src/HOL/Groups_Big.thy
 author hoelzl Fri May 30 14:55:10 2014 +0200 (2014-05-30) changeset 57129 7edb7550663e parent 56545 8f1e7596deb7 child 57275 0ddb5b755cdc permissions -rw-r--r--
introduce more powerful reindexing rules for big operators
```     1 (*  Title:      HOL/Groups_Big.thy
```
```     2     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
```
```     3                 with contributions by Jeremy Avigad
```
```     4 *)
```
```     5
```
```     6 header {* Big sum and product over finite (non-empty) sets *}
```
```     7
```
```     8 theory Groups_Big
```
```     9 imports Finite_Set
```
```    10 begin
```
```    11
```
```    12 subsection {* Generic monoid operation over a set *}
```
```    13
```
```    14 no_notation times (infixl "*" 70)
```
```    15 no_notation Groups.one ("1")
```
```    16
```
```    17 locale comm_monoid_set = comm_monoid
```
```    18 begin
```
```    19
```
```    20 interpretation comp_fun_commute f
```
```    21   by default (simp add: fun_eq_iff left_commute)
```
```    22
```
```    23 interpretation comp?: comp_fun_commute "f \<circ> g"
```
```    24   by (fact comp_comp_fun_commute)
```
```    25
```
```    26 definition F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
```
```    27 where
```
```    28   eq_fold: "F g A = Finite_Set.fold (f \<circ> g) 1 A"
```
```    29
```
```    30 lemma infinite [simp]:
```
```    31   "\<not> finite A \<Longrightarrow> F g A = 1"
```
```    32   by (simp add: eq_fold)
```
```    33
```
```    34 lemma empty [simp]:
```
```    35   "F g {} = 1"
```
```    36   by (simp add: eq_fold)
```
```    37
```
```    38 lemma insert [simp]:
```
```    39   assumes "finite A" and "x \<notin> A"
```
```    40   shows "F g (insert x A) = g x * F g A"
```
```    41   using assms by (simp add: eq_fold)
```
```    42
```
```    43 lemma remove:
```
```    44   assumes "finite A" and "x \<in> A"
```
```    45   shows "F g A = g x * F g (A - {x})"
```
```    46 proof -
```
```    47   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
```
```    48     by (auto dest: mk_disjoint_insert)
```
```    49   moreover from `finite A` A have "finite B" by simp
```
```    50   ultimately show ?thesis by simp
```
```    51 qed
```
```    52
```
```    53 lemma insert_remove:
```
```    54   assumes "finite A"
```
```    55   shows "F g (insert x A) = g x * F g (A - {x})"
```
```    56   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
```
```    57
```
```    58 lemma neutral:
```
```    59   assumes "\<forall>x\<in>A. g x = 1"
```
```    60   shows "F g A = 1"
```
```    61   using assms by (induct A rule: infinite_finite_induct) simp_all
```
```    62
```
```    63 lemma neutral_const [simp]:
```
```    64   "F (\<lambda>_. 1) A = 1"
```
```    65   by (simp add: neutral)
```
```    66
```
```    67 lemma union_inter:
```
```    68   assumes "finite A" and "finite B"
```
```    69   shows "F g (A \<union> B) * F g (A \<inter> B) = F g A * F g B"
```
```    70   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
```
```    71 using assms proof (induct A)
```
```    72   case empty then show ?case by simp
```
```    73 next
```
```    74   case (insert x A) then show ?case
```
```    75     by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
```
```    76 qed
```
```    77
```
```    78 corollary union_inter_neutral:
```
```    79   assumes "finite A" and "finite B"
```
```    80   and I0: "\<forall>x \<in> A \<inter> B. g x = 1"
```
```    81   shows "F g (A \<union> B) = F g A * F g B"
```
```    82   using assms by (simp add: union_inter [symmetric] neutral)
```
```    83
```
```    84 corollary union_disjoint:
```
```    85   assumes "finite A" and "finite B"
```
```    86   assumes "A \<inter> B = {}"
```
```    87   shows "F g (A \<union> B) = F g A * F g B"
```
```    88   using assms by (simp add: union_inter_neutral)
```
```    89
```
```    90 lemma subset_diff:
```
```    91   assumes "B \<subseteq> A" and "finite A"
```
```    92   shows "F g A = F g (A - B) * F g B"
```
```    93 proof -
```
```    94   from assms have "finite (A - B)" by auto
```
```    95   moreover from assms have "finite B" by (rule finite_subset)
```
```    96   moreover from assms have "(A - B) \<inter> B = {}" by auto
```
```    97   ultimately have "F g (A - B \<union> B) = F g (A - B) * F g B" by (rule union_disjoint)
```
```    98   moreover from assms have "A \<union> B = A" by auto
```
```    99   ultimately show ?thesis by simp
```
```   100 qed
```
```   101
```
```   102 lemma setdiff_irrelevant:
```
```   103   assumes "finite A"
```
```   104   shows "F g (A - {x. g x = z}) = F g A"
```
```   105   using assms by (induct A) (simp_all add: insert_Diff_if)
```
```   106
```
```   107 lemma not_neutral_contains_not_neutral:
```
```   108   assumes "F g A \<noteq> z"
```
```   109   obtains a where "a \<in> A" and "g a \<noteq> z"
```
```   110 proof -
```
```   111   from assms have "\<exists>a\<in>A. g a \<noteq> z"
```
```   112   proof (induct A rule: infinite_finite_induct)
```
```   113     case (insert a A)
```
```   114     then show ?case by simp (rule, simp)
```
```   115   qed simp_all
```
```   116   with that show thesis by blast
```
```   117 qed
```
```   118
```
```   119 lemma reindex:
```
```   120   assumes "inj_on h A"
```
```   121   shows "F g (h ` A) = F (g \<circ> h) A"
```
```   122 proof (cases "finite A")
```
```   123   case True
```
```   124   with assms show ?thesis by (simp add: eq_fold fold_image comp_assoc)
```
```   125 next
```
```   126   case False with assms have "\<not> finite (h ` A)" by (blast dest: finite_imageD)
```
```   127   with False show ?thesis by simp
```
```   128 qed
```
```   129
```
```   130 lemma cong:
```
```   131   assumes "A = B"
```
```   132   assumes g_h: "\<And>x. x \<in> B \<Longrightarrow> g x = h x"
```
```   133   shows "F g A = F h B"
```
```   134   using g_h unfolding `A = B`
```
```   135   by (induct B rule: infinite_finite_induct) auto
```
```   136
```
```   137 lemma strong_cong [cong]:
```
```   138   assumes "A = B" "\<And>x. x \<in> B =simp=> g x = h x"
```
```   139   shows "F (\<lambda>x. g x) A = F (\<lambda>x. h x) B"
```
```   140   by (rule cong) (insert assms, simp_all add: simp_implies_def)
```
```   141
```
```   142 lemma UNION_disjoint:
```
```   143   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
```
```   144   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
```
```   145   shows "F g (UNION I A) = F (\<lambda>x. F g (A x)) I"
```
```   146 apply (insert assms)
```
```   147 apply (induct rule: finite_induct)
```
```   148 apply simp
```
```   149 apply atomize
```
```   150 apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
```
```   151  prefer 2 apply blast
```
```   152 apply (subgoal_tac "A x Int UNION Fa A = {}")
```
```   153  prefer 2 apply blast
```
```   154 apply (simp add: union_disjoint)
```
```   155 done
```
```   156
```
```   157 lemma Union_disjoint:
```
```   158   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
```
```   159   shows "F g (Union C) = F (F g) C"
```
```   160 proof cases
```
```   161   assume "finite C"
```
```   162   from UNION_disjoint [OF this assms]
```
```   163   show ?thesis by simp
```
```   164 qed (auto dest: finite_UnionD intro: infinite)
```
```   165
```
```   166 lemma distrib:
```
```   167   "F (\<lambda>x. g x * h x) A = F g A * F h A"
```
```   168   using assms by (induct A rule: infinite_finite_induct) (simp_all add: assoc commute left_commute)
```
```   169
```
```   170 lemma Sigma:
```
```   171   "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)"
```
```   172 apply (subst Sigma_def)
```
```   173 apply (subst UNION_disjoint, assumption, simp)
```
```   174  apply blast
```
```   175 apply (rule cong)
```
```   176 apply rule
```
```   177 apply (simp add: fun_eq_iff)
```
```   178 apply (subst UNION_disjoint, simp, simp)
```
```   179  apply blast
```
```   180 apply (simp add: comp_def)
```
```   181 done
```
```   182
```
```   183 lemma related:
```
```   184   assumes Re: "R 1 1"
```
```   185   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
```
```   186   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
```
```   187   shows "R (F h S) (F g S)"
```
```   188   using fS by (rule finite_subset_induct) (insert assms, auto)
```
```   189
```
```   190 lemma mono_neutral_cong_left:
```
```   191   assumes "finite T" and "S \<subseteq> T" and "\<forall>i \<in> T - S. h i = 1"
```
```   192   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g S = F h T"
```
```   193 proof-
```
```   194   have eq: "T = S \<union> (T - S)" using `S \<subseteq> T` by blast
```
```   195   have d: "S \<inter> (T - S) = {}" using `S \<subseteq> T` by blast
```
```   196   from `finite T` `S \<subseteq> T` have f: "finite S" "finite (T - S)"
```
```   197     by (auto intro: finite_subset)
```
```   198   show ?thesis using assms(4)
```
```   199     by (simp add: union_disjoint [OF f d, unfolded eq [symmetric]] neutral [OF assms(3)])
```
```   200 qed
```
```   201
```
```   202 lemma mono_neutral_cong_right:
```
```   203   "\<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>
```
```   204    \<Longrightarrow> F g T = F h S"
```
```   205   by (auto intro!: mono_neutral_cong_left [symmetric])
```
```   206
```
```   207 lemma mono_neutral_left:
```
```   208   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g S = F g T"
```
```   209   by (blast intro: mono_neutral_cong_left)
```
```   210
```
```   211 lemma mono_neutral_right:
```
```   212   "\<lbrakk> finite T;  S \<subseteq> T;  \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g T = F g S"
```
```   213   by (blast intro!: mono_neutral_left [symmetric])
```
```   214
```
```   215 lemma reindex_bij_betw: "bij_betw h S T \<Longrightarrow> F (\<lambda>x. g (h x)) S = F g T"
```
```   216   by (auto simp: bij_betw_def reindex)
```
```   217
```
```   218 lemma reindex_bij_witness:
```
```   219   assumes witness:
```
```   220     "\<And>a. a \<in> S \<Longrightarrow> i (j a) = a"
```
```   221     "\<And>a. a \<in> S \<Longrightarrow> j a \<in> T"
```
```   222     "\<And>b. b \<in> T \<Longrightarrow> j (i b) = b"
```
```   223     "\<And>b. b \<in> T \<Longrightarrow> i b \<in> S"
```
```   224   assumes eq:
```
```   225     "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a"
```
```   226   shows "F g S = F h T"
```
```   227 proof -
```
```   228   have "bij_betw j S T"
```
```   229     using bij_betw_byWitness[where A=S and f=j and f'=i and A'=T] witness by auto
```
```   230   moreover have "F g S = F (\<lambda>x. h (j x)) S"
```
```   231     by (intro cong) (auto simp: eq)
```
```   232   ultimately show ?thesis
```
```   233     by (simp add: reindex_bij_betw)
```
```   234 qed
```
```   235
```
```   236 lemma reindex_bij_betw_not_neutral:
```
```   237   assumes fin: "finite S'" "finite T'"
```
```   238   assumes bij: "bij_betw h (S - S') (T - T')"
```
```   239   assumes nn:
```
```   240     "\<And>a. a \<in> S' \<Longrightarrow> g (h a) = z"
```
```   241     "\<And>b. b \<in> T' \<Longrightarrow> g b = z"
```
```   242   shows "F (\<lambda>x. g (h x)) S = F g T"
```
```   243 proof -
```
```   244   have [simp]: "finite S \<longleftrightarrow> finite T"
```
```   245     using bij_betw_finite[OF bij] fin by auto
```
```   246
```
```   247   show ?thesis
```
```   248   proof cases
```
```   249     assume "finite S"
```
```   250     with nn have "F (\<lambda>x. g (h x)) S = F (\<lambda>x. g (h x)) (S - S')"
```
```   251       by (intro mono_neutral_cong_right) auto
```
```   252     also have "\<dots> = F g (T - T')"
```
```   253       using bij by (rule reindex_bij_betw)
```
```   254     also have "\<dots> = F g T"
```
```   255       using nn `finite S` by (intro mono_neutral_cong_left) auto
```
```   256     finally show ?thesis .
```
```   257   qed simp
```
```   258 qed
```
```   259
```
```   260 lemma reindex_bij_witness_not_neutral:
```
```   261   assumes fin: "finite S'" "finite T'"
```
```   262   assumes witness:
```
```   263     "\<And>a. a \<in> S - S' \<Longrightarrow> i (j a) = a"
```
```   264     "\<And>a. a \<in> S - S' \<Longrightarrow> j a \<in> T - T'"
```
```   265     "\<And>b. b \<in> T - T' \<Longrightarrow> j (i b) = b"
```
```   266     "\<And>b. b \<in> T - T' \<Longrightarrow> i b \<in> S - S'"
```
```   267   assumes nn:
```
```   268     "\<And>a. a \<in> S' \<Longrightarrow> g a = z"
```
```   269     "\<And>b. b \<in> T' \<Longrightarrow> h b = z"
```
```   270   assumes eq:
```
```   271     "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a"
```
```   272   shows "F g S = F h T"
```
```   273 proof -
```
```   274   have bij: "bij_betw j (S - (S' \<inter> S)) (T - (T' \<inter> T))"
```
```   275     using witness by (intro bij_betw_byWitness[where f'=i]) auto
```
```   276   have F_eq: "F g S = F (\<lambda>x. h (j x)) S"
```
```   277     by (intro cong) (auto simp: eq)
```
```   278   show ?thesis
```
```   279     unfolding F_eq using fin nn eq
```
```   280     by (intro reindex_bij_betw_not_neutral[OF _ _ bij]) auto
```
```   281 qed
```
```   282
```
```   283 lemma delta:
```
```   284   assumes fS: "finite S"
```
```   285   shows "F (\<lambda>k. if k = a then b k else 1) S = (if a \<in> S then b a else 1)"
```
```   286 proof-
```
```   287   let ?f = "(\<lambda>k. if k=a then b k else 1)"
```
```   288   { assume a: "a \<notin> S"
```
```   289     hence "\<forall>k\<in>S. ?f k = 1" by simp
```
```   290     hence ?thesis  using a by simp }
```
```   291   moreover
```
```   292   { assume a: "a \<in> S"
```
```   293     let ?A = "S - {a}"
```
```   294     let ?B = "{a}"
```
```   295     have eq: "S = ?A \<union> ?B" using a by blast
```
```   296     have dj: "?A \<inter> ?B = {}" by simp
```
```   297     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```   298     have "F ?f S = F ?f ?A * F ?f ?B"
```
```   299       using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]]
```
```   300       by simp
```
```   301     then have ?thesis using a by simp }
```
```   302   ultimately show ?thesis by blast
```
```   303 qed
```
```   304
```
```   305 lemma delta':
```
```   306   assumes fS: "finite S"
```
```   307   shows "F (\<lambda>k. if a = k then b k else 1) S = (if a \<in> S then b a else 1)"
```
```   308   using delta [OF fS, of a b, symmetric] by (auto intro: cong)
```
```   309
```
```   310 lemma If_cases:
```
```   311   fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
```
```   312   assumes fA: "finite A"
```
```   313   shows "F (\<lambda>x. if P x then h x else g x) A =
```
```   314     F h (A \<inter> {x. P x}) * F g (A \<inter> - {x. P x})"
```
```   315 proof -
```
```   316   have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}"
```
```   317           "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}"
```
```   318     by blast+
```
```   319   from fA
```
```   320   have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
```
```   321   let ?g = "\<lambda>x. if P x then h x else g x"
```
```   322   from union_disjoint [OF f a(2), of ?g] a(1)
```
```   323   show ?thesis
```
```   324     by (subst (1 2) cong) simp_all
```
```   325 qed
```
```   326
```
```   327 lemma cartesian_product:
```
```   328    "F (\<lambda>x. F (g x) B) A = F (split g) (A <*> B)"
```
```   329 apply (rule sym)
```
```   330 apply (cases "finite A")
```
```   331  apply (cases "finite B")
```
```   332   apply (simp add: Sigma)
```
```   333  apply (cases "A={}", simp)
```
```   334  apply simp
```
```   335 apply (auto intro: infinite dest: finite_cartesian_productD2)
```
```   336 apply (cases "B = {}") apply (auto intro: infinite dest: finite_cartesian_productD1)
```
```   337 done
```
```   338
```
```   339 end
```
```   340
```
```   341 notation times (infixl "*" 70)
```
```   342 notation Groups.one ("1")
```
```   343
```
```   344
```
```   345 subsection {* Generalized summation over a set *}
```
```   346
```
```   347 context comm_monoid_add
```
```   348 begin
```
```   349
```
```   350 definition setsum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
```
```   351 where
```
```   352   "setsum = comm_monoid_set.F plus 0"
```
```   353
```
```   354 sublocale setsum!: comm_monoid_set plus 0
```
```   355 where
```
```   356   "comm_monoid_set.F plus 0 = setsum"
```
```   357 proof -
```
```   358   show "comm_monoid_set plus 0" ..
```
```   359   then interpret setsum!: comm_monoid_set plus 0 .
```
```   360   from setsum_def show "comm_monoid_set.F plus 0 = setsum" by rule
```
```   361 qed
```
```   362
```
```   363 abbreviation
```
```   364   Setsum ("\<Sum>_" [1000] 999) where
```
```   365   "\<Sum>A \<equiv> setsum (%x. x) A"
```
```   366
```
```   367 end
```
```   368
```
```   369 text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
```
```   370 written @{text"\<Sum>x\<in>A. e"}. *}
```
```   371
```
```   372 syntax
```
```   373   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
```
```   374 syntax (xsymbols)
```
```   375   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```   376 syntax (HTML output)
```
```   377   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```   378
```
```   379 translations -- {* Beware of argument permutation! *}
```
```   380   "SUM i:A. b" == "CONST setsum (%i. b) A"
```
```   381   "\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A"
```
```   382
```
```   383 text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
```
```   384  @{text"\<Sum>x|P. e"}. *}
```
```   385
```
```   386 syntax
```
```   387   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
```
```   388 syntax (xsymbols)
```
```   389   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   390 syntax (HTML output)
```
```   391   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   392
```
```   393 translations
```
```   394   "SUM x|P. t" => "CONST setsum (%x. t) {x. P}"
```
```   395   "\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}"
```
```   396
```
```   397 print_translation {*
```
```   398 let
```
```   399   fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) \$ Abs (y, Ty, P)] =
```
```   400         if x <> y then raise Match
```
```   401         else
```
```   402           let
```
```   403             val x' = Syntax_Trans.mark_bound_body (x, Tx);
```
```   404             val t' = subst_bound (x', t);
```
```   405             val P' = subst_bound (x', P);
```
```   406           in
```
```   407             Syntax.const @{syntax_const "_qsetsum"} \$ Syntax_Trans.mark_bound_abs (x, Tx) \$ P' \$ t'
```
```   408           end
```
```   409     | setsum_tr' _ = raise Match;
```
```   410 in [(@{const_syntax setsum}, K setsum_tr')] end
```
```   411 *}
```
```   412
```
```   413 text {* TODO These are candidates for generalization *}
```
```   414
```
```   415 context comm_monoid_add
```
```   416 begin
```
```   417
```
```   418 lemma setsum_reindex_id:
```
```   419   "inj_on f B \<Longrightarrow> setsum f B = setsum id (f ` B)"
```
```   420   by (simp add: setsum.reindex)
```
```   421
```
```   422 lemma setsum_reindex_nonzero:
```
```   423   assumes fS: "finite S"
```
```   424   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"
```
```   425   shows "setsum h (f ` S) = setsum (h \<circ> f) S"
```
```   426 proof (subst setsum.reindex_bij_betw_not_neutral[symmetric])
```
```   427   show "bij_betw f (S - {x\<in>S. h (f x) = 0}) (f`S - f`{x\<in>S. h (f x) = 0})"
```
```   428     using nz by (auto intro!: inj_onI simp: bij_betw_def)
```
```   429 qed (insert fS, auto)
```
```   430
```
```   431 lemma setsum_cong2:
```
```   432   "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> setsum f A = setsum g A"
```
```   433   by (auto intro: setsum.cong)
```
```   434
```
```   435 lemma setsum_reindex_cong:
```
```   436    "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|]
```
```   437     ==> setsum h B = setsum g A"
```
```   438   by (simp add: setsum.reindex)
```
```   439
```
```   440 lemma setsum_restrict_set:
```
```   441   assumes fA: "finite A"
```
```   442   shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"
```
```   443 proof-
```
```   444   from fA have fab: "finite (A \<inter> B)" by auto
```
```   445   have aba: "A \<inter> B \<subseteq> A" by blast
```
```   446   let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"
```
```   447   from setsum.mono_neutral_left [OF fA aba, of ?g]
```
```   448   show ?thesis by simp
```
```   449 qed
```
```   450
```
```   451 lemma setsum_Union_disjoint:
```
```   452   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"
```
```   453   shows "setsum f (Union C) = setsum (setsum f) C"
```
```   454   using assms by (fact setsum.Union_disjoint)
```
```   455
```
```   456 lemma setsum_cartesian_product:
```
```   457   "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
```
```   458   by (fact setsum.cartesian_product)
```
```   459
```
```   460 lemma setsum_UNION_zero:
```
```   461   assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"
```
```   462   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"
```
```   463   shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"
```
```   464   using fSS f0
```
```   465 proof(induct rule: finite_induct[OF fS])
```
```   466   case 1 thus ?case by simp
```
```   467 next
```
```   468   case (2 T F)
```
```   469   then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F"
```
```   470     and H: "setsum f (\<Union> F) = setsum (setsum f) F" by auto
```
```   471   from fTF have fUF: "finite (\<Union>F)" by auto
```
```   472   from "2.prems" TF fTF
```
```   473   show ?case
```
```   474     by (auto simp add: H [symmetric] intro: setsum.union_inter_neutral [OF fTF(1) fUF, of f])
```
```   475 qed
```
```   476
```
```   477 text {* Commuting outer and inner summation *}
```
```   478
```
```   479 lemma setsum_commute:
```
```   480   "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
```
```   481   unfolding setsum_cartesian_product
```
```   482   by (rule setsum.reindex_bij_witness[where i="\<lambda>(i, j). (j, i)" and j="\<lambda>(i, j). (j, i)"]) auto
```
```   483
```
```   484 lemma setsum_Plus:
```
```   485   fixes A :: "'a set" and B :: "'b set"
```
```   486   assumes fin: "finite A" "finite B"
```
```   487   shows "setsum f (A <+> B) = setsum (f \<circ> Inl) A + setsum (f \<circ> Inr) B"
```
```   488 proof -
```
```   489   have "A <+> B = Inl ` A \<union> Inr ` B" by auto
```
```   490   moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"
```
```   491     by auto
```
```   492   moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto
```
```   493   moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)
```
```   494   ultimately show ?thesis using fin by(simp add: setsum.union_disjoint setsum.reindex)
```
```   495 qed
```
```   496
```
```   497 end
```
```   498
```
```   499 text {* TODO These are legacy *}
```
```   500
```
```   501 lemma setsum_empty:
```
```   502   "setsum f {} = 0"
```
```   503   by (fact setsum.empty)
```
```   504
```
```   505 lemma setsum_insert:
```
```   506   "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
```
```   507   by (fact setsum.insert)
```
```   508
```
```   509 lemma setsum_infinite:
```
```   510   "~ finite A ==> setsum f A = 0"
```
```   511   by (fact setsum.infinite)
```
```   512
```
```   513 lemma setsum_reindex:
```
```   514   "inj_on f B \<Longrightarrow> setsum h (f ` B) = setsum (h \<circ> f) B"
```
```   515   by (fact setsum.reindex)
```
```   516
```
```   517 lemma setsum_cong:
```
```   518   "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
```
```   519   by (fact setsum.cong)
```
```   520
```
```   521 lemma strong_setsum_cong:
```
```   522   "A = B ==> (!!x. x:B =simp=> f x = g x)
```
```   523    ==> setsum (%x. f x) A = setsum (%x. g x) B"
```
```   524   by (fact setsum.strong_cong)
```
```   525
```
```   526 lemmas setsum_0 = setsum.neutral_const
```
```   527 lemmas setsum_0' = setsum.neutral
```
```   528
```
```   529 lemma setsum_Un_Int: "finite A ==> finite B ==>
```
```   530   setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
```
```   531   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
```
```   532   by (fact setsum.union_inter)
```
```   533
```
```   534 lemma setsum_Un_disjoint: "finite A ==> finite B
```
```   535   ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
```
```   536   by (fact setsum.union_disjoint)
```
```   537
```
```   538 lemma setsum_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
```
```   539     setsum f A = setsum f (A - B) + setsum f B"
```
```   540   by (fact setsum.subset_diff)
```
```   541
```
```   542 lemma setsum_mono_zero_left:
```
```   543   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 0 \<rbrakk> \<Longrightarrow> setsum f S = setsum f T"
```
```   544   by (fact setsum.mono_neutral_left)
```
```   545
```
```   546 lemmas setsum_mono_zero_right = setsum.mono_neutral_right
```
```   547
```
```   548 lemma setsum_mono_zero_cong_left:
```
```   549   "\<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>
```
```   550   \<Longrightarrow> setsum f S = setsum g T"
```
```   551   by (fact setsum.mono_neutral_cong_left)
```
```   552
```
```   553 lemmas setsum_mono_zero_cong_right = setsum.mono_neutral_cong_right
```
```   554
```
```   555 lemma setsum_delta: "finite S \<Longrightarrow>
```
```   556   setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
```
```   557   by (fact setsum.delta)
```
```   558
```
```   559 lemma setsum_delta': "finite S \<Longrightarrow>
```
```   560   setsum (\<lambda>k. if a = k then b k else 0) S = (if a\<in> S then b a else 0)"
```
```   561   by (fact setsum.delta')
```
```   562
```
```   563 lemma setsum_cases:
```
```   564   assumes "finite A"
```
```   565   shows "setsum (\<lambda>x. if P x then f x else g x) A =
```
```   566          setsum f (A \<inter> {x. P x}) + setsum g (A \<inter> - {x. P x})"
```
```   567   using assms by (fact setsum.If_cases)
```
```   568
```
```   569 (*But we can't get rid of finite I. If infinite, although the rhs is 0,
```
```   570   the lhs need not be, since UNION I A could still be finite.*)
```
```   571 lemma setsum_UN_disjoint:
```
```   572   assumes "finite I" and "ALL i:I. finite (A i)"
```
```   573     and "ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}"
```
```   574   shows "setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
```
```   575   using assms by (fact setsum.UNION_disjoint)
```
```   576
```
```   577 (*But we can't get rid of finite A. If infinite, although the lhs is 0,
```
```   578   the rhs need not be, since SIGMA A B could still be finite.*)
```
```   579 lemma setsum_Sigma:
```
```   580   assumes "finite A" and  "ALL x:A. finite (B x)"
```
```   581   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)"
```
```   582   using assms by (fact setsum.Sigma)
```
```   583
```
```   584 lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
```
```   585   by (fact setsum.distrib)
```
```   586
```
```   587 lemma setsum_Un_zero:
```
```   588   "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 0 \<rbrakk> \<Longrightarrow>
```
```   589   setsum f (S \<union> T) = setsum f S + setsum f T"
```
```   590   by (fact setsum.union_inter_neutral)
```
```   591
```
```   592 subsubsection {* Properties in more restricted classes of structures *}
```
```   593
```
```   594 lemma setsum_Un: "finite A ==> finite B ==>
```
```   595   (setsum f (A Un B) :: 'a :: ab_group_add) =
```
```   596    setsum f A + setsum f B - setsum f (A Int B)"
```
```   597 by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
```
```   598
```
```   599 lemma setsum_Un2:
```
```   600   assumes "finite (A \<union> B)"
```
```   601   shows "setsum f (A \<union> B) = setsum f (A - B) + setsum f (B - A) + setsum f (A \<inter> B)"
```
```   602 proof -
```
```   603   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
```
```   604     by auto
```
```   605   with assms show ?thesis by simp (subst setsum_Un_disjoint, auto)+
```
```   606 qed
```
```   607
```
```   608 lemma setsum_diff1: "finite A \<Longrightarrow>
```
```   609   (setsum f (A - {a}) :: ('a::ab_group_add)) =
```
```   610   (if a:A then setsum f A - f a else setsum f A)"
```
```   611 by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```   612
```
```   613 lemma setsum_diff:
```
```   614   assumes le: "finite A" "B \<subseteq> A"
```
```   615   shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
```
```   616 proof -
```
```   617   from le have finiteB: "finite B" using finite_subset by auto
```
```   618   show ?thesis using finiteB le
```
```   619   proof induct
```
```   620     case empty
```
```   621     thus ?case by auto
```
```   622   next
```
```   623     case (insert x F)
```
```   624     thus ?case using le finiteB
```
```   625       by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
```
```   626   qed
```
```   627 qed
```
```   628
```
```   629 lemma setsum_mono:
```
```   630   assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))"
```
```   631   shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
```
```   632 proof (cases "finite K")
```
```   633   case True
```
```   634   thus ?thesis using le
```
```   635   proof induct
```
```   636     case empty
```
```   637     thus ?case by simp
```
```   638   next
```
```   639     case insert
```
```   640     thus ?case using add_mono by fastforce
```
```   641   qed
```
```   642 next
```
```   643   case False then show ?thesis by simp
```
```   644 qed
```
```   645
```
```   646 lemma setsum_strict_mono:
```
```   647   fixes f :: "'a \<Rightarrow> 'b::{ordered_cancel_ab_semigroup_add,comm_monoid_add}"
```
```   648   assumes "finite A"  "A \<noteq> {}"
```
```   649     and "!!x. x:A \<Longrightarrow> f x < g x"
```
```   650   shows "setsum f A < setsum g A"
```
```   651   using assms
```
```   652 proof (induct rule: finite_ne_induct)
```
```   653   case singleton thus ?case by simp
```
```   654 next
```
```   655   case insert thus ?case by (auto simp: add_strict_mono)
```
```   656 qed
```
```   657
```
```   658 lemma setsum_strict_mono_ex1:
```
```   659 fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_cancel_ab_semigroup_add}"
```
```   660 assumes "finite A" and "ALL x:A. f x \<le> g x" and "EX a:A. f a < g a"
```
```   661 shows "setsum f A < setsum g A"
```
```   662 proof-
```
```   663   from assms(3) obtain a where a: "a:A" "f a < g a" by blast
```
```   664   have "setsum f A = setsum f ((A-{a}) \<union> {a})"
```
```   665     by(simp add:insert_absorb[OF `a:A`])
```
```   666   also have "\<dots> = setsum f (A-{a}) + setsum f {a}"
```
```   667     using `finite A` by(subst setsum_Un_disjoint) auto
```
```   668   also have "setsum f (A-{a}) \<le> setsum g (A-{a})"
```
```   669     by(rule setsum_mono)(simp add: assms(2))
```
```   670   also have "setsum f {a} < setsum g {a}" using a by simp
```
```   671   also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"
```
```   672     using `finite A` by(subst setsum_Un_disjoint[symmetric]) auto
```
```   673   also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF `a:A`])
```
```   674   finally show ?thesis by (auto simp add: add_right_mono add_strict_left_mono)
```
```   675 qed
```
```   676
```
```   677 lemma setsum_negf:
```
```   678   "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
```
```   679 proof (cases "finite A")
```
```   680   case True thus ?thesis by (induct set: finite) auto
```
```   681 next
```
```   682   case False thus ?thesis by simp
```
```   683 qed
```
```   684
```
```   685 lemma setsum_subtractf:
```
```   686   "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
```
```   687     setsum f A - setsum g A"
```
```   688   using setsum_addf [of f "- g" A] by (simp add: setsum_negf)
```
```   689
```
```   690 lemma setsum_nonneg:
```
```   691   assumes nn: "\<forall>x\<in>A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
```
```   692   shows "0 \<le> setsum f A"
```
```   693 proof (cases "finite A")
```
```   694   case True thus ?thesis using nn
```
```   695   proof induct
```
```   696     case empty then show ?case by simp
```
```   697   next
```
```   698     case (insert x F)
```
```   699     then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
```
```   700     with insert show ?case by simp
```
```   701   qed
```
```   702 next
```
```   703   case False thus ?thesis by simp
```
```   704 qed
```
```   705
```
```   706 lemma setsum_nonpos:
```
```   707   assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})"
```
```   708   shows "setsum f A \<le> 0"
```
```   709 proof (cases "finite A")
```
```   710   case True thus ?thesis using np
```
```   711   proof induct
```
```   712     case empty then show ?case by simp
```
```   713   next
```
```   714     case (insert x F)
```
```   715     then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
```
```   716     with insert show ?case by simp
```
```   717   qed
```
```   718 next
```
```   719   case False thus ?thesis by simp
```
```   720 qed
```
```   721
```
```   722 lemma setsum_nonneg_leq_bound:
```
```   723   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
```
```   724   assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"
```
```   725   shows "f i \<le> B"
```
```   726 proof -
```
```   727   have "0 \<le> (\<Sum> i \<in> s - {i}. f i)" and "0 \<le> f i"
```
```   728     using assms by (auto intro!: setsum_nonneg)
```
```   729   moreover
```
```   730   have "(\<Sum> i \<in> s - {i}. f i) + f i = B"
```
```   731     using assms by (simp add: setsum_diff1)
```
```   732   ultimately show ?thesis by auto
```
```   733 qed
```
```   734
```
```   735 lemma setsum_nonneg_0:
```
```   736   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
```
```   737   assumes "finite s" and pos: "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"
```
```   738   and "(\<Sum> i \<in> s. f i) = 0" and i: "i \<in> s"
```
```   739   shows "f i = 0"
```
```   740   using setsum_nonneg_leq_bound[OF assms] pos[OF i] by auto
```
```   741
```
```   742 lemma setsum_mono2:
```
```   743 fixes f :: "'a \<Rightarrow> 'b :: ordered_comm_monoid_add"
```
```   744 assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
```
```   745 shows "setsum f A \<le> setsum f B"
```
```   746 proof -
```
```   747   have "setsum f A \<le> setsum f A + setsum f (B-A)"
```
```   748     by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
```
```   749   also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
```
```   750     by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
```
```   751   also have "A \<union> (B-A) = B" using sub by blast
```
```   752   finally show ?thesis .
```
```   753 qed
```
```   754
```
```   755 lemma setsum_mono3: "finite B ==> A <= B ==>
```
```   756     ALL x: B - A.
```
```   757       0 <= ((f x)::'a::{comm_monoid_add,ordered_ab_semigroup_add}) ==>
```
```   758         setsum f A <= setsum f B"
```
```   759   apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
```
```   760   apply (erule ssubst)
```
```   761   apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
```
```   762   apply simp
```
```   763   apply (rule add_left_mono)
```
```   764   apply (erule setsum_nonneg)
```
```   765   apply (subst setsum_Un_disjoint [THEN sym])
```
```   766   apply (erule finite_subset, assumption)
```
```   767   apply (rule finite_subset)
```
```   768   prefer 2
```
```   769   apply assumption
```
```   770   apply (auto simp add: sup_absorb2)
```
```   771 done
```
```   772
```
```   773 lemma setsum_right_distrib:
```
```   774   fixes f :: "'a => ('b::semiring_0)"
```
```   775   shows "r * setsum f A = setsum (%n. r * f n) A"
```
```   776 proof (cases "finite A")
```
```   777   case True
```
```   778   thus ?thesis
```
```   779   proof induct
```
```   780     case empty thus ?case by simp
```
```   781   next
```
```   782     case (insert x A) thus ?case by (simp add: distrib_left)
```
```   783   qed
```
```   784 next
```
```   785   case False thus ?thesis by simp
```
```   786 qed
```
```   787
```
```   788 lemma setsum_left_distrib:
```
```   789   "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
```
```   790 proof (cases "finite A")
```
```   791   case True
```
```   792   then show ?thesis
```
```   793   proof induct
```
```   794     case empty thus ?case by simp
```
```   795   next
```
```   796     case (insert x A) thus ?case by (simp add: distrib_right)
```
```   797   qed
```
```   798 next
```
```   799   case False thus ?thesis by simp
```
```   800 qed
```
```   801
```
```   802 lemma setsum_divide_distrib:
```
```   803   "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
```
```   804 proof (cases "finite A")
```
```   805   case True
```
```   806   then show ?thesis
```
```   807   proof induct
```
```   808     case empty thus ?case by simp
```
```   809   next
```
```   810     case (insert x A) thus ?case by (simp add: add_divide_distrib)
```
```   811   qed
```
```   812 next
```
```   813   case False thus ?thesis by simp
```
```   814 qed
```
```   815
```
```   816 lemma setsum_abs[iff]:
```
```   817   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   818   shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
```
```   819 proof (cases "finite A")
```
```   820   case True
```
```   821   thus ?thesis
```
```   822   proof induct
```
```   823     case empty thus ?case by simp
```
```   824   next
```
```   825     case (insert x A)
```
```   826     thus ?case by (auto intro: abs_triangle_ineq order_trans)
```
```   827   qed
```
```   828 next
```
```   829   case False thus ?thesis by simp
```
```   830 qed
```
```   831
```
```   832 lemma setsum_abs_ge_zero[iff]:
```
```   833   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   834   shows "0 \<le> setsum (%i. abs(f i)) A"
```
```   835 proof (cases "finite A")
```
```   836   case True
```
```   837   thus ?thesis
```
```   838   proof induct
```
```   839     case empty thus ?case by simp
```
```   840   next
```
```   841     case (insert x A) thus ?case by auto
```
```   842   qed
```
```   843 next
```
```   844   case False thus ?thesis by simp
```
```   845 qed
```
```   846
```
```   847 lemma abs_setsum_abs[simp]:
```
```   848   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   849   shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
```
```   850 proof (cases "finite A")
```
```   851   case True
```
```   852   thus ?thesis
```
```   853   proof induct
```
```   854     case empty thus ?case by simp
```
```   855   next
```
```   856     case (insert a A)
```
```   857     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
```
```   858     also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
```
```   859     also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
```
```   860       by (simp del: abs_of_nonneg)
```
```   861     also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
```
```   862     finally show ?case .
```
```   863   qed
```
```   864 next
```
```   865   case False thus ?thesis by simp
```
```   866 qed
```
```   867
```
```   868 lemma setsum_diff1'[rule_format]:
```
```   869   "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
```
```   870 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))"])
```
```   871 apply (auto simp add: insert_Diff_if add_ac)
```
```   872 done
```
```   873
```
```   874 lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
```
```   875   shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
```
```   876 unfolding setsum_diff1'[OF assms] by auto
```
```   877
```
```   878 lemma setsum_product:
```
```   879   fixes f :: "'a => ('b::semiring_0)"
```
```   880   shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
```
```   881   by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)
```
```   882
```
```   883 lemma setsum_mult_setsum_if_inj:
```
```   884 fixes f :: "'a => ('b::semiring_0)"
```
```   885 shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
```
```   886   setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
```
```   887 by(auto simp: setsum_product setsum_cartesian_product
```
```   888         intro!:  setsum_reindex_cong[symmetric])
```
```   889
```
```   890 lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
```
```   891 apply (case_tac "finite A")
```
```   892  prefer 2 apply simp
```
```   893 apply (erule rev_mp)
```
```   894 apply (erule finite_induct, auto)
```
```   895 done
```
```   896
```
```   897 lemma setsum_eq_0_iff [simp]:
```
```   898   "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
```
```   899   by (induct set: finite) auto
```
```   900
```
```   901 lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
```
```   902   setsum f A = Suc 0 \<longleftrightarrow> (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
```
```   903 apply(erule finite_induct)
```
```   904 apply (auto simp add:add_is_1)
```
```   905 done
```
```   906
```
```   907 lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
```
```   908
```
```   909 lemma setsum_Un_nat: "finite A ==> finite B ==>
```
```   910   (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
```
```   911   -- {* For the natural numbers, we have subtraction. *}
```
```   912 by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
```
```   913
```
```   914 lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
```
```   915   (if a:A then setsum f A - f a else setsum f A)"
```
```   916 apply (case_tac "finite A")
```
```   917  prefer 2 apply simp
```
```   918 apply (erule finite_induct)
```
```   919  apply (auto simp add: insert_Diff_if)
```
```   920 apply (drule_tac a = a in mk_disjoint_insert, auto)
```
```   921 done
```
```   922
```
```   923 lemma setsum_diff_nat:
```
```   924 assumes "finite B" and "B \<subseteq> A"
```
```   925 shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
```
```   926 using assms
```
```   927 proof induct
```
```   928   show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
```
```   929 next
```
```   930   fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
```
```   931     and xFinA: "insert x F \<subseteq> A"
```
```   932     and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
```
```   933   from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
```
```   934   from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
```
```   935     by (simp add: setsum_diff1_nat)
```
```   936   from xFinA have "F \<subseteq> A" by simp
```
```   937   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
```
```   938   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
```
```   939     by simp
```
```   940   from xnotinF have "A - insert x F = (A - F) - {x}" by auto
```
```   941   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
```
```   942     by simp
```
```   943   from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
```
```   944   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
```
```   945     by simp
```
```   946   thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
```
```   947 qed
```
```   948
```
```   949 lemma setsum_comp_morphism:
```
```   950   assumes "h 0 = 0" and "\<And>x y. h (x + y) = h x + h y"
```
```   951   shows "setsum (h \<circ> g) A = h (setsum g A)"
```
```   952 proof (cases "finite A")
```
```   953   case False then show ?thesis by (simp add: assms)
```
```   954 next
```
```   955   case True then show ?thesis by (induct A) (simp_all add: assms)
```
```   956 qed
```
```   957
```
```   958
```
```   959 subsubsection {* Cardinality as special case of @{const setsum} *}
```
```   960
```
```   961 lemma card_eq_setsum:
```
```   962   "card A = setsum (\<lambda>x. 1) A"
```
```   963 proof -
```
```   964   have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)"
```
```   965     by (simp add: fun_eq_iff)
```
```   966   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) = Finite_Set.fold (\<lambda>_. Suc)"
```
```   967     by (rule arg_cong)
```
```   968   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
```
```   969     by (blast intro: fun_cong)
```
```   970   then show ?thesis by (simp add: card.eq_fold setsum.eq_fold)
```
```   971 qed
```
```   972
```
```   973 lemma setsum_constant [simp]:
```
```   974   "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
```
```   975 apply (cases "finite A")
```
```   976 apply (erule finite_induct)
```
```   977 apply (auto simp add: algebra_simps)
```
```   978 done
```
```   979
```
```   980 lemma setsum_bounded:
```
```   981   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"
```
```   982   shows "setsum f A \<le> of_nat (card A) * K"
```
```   983 proof (cases "finite A")
```
```   984   case True
```
```   985   thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
```
```   986 next
```
```   987   case False thus ?thesis by simp
```
```   988 qed
```
```   989
```
```   990 lemma card_UN_disjoint:
```
```   991   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
```
```   992     and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
```
```   993   shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
```
```   994 proof -
```
```   995   have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)" by simp
```
```   996   with assms show ?thesis by (simp add: card_eq_setsum setsum_UN_disjoint del: setsum_constant)
```
```   997 qed
```
```   998
```
```   999 lemma card_Union_disjoint:
```
```  1000   "finite C ==> (ALL A:C. finite A) ==>
```
```  1001    (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {})
```
```  1002    ==> card (Union C) = setsum card C"
```
```  1003 apply (frule card_UN_disjoint [of C id])
```
```  1004 apply simp_all
```
```  1005 done
```
```  1006
```
```  1007
```
```  1008 subsubsection {* Cardinality of products *}
```
```  1009
```
```  1010 lemma card_SigmaI [simp]:
```
```  1011   "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
```
```  1012   \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
```
```  1013 by(simp add: card_eq_setsum setsum_Sigma del:setsum_constant)
```
```  1014
```
```  1015 (*
```
```  1016 lemma SigmaI_insert: "y \<notin> A ==>
```
```  1017   (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
```
```  1018   by auto
```
```  1019 *)
```
```  1020
```
```  1021 lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
```
```  1022   by (cases "finite A \<and> finite B")
```
```  1023     (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1024
```
```  1025 lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
```
```  1026 by (simp add: card_cartesian_product)
```
```  1027
```
```  1028
```
```  1029 subsection {* Generalized product over a set *}
```
```  1030
```
```  1031 context comm_monoid_mult
```
```  1032 begin
```
```  1033
```
```  1034 definition setprod :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
```
```  1035 where
```
```  1036   "setprod = comm_monoid_set.F times 1"
```
```  1037
```
```  1038 sublocale setprod!: comm_monoid_set times 1
```
```  1039 where
```
```  1040   "comm_monoid_set.F times 1 = setprod"
```
```  1041 proof -
```
```  1042   show "comm_monoid_set times 1" ..
```
```  1043   then interpret setprod!: comm_monoid_set times 1 .
```
```  1044   from setprod_def show "comm_monoid_set.F times 1 = setprod" by rule
```
```  1045 qed
```
```  1046
```
```  1047 abbreviation
```
```  1048   Setprod ("\<Prod>_" [1000] 999) where
```
```  1049   "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"
```
```  1050
```
```  1051 end
```
```  1052
```
```  1053 syntax
```
```  1054   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
```
```  1055 syntax (xsymbols)
```
```  1056   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```  1057 syntax (HTML output)
```
```  1058   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```  1059
```
```  1060 translations -- {* Beware of argument permutation! *}
```
```  1061   "PROD i:A. b" == "CONST setprod (%i. b) A"
```
```  1062   "\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A"
```
```  1063
```
```  1064 text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
```
```  1065  @{text"\<Prod>x|P. e"}. *}
```
```  1066
```
```  1067 syntax
```
```  1068   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
```
```  1069 syntax (xsymbols)
```
```  1070   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```  1071 syntax (HTML output)
```
```  1072   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```  1073
```
```  1074 translations
```
```  1075   "PROD x|P. t" => "CONST setprod (%x. t) {x. P}"
```
```  1076   "\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"
```
```  1077
```
```  1078 text {* TODO These are candidates for generalization *}
```
```  1079
```
```  1080 context comm_monoid_mult
```
```  1081 begin
```
```  1082
```
```  1083 lemma setprod_reindex_id:
```
```  1084   "inj_on f B ==> setprod f B = setprod id (f ` B)"
```
```  1085   by (auto simp add: setprod.reindex)
```
```  1086
```
```  1087 lemma setprod_reindex_cong:
```
```  1088   "inj_on f A ==> B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
```
```  1089   by (frule setprod.reindex, simp)
```
```  1090
```
```  1091 lemma strong_setprod_reindex_cong:
```
```  1092   "inj_on f A \<Longrightarrow> B = f ` A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x) \<Longrightarrow> setprod h B = setprod g A"
```
```  1093   by (subst setprod.reindex_bij_betw[symmetric, where h=f])
```
```  1094      (auto simp: bij_betw_def)
```
```  1095
```
```  1096 lemma setprod_Union_disjoint:
```
```  1097   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"
```
```  1098   shows "setprod f (Union C) = setprod (setprod f) C"
```
```  1099   using assms by (fact setprod.Union_disjoint)
```
```  1100
```
```  1101 text{*Here we can eliminate the finiteness assumptions, by cases.*}
```
```  1102 lemma setprod_cartesian_product:
```
```  1103   "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
```
```  1104   by (fact setprod.cartesian_product)
```
```  1105
```
```  1106 lemma setprod_Un2:
```
```  1107   assumes "finite (A \<union> B)"
```
```  1108   shows "setprod f (A \<union> B) = setprod f (A - B) * setprod f (B - A) * setprod f (A \<inter> B)"
```
```  1109 proof -
```
```  1110   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
```
```  1111     by auto
```
```  1112   with assms show ?thesis by simp (subst setprod.union_disjoint, auto)+
```
```  1113 qed
```
```  1114
```
```  1115 end
```
```  1116
```
```  1117 text {* TODO These are legacy *}
```
```  1118
```
```  1119 lemma setprod_empty: "setprod f {} = 1"
```
```  1120   by (fact setprod.empty)
```
```  1121
```
```  1122 lemma setprod_insert: "[| finite A; a \<notin> A |] ==>
```
```  1123     setprod f (insert a A) = f a * setprod f A"
```
```  1124   by (fact setprod.insert)
```
```  1125
```
```  1126 lemma setprod_infinite: "~ finite A ==> setprod f A = 1"
```
```  1127   by (fact setprod.infinite)
```
```  1128
```
```  1129 lemma setprod_reindex:
```
```  1130   "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
```
```  1131   by (fact setprod.reindex)
```
```  1132
```
```  1133 lemma setprod_cong:
```
```  1134   "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
```
```  1135   by (fact setprod.cong)
```
```  1136
```
```  1137 lemma strong_setprod_cong:
```
```  1138   "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
```
```  1139   by (fact setprod.strong_cong)
```
```  1140
```
```  1141 lemma setprod_Un_one:
```
```  1142   "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 1 \<rbrakk>
```
```  1143   \<Longrightarrow> setprod f (S \<union> T) = setprod f S  * setprod f T"
```
```  1144   by (fact setprod.union_inter_neutral)
```
```  1145
```
```  1146 lemmas setprod_1 = setprod.neutral_const
```
```  1147 lemmas setprod_1' = setprod.neutral
```
```  1148
```
```  1149 lemma setprod_Un_Int: "finite A ==> finite B
```
```  1150     ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
```
```  1151   by (fact setprod.union_inter)
```
```  1152
```
```  1153 lemma setprod_Un_disjoint: "finite A ==> finite B
```
```  1154   ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
```
```  1155   by (fact setprod.union_disjoint)
```
```  1156
```
```  1157 lemma setprod_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
```
```  1158     setprod f A = setprod f (A - B) * setprod f B"
```
```  1159   by (fact setprod.subset_diff)
```
```  1160
```
```  1161 lemma setprod_mono_one_left:
```
```  1162   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 1 \<rbrakk> \<Longrightarrow> setprod f S = setprod f T"
```
```  1163   by (fact setprod.mono_neutral_left)
```
```  1164
```
```  1165 lemmas setprod_mono_one_right = setprod.mono_neutral_right
```
```  1166
```
```  1167 lemma setprod_mono_one_cong_left:
```
```  1168   "\<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>
```
```  1169   \<Longrightarrow> setprod f S = setprod g T"
```
```  1170   by (fact setprod.mono_neutral_cong_left)
```
```  1171
```
```  1172 lemmas setprod_mono_one_cong_right = setprod.mono_neutral_cong_right
```
```  1173
```
```  1174 lemma setprod_delta: "finite S \<Longrightarrow>
```
```  1175   setprod (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"
```
```  1176   by (fact setprod.delta)
```
```  1177
```
```  1178 lemma setprod_delta': "finite S \<Longrightarrow>
```
```  1179   setprod (\<lambda>k. if a = k then b k else 1) S = (if a\<in> S then b a else 1)"
```
```  1180   by (fact setprod.delta')
```
```  1181
```
```  1182 lemma setprod_UN_disjoint:
```
```  1183     "finite I ==> (ALL i:I. finite (A i)) ==>
```
```  1184         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
```
```  1185       setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
```
```  1186   by (fact setprod.UNION_disjoint)
```
```  1187
```
```  1188 lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```  1189     (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
```
```  1190     (\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
```
```  1191   by (fact setprod.Sigma)
```
```  1192
```
```  1193 lemma setprod_timesf: "setprod (\<lambda>x. f x * g x) A = setprod f A * setprod g A"
```
```  1194   by (fact setprod.distrib)
```
```  1195
```
```  1196
```
```  1197 subsubsection {* Properties in more restricted classes of structures *}
```
```  1198
```
```  1199 lemma setprod_zero:
```
```  1200      "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0"
```
```  1201 apply (induct set: finite, force, clarsimp)
```
```  1202 apply (erule disjE, auto)
```
```  1203 done
```
```  1204
```
```  1205 lemma setprod_zero_iff[simp]: "finite A ==>
```
```  1206   (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) =
```
```  1207   (EX x: A. f x = 0)"
```
```  1208 by (erule finite_induct, auto simp:no_zero_divisors)
```
```  1209
```
```  1210 lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
```
```  1211   (setprod f (A Un B) :: 'a ::{field})
```
```  1212    = setprod f A * setprod f B / setprod f (A Int B)"
```
```  1213 by (subst setprod_Un_Int [symmetric], auto)
```
```  1214
```
```  1215 lemma setprod_nonneg [rule_format]:
```
```  1216    "(ALL x: A. (0::'a::linordered_semidom) \<le> f x) --> 0 \<le> setprod f A"
```
```  1217 by (cases "finite A", induct set: finite, simp_all)
```
```  1218
```
```  1219 lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::linordered_semidom) < f x)
```
```  1220   --> 0 < setprod f A"
```
```  1221 by (cases "finite A", induct set: finite, simp_all)
```
```  1222
```
```  1223 lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
```
```  1224   (setprod f (A - {a}) :: 'a :: {field}) =
```
```  1225   (if a:A then setprod f A / f a else setprod f A)"
```
```  1226   by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```  1227
```
```  1228 lemma setprod_inversef:
```
```  1229   fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
```
```  1230   shows "finite A ==> setprod (inverse \<circ> f) A = inverse (setprod f A)"
```
```  1231 by (erule finite_induct) auto
```
```  1232
```
```  1233 lemma setprod_dividef:
```
```  1234   fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
```
```  1235   shows "finite A
```
```  1236     ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
```
```  1237 apply (subgoal_tac
```
```  1238          "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
```
```  1239 apply (erule ssubst)
```
```  1240 apply (subst divide_inverse)
```
```  1241 apply (subst setprod_timesf)
```
```  1242 apply (subst setprod_inversef, assumption+, rule refl)
```
```  1243 apply (rule setprod_cong, rule refl)
```
```  1244 apply (subst divide_inverse, auto)
```
```  1245 done
```
```  1246
```
```  1247 lemma setprod_dvd_setprod [rule_format]:
```
```  1248     "(ALL x : A. f x dvd g x) \<longrightarrow> setprod f A dvd setprod g A"
```
```  1249   apply (cases "finite A")
```
```  1250   apply (induct set: finite)
```
```  1251   apply (auto simp add: dvd_def)
```
```  1252   apply (rule_tac x = "k * ka" in exI)
```
```  1253   apply (simp add: algebra_simps)
```
```  1254 done
```
```  1255
```
```  1256 lemma setprod_dvd_setprod_subset:
```
```  1257   "finite B \<Longrightarrow> A <= B \<Longrightarrow> setprod f A dvd setprod f B"
```
```  1258   apply (subgoal_tac "setprod f B = setprod f A * setprod f (B - A)")
```
```  1259   apply (unfold dvd_def, blast)
```
```  1260   apply (subst setprod_Un_disjoint [symmetric])
```
```  1261   apply (auto elim: finite_subset intro: setprod_cong)
```
```  1262 done
```
```  1263
```
```  1264 lemma setprod_dvd_setprod_subset2:
```
```  1265   "finite B \<Longrightarrow> A <= B \<Longrightarrow> ALL x : A. (f x::'a::comm_semiring_1) dvd g x \<Longrightarrow>
```
```  1266       setprod f A dvd setprod g B"
```
```  1267   apply (rule dvd_trans)
```
```  1268   apply (rule setprod_dvd_setprod, erule (1) bspec)
```
```  1269   apply (erule (1) setprod_dvd_setprod_subset)
```
```  1270 done
```
```  1271
```
```  1272 lemma dvd_setprod: "finite A \<Longrightarrow> i:A \<Longrightarrow>
```
```  1273     (f i ::'a::comm_semiring_1) dvd setprod f A"
```
```  1274 by (induct set: finite) (auto intro: dvd_mult)
```
```  1275
```
```  1276 lemma dvd_setsum [rule_format]: "(ALL i : A. d dvd f i) \<longrightarrow>
```
```  1277     (d::'a::comm_semiring_1) dvd (SUM x : A. f x)"
```
```  1278   apply (cases "finite A")
```
```  1279   apply (induct set: finite)
```
```  1280   apply auto
```
```  1281 done
```
```  1282
```
```  1283 lemma setprod_mono:
```
```  1284   fixes f :: "'a \<Rightarrow> 'b\<Colon>linordered_semidom"
```
```  1285   assumes "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i"
```
```  1286   shows "setprod f A \<le> setprod g A"
```
```  1287 proof (cases "finite A")
```
```  1288   case True
```
```  1289   hence ?thesis "setprod f A \<ge> 0" using subset_refl[of A]
```
```  1290   proof (induct A rule: finite_subset_induct)
```
```  1291     case (insert a F)
```
```  1292     thus "setprod f (insert a F) \<le> setprod g (insert a F)" "0 \<le> setprod f (insert a F)"
```
```  1293       unfolding setprod_insert[OF insert(1,3)]
```
```  1294       using assms[rule_format,OF insert(2)] insert
```
```  1295       by (auto intro: mult_mono)
```
```  1296   qed auto
```
```  1297   thus ?thesis by simp
```
```  1298 qed auto
```
```  1299
```
```  1300 lemma abs_setprod:
```
```  1301   fixes f :: "'a \<Rightarrow> 'b\<Colon>{linordered_field,abs}"
```
```  1302   shows "abs (setprod f A) = setprod (\<lambda>x. abs (f x)) A"
```
```  1303 proof (cases "finite A")
```
```  1304   case True thus ?thesis
```
```  1305     by induct (auto simp add: field_simps abs_mult)
```
```  1306 qed auto
```
```  1307
```
```  1308 lemma setprod_eq_1_iff [simp]:
```
```  1309   "finite F ==> setprod f F = 1 \<longleftrightarrow> (ALL a:F. f a = (1::nat))"
```
```  1310   by (induct set: finite) auto
```
```  1311
```
```  1312 lemma setprod_pos_nat:
```
```  1313   "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0"
```
```  1314 using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
```
```  1315
```
```  1316 lemma setprod_pos_nat_iff[simp]:
```
```  1317   "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))"
```
```  1318 using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
```
```  1319
```
```  1320 end
```