum.subset_diff)
```
```   551
```
```   552 lemma setsum_mono_zero_left:
```
```   553   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 0 \<rbrakk> \<Longrightarrow> setsum f S = setsum f T"
```
```   554   by (fact setsum.mono_neutral_left)
```
```   555
```
```   556 lemmas setsum_mono_zero_right = setsum.mono_neutral_right
```
```   557
```
```   558 lemma setsum_mono_zero_cong_left:
```
```   559   "\<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>
```
```   560   \<Longrightarrow> setsum f S = setsum g T"
```
```   561   by (fact setsum.mono_neutral_cong_left)
```
```   562
```
```   563 lemmas setsum_mono_zero_cong_right = setsum.mono_neutral_cong_right
```
```   564
```
```   565 lemma setsum_delta: "finite S \<Longrightarrow>
```
```   566   setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
```
```   567   by (fact setsum.delta)
```
```   568
```
```   569 lemma setsum_delta': "finite S \<Longrightarrow>
```
```   570   setsum (\<lambda>k. if a = k then b k else 0) S = (if a\<in> S then b a else 0)"
```
```   571   by (fact setsum.delta')
```
```   572
```
```   573 lemma setsum_cases:
```
```   574   assumes "finite A"
```
```   575   shows "setsum (\<lambda>x. if P x then f x else g x) A =
```
```   576          setsum f (A \<inter> {x. P x}) + setsum g (A \<inter> - {x. P x})"
```
```   577   using assms by (fact setsum.If_cases)
```
```   578
```
```   579 (*But we can't get rid of finite I. If infinite, although the rhs is 0,
```
```   580   the lhs need not be, since UNION I A could still be finite.*)
```
```   581 lemma setsum_UN_disjoint:
```
```   582   assumes "finite I" and "ALL i:I. finite (A i)"
```
```   583     and "ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}"
```
```   584   shows "setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
```
```   585   using assms by (fact setsum.UNION_disjoint)
```
```   586
```
```   587 (*But we can't get rid of finite A. If infinite, although the lhs is 0,
```
```   588   the rhs need not be, since SIGMA A B could still be finite.*)
```
```   589 lemma setsum_Sigma:
```
```   590   assumes "finite A" and  "ALL x:A. finite (B x)"
```
```   591   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)"
```
```   592   using assms by (fact setsum.Sigma)
```
```   593
```
```   594 lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
```
```   595   by (fact setsum.distrib)
```
```   596
```
```   597 lemma setsum_Un_zero:
```
```   598   "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 0 \<rbrakk> \<Longrightarrow>
```
```   599   setsum f (S \<union> T) = setsum f S + setsum f T"
```
```   600   by (fact setsum.union_inter_neutral)
```
```   601
```
```   602 lemma setsum_eq_general_reverses:
```
```   603   assumes fS: "finite S" and fT: "finite T"
```
```   604   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
```
```   605   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"
```
```   606   shows "setsum f S = setsum g T"
```
```   607   using kh hk by (fact setsum.eq_general_reverses)
```
```   608
```
```   609
```
```   610 subsubsection {* Properties in more restricted classes of structures *}
```
```   611
```
```   612 lemma setsum_Un: "finite A ==> finite B ==>
```
```   613   (setsum f (A Un B) :: 'a :: ab_group_add) =
```
```   614    setsum f A + setsum f B - setsum f (A Int B)"
```
```   615 by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
```
```   616
```
```   617 lemma setsum_Un2:
```
```   618   assumes "finite (A \<union> B)"
```
```   619   shows "setsum f (A \<union> B) = setsum f (A - B) + setsum f (B - A) + setsum f (A \<inter> B)"
```
```   620 proof -
```
```   621   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
```
```   622     by auto
```
```   623   with assms show ?thesis by simp (subst setsum_Un_disjoint, auto)+
```
```   624 qed
```
```   625
```
```   626 lemma setsum_diff1: "finite A \<Longrightarrow>
```
```   627   (setsum f (A - {a}) :: ('a::ab_group_add)) =
```
```   628   (if a:A then setsum f A - f a else setsum f A)"
```
```   629 by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```   630
```
```   631 lemma setsum_diff:
```
```   632   assumes le: "finite A" "B \<subseteq> A"
```
```   633   shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
```
```   634 proof -
```
```   635   from le have finiteB: "finite B" using finite_subset by auto
```
```   636   show ?thesis using finiteB le
```
```   637   proof induct
```
```   638     case empty
```
```   639     thus ?case by auto
```
```   640   next
```
```   641     case (insert x F)
```
```   642     thus ?case using le finiteB
```
```   643       by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
```
```   644   qed
```
```   645 qed
```
```   646
```
```   647 lemma setsum_mono:
```
```   648   assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))"
```
```   649   shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
```
```   650 proof (cases "finite K")
```
```   651   case True
```
```   652   thus ?thesis using le
```
```   653   proof induct
```
```   654     case empty
```
```   655     thus ?case by simp
```
```   656   next
```
```   657     case insert
```
```   658     thus ?case using add_mono by fastforce
```
```   659   qed
```
```   660 next
```
```   661   case False then show ?thesis by simp
```
```   662 qed
```
```   663
```
```   664 lemma setsum_strict_mono:
```
```   665   fixes f :: "'a \<Rightarrow> 'b::{ordered_cancel_ab_semigroup_add,comm_monoid_add}"
```
```   666   assumes "finite A"  "A \<noteq> {}"
```
```   667     and "!!x. x:A \<Longrightarrow> f x < g x"
```
```   668   shows "setsum f A < setsum g A"
```
```   669   using assms
```
```   670 proof (induct rule: finite_ne_induct)
```
```   671   case singleton thus ?case by simp
```
```   672 next
```
```   673   case insert thus ?case by (auto simp: add_strict_mono)
```
```   674 qed
```
```   675
```
```   676 lemma setsum_strict_mono_ex1:
```
```   677 fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_cancel_ab_semigroup_add}"
```
```   678 assumes "finite A" and "ALL x:A. f x \<le> g x" and "EX a:A. f a < g a"
```
```   679 shows "setsum f A < setsum g A"
```
```   680 proof-
```
```   681   from assms(3) obtain a where a: "a:A" "f a < g a" by blast
```
```   682   have "setsum f A = setsum f ((A-{a}) \<union> {a})"
```
```   683     by(simp add:insert_absorb[OF `a:A`])
```
```   684   also have "\<dots> = setsum f (A-{a}) + setsum f {a}"
```
```   685     using `finite A` by(subst setsum_Un_disjoint) auto
```
```   686   also have "setsum f (A-{a}) \<le> setsum g (A-{a})"
```
```   687     by(rule setsum_mono)(simp add: assms(2))
```
```   688   also have "setsum f {a} < setsum g {a}" using a by simp
```
```   689   also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"
```
```   690     using `finite A` by(subst setsum_Un_disjoint[symmetric]) auto
```
```   691   also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF `a:A`])
```
```   692   finally show ?thesis by (metis add_right_mono add_strict_left_mono)
```
```   693 qed
```
```   694
```
```   695 lemma setsum_negf:
```
```   696   "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
```
```   697 proof (cases "finite A")
```
```   698   case True thus ?thesis by (induct set: finite) auto
```
```   699 next
```
```   700   case False thus ?thesis by simp
```
```   701 qed
```
```   702
```
```   703 lemma setsum_subtractf:
```
```   704   "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
```
```   705     setsum f A - setsum g A"
```
```   706 proof (cases "finite A")
```
```   707   case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
```
```   708 next
```
```   709   case False thus ?thesis by simp
```
```   710 qed
```
```   711
```
```   712 lemma setsum_nonneg:
```
```   713   assumes nn: "\<forall>x\<in>A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
```
```   714   shows "0 \<le> setsum f A"
```
```   715 proof (cases "finite A")
```
```   716   case True thus ?thesis using nn
```
```   717   proof induct
```
```   718     case empty then show ?case by simp
```
```   719   next
```
```   720     case (insert x F)
```
```   721     then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
```
```   722     with insert show ?case by simp
```
```   723   qed
```
```   724 next
```
```   725   case False thus ?thesis by simp
```
```   726 qed
```
```   727
```
```   728 lemma setsum_nonpos:
```
```   729   assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})"
```
```   730   shows "setsum f A \<le> 0"
```
```   731 proof (cases "finite A")
```
```   732   case True thus ?thesis using np
```
```   733   proof induct
```
```   734     case empty then show ?case by simp
```
```   735   next
```
```   736     case (insert x F)
```
```   737     then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
```
```   738     with insert show ?case by simp
```
```   739   qed
```
```   740 next
```
```   741   case False thus ?thesis by simp
```
```   742 qed
```
```   743
```
```   744 lemma setsum_nonneg_leq_bound:
```
```   745   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
```
```   746   assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"
```
```   747   shows "f i \<le> B"
```
```   748 proof -
```
```   749   have "0 \<le> (\<Sum> i \<in> s - {i}. f i)" and "0 \<le> f i"
```
```   750     using assms by (auto intro!: setsum_nonneg)
```
```   751   moreover
```
```   752   have "(\<Sum> i \<in> s - {i}. f i) + f i = B"
```
```   753     using assms by (simp add: setsum_diff1)
```
```   754   ultimately show ?thesis by auto
```
```   755 qed
```
```   756
```
```   757 lemma setsum_nonneg_0:
```
```   758   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
```
```   759   assumes "finite s" and pos: "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"
```
```   760   and "(\<Sum> i \<in> s. f i) = 0" and i: "i \<in> s"
```
```   761   shows "f i = 0"
```
```   762   using setsum_nonneg_leq_bound[OF assms] pos[OF i] by auto
```
```   763
```
```   764 lemma setsum_mono2:
```
```   765 fixes f :: "'a \<Rightarrow> 'b :: ordered_comm_monoid_add"
```
```   766 assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
```
```   767 shows "setsum f A \<le> setsum f B"
```
```   768 proof -
```
```   769   have "setsum f A \<le> setsum f A + setsum f (B-A)"
```
```   770     by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
```
```   771   also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
```
```   772     by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
```
```   773   also have "A \<union> (B-A) = B" using sub by blast
```
```   774   finally show ?thesis .
```
```   775 qed
```
```   776
```
```   777 lemma setsum_mono3: "finite B ==> A <= B ==>
```
```   778     ALL x: B - A.
```
```   779       0 <= ((f x)::'a::{comm_monoid_add,ordered_ab_semigroup_add}) ==>
```
```   780         setsum f A <= setsum f B"
```
```   781   apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
```
```   782   apply (erule ssubst)
```
```   783   apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
```
```   784   apply simp
```
```   785   apply (rule add_left_mono)
```
```   786   apply (erule setsum_nonneg)
```
```   787   apply (subst setsum_Un_disjoint [THEN sym])
```
```   788   apply (erule finite_subset, assumption)
```
```   789   apply (rule finite_subset)
```
```   790   prefer 2
```
```   791   apply assumption
```
```   792   apply (auto simp add: sup_absorb2)
```
```   793 done
```
```   794
```
```   795 lemma setsum_right_distrib:
```
```   796   fixes f :: "'a => ('b::semiring_0)"
```
```   797   shows "r * setsum f A = setsum (%n. r * f n) A"
```
```   798 proof (cases "finite A")
```
```   799   case True
```
```   800   thus ?thesis
```
```   801   proof induct
```
```   802     case empty thus ?case by simp
```
```   803   next
```
```   804     case (insert x A) thus ?case by (simp add: distrib_left)
```
```   805   qed
```
```   806 next
```
```   807   case False thus ?thesis by simp
```
```   808 qed
```
```   809
```
```   810 lemma setsum_left_distrib:
```
```   811   "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
```
```   812 proof (cases "finite A")
```
```   813   case True
```
```   814   then show ?thesis
```
```   815   proof induct
```
```   816     case empty thus ?case by simp
```
```   817   next
```
```   818     case (insert x A) thus ?case by (simp add: distrib_right)
```
```   819   qed
```
```   820 next
```
```   821   case False thus ?thesis by simp
```
```   822 qed
```
```   823
```
```   824 lemma setsum_divide_distrib:
```
```   825   "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
```
```   826 proof (cases "finite A")
```
```   827   case True
```
```   828   then show ?thesis
```
```   829   proof induct
```
```   830     case empty thus ?case by simp
```
```   831   next
```
```   832     case (insert x A) thus ?case by (simp add: add_divide_distrib)
```
```   833   qed
```
```   834 next
```
```   835   case False thus ?thesis by simp
```
```   836 qed
```
```   837
```
```   838 lemma setsum_abs[iff]:
```
```   839   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   840   shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
```
```   841 proof (cases "finite A")
```
```   842   case True
```
```   843   thus ?thesis
```
```   844   proof induct
```
```   845     case empty thus ?case by simp
```
```   846   next
```
```   847     case (insert x A)
```
```   848     thus ?case by (auto intro: abs_triangle_ineq order_trans)
```
```   849   qed
```
```   850 next
```
```   851   case False thus ?thesis by simp
```
```   852 qed
```
```   853
```
```   854 lemma setsum_abs_ge_zero[iff]:
```
```   855   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   856   shows "0 \<le> setsum (%i. abs(f i)) A"
```
```   857 proof (cases "finite A")
```
```   858   case True
```
```   859   thus ?thesis
```
```   860   proof induct
```
```   861     case empty thus ?case by simp
```
```   862   next
```
```   863     case (insert x A) thus ?case by auto
```
```   864   qed
```
```   865 next
```
```   866   case False thus ?thesis by simp
```
```   867 qed
```
```   868
```
```   869 lemma abs_setsum_abs[simp]:
```
```   870   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   871   shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
```
```   872 proof (cases "finite A")
```
```   873   case True
```
```   874   thus ?thesis
```
```   875   proof induct
```
```   876     case empty thus ?case by simp
```
```   877   next
```
```   878     case (insert a A)
```
```   879     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
```
```   880     also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
```
```   881     also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
```
```   882       by (simp del: abs_of_nonneg)
```
```   883     also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
```
```   884     finally show ?case .
```
```   885   qed
```
```   886 next
```
```   887   case False thus ?thesis by simp
```
```   888 qed
```
```   889
```
```   890 lemma setsum_diff1'[rule_format]:
```
```   891   "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
```
```   892 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))"])
```
```   893 apply (auto simp add: insert_Diff_if add_ac)
```
```   894 done
```
```   895
```
```   896 lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
```
```   897   shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
```
```   898 unfolding setsum_diff1'[OF assms] by auto
```
```   899
```
```   900 lemma setsum_product:
```
```   901   fixes f :: "'a => ('b::semiring_0)"
```
```   902   shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
```
```   903   by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)
```
```   904
```
```   905 lemma setsum_mult_setsum_if_inj:
```
```   906 fixes f :: "'a => ('b::semiring_0)"
```
```   907 shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
```
```   908   setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
```
```   909 by(auto simp: setsum_product setsum_cartesian_product
```
```   910         intro!:  setsum_reindex_cong[symmetric])
```
```   911
```
```   912 lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
```
```   913 apply (case_tac "finite A")
```
```   914  prefer 2 apply simp
```
```   915 apply (erule rev_mp)
```
```   916 apply (erule finite_induct, auto)
```
```   917 done
```
```   918
```
```   919 lemma setsum_eq_0_iff [simp]:
```
```   920   "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
```
```   921   by (induct set: finite) auto
```
```   922
```
```   923 lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
```
```   924   setsum f A = Suc 0 \<longleftrightarrow> (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
```
```   925 apply(erule finite_induct)
```
```   926 apply (auto simp add:add_is_1)
```
```   927 done
```
```   928
```
```   929 lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
```
```   930
```
```   931 lemma setsum_Un_nat: "finite A ==> finite B ==>
```
```   932   (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
```
```   933   -- {* For the natural numbers, we have subtraction. *}
```
```   934 by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
```
```   935
```
```   936 lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
```
```   937   (if a:A then setsum f A - f a else setsum f A)"
```
```   938 apply (case_tac "finite A")
```
```   939  prefer 2 apply simp
```
```   940 apply (erule finite_induct)
```
```   941  apply (auto simp add: insert_Diff_if)
```
```   942 apply (drule_tac a = a in mk_disjoint_insert, auto)
```
```   943 done
```
```   944
```
```   945 lemma setsum_diff_nat:
```
```   946 assumes "finite B" and "B \<subseteq> A"
```
```   947 shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
```
```   948 using assms
```
```   949 proof induct
```
```   950   show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
```
```   951 next
```
```   952   fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
```
```   953     and xFinA: "insert x F \<subseteq> A"
```
```   954     and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
```
```   955   from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
```
```   956   from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
```
```   957     by (simp add: setsum_diff1_nat)
```
```   958   from xFinA have "F \<subseteq> A" by simp
```
```   959   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
```
```   960   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
```
```   961     by simp
```
```   962   from xnotinF have "A - insert x F = (A - F) - {x}" by auto
```
```   963   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
```
```   964     by simp
```
```   965   from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
```
```   966   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
```
```   967     by simp
```
```   968   thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
```
```   969 qed
```
```   970
```
```   971
```
```   972 subsubsection {* Cardinality as special case of @{const setsum} *}
```
```   973
```
```   974 lemma card_eq_setsum:
```
```   975   "card A = setsum (\<lambda>x. 1) A"
```
```   976 proof -
```
```   977   have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)"
```
```   978     by (simp add: fun_eq_iff)
```
```   979   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) = Finite_Set.fold (\<lambda>_. Suc)"
```
```   980     by (rule arg_cong)
```
```   981   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
```
```   982     by (blast intro: fun_cong)
```
```   983   then show ?thesis by (simp add: card.eq_fold setsum.eq_fold)
```
```   984 qed
```
```   985
```
```   986 lemma setsum_constant [simp]:
```
```   987   "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
```
```   988 apply (cases "finite A")
```
```   989 apply (erule finite_induct)
```
```   990 apply (auto simp add: algebra_simps)
```
```   991 done
```
```   992
```
```   993 lemma setsum_bounded:
```
```   994   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"
```
```   995   shows "setsum f A \<le> of_nat (card A) * K"
```
```   996 proof (cases "finite A")
```
```   997   case True
```
```   998   thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
```
```   999 next
```
```  1000   case False thus ?thesis by simp
```
```  1001 qed
```
```  1002
```
```  1003 lemma card_UN_disjoint:
```
```  1004   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
```
```  1005     and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
```
```  1006   shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
```
```  1007 proof -
```
```  1008   have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)" by simp
```
```  1009   with assms show ?thesis by (simp add: card_eq_setsum setsum_UN_disjoint del: setsum_constant)
```
```  1010 qed
```
```  1011
```
```  1012 lemma card_Union_disjoint:
```
```  1013   "finite C ==> (ALL A:C. finite A) ==>
```
```  1014    (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {})
```
```  1015    ==> card (Union C) = setsum card C"
```
```  1016 apply (frule card_UN_disjoint [of C id])
```
```  1017 apply (simp_all add: SUP_def id_def)
```
```  1018 done
```
```  1019
```
```  1020
```
```  1021 subsubsection {* Cardinality of products *}
```
```  1022
```
```  1023 lemma card_SigmaI [simp]:
```
```  1024   "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
```
```  1025   \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
```
```  1026 by(simp add: card_eq_setsum setsum_Sigma del:setsum_constant)
```
```  1027
```
```  1028 (*
```
```  1029 lemma SigmaI_insert: "y \<notin> A ==>
```
```  1030   (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
```
```  1031   by auto
```
```  1032 *)
```
```  1033
```
```  1034 lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
```
```  1035   by (cases "finite A \<and> finite B")
```
```  1036     (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1037
```
```  1038 lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
```
```  1039 by (simp add: card_cartesian_product)
```
```  1040
```
```  1041
```
```  1042 subsection {* Generalized product over a set *}
```
```  1043
```
```  1044 definition (in comm_monoid_mult) setprod :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
```
```  1045 where
```
```  1046   "setprod = comm_monoid_set.F times 1"
```
```  1047
```
```  1048 sublocale comm_monoid_mult < setprod!: comm_monoid_set times 1
```
```  1049 where
```
```  1050   "comm_monoid_set.F times 1 = setprod"
```
```  1051 proof -
```
```  1052   show "comm_monoid_set times 1" ..
```
```  1053   then interpret setprod!: comm_monoid_set times 1 .
```
```  1054   from setprod_def show "comm_monoid_set.F times 1 = setprod" by rule
```
```  1055 qed
```
```  1056
```
```  1057 abbreviation
```
```  1058   Setprod ("\<Prod>_" [1000] 999) where
```
```  1059   "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"
```
```  1060
```
```  1061 syntax
```
```  1062   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
```
```  1063 syntax (xsymbols)
```
```  1064   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```  1065 syntax (HTML output)
```
```  1066   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```  1067
```
```  1068 translations -- {* Beware of argument permutation! *}
```
```  1069   "PROD i:A. b" == "CONST setprod (%i. b) A"
```
```  1070   "\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A"
```
```  1071
```
```  1072 text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
```
```  1073  @{text"\<Prod>x|P. e"}. *}
```
```  1074
```
```  1075 syntax
```
```  1076   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
```
```  1077 syntax (xsymbols)
```
```  1078   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```  1079 syntax (HTML output)
```
```  1080   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```  1081
```
```  1082 translations
```
```  1083   "PROD x|P. t" => "CONST setprod (%x. t) {x. P}"
```
```  1084   "\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"
```
```  1085
```
```  1086 text {* TODO These are candidates for generalization *}
```
```  1087
```
```  1088 context comm_monoid_mult
```
```  1089 begin
```
```  1090
```
```  1091 lemma setprod_reindex_id:
```
```  1092   "inj_on f B ==> setprod f B = setprod id (f ` B)"
```
```  1093   by (auto simp add: setprod.reindex)
```
```  1094
```
```  1095 lemma setprod_reindex_cong:
```
```  1096   "inj_on f A ==> B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
```
```  1097   by (frule setprod.reindex, simp)
```
```  1098
```
```  1099 lemma strong_setprod_reindex_cong:
```
```  1100   assumes i: "inj_on f A"
```
```  1101   and B: "B = f ` A" and eq: "\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x"
```
```  1102   shows "setprod h B = setprod g A"
```
```  1103 proof-
```
```  1104   have "setprod h B = setprod (h o f) A"
```
```  1105     by (simp add: B setprod.reindex [OF i, of h])
```
```  1106   then show ?thesis apply simp
```
```  1107     apply (rule setprod.cong)
```
```  1108     apply simp
```
```  1109     by (simp add: eq)
```
```  1110 qed
```
```  1111
```
```  1112 lemma setprod_Union_disjoint:
```
```  1113   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"
```
```  1114   shows "setprod f (Union C) = setprod (setprod f) C"
```
```  1115   using assms by (fact setprod.Union_disjoint)
```
```  1116
```
```  1117 text{*Here we can eliminate the finiteness assumptions, by cases.*}
```
```  1118 lemma setprod_cartesian_product:
```
```  1119   "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
```
```  1120   by (fact setprod.cartesian_product)
```
```  1121
```
```  1122 lemma setprod_Un2:
```
```  1123   assumes "finite (A \<union> B)"
```
```  1124   shows "setprod f (A \<union> B) = setprod f (A - B) * setprod f (B - A) * setprod f (A \<inter> B)"
```
```  1125 proof -
```
```  1126   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
```
```  1127     by auto
```
```  1128   with assms show ?thesis by simp (subst setprod.union_disjoint, auto)+
```
```  1129 qed
```
```  1130
```
```  1131 end
```
```  1132
```
```  1133 text {* TODO These are legacy *}
```
```  1134
```
```  1135 lemma setprod_empty: "setprod f {} = 1"
```
```  1136   by (fact setprod.empty)
```
```  1137
```
```  1138 lemma setprod_insert: "[| finite A; a \<notin> A |] ==>
```
```  1139     setprod f (insert a A) = f a * setprod f A"
```
```  1140   by (fact setprod.insert)
```
```  1141
```
```  1142 lemma setprod_infinite: "~ finite A ==> setprod f A = 1"
```
```  1143   by (fact setprod.infinite)
```
```  1144
```
```  1145 lemma setprod_reindex:
```
```  1146   "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
```
```  1147   by (fact setprod.reindex)
```
```  1148
```
```  1149 lemma setprod_cong:
```
```  1150   "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
```
```  1151   by (fact setprod.cong)
```
```  1152
```
```  1153 lemma strong_setprod_cong:
```
```  1154   "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
```
```  1155   by (fact setprod.strong_cong)
```
```  1156
```
```  1157 lemma setprod_Un_one:
```
```  1158   "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 1 \<rbrakk>
```
```  1159   \<Longrightarrow> setprod f (S \<union> T) = setprod f S  * setprod f T"
```
```  1160   by (fact setprod.union_inter_neutral)
```
```  1161
```
```  1162 lemmas setprod_1 = setprod.neutral_const
```
```  1163 lemmas setprod_1' = setprod.neutral
```
```  1164
```
```  1165 lemma setprod_Un_Int: "finite A ==> finite B
```
```  1166     ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
```
```  1167   by (fact setprod.union_inter)
```
```  1168
```
```  1169 lemma setprod_Un_disjoint: "finite A ==> finite B
```
```  1170   ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
```
```  1171   by (fact setprod.union_disjoint)
```
```  1172
```
```  1173 lemma setprod_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
```
```  1174     setprod f A = setprod f (A - B) * setprod f B"
```
```  1175   by (fact setprod.subset_diff)
```
```  1176
```
```  1177 lemma setprod_mono_one_left:
```
```  1178   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 1 \<rbrakk> \<Longrightarrow> setprod f S = setprod f T"
```
```  1179   by (fact setprod.mono_neutral_left)
```
```  1180
```
```  1181 lemmas setprod_mono_one_right = setprod.mono_neutral_right
```
```  1182
```
```  1183 lemma setprod_mono_one_cong_left:
```
```  1184   "\<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>
```
```  1185   \<Longrightarrow> setprod f S = setprod g T"
```
```  1186   by (fact setprod.mono_neutral_cong_left)
```
```  1187
```
```  1188 lemmas setprod_mono_one_cong_right = setprod.mono_neutral_cong_right
```
```  1189
```
```  1190 lemma setprod_delta: "finite S \<Longrightarrow>
```
```  1191   setprod (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"
```
```  1192   by (fact setprod.delta)
```
```  1193
```
```  1194 lemma setprod_delta': "finite S \<Longrightarrow>
```
```  1195   setprod (\<lambda>k. if a = k then b k else 1) S = (if a\<in> S then b a else 1)"
```
```  1196   by (fact setprod.delta')
```
```  1197
```
```  1198 lemma setprod_UN_disjoint:
```
```  1199     "finite I ==> (ALL i:I. finite (A i)) ==>
```
```  1200         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
```
```  1201       setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
```
```  1202   by (fact setprod.UNION_disjoint)
```
```  1203
```
```  1204 lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```  1205     (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
```
```  1206     (\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
```
```  1207   by (fact setprod.Sigma)
```
```  1208
```
```  1209 lemma setprod_timesf: "setprod (\<lambda>x. f x * g x) A = setprod f A * setprod g A"
```
```  1210   by (fact setprod.distrib)
```
```  1211
```
```  1212
```
```  1213 subsubsection {* Properties in more restricted classes of structures *}
```
```  1214
```
```  1215 lemma setprod_zero:
```
```  1216      "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0"
```
```  1217 apply (induct set: finite, force, clarsimp)
```
```  1218 apply (erule disjE, auto)
```
```  1219 done
```
```  1220
```
```  1221 lemma setprod_zero_iff[simp]: "finite A ==>
```
```  1222   (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) =
```
```  1223   (EX x: A. f x = 0)"
```
```  1224 by (erule finite_induct, auto simp:no_zero_divisors)
```
```  1225
```
```  1226 lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
```
```  1227   (setprod f (A Un B) :: 'a ::{field})
```
```  1228    = setprod f A * setprod f B / setprod f (A Int B)"
```
```  1229 by (subst setprod_Un_Int [symmetric], auto)
```
```  1230
```
```  1231 lemma setprod_nonneg [rule_format]:
```
```  1232    "(ALL x: A. (0::'a::linordered_semidom) \<le> f x) --> 0 \<le> setprod f A"
```
```  1233 by (cases "finite A", induct set: finite, simp_all add: mult_nonneg_nonneg)
```
```  1234
```
```  1235 lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::linordered_semidom) < f x)
```
```  1236   --> 0 < setprod f A"
```
```  1237 by (cases "finite A", induct set: finite, simp_all add: mult_pos_pos)
```
```  1238
```
```  1239 lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
```
```  1240   (setprod f (A - {a}) :: 'a :: {field}) =
```
```  1241   (if a:A then setprod f A / f a else setprod f A)"
```
```  1242   by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```  1243
```
```  1244 lemma setprod_inversef:
```
```  1245   fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
```
```  1246   shows "finite A ==> setprod (inverse \<circ> f) A = inverse (setprod f A)"
```
```  1247 by (erule finite_induct) auto
```
```  1248
```
```  1249 lemma setprod_dividef:
```
```  1250   fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
```
```  1251   shows "finite A
```
```  1252     ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
```
```  1253 apply (subgoal_tac
```
```  1254          "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
```
```  1255 apply (erule ssubst)
```
```  1256 apply (subst divide_inverse)
```
```  1257 apply (subst setprod_timesf)
```
```  1258 apply (subst setprod_inversef, assumption+, rule refl)
```
```  1259 apply (rule setprod_cong, rule refl)
```
```  1260 apply (subst divide_inverse, auto)
```
```  1261 done
```
```  1262
```
```  1263 lemma setprod_dvd_setprod [rule_format]:
```
```  1264     "(ALL x : A. f x dvd g x) \<longrightarrow> setprod f A dvd setprod g A"
```
```  1265   apply (cases "finite A")
```
```  1266   apply (induct set: finite)
```
```  1267   apply (auto simp add: dvd_def)
```
```  1268   apply (rule_tac x = "k * ka" in exI)
```
```  1269   apply (simp add: algebra_simps)
```
```  1270 done
```
```  1271
```
```  1272 lemma setprod_dvd_setprod_subset:
```
```  1273   "finite B \<Longrightarrow> A <= B \<Longrightarrow> setprod f A dvd setprod f B"
```
```  1274   apply (subgoal_tac "setprod f B = setprod f A * setprod f (B - A)")
```
```  1275   apply (unfold dvd_def, blast)
```
```  1276   apply (subst setprod_Un_disjoint [symmetric])
```
```  1277   apply (auto elim: finite_subset intro: setprod_cong)
```
```  1278 done
```
```  1279
```
```  1280 lemma setprod_dvd_setprod_subset2:
```
```  1281   "finite B \<Longrightarrow> A <= B \<Longrightarrow> ALL x : A. (f x::'a::comm_semiring_1) dvd g x \<Longrightarrow>
```
```  1282       setprod f A dvd setprod g B"
```
```  1283   apply (rule dvd_trans)
```
```  1284   apply (rule setprod_dvd_setprod, erule (1) bspec)
```
```  1285   apply (erule (1) setprod_dvd_setprod_subset)
```
```  1286 done
```
```  1287
```
```  1288 lemma dvd_setprod: "finite A \<Longrightarrow> i:A \<Longrightarrow>
```
```  1289     (f i ::'a::comm_semiring_1) dvd setprod f A"
```
```  1290 by (induct set: finite) (auto intro: dvd_mult)
```
```  1291
```
```  1292 lemma dvd_setsum [rule_format]: "(ALL i : A. d dvd f i) \<longrightarrow>
```
```  1293     (d::'a::comm_semiring_1) dvd (SUM x : A. f x)"
```
```  1294   apply (cases "finite A")
```
```  1295   apply (induct set: finite)
```
```  1296   apply auto
```
```  1297 done
```
```  1298
```
```  1299 lemma setprod_mono:
```
```  1300   fixes f :: "'a \<Rightarrow> 'b\<Colon>linordered_semidom"
```
```  1301   assumes "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i"
```
```  1302   shows "setprod f A \<le> setprod g A"
```
```  1303 proof (cases "finite A")
```
```  1304   case True
```
```  1305   hence ?thesis "setprod f A \<ge> 0" using subset_refl[of A]
```
```  1306   proof (induct A rule: finite_subset_induct)
```
```  1307     case (insert a F)
```
```  1308     thus "setprod f (insert a F) \<le> setprod g (insert a F)" "0 \<le> setprod f (insert a F)"
```
```  1309       unfolding setprod_insert[OF insert(1,3)]
```
```  1310       using assms[rule_format,OF insert(2)] insert
```
```  1311       by (auto intro: mult_mono mult_nonneg_nonneg)
```
```  1312   qed auto
```
```  1313   thus ?thesis by simp
```
```  1314 qed auto
```
```  1315
```
```  1316 lemma abs_setprod:
```
```  1317   fixes f :: "'a \<Rightarrow> 'b\<Colon>{linordered_field,abs}"
```
```  1318   shows "abs (setprod f A) = setprod (\<lambda>x. abs (f x)) A"
```
```  1319 proof (cases "finite A")
```
```  1320   case True thus ?thesis
```
```  1321     by induct (auto simp add: field_simps abs_mult)
```
```  1322 qed auto
```
```  1323
```
```  1324 lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
```
```  1325 apply (erule finite_induct)
```
```  1326 apply auto
```
```  1327 done
```
```  1328
```
```  1329 lemma setprod_gen_delta:
```
```  1330   assumes fS: "finite S"
```
```  1331   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)"
```
```  1332 proof-
```
```  1333   let ?f = "(\<lambda>k. if k=a then b k else c)"
```
```  1334   {assume a: "a \<notin> S"
```
```  1335     hence "\<forall> k\<in> S. ?f k = c" by simp
```
```  1336     hence ?thesis  using a setprod_constant[OF fS, of c] by simp }
```
```  1337   moreover
```
```  1338   {assume a: "a \<in> S"
```
```  1339     let ?A = "S - {a}"
```
```  1340     let ?B = "{a}"
```
```  1341     have eq: "S = ?A \<union> ?B" using a by blast
```
```  1342     have dj: "?A \<inter> ?B = {}" by simp
```
```  1343     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```  1344     have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
```
```  1345       apply (rule setprod_cong) by auto
```
```  1346     have cA: "card ?A = card S - 1" using fS a by auto
```
```  1347     have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
```
```  1348     have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
```
```  1349       using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
```
```  1350       by simp
```
```  1351     then have ?thesis using a cA
```
```  1352       by (simp add: fA1 field_simps cong add: setprod_cong cong del: if_weak_cong)}
```
```  1353   ultimately show ?thesis by blast
```
```  1354 qed
```
```  1355
```
```  1356 lemma setprod_eq_1_iff [simp]:
```
```  1357   "finite F ==> setprod f F = 1 \<longleftrightarrow> (ALL a:F. f a = (1::nat))"
```
```  1358   by (induct set: finite) auto
```
```  1359
```
```  1360 lemma setprod_pos_nat:
```
```  1361   "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0"
```
```  1362 using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
```
```  1363
```
```  1364 lemma setprod_pos_nat_iff[simp]:
```
```  1365   "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))"
```
```  1366 using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
```
```  1367
```
```  1368
```
```  1369 subsection {* Generic lattice operations over a set *}
```
```  1370
```
```  1371 no_notation times (infixl "*" 70)
```
```  1372 no_notation Groups.one ("1")
```
```  1373
```
```  1374
```
```  1375 subsubsection {* Without neutral element *}
```
```  1376
```
```  1377 locale semilattice_set = semilattice
```
```  1378 begin
```
```  1379
```
```  1380 definition F :: "'a set \<Rightarrow> 'a"
```
```  1381 where
```
```  1382   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)"
```
```  1383
```
```  1384 lemma eq_fold:
```
```  1385   assumes "finite A"
```
```  1386   shows "F (insert x A) = Finite_Set.fold f x A"
```
```  1387 proof (rule sym)
```
```  1388   let ?f = "\<lambda>x y. Some (case y of None \<Rightarrow> x | Some z \<Rightarrow> f x z)"
```
```  1389   interpret comp_fun_idem f
```
```  1390     by default (simp_all add: fun_eq_iff left_commute)
```
```  1391   interpret comp_fun_idem "?f"
```
```  1392     by default (simp_all add: fun_eq_iff commute left_commute split: option.split)
```
```  1393   from assms show "Finite_Set.fold f x A = F (insert x A)"
```
```  1394   proof induct
```
```  1395     case empty then show ?case by (simp add: eq_fold')
```
```  1396   next
```
```  1397     case (insert y B) then show ?case by (simp add: insert_commute [of x] eq_fold')
```
```  1398   qed
```
```  1399 qed
```
```  1400
```
```  1401 lemma singleton [simp]:
```
```  1402   "F {x} = x"
```
```  1403   by (simp add: eq_fold)
```
```  1404
```
```  1405 lemma insert_not_elem:
```
```  1406   assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
```
```  1407   shows "F (insert x A) = x * F A"
```
```  1408 proof -
```
```  1409   interpret comp_fun_idem f
```
```  1410     by default (simp_all add: fun_eq_iff left_commute)
```
```  1411   from `A \<noteq> {}` obtain b where "b \<in> A" by blast
```
```  1412   then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
```
```  1413   with `finite A` and `x \<notin> A`
```
```  1414     have "finite (insert x B)" and "b \<notin> insert x B" by auto
```
```  1415   then have "F (insert b (insert x B)) = x * F (insert b B)"
```
```  1416     by (simp add: eq_fold)
```
```  1417   then show ?thesis by (simp add: * insert_commute)
```
```  1418 qed
```
```  1419
```
```  1420 lemma subsumption:
```
```  1421   assumes "finite A" and "x \<in> A"
```
```  1422   shows "x * F A = F A"
```
```  1423 proof -
```
```  1424   from assms have "A \<noteq> {}" by auto
```
```  1425   with `finite A` show ?thesis using `x \<in> A`
```
```  1426     by (induct A rule: finite_ne_induct) (auto simp add: ac_simps insert_not_elem)
```
```  1427 qed
```
```  1428
```
```  1429 lemma insert [simp]:
```
```  1430   assumes "finite A" and "A \<noteq> {}"
```
```  1431   shows "F (insert x A) = x * F A"
```
```  1432   using assms by (cases "x \<in> A") (simp_all add: insert_absorb subsumption insert_not_elem)
```
```  1433
```
```  1434 lemma union:
```
```  1435   assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
```
```  1436   shows "F (A \<union> B) = F A * F B"
```
```  1437   using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
```
```  1438
```
```  1439 lemma remove:
```
```  1440   assumes "finite A" and "x \<in> A"
```
```  1441   shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
```
```  1442 proof -
```
```  1443   from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
```
```  1444   with assms show ?thesis by simp
```
```  1445 qed
```
```  1446
```
```  1447 lemma insert_remove:
```
```  1448   assumes "finite A"
```
```  1449   shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
```
```  1450   using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
```
```  1451
```
```  1452 lemma subset:
```
```  1453   assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
```
```  1454   shows "F B * F A = F A"
```
```  1455 proof -
```
```  1456   from assms have "A \<noteq> {}" and "finite B" by (auto dest: finite_subset)
```
```  1457   with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
```
```  1458 qed
```
```  1459
```
```  1460 lemma closed:
```
```  1461   assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
```
```  1462   shows "F A \<in> A"
```
```  1463 using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
```
```  1464   case singleton then show ?case by simp
```
```  1465 next
```
```  1466   case insert with elem show ?case by force
```
```  1467 qed
```
```  1468
```
```  1469 lemma hom_commute:
```
```  1470   assumes hom: "\<And>x y. h (x * y) = h x * h y"
```
```  1471   and N: "finite N" "N \<noteq> {}"
```
```  1472   shows "h (F N) = F (h ` N)"
```
```  1473 using N proof (induct rule: finite_ne_induct)
```
```  1474   case singleton thus ?case by simp
```
```  1475 next
```
```  1476   case (insert n N)
```
```  1477   then have "h (F (insert n N)) = h (n * F N)" by simp
```
```  1478   also have "\<dots> = h n * h (F N)" by (rule hom)
```
```  1479   also have "h (F N) = F (h ` N)" by (rule insert)
```
```  1480   also have "h n * \<dots> = F (insert (h n) (h ` N))"
```
```  1481     using insert by simp
```
```  1482   also have "insert (h n) (h ` N) = h ` insert n N" by simp
```
```  1483   finally show ?case .
```
```  1484 qed
```
```  1485
```
```  1486 end
```
```  1487
```
```  1488 locale semilattice_order_set = semilattice_order + semilattice_set
```
```  1489 begin
```
```  1490
```
```  1491 lemma bounded_iff:
```
```  1492   assumes "finite A" and "A \<noteq> {}"
```
```  1493   shows "x \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> a)"
```
```  1494   using assms by (induct rule: finite_ne_induct) (simp_all add: bounded_iff)
```
```  1495
```
```  1496 lemma boundedI:
```
```  1497   assumes "finite A"
```
```  1498   assumes "A \<noteq> {}"
```
```  1499   assumes "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
```
```  1500   shows "x \<preceq> F A"
```
```  1501   using assms by (simp add: bounded_iff)
```
```  1502
```
```  1503 lemma boundedE:
```
```  1504   assumes "finite A" and "A \<noteq> {}" and "x \<preceq> F A"
```
```  1505   obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
```
```  1506   using assms by (simp add: bounded_iff)
```
```  1507
```
```  1508 lemma coboundedI:
```
```  1509   assumes "finite A"
```
```  1510     and "a \<in> A"
```
```  1511   shows "F A \<preceq> a"
```
```  1512 proof -
```
```  1513   from assms have "A \<noteq> {}" by auto
```
```  1514   from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
```
```  1515   proof (induct rule: finite_ne_induct)
```
```  1516     case singleton thus ?case by (simp add: refl)
```
```  1517   next
```
```  1518     case (insert x B)
```
```  1519     from insert have "a = x \<or> a \<in> B" by simp
```
```  1520     then show ?case using insert by (auto intro: coboundedI2)
```
```  1521   qed
```
```  1522 qed
```
```  1523
```
```  1524 lemma antimono:
```
```  1525   assumes "A \<subseteq> B" and "A \<noteq> {}" and "finite B"
```
```  1526   shows "F B \<preceq> F A"
```
```  1527 proof (cases "A = B")
```
```  1528   case True then show ?thesis by (simp add: refl)
```
```  1529 next
```
```  1530   case False
```
```  1531   have B: "B = A \<union> (B - A)" using `A \<subseteq> B` by blast
```
```  1532   then have "F B = F (A \<union> (B - A))" by simp
```
```  1533   also have "\<dots> = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
```
```  1534   also have "\<dots> \<preceq> F A" by simp
```
```  1535   finally show ?thesis .
```
```  1536 qed
```
```  1537
```
```  1538 end
```
```  1539
```
```  1540
```
```  1541 subsubsection {* With neutral element *}
```
```  1542
```
```  1543 locale semilattice_neutr_set = semilattice_neutr
```
```  1544 begin
```
```  1545
```
```  1546 definition F :: "'a set \<Rightarrow> 'a"
```
```  1547 where
```
```  1548   eq_fold: "F A = Finite_Set.fold f 1 A"
```
```  1549
```
```  1550 lemma infinite [simp]:
```
```  1551   "\<not> finite A \<Longrightarrow> F A = 1"
```
```  1552   by (simp add: eq_fold)
```
```  1553
```
```  1554 lemma empty [simp]:
```
```  1555   "F {} = 1"
```
```  1556   by (simp add: eq_fold)
```
```  1557
```
```  1558 lemma insert [simp]:
```
```  1559   assumes "finite A"
```
```  1560   shows "F (insert x A) = x * F A"
```
```  1561 proof -
```
```  1562   interpret comp_fun_idem f
```
```  1563     by default (simp_all add: fun_eq_iff left_commute)
```
```  1564   from assms show ?thesis by (simp add: eq_fold)
```
```  1565 qed
```
```  1566
```
```  1567 lemma subsumption:
```
```  1568   assumes "finite A" and "x \<in> A"
```
```  1569   shows "x * F A = F A"
```
```  1570 proof -
```
```  1571   from assms have "A \<noteq> {}" by auto
```
```  1572   with `finite A` show ?thesis using `x \<in> A`
```
```  1573     by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
```
```  1574 qed
```
```  1575
```
```  1576 lemma union:
```
```  1577   assumes "finite A" and "finite B"
```
```  1578   shows "F (A \<union> B) = F A * F B"
```
```  1579   using assms by (induct A) (simp_all add: ac_simps)
```
```  1580
```
```  1581 lemma remove:
```
```  1582   assumes "finite A" and "x \<in> A"
```
```  1583   shows "F A = x * F (A - {x})"
```
```  1584 proof -
```
```  1585   from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
```
```  1586   with assms show ?thesis by simp
```
```  1587 qed
```
```  1588
```
```  1589 lemma insert_remove:
```
```  1590   assumes "finite A"
```
```  1591   shows "F (insert x A) = x * F (A - {x})"
```
```  1592   using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
```
```  1593
```
```  1594 lemma subset:
```
```  1595   assumes "finite A" and "B \<subseteq> A"
```
```  1596   shows "F B * F A = F A"
```
```  1597 proof -
```
```  1598   from assms have "finite B" by (auto dest: finite_subset)
```
```  1599   with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
```
```  1600 qed
```
```  1601
```
```  1602 lemma closed:
```
```  1603   assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
```
```  1604   shows "F A \<in> A"
```
```  1605 using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
```
```  1606   case singleton then show ?case by simp
```
```  1607 next
```
```  1608   case insert with elem show ?case by force
```
```  1609 qed
```
```  1610
```
```  1611 end
```
```  1612
```
```  1613 locale semilattice_order_neutr_set = semilattice_neutr_order + semilattice_neutr_set
```
```  1614 begin
```
```  1615
```
```  1616 lemma bounded_iff:
```
```  1617   assumes "finite A"
```
```  1618   shows "x \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> a)"
```
```  1619   using assms by (induct A) (simp_all add: bounded_iff)
```
```  1620
```
```  1621 lemma boundedI:
```
```  1622   assumes "finite A"
```
```  1623   assumes "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
```
```  1624   shows "x \<preceq> F A"
```
```  1625   using assms by (simp add: bounded_iff)
```
```  1626
```
```  1627 lemma boundedE:
```
```  1628   assumes "finite A" and "x \<preceq> F A"
```
```  1629   obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
```
```  1630   using assms by (simp add: bounded_iff)
```
```  1631
```
```  1632 lemma coboundedI:
```
```  1633   assumes "finite A"
```
```  1634     and "a \<in> A"
```
```  1635   shows "F A \<preceq> a"
```
```  1636 proof -
```
```  1637   from assms have "A \<noteq> {}" by auto
```
```  1638   from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
```
```  1639   proof (induct rule: finite_ne_induct)
```
```  1640     case singleton thus ?case by (simp add: refl)
```
```  1641   next
```
```  1642     case (insert x B)
```
```  1643     from insert have "a = x \<or> a \<in> B" by simp
```
```  1644     then show ?case using insert by (auto intro: coboundedI2)
```
```  1645   qed
```
```  1646 qed
```
```  1647
```
```  1648 lemma antimono:
```
```  1649   assumes "A \<subseteq> B" and "finite B"
```
```  1650   shows "F B \<preceq> F A"
```
```  1651 proof (cases "A = B")
```
```  1652   case True then show ?thesis by (simp add: refl)
```
```  1653 next
```
```  1654   case False
```
```  1655   have B: "B = A \<union> (B - A)" using `A \<subseteq> B` by blast
```
```  1656   then have "F B = F (A \<union> (B - A))" by simp
```
```  1657   also have "\<dots> = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
```
```  1658   also have "\<dots> \<preceq> F A" by simp
```
```  1659   finally show ?thesis .
```
```  1660 qed
```
```  1661
```
```  1662 end
```
```  1663
```
```  1664 notation times (infixl "*" 70)
```
```  1665 notation Groups.one ("1")
```
```  1666
```
```  1667
```
```  1668 subsection {* Lattice operations on finite sets *}
```
```  1669
```
```  1670 text {*
```
```  1671   For historic reasons, there is the sublocale dependency from @{class distrib_lattice}
```
```  1672   to @{class linorder}.  This is badly designed: both should depend on a common abstract
```
```  1673   distributive lattice rather than having this non-subclass dependecy between two
```
```  1674   classes.  But for the moment we have to live with it.  This forces us to setup
```
```  1675   this sublocale dependency simultaneously with the lattice operations on finite
```
```  1676   sets, to avoid garbage.
```
```  1677 *}
```
```  1678
```
```  1679 definition (in semilattice_inf) Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>\<^bsub>fin\<^esub>_" [900] 900)
```
```  1680 where
```
```  1681   "Inf_fin = semilattice_set.F inf"
```
```  1682
```
```  1683 definition (in semilattice_sup) Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>\<^bsub>fin\<^esub>_" [900] 900)
```
```  1684 where
```
```  1685   "Sup_fin = semilattice_set.F sup"
```
```  1686
```
```  1687 definition (in linorder) Min :: "'a set \<Rightarrow> 'a"
```
```  1688 where
```
```  1689   "Min = semilattice_set.F min"
```
```  1690
```
```  1691 definition (in linorder) Max :: "'a set \<Rightarrow> 'a"
```
```  1692 where
```
```  1693   "Max = semilattice_set.F max"
```
```  1694
```
```  1695 sublocale linorder < Min!: semilattice_order_set min less_eq less
```
```  1696   + Max!: semilattice_order_set max greater_eq greater
```
```  1697 where
```
```  1698   "semilattice_set.F min = Min"
```
```  1699   and "semilattice_set.F max = Max"
```
```  1700 proof -
```
```  1701   show "semilattice_order_set min less_eq less" by default (auto simp add: min_def)
```
```  1702   then interpret Min!: semilattice_order_set min less_eq less.
```
```  1703   show "semilattice_order_set max greater_eq greater" by default (auto simp add: max_def)
```
```  1704   then interpret Max!: semilattice_order_set max greater_eq greater .
```
```  1705   from Min_def show "semilattice_set.F min = Min" by rule
```
```  1706   from Max_def show "semilattice_set.F max = Max" by rule
```
```  1707 qed
```
```  1708
```
```  1709
```
```  1710 text {* An aside: @{const min}/@{const max} on linear orders as special case of @{const inf}/@{const sup} *}
```
```  1711
```
```  1712 sublocale linorder < min_max!: distrib_lattice min less_eq less max
```
```  1713 where
```
```  1714   "semilattice_inf.Inf_fin min = Min"
```
```  1715   and "semilattice_sup.Sup_fin max = Max"
```
```  1716 proof -
```
```  1717   show "class.distrib_lattice min less_eq less max"
```
```  1718   proof
```
```  1719     fix x y z
```
```  1720     show "max x (min y z) = min (max x y) (max x z)"
```
```  1721       by (auto simp add: min_def max_def)
```
```  1722   qed (auto simp add: min_def max_def not_le less_imp_le)
```
```  1723   then interpret min_max!: distrib_lattice min less_eq less max .
```
```  1724   show "semilattice_inf.Inf_fin min = Min"
```
```  1725     by (simp only: min_max.Inf_fin_def Min_def)
```
```  1726   show "semilattice_sup.Sup_fin max = Max"
```
```  1727     by (simp only: min_max.Sup_fin_def Max_def)
```
```  1728 qed
```
```  1729
```
```  1730 lemmas le_maxI1 = min_max.sup_ge1
```
```  1731 lemmas le_maxI2 = min_max.sup_ge2
```
```  1732
```
```  1733 lemmas min_ac = min_max.inf_assoc min_max.inf_commute
```
```  1734   min.left_commute
```
```  1735
```
```  1736 lemmas max_ac = min_max.sup_assoc min_max.sup_commute
```
```  1737   max.left_commute
```
```  1738
```
```  1739
```
```  1740 text {* Lattice operations proper *}
```
```  1741
```
```  1742 sublocale semilattice_inf < Inf_fin!: semilattice_order_set inf less_eq less
```
```  1743 where
```
```  1744   "semilattice_set.F inf = Inf_fin"
```
```  1745 proof -
```
```  1746   show "semilattice_order_set inf less_eq less" ..
```
```  1747   then interpret Inf_fin!: semilattice_order_set inf less_eq less.
```
```  1748   from Inf_fin_def show "semilattice_set.F inf = Inf_fin" by rule
```
```  1749 qed
```
```  1750
```
```  1751 sublocale semilattice_sup < Sup_fin!: semilattice_order_set sup greater_eq greater
```
```  1752 where
```
```  1753   "semilattice_set.F sup = Sup_fin"
```
```  1754 proof -
```
```  1755   show "semilattice_order_set sup greater_eq greater" ..
```
```  1756   then interpret Sup_fin!: semilattice_order_set sup greater_eq greater .
```
```  1757   from Sup_fin_def show "semilattice_set.F sup = Sup_fin" by rule
```
```  1758 qed
```
```  1759
```
```  1760
```
```  1761 text {* An aside again: @{const Min}/@{const Max} on linear orders as special case of @{const Inf_fin}/@{const Sup_fin} *}
```
```  1762
```
```  1763 lemma Inf_fin_Min:
```
```  1764   "Inf_fin = (Min :: 'a::{semilattice_inf, linorder} set \<Rightarrow> 'a)"
```
```  1765   by (simp add: Inf_fin_def Min_def inf_min)
```
```  1766
```
```  1767 lemma Sup_fin_Max:
```
```  1768   "Sup_fin = (Max :: 'a::{semilattice_sup, linorder} set \<Rightarrow> 'a)"
```
```  1769   by (simp add: Sup_fin_def Max_def sup_max)
```
```  1770
```
```  1771
```
```  1772
```
```  1773 subsection {* Infimum and Supremum over non-empty sets *}
```
```  1774
```
```  1775 text {*
```
```  1776   After this non-regular bootstrap, things continue canonically.
```
```  1777 *}
```
```  1778
```
```  1779 context lattice
```
```  1780 begin
```
```  1781
```
```  1782 lemma Inf_le_Sup [simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>\<^bsub>fin\<^esub>A \<le> \<Squnion>\<^bsub>fin\<^esub>A"
```
```  1783 apply(subgoal_tac "EX a. a:A")
```
```  1784 prefer 2 apply blast
```
```  1785 apply(erule exE)
```
```  1786 apply(rule order_trans)
```
```  1787 apply(erule (1) Inf_fin.coboundedI)
```
```  1788 apply(erule (1) Sup_fin.coboundedI)
```
```  1789 done
```
```  1790
```
```  1791 lemma sup_Inf_absorb [simp]:
```
```  1792   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a (\<Sqinter>\<^bsub>fin\<^esub>A) = a"
```
```  1793 apply(subst sup_commute)
```
```  1794 apply(simp add: sup_absorb2 Inf_fin.coboundedI)
```
```  1795 done
```
```  1796
```
```  1797 lemma inf_Sup_absorb [simp]:
```
```  1798   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>\<^bsub>fin\<^esub>A) = a"
```
```  1799 by (simp add: inf_absorb1 Sup_fin.coboundedI)
```
```  1800
```
```  1801 end
```
```  1802
```
```  1803 context distrib_lattice
```
```  1804 begin
```
```  1805
```
```  1806 lemma sup_Inf1_distrib:
```
```  1807   assumes "finite A"
```
```  1808     and "A \<noteq> {}"
```
```  1809   shows "sup x (\<Sqinter>\<^bsub>fin\<^esub>A) = \<Sqinter>\<^bsub>fin\<^esub>{sup x a|a. a \<in> A}"
```
```  1810 using assms by (simp add: image_def Inf_fin.hom_commute [where h="sup x", OF sup_inf_distrib1])
```
```  1811   (rule arg_cong [where f="Inf_fin"], blast)
```
```  1812
```
```  1813 lemma sup_Inf2_distrib:
```
```  1814   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
```
```  1815   shows "sup (\<Sqinter>\<^bsub>fin\<^esub>A) (\<Sqinter>\<^bsub>fin\<^esub>B) = \<Sqinter>\<^bsub>fin\<^esub>{sup a b|a b. a \<in> A \<and> b \<in> B}"
```
```  1816 using A proof (induct rule: finite_ne_induct)
```
```  1817   case singleton then show ?case
```
```  1818     by (simp add: sup_Inf1_distrib [OF B])
```
```  1819 next
```
```  1820   case (insert x A)
```
```  1821   have finB: "finite {sup x b |b. b \<in> B}"
```
```  1822     by (rule finite_surj [where f = "sup x", OF B(1)], auto)
```
```  1823   have finAB: "finite {sup a b |a b. a \<in> A \<and> b \<in> B}"
```
```  1824   proof -
```
```  1825     have "{sup a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {sup a b})"
```
```  1826       by blast
```
```  1827     thus ?thesis by(simp add: insert(1) B(1))
```
```  1828   qed
```
```  1829   have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
```
```  1830   have "sup (\<Sqinter>\<^bsub>fin\<^esub>(insert x A)) (\<Sqinter>\<^bsub>fin\<^esub>B) = sup (inf x (\<Sqinter>\<^bsub>fin\<^esub>A)) (\<Sqinter>\<^bsub>fin\<^esub>B)"
```
```  1831     using insert by simp
```
```  1832   also have "\<dots> = inf (sup x (\<Sqinter>\<^bsub>fin\<^esub>B)) (sup (\<Sqinter>\<^bsub>fin\<^esub>A) (\<Sqinter>\<^bsub>fin\<^esub>B))" by(rule sup_inf_distrib2)
```
```  1833   also have "\<dots> = inf (\<Sqinter>\<^bsub>fin\<^esub>{sup x b|b. b \<in> B}) (\<Sqinter>\<^bsub>fin\<^esub>{sup a b|a b. a \<in> A \<and> b \<in> B})"
```
```  1834     using insert by(simp add:sup_Inf1_distrib[OF B])
```
```  1835   also have "\<dots> = \<Sqinter>\<^bsub>fin\<^esub>({sup x b |b. b \<in> B} \<union> {sup a b |a b. a \<in> A \<and> b \<in> B})"
```
```  1836     (is "_ = \<Sqinter>\<^bsub>fin\<^esub>?M")
```
```  1837     using B insert
```
```  1838     by (simp add: Inf_fin.union [OF finB _ finAB ne])
```
```  1839   also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"
```
```  1840     by blast
```
```  1841   finally show ?case .
```
```  1842 qed
```
```  1843
```
```  1844 lemma inf_Sup1_distrib:
```
```  1845   assumes "finite A" and "A \<noteq> {}"
```
```  1846   shows "inf x (\<Squnion>\<^bsub>fin\<^esub>A) = \<Squnion>\<^bsub>fin\<^esub>{inf x a|a. a \<in> A}"
```
```  1847 using assms by (simp add: image_def Sup_fin.hom_commute [where h="inf x", OF inf_sup_distrib1])
```
```  1848   (rule arg_cong [where f="Sup_fin"], blast)
```
```  1849
```
```  1850 lemma inf_Sup2_distrib:
```
```  1851   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
```
```  1852   shows "inf (\<Squnion>\<^bsub>fin\<^esub>A) (\<Squnion>\<^bsub>fin\<^esub>B) = \<Squnion>\<^bsub>fin\<^esub>{inf a b|a b. a \<in> A \<and> b \<in> B}"
```
```  1853 using A proof (induct rule: finite_ne_induct)
```
```  1854   case singleton thus ?case
```
```  1855     by(simp add: inf_Sup1_distrib [OF B])
```
```  1856 next
```
```  1857   case (insert x A)
```
```  1858   have finB: "finite {inf x b |b. b \<in> B}"
```
```  1859     by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto)
```
```  1860   have finAB: "finite {inf a b |a b. a \<in> A \<and> b \<in> B}"
```
```  1861   proof -
```
```  1862     have "{inf a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {inf a b})"
```
```  1863       by blast
```
```  1864     thus ?thesis by(simp add: insert(1) B(1))
```
```  1865   qed
```
```  1866   have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
```
```  1867   have "inf (\<Squnion>\<^bsub>fin\<^esub>(insert x A)) (\<Squnion>\<^bsub>fin\<^esub>B) = inf (sup x (\<Squnion>\<^bsub>fin\<^esub>A)) (\<Squnion>\<^bsub>fin\<^esub>B)"
```
```  1868     using insert by simp
```
```  1869   also have "\<dots> = sup (inf x (\<Squnion>\<^bsub>fin\<^esub>B)) (inf (\<Squnion>\<^bsub>fin\<^esub>A) (\<Squnion>\<^bsub>fin\<^esub>B))" by(rule inf_sup_distrib2)
```
```  1870   also have "\<dots> = sup (\<Squnion>\<^bsub>fin\<^esub>{inf x b|b. b \<in> B}) (\<Squnion>\<^bsub>fin\<^esub>{inf a b|a b. a \<in> A \<and> b \<in> B})"
```
```  1871     using insert by(simp add:inf_Sup1_distrib[OF B])
```
```  1872   also have "\<dots> = \<Squnion>\<^bsub>fin\<^esub>({inf x b |b. b \<in> B} \<union> {inf a b |a b. a \<in> A \<and> b \<in> B})"
```
```  1873     (is "_ = \<Squnion>\<^bsub>fin\<^esub>?M")
```
```  1874     using B insert
```
```  1875     by (simp add: Sup_fin.union [OF finB _ finAB ne])
```
```  1876   also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"
```
```  1877     by blast
```
```  1878   finally show ?case .
```
```  1879 qed
```
```  1880
```
```  1881 end
```
```  1882
```
```  1883 context complete_lattice
```
```  1884 begin
```
```  1885
```
```  1886 lemma Inf_fin_Inf:
```
```  1887   assumes "finite A" and "A \<noteq> {}"
```
```  1888   shows "\<Sqinter>\<^bsub>fin\<^esub>A = Inf A"
```
```  1889 proof -
```
```  1890   from assms obtain b B where "A = insert b B" and "finite B" by auto
```
```  1891   then show ?thesis
```
```  1892     by (simp add: Inf_fin.eq_fold inf_Inf_fold_inf inf.commute [of b])
```
```  1893 qed
```
```  1894
```
```  1895 lemma Sup_fin_Sup:
```
```  1896   assumes "finite A" and "A \<noteq> {}"
```
```  1897   shows "\<Squnion>\<^bsub>fin\<^esub>A = Sup A"
```
```  1898 proof -
```
```  1899   from assms obtain b B where "A = insert b B" and "finite B" by auto
```
```  1900   then show ?thesis
```
```  1901     by (simp add: Sup_fin.eq_fold sup_Sup_fold_sup sup.commute [of b])
```
```  1902 qed
```
```  1903
```
```  1904 end
```
```  1905
```
```  1906
```
```  1907 subsection {* Minimum and Maximum over non-empty sets *}
```
```  1908
```
```  1909 context linorder
```
```  1910 begin
```
```  1911
```
```  1912 lemma dual_min:
```
```  1913   "ord.min greater_eq = max"
```
```  1914   by (auto simp add: ord.min_def max_def fun_eq_iff)
```
```  1915
```
```  1916 lemma dual_max:
```
```  1917   "ord.max greater_eq = min"
```
```  1918   by (auto simp add: ord.max_def min_def fun_eq_iff)
```
```  1919
```
```  1920 lemma dual_Min:
```
```  1921   "linorder.Min greater_eq = Max"
```
```  1922 proof -
```
```  1923   interpret dual!: linorder greater_eq greater by (fact dual_linorder)
```
```  1924   show ?thesis by (simp add: dual.Min_def dual_min Max_def)
```
```  1925 qed
```
```  1926
```
```  1927 lemma dual_Max:
```
```  1928   "linorder.Max greater_eq = Min"
```
```  1929 proof -
```
```  1930   interpret dual!: linorder greater_eq greater by (fact dual_linorder)
```
```  1931   show ?thesis by (simp add: dual.Max_def dual_max Min_def)
```
```  1932 qed
```
```  1933
```
```  1934 lemmas Min_singleton = Min.singleton
```
```  1935 lemmas Max_singleton = Max.singleton
```
```  1936 lemmas Min_insert = Min.insert
```
```  1937 lemmas Max_insert = Max.insert
```
```  1938 lemmas Min_Un = Min.union
```
```  1939 lemmas Max_Un = Max.union
```
```  1940 lemmas hom_Min_commute = Min.hom_commute
```
```  1941 lemmas hom_Max_commute = Max.hom_commute
```
```  1942
```
```  1943 lemma Min_in [simp]:
```
```  1944   assumes "finite A" and "A \<noteq> {}"
```
```  1945   shows "Min A \<in> A"
```
```  1946   using assms by (auto simp add: min_def Min.closed)
```
```  1947
```
```  1948 lemma Max_in [simp]:
```
```  1949   assumes "finite A" and "A \<noteq> {}"
```
```  1950   shows "Max A \<in> A"
```
```  1951   using assms by (auto simp add: max_def Max.closed)
```
```  1952
```
```  1953 lemma Min_le [simp]:
```
```  1954   assumes "finite A" and "x \<in> A"
```
```  1955   shows "Min A \<le> x"
```
```  1956   using assms by (fact Min.coboundedI)
```
```  1957
```
```  1958 lemma Max_ge [simp]:
```
```  1959   assumes "finite A" and "x \<in> A"
```
```  1960   shows "x \<le> Max A"
```
```  1961   using assms by (fact Max.coboundedI)
```
```  1962
```
```  1963 lemma Min_eqI:
```
```  1964   assumes "finite A"
```
```  1965   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<ge> x"
```
```  1966     and "x \<in> A"
```
```  1967   shows "Min A = x"
```
```  1968 proof (rule antisym)
```
```  1969   from `x \<in> A` have "A \<noteq> {}" by auto
```
```  1970   with assms show "Min A \<ge> x" by simp
```
```  1971 next
```
```  1972   from assms show "x \<ge> Min A" by simp
```
```  1973 qed
```
```  1974
```
```  1975 lemma Max_eqI:
```
```  1976   assumes "finite A"
```
```  1977   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
```
```  1978     and "x \<in> A"
```
```  1979   shows "Max A = x"
```
```  1980 proof (rule antisym)
```
```  1981   from `x \<in> A` have "A \<noteq> {}" by auto
```
```  1982   with assms show "Max A \<le> x" by simp
```
```  1983 next
```
```  1984   from assms show "x \<le> Max A" by simp
```
```  1985 qed
```
```  1986
```
```  1987 lemma Min_ge_iff [simp, no_atp]:
```
```  1988   assumes "finite A" and "A \<noteq> {}"
```
```  1989   shows "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
```
```  1990   using assms by (fact Min.bounded_iff)
```
```  1991
```
```  1992 lemma Max_le_iff [simp, no_atp]:
```
```  1993   assumes "finite A" and "A \<noteq> {}"
```
```  1994   shows "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
```
```  1995   using assms by (fact Max.bounded_iff)
```
```  1996
```
```  1997 lemma Min_gr_iff [simp, no_atp]:
```
```  1998   assumes "finite A" and "A \<noteq> {}"
```
```  1999   shows "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
```
```  2000   using assms by (induct rule: finite_ne_induct) simp_all
```
```  2001
```
```  2002 lemma Max_less_iff [simp, no_atp]:
```
```  2003   assumes "finite A" and "A \<noteq> {}"
```
```  2004   shows "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
```
```  2005   using assms by (induct rule: finite_ne_induct) simp_all
```
```  2006
```
```  2007 lemma Min_le_iff [no_atp]:
```
```  2008   assumes "finite A" and "A \<noteq> {}"
```
```  2009   shows "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
```
```  2010   using assms by (induct rule: finite_ne_induct) (simp_all add: min_le_iff_disj)
```
```  2011
```
```  2012 lemma Max_ge_iff [no_atp]:
```
```  2013   assumes "finite A" and "A \<noteq> {}"
```
```  2014   shows "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
```
```  2015   using assms by (induct rule: finite_ne_induct) (simp_all add: le_max_iff_disj)
```
```  2016
```
```  2017 lemma Min_less_iff [no_atp]:
```
```  2018   assumes "finite A" and "A \<noteq> {}"
```
```  2019   shows "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
```
```  2020   using assms by (induct rule: finite_ne_induct) (simp_all add: min_less_iff_disj)
```
```  2021
```
```  2022 lemma Max_gr_iff [no_atp]:
```
```  2023   assumes "finite A" and "A \<noteq> {}"
```
```  2024   shows "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
```
```  2025   using assms by (induct rule: finite_ne_induct) (simp_all add: less_max_iff_disj)
```
```  2026
```
```  2027 lemma Min_antimono:
```
```  2028   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
```
```  2029   shows "Min N \<le> Min M"
```
```  2030   using assms by (fact Min.antimono)
```
```  2031
```
```  2032 lemma Max_mono:
```
```  2033   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
```
```  2034   shows "Max M \<le> Max N"
```
```  2035   using assms by (fact Max.antimono)
```
```  2036
```
```  2037 lemma mono_Min_commute:
```
```  2038   assumes "mono f"
```
```  2039   assumes "finite A" and "A \<noteq> {}"
```
```  2040   shows "f (Min A) = Min (f ` A)"
```
```  2041 proof (rule linorder_class.Min_eqI [symmetric])
```
```  2042   from `finite A` show "finite (f ` A)" by simp
```
```  2043   from assms show "f (Min A) \<in> f ` A" by simp
```
```  2044   fix x
```
```  2045   assume "x \<in> f ` A"
```
```  2046   then obtain y where "y \<in> A" and "x = f y" ..
```
```  2047   with assms have "Min A \<le> y" by auto
```
```  2048   with `mono f` have "f (Min A) \<le> f y" by (rule monoE)
```
```  2049   with `x = f y` show "f (Min A) \<le> x" by simp
```
```  2050 qed
```
```  2051
```
```  2052 lemma mono_Max_commute:
```
```  2053   assumes "mono f"
```
```  2054   assumes "finite A" and "A \<noteq> {}"
```
```  2055   shows "f (Max A) = Max (f ` A)"
```
```  2056 proof (rule linorder_class.Max_eqI [symmetric])
```
```  2057   from `finite A` show "finite (f ` A)" by simp
```
```  2058   from assms show "f (Max A) \<in> f ` A" by simp
```
```  2059   fix x
```
```  2060   assume "x \<in> f ` A"
```
```  2061   then obtain y where "y \<in> A" and "x = f y" ..
```
```  2062   with assms have "y \<le> Max A" by auto
```
```  2063   with `mono f` have "f y \<le> f (Max A)" by (rule monoE)
```
```  2064   with `x = f y` show "x \<le> f (Max A)" by simp
```
```  2065 qed
```
```  2066
```
```  2067 lemma finite_linorder_max_induct [consumes 1, case_names empty insert]:
```
```  2068   assumes fin: "finite A"
```
```  2069   and empty: "P {}"
```
```  2070   and insert: "\<And>b A. finite A \<Longrightarrow> \<forall>a\<in>A. a < b \<Longrightarrow> P A \<Longrightarrow> P (insert b A)"
```
```  2071   shows "P A"
```
```  2072 using fin empty insert
```
```  2073 proof (induct rule: finite_psubset_induct)
```
```  2074   case (psubset A)
```
```  2075   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
```
```  2076   have fin: "finite A" by fact
```
```  2077   have empty: "P {}" by fact
```
```  2078   have step: "\<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. a < b; P A\<rbrakk> \<Longrightarrow> P (insert b A)" by fact
```
```  2079   show "P A"
```
```  2080   proof (cases "A = {}")
```
```  2081     assume "A = {}"
```
```  2082     then show "P A" using `P {}` by simp
```
```  2083   next
```
```  2084     let ?B = "A - {Max A}"
```
```  2085     let ?A = "insert (Max A) ?B"
```
```  2086     have "finite ?B" using `finite A` by simp
```
```  2087     assume "A \<noteq> {}"
```
```  2088     with `finite A` have "Max A : A" by auto
```
```  2089     then have A: "?A = A" using insert_Diff_single insert_absorb by auto
```
```  2090     then have "P ?B" using `P {}` step IH [of ?B] by blast
```
```  2091     moreover
```
```  2092     have "\<forall>a\<in>?B. a < Max A" using Max_ge [OF `finite A`] by fastforce
```
```  2093     ultimately show "P A" using A insert_Diff_single step [OF `finite ?B`] by fastforce
```
```  2094   qed
```
```  2095 qed
```
```  2096
```
```  2097 lemma finite_linorder_min_induct [consumes 1, case_names empty insert]:
```
```  2098   "\<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"
```
```  2099   by (rule linorder.finite_linorder_max_induct [OF dual_linorder])
```
```  2100
```
```  2101 end
```
```  2102
```
```  2103 context linordered_ab_semigroup_add
```
```  2104 begin
```
```  2105
```
```  2106 lemma add_Min_commute:
```
```  2107   fixes k
```
```  2108   assumes "finite N" and "N \<noteq> {}"
```
```  2109   shows "k + Min N = Min {k + m | m. m \<in> N}"
```
```  2110 proof -
```
```  2111   have "\<And>x y. k + min x y = min (k + x) (k + y)"
```
```  2112     by (simp add: min_def not_le)
```
```  2113       (blast intro: antisym less_imp_le add_left_mono)
```
```  2114   with assms show ?thesis
```
```  2115     using hom_Min_commute [of "plus k" N]
```
```  2116     by simp (blast intro: arg_cong [where f = Min])
```
```  2117 qed
```
```  2118
```
```  2119 lemma add_Max_commute:
```
```  2120   fixes k
```
```  2121   assumes "finite N" and "N \<noteq> {}"
```
```  2122   shows "k + Max N = Max {k + m | m. m \<in> N}"
```
```  2123 proof -
```
```  2124   have "\<And>x y. k + max x y = max (k + x) (k + y)"
```
```  2125     by (simp add: max_def not_le)
```
```  2126       (blast intro: antisym less_imp_le add_left_mono)
```
```  2127   with assms show ?thesis
```
```  2128     using hom_Max_commute [of "plus k" N]
```
```  2129     by simp (blast intro: arg_cong [where f = Max])
```
```  2130 qed
```
```  2131
```
```  2132 end
```
```  2133
```
```  2134 context linordered_ab_group_add
```
```  2135 begin
```
```  2136
```
```  2137 lemma minus_Max_eq_Min [simp]:
```
```  2138   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Max S = Min (uminus ` S)"
```
```  2139   by (induct S rule: finite_ne_induct) (simp_all add: minus_max_eq_min)
```
```  2140
```
```  2141 lemma minus_Min_eq_Max [simp]:
```
```  2142   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Min S = Max (uminus ` S)"
```
```  2143   by (induct S rule: finite_ne_induct) (simp_all add: minus_min_eq_max)
```
```  2144
```
```  2145 end
```
```  2146
```
```  2147 context complete_linorder
```
```  2148 begin
```
```  2149
```
```  2150 lemma Min_Inf:
```
```  2151   assumes "finite A" and "A \<noteq> {}"
```
```  2152   shows "Min A = Inf A"
```
```  2153 proof -
```
```  2154   from assms obtain b B where "A = insert b B" and "finite B" by auto
```
```  2155   then show ?thesis
```
```  2156     by (simp add: Min.eq_fold complete_linorder_inf_min [symmetric] inf_Inf_fold_inf inf.commute [of b])
```
```  2157 qed
```
```  2158
```
```  2159 lemma Max_Sup:
```
```  2160   assumes "finite A" and "A \<noteq> {}"
```
```  2161   shows "Max A = Sup A"
```
```  2162 proof -
```
```  2163   from assms obtain b B where "A = insert b B" and "finite B" by auto
```
```  2164   then show ?thesis
```
```  2165     by (simp add: Max.eq_fold complete_linorder_sup_max [symmetric] sup_Sup_fold_sup sup.commute [of b])
```
```  2166 qed
```
```  2167
```
```  2168 end
```
```  2169
```
```  2170 end
```
```  2171
```