src/HOL/Groups_Big.thy
 author wenzelm Sun Sep 18 20:33:48 2016 +0200 (2016-09-18) changeset 63915 bab633745c7f parent 63654 f90e3926e627 child 63918 6bf55e6e0b75 permissions -rw-r--r--
tuned proofs;
```     1 (*  Title:      HOL/Groups_Big.thy
```
```     2     Author:     Tobias Nipkow
```
```     3     Author:     Lawrence C Paulson
```
```     4     Author:     Markus Wenzel
```
```     5     Author:     Jeremy Avigad
```
```     6 *)
```
```     7
```
```     8 section \<open>Big sum and product over finite (non-empty) sets\<close>
```
```     9
```
```    10 theory Groups_Big
```
```    11   imports Finite_Set Power
```
```    12 begin
```
```    13
```
```    14 subsection \<open>Generic monoid operation over a set\<close>
```
```    15
```
```    16 locale comm_monoid_set = comm_monoid
```
```    17 begin
```
```    18
```
```    19 interpretation comp_fun_commute f
```
```    20   by standard (simp add: fun_eq_iff left_commute)
```
```    21
```
```    22 interpretation comp?: comp_fun_commute "f \<circ> g"
```
```    23   by (fact comp_comp_fun_commute)
```
```    24
```
```    25 definition F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
```
```    26   where eq_fold: "F g A = Finite_Set.fold (f \<circ> g) \<^bold>1 A"
```
```    27
```
```    28 lemma infinite [simp]: "\<not> finite A \<Longrightarrow> F g A = \<^bold>1"
```
```    29   by (simp add: eq_fold)
```
```    30
```
```    31 lemma empty [simp]: "F g {} = \<^bold>1"
```
```    32   by (simp add: eq_fold)
```
```    33
```
```    34 lemma insert [simp]: "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> F g (insert x A) = g x \<^bold>* F g A"
```
```    35   by (simp add: eq_fold)
```
```    36
```
```    37 lemma remove:
```
```    38   assumes "finite A" and "x \<in> A"
```
```    39   shows "F g A = g x \<^bold>* F g (A - {x})"
```
```    40 proof -
```
```    41   from \<open>x \<in> A\<close> obtain B where B: "A = insert x B" and "x \<notin> B"
```
```    42     by (auto dest: mk_disjoint_insert)
```
```    43   moreover from \<open>finite A\<close> B have "finite B" by simp
```
```    44   ultimately show ?thesis by simp
```
```    45 qed
```
```    46
```
```    47 lemma insert_remove: "finite A \<Longrightarrow> F g (insert x A) = g x \<^bold>* F g (A - {x})"
```
```    48   by (cases "x \<in> A") (simp_all add: remove insert_absorb)
```
```    49
```
```    50 lemma neutral: "\<forall>x\<in>A. g x = \<^bold>1 \<Longrightarrow> F g A = \<^bold>1"
```
```    51   by (induct A rule: infinite_finite_induct) simp_all
```
```    52
```
```    53 lemma neutral_const [simp]: "F (\<lambda>_. \<^bold>1) A = \<^bold>1"
```
```    54   by (simp add: neutral)
```
```    55
```
```    56 lemma union_inter:
```
```    57   assumes "finite A" and "finite B"
```
```    58   shows "F g (A \<union> B) \<^bold>* F g (A \<inter> B) = F g A \<^bold>* F g B"
```
```    59   \<comment> \<open>The reversed orientation looks more natural, but LOOPS as a simprule!\<close>
```
```    60   using assms
```
```    61 proof (induct A)
```
```    62   case empty
```
```    63   then show ?case by simp
```
```    64 next
```
```    65   case (insert x A)
```
```    66   then show ?case
```
```    67     by (auto simp: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
```
```    68 qed
```
```    69
```
```    70 corollary union_inter_neutral:
```
```    71   assumes "finite A" and "finite B"
```
```    72     and "\<forall>x \<in> A \<inter> B. g x = \<^bold>1"
```
```    73   shows "F g (A \<union> B) = F g A \<^bold>* F g B"
```
```    74   using assms by (simp add: union_inter [symmetric] neutral)
```
```    75
```
```    76 corollary union_disjoint:
```
```    77   assumes "finite A" and "finite B"
```
```    78   assumes "A \<inter> B = {}"
```
```    79   shows "F g (A \<union> B) = F g A \<^bold>* F g B"
```
```    80   using assms by (simp add: union_inter_neutral)
```
```    81
```
```    82 lemma union_diff2:
```
```    83   assumes "finite A" and "finite B"
```
```    84   shows "F g (A \<union> B) = F g (A - B) \<^bold>* F g (B - A) \<^bold>* F g (A \<inter> B)"
```
```    85 proof -
```
```    86   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
```
```    87     by auto
```
```    88   with assms show ?thesis
```
```    89     by simp (subst union_disjoint, auto)+
```
```    90 qed
```
```    91
```
```    92 lemma subset_diff:
```
```    93   assumes "B \<subseteq> A" and "finite A"
```
```    94   shows "F g A = F g (A - B) \<^bold>* F g B"
```
```    95 proof -
```
```    96   from assms have "finite (A - B)" by auto
```
```    97   moreover from assms have "finite B" by (rule finite_subset)
```
```    98   moreover from assms have "(A - B) \<inter> B = {}" by auto
```
```    99   ultimately have "F g (A - B \<union> B) = F g (A - B) \<^bold>* F g B" by (rule union_disjoint)
```
```   100   moreover from assms have "A \<union> B = A" by auto
```
```   101   ultimately show ?thesis by simp
```
```   102 qed
```
```   103
```
```   104 lemma setdiff_irrelevant:
```
```   105   assumes "finite A"
```
```   106   shows "F g (A - {x. g x = z}) = F g A"
```
```   107   using assms by (induct A) (simp_all add: insert_Diff_if)
```
```   108
```
```   109 lemma not_neutral_contains_not_neutral:
```
```   110   assumes "F g A \<noteq> \<^bold>1"
```
```   111   obtains a where "a \<in> A" and "g a \<noteq> \<^bold>1"
```
```   112 proof -
```
```   113   from assms have "\<exists>a\<in>A. g a \<noteq> \<^bold>1"
```
```   114   proof (induct A rule: infinite_finite_induct)
```
```   115     case infinite
```
```   116     then show ?case by simp
```
```   117   next
```
```   118     case empty
```
```   119     then show ?case by simp
```
```   120   next
```
```   121     case (insert a A)
```
```   122     then show ?case by fastforce
```
```   123   qed
```
```   124   with that show thesis by blast
```
```   125 qed
```
```   126
```
```   127 lemma reindex:
```
```   128   assumes "inj_on h A"
```
```   129   shows "F g (h ` A) = F (g \<circ> h) A"
```
```   130 proof (cases "finite A")
```
```   131   case True
```
```   132   with assms show ?thesis
```
```   133     by (simp add: eq_fold fold_image comp_assoc)
```
```   134 next
```
```   135   case False
```
```   136   with assms have "\<not> finite (h ` A)" by (blast dest: finite_imageD)
```
```   137   with False show ?thesis by simp
```
```   138 qed
```
```   139
```
```   140 lemma cong [fundef_cong]:
```
```   141   assumes "A = B"
```
```   142   assumes g_h: "\<And>x. x \<in> B \<Longrightarrow> g x = h x"
```
```   143   shows "F g A = F h B"
```
```   144   using g_h unfolding \<open>A = B\<close>
```
```   145   by (induct B rule: infinite_finite_induct) auto
```
```   146
```
```   147 lemma strong_cong [cong]:
```
```   148   assumes "A = B" "\<And>x. x \<in> B =simp=> g x = h x"
```
```   149   shows "F (\<lambda>x. g x) A = F (\<lambda>x. h x) B"
```
```   150   by (rule cong) (use assms in \<open>simp_all add: simp_implies_def\<close>)
```
```   151
```
```   152 lemma reindex_cong:
```
```   153   assumes "inj_on l B"
```
```   154   assumes "A = l ` B"
```
```   155   assumes "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x"
```
```   156   shows "F g A = F h B"
```
```   157   using assms by (simp add: reindex)
```
```   158
```
```   159 lemma UNION_disjoint:
```
```   160   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
```
```   161     and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
```
```   162   shows "F g (UNION I A) = F (\<lambda>x. F g (A x)) I"
```
```   163   apply (insert assms)
```
```   164   apply (induct rule: finite_induct)
```
```   165    apply simp
```
```   166   apply atomize
```
```   167   apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
```
```   168    prefer 2 apply blast
```
```   169   apply (subgoal_tac "A x \<inter> UNION Fa A = {}")
```
```   170    prefer 2 apply blast
```
```   171   apply (simp add: union_disjoint)
```
```   172   done
```
```   173
```
```   174 lemma Union_disjoint:
```
```   175   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
```
```   176   shows "F g (\<Union>C) = (F \<circ> F) g C"
```
```   177 proof (cases "finite C")
```
```   178   case True
```
```   179   from UNION_disjoint [OF this assms] show ?thesis by simp
```
```   180 next
```
```   181   case False
```
```   182   then show ?thesis by (auto dest: finite_UnionD intro: infinite)
```
```   183 qed
```
```   184
```
```   185 lemma distrib: "F (\<lambda>x. g x \<^bold>* h x) A = F g A \<^bold>* F h A"
```
```   186   by (induct A rule: infinite_finite_induct) (simp_all add: assoc commute left_commute)
```
```   187
```
```   188 lemma Sigma:
```
```   189   "finite A \<Longrightarrow> \<forall>x\<in>A. finite (B x) \<Longrightarrow> F (\<lambda>x. F (g x) (B x)) A = F (case_prod g) (SIGMA x:A. B x)"
```
```   190   apply (subst Sigma_def)
```
```   191   apply (subst UNION_disjoint)
```
```   192      apply assumption
```
```   193     apply simp
```
```   194    apply blast
```
```   195   apply (rule cong)
```
```   196    apply rule
```
```   197   apply (simp add: fun_eq_iff)
```
```   198   apply (subst UNION_disjoint)
```
```   199      apply simp
```
```   200     apply simp
```
```   201    apply blast
```
```   202   apply (simp add: comp_def)
```
```   203   done
```
```   204
```
```   205 lemma related:
```
```   206   assumes Re: "R \<^bold>1 \<^bold>1"
```
```   207     and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 \<^bold>* y1) (x2 \<^bold>* y2)"
```
```   208     and fin: "finite S"
```
```   209     and R_h_g: "\<forall>x\<in>S. R (h x) (g x)"
```
```   210   shows "R (F h S) (F g S)"
```
```   211   using fin by (rule finite_subset_induct) (use assms in auto)
```
```   212
```
```   213 lemma mono_neutral_cong_left:
```
```   214   assumes "finite T"
```
```   215     and "S \<subseteq> T"
```
```   216     and "\<forall>i \<in> T - S. h i = \<^bold>1"
```
```   217     and "\<And>x. x \<in> S \<Longrightarrow> g x = h x"
```
```   218   shows "F g S = F h T"
```
```   219 proof-
```
```   220   have eq: "T = S \<union> (T - S)" using \<open>S \<subseteq> T\<close> by blast
```
```   221   have d: "S \<inter> (T - S) = {}" using \<open>S \<subseteq> T\<close> by blast
```
```   222   from \<open>finite T\<close> \<open>S \<subseteq> T\<close> have f: "finite S" "finite (T - S)"
```
```   223     by (auto intro: finite_subset)
```
```   224   show ?thesis using assms(4)
```
```   225     by (simp add: union_disjoint [OF f d, unfolded eq [symmetric]] neutral [OF assms(3)])
```
```   226 qed
```
```   227
```
```   228 lemma mono_neutral_cong_right:
```
```   229   "finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. g i = \<^bold>1 \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> g x = h x) \<Longrightarrow>
```
```   230     F g T = F h S"
```
```   231   by (auto intro!: mono_neutral_cong_left [symmetric])
```
```   232
```
```   233 lemma mono_neutral_left: "finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. g i = \<^bold>1 \<Longrightarrow> F g S = F g T"
```
```   234   by (blast intro: mono_neutral_cong_left)
```
```   235
```
```   236 lemma mono_neutral_right: "finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. g i = \<^bold>1 \<Longrightarrow> F g T = F g S"
```
```   237   by (blast intro!: mono_neutral_left [symmetric])
```
```   238
```
```   239 lemma reindex_bij_betw: "bij_betw h S T \<Longrightarrow> F (\<lambda>x. g (h x)) S = F g T"
```
```   240   by (auto simp: bij_betw_def reindex)
```
```   241
```
```   242 lemma reindex_bij_witness:
```
```   243   assumes witness:
```
```   244     "\<And>a. a \<in> S \<Longrightarrow> i (j a) = a"
```
```   245     "\<And>a. a \<in> S \<Longrightarrow> j a \<in> T"
```
```   246     "\<And>b. b \<in> T \<Longrightarrow> j (i b) = b"
```
```   247     "\<And>b. b \<in> T \<Longrightarrow> i b \<in> S"
```
```   248   assumes eq:
```
```   249     "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a"
```
```   250   shows "F g S = F h T"
```
```   251 proof -
```
```   252   have "bij_betw j S T"
```
```   253     using bij_betw_byWitness[where A=S and f=j and f'=i and A'=T] witness by auto
```
```   254   moreover have "F g S = F (\<lambda>x. h (j x)) S"
```
```   255     by (intro cong) (auto simp: eq)
```
```   256   ultimately show ?thesis
```
```   257     by (simp add: reindex_bij_betw)
```
```   258 qed
```
```   259
```
```   260 lemma reindex_bij_betw_not_neutral:
```
```   261   assumes fin: "finite S'" "finite T'"
```
```   262   assumes bij: "bij_betw h (S - S') (T - T')"
```
```   263   assumes nn:
```
```   264     "\<And>a. a \<in> S' \<Longrightarrow> g (h a) = z"
```
```   265     "\<And>b. b \<in> T' \<Longrightarrow> g b = z"
```
```   266   shows "F (\<lambda>x. g (h x)) S = F g T"
```
```   267 proof -
```
```   268   have [simp]: "finite S \<longleftrightarrow> finite T"
```
```   269     using bij_betw_finite[OF bij] fin by auto
```
```   270   show ?thesis
```
```   271   proof (cases "finite S")
```
```   272     case True
```
```   273     with nn have "F (\<lambda>x. g (h x)) S = F (\<lambda>x. g (h x)) (S - S')"
```
```   274       by (intro mono_neutral_cong_right) auto
```
```   275     also have "\<dots> = F g (T - T')"
```
```   276       using bij by (rule reindex_bij_betw)
```
```   277     also have "\<dots> = F g T"
```
```   278       using nn \<open>finite S\<close> by (intro mono_neutral_cong_left) auto
```
```   279     finally show ?thesis .
```
```   280   next
```
```   281     case False
```
```   282     then show ?thesis by simp
```
```   283   qed
```
```   284 qed
```
```   285
```
```   286 lemma reindex_nontrivial:
```
```   287   assumes "finite A"
```
```   288     and nz: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> h x = h y \<Longrightarrow> g (h x) = \<^bold>1"
```
```   289   shows "F g (h ` A) = F (g \<circ> h) A"
```
```   290 proof (subst reindex_bij_betw_not_neutral [symmetric])
```
```   291   show "bij_betw h (A - {x \<in> A. (g \<circ> h) x = \<^bold>1}) (h ` A - h ` {x \<in> A. (g \<circ> h) x = \<^bold>1})"
```
```   292     using nz by (auto intro!: inj_onI simp: bij_betw_def)
```
```   293 qed (use \<open>finite A\<close> in auto)
```
```   294
```
```   295 lemma reindex_bij_witness_not_neutral:
```
```   296   assumes fin: "finite S'" "finite T'"
```
```   297   assumes witness:
```
```   298     "\<And>a. a \<in> S - S' \<Longrightarrow> i (j a) = a"
```
```   299     "\<And>a. a \<in> S - S' \<Longrightarrow> j a \<in> T - T'"
```
```   300     "\<And>b. b \<in> T - T' \<Longrightarrow> j (i b) = b"
```
```   301     "\<And>b. b \<in> T - T' \<Longrightarrow> i b \<in> S - S'"
```
```   302   assumes nn:
```
```   303     "\<And>a. a \<in> S' \<Longrightarrow> g a = z"
```
```   304     "\<And>b. b \<in> T' \<Longrightarrow> h b = z"
```
```   305   assumes eq:
```
```   306     "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a"
```
```   307   shows "F g S = F h T"
```
```   308 proof -
```
```   309   have bij: "bij_betw j (S - (S' \<inter> S)) (T - (T' \<inter> T))"
```
```   310     using witness by (intro bij_betw_byWitness[where f'=i]) auto
```
```   311   have F_eq: "F g S = F (\<lambda>x. h (j x)) S"
```
```   312     by (intro cong) (auto simp: eq)
```
```   313   show ?thesis
```
```   314     unfolding F_eq using fin nn eq
```
```   315     by (intro reindex_bij_betw_not_neutral[OF _ _ bij]) auto
```
```   316 qed
```
```   317
```
```   318 lemma delta:
```
```   319   assumes fS: "finite S"
```
```   320   shows "F (\<lambda>k. if k = a then b k else \<^bold>1) S = (if a \<in> S then b a else \<^bold>1)"
```
```   321 proof -
```
```   322   let ?f = "(\<lambda>k. if k = a then b k else \<^bold>1)"
```
```   323   show ?thesis
```
```   324   proof (cases "a \<in> S")
```
```   325     case False
```
```   326     then have "\<forall>k\<in>S. ?f k = \<^bold>1" by simp
```
```   327     with False show ?thesis by simp
```
```   328   next
```
```   329     case True
```
```   330     let ?A = "S - {a}"
```
```   331     let ?B = "{a}"
```
```   332     from True have eq: "S = ?A \<union> ?B" by blast
```
```   333     have dj: "?A \<inter> ?B = {}" by simp
```
```   334     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```   335     have "F ?f S = F ?f ?A \<^bold>* F ?f ?B"
```
```   336       using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]] by simp
```
```   337     with True show ?thesis by simp
```
```   338   qed
```
```   339 qed
```
```   340
```
```   341 lemma delta':
```
```   342   assumes fin: "finite S"
```
```   343   shows "F (\<lambda>k. if a = k then b k else \<^bold>1) S = (if a \<in> S then b a else \<^bold>1)"
```
```   344   using delta [OF fin, of a b, symmetric] by (auto intro: cong)
```
```   345
```
```   346 lemma If_cases:
```
```   347   fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
```
```   348   assumes fin: "finite A"
```
```   349   shows "F (\<lambda>x. if P x then h x else g x) A = F h (A \<inter> {x. P x}) \<^bold>* F g (A \<inter> - {x. P x})"
```
```   350 proof -
```
```   351   have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}" "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}"
```
```   352     by blast+
```
```   353   from fin have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
```
```   354   let ?g = "\<lambda>x. if P x then h x else g x"
```
```   355   from union_disjoint [OF f a(2), of ?g] a(1) show ?thesis
```
```   356     by (subst (1 2) cong) simp_all
```
```   357 qed
```
```   358
```
```   359 lemma cartesian_product: "F (\<lambda>x. F (g x) B) A = F (case_prod g) (A \<times> B)"
```
```   360   apply (rule sym)
```
```   361   apply (cases "finite A")
```
```   362    apply (cases "finite B")
```
```   363     apply (simp add: Sigma)
```
```   364    apply (cases "A = {}")
```
```   365     apply simp
```
```   366    apply simp
```
```   367    apply (auto intro: infinite dest: finite_cartesian_productD2)
```
```   368   apply (cases "B = {}")
```
```   369    apply (auto intro: infinite dest: finite_cartesian_productD1)
```
```   370   done
```
```   371
```
```   372 lemma inter_restrict:
```
```   373   assumes "finite A"
```
```   374   shows "F g (A \<inter> B) = F (\<lambda>x. if x \<in> B then g x else \<^bold>1) A"
```
```   375 proof -
```
```   376   let ?g = "\<lambda>x. if x \<in> A \<inter> B then g x else \<^bold>1"
```
```   377   have "\<forall>i\<in>A - A \<inter> B. (if i \<in> A \<inter> B then g i else \<^bold>1) = \<^bold>1" by simp
```
```   378   moreover have "A \<inter> B \<subseteq> A" by blast
```
```   379   ultimately have "F ?g (A \<inter> B) = F ?g A"
```
```   380     using \<open>finite A\<close> by (intro mono_neutral_left) auto
```
```   381   then show ?thesis by simp
```
```   382 qed
```
```   383
```
```   384 lemma inter_filter:
```
```   385   "finite A \<Longrightarrow> F g {x \<in> A. P x} = F (\<lambda>x. if P x then g x else \<^bold>1) A"
```
```   386   by (simp add: inter_restrict [symmetric, of A "{x. P x}" g, simplified mem_Collect_eq] Int_def)
```
```   387
```
```   388 lemma Union_comp:
```
```   389   assumes "\<forall>A \<in> B. finite A"
```
```   390     and "\<And>A1 A2 x. A1 \<in> B \<Longrightarrow> A2 \<in> B \<Longrightarrow> A1 \<noteq> A2 \<Longrightarrow> x \<in> A1 \<Longrightarrow> x \<in> A2 \<Longrightarrow> g x = \<^bold>1"
```
```   391   shows "F g (\<Union>B) = (F \<circ> F) g B"
```
```   392   using assms
```
```   393 proof (induct B rule: infinite_finite_induct)
```
```   394   case (infinite A)
```
```   395   then have "\<not> finite (\<Union>A)" by (blast dest: finite_UnionD)
```
```   396   with infinite show ?case by simp
```
```   397 next
```
```   398   case empty
```
```   399   then show ?case by simp
```
```   400 next
```
```   401   case (insert A B)
```
```   402   then have "finite A" "finite B" "finite (\<Union>B)" "A \<notin> B"
```
```   403     and "\<forall>x\<in>A \<inter> \<Union>B. g x = \<^bold>1"
```
```   404     and H: "F g (\<Union>B) = (F \<circ> F) g B" by auto
```
```   405   then have "F g (A \<union> \<Union>B) = F g A \<^bold>* F g (\<Union>B)"
```
```   406     by (simp add: union_inter_neutral)
```
```   407   with \<open>finite B\<close> \<open>A \<notin> B\<close> show ?case
```
```   408     by (simp add: H)
```
```   409 qed
```
```   410
```
```   411 lemma commute: "F (\<lambda>i. F (g i) B) A = F (\<lambda>j. F (\<lambda>i. g i j) A) B"
```
```   412   unfolding cartesian_product
```
```   413   by (rule reindex_bij_witness [where i = "\<lambda>(i, j). (j, i)" and j = "\<lambda>(i, j). (j, i)"]) auto
```
```   414
```
```   415 lemma commute_restrict:
```
```   416   "finite A \<Longrightarrow> finite B \<Longrightarrow>
```
```   417     F (\<lambda>x. F (g x) {y. y \<in> B \<and> R x y}) A = F (\<lambda>y. F (\<lambda>x. g x y) {x. x \<in> A \<and> R x y}) B"
```
```   418   by (simp add: inter_filter) (rule commute)
```
```   419
```
```   420 lemma Plus:
```
```   421   fixes A :: "'b set" and B :: "'c set"
```
```   422   assumes fin: "finite A" "finite B"
```
```   423   shows "F g (A <+> B) = F (g \<circ> Inl) A \<^bold>* F (g \<circ> Inr) B"
```
```   424 proof -
```
```   425   have "A <+> B = Inl ` A \<union> Inr ` B" by auto
```
```   426   moreover from fin have "finite (Inl ` A)" "finite (Inr ` B)" by auto
```
```   427   moreover have "Inl ` A \<inter> Inr ` B = {}" by auto
```
```   428   moreover have "inj_on Inl A" "inj_on Inr B" by (auto intro: inj_onI)
```
```   429   ultimately show ?thesis
```
```   430     using fin by (simp add: union_disjoint reindex)
```
```   431 qed
```
```   432
```
```   433 lemma same_carrier:
```
```   434   assumes "finite C"
```
```   435   assumes subset: "A \<subseteq> C" "B \<subseteq> C"
```
```   436   assumes trivial: "\<And>a. a \<in> C - A \<Longrightarrow> g a = \<^bold>1" "\<And>b. b \<in> C - B \<Longrightarrow> h b = \<^bold>1"
```
```   437   shows "F g A = F h B \<longleftrightarrow> F g C = F h C"
```
```   438 proof -
```
```   439   have "finite A" and "finite B" and "finite (C - A)" and "finite (C - B)"
```
```   440     using \<open>finite C\<close> subset by (auto elim: finite_subset)
```
```   441   from subset have [simp]: "A - (C - A) = A" by auto
```
```   442   from subset have [simp]: "B - (C - B) = B" by auto
```
```   443   from subset have "C = A \<union> (C - A)" by auto
```
```   444   then have "F g C = F g (A \<union> (C - A))" by simp
```
```   445   also have "\<dots> = F g (A - (C - A)) \<^bold>* F g (C - A - A) \<^bold>* F g (A \<inter> (C - A))"
```
```   446     using \<open>finite A\<close> \<open>finite (C - A)\<close> by (simp only: union_diff2)
```
```   447   finally have *: "F g C = F g A" using trivial by simp
```
```   448   from subset have "C = B \<union> (C - B)" by auto
```
```   449   then have "F h C = F h (B \<union> (C - B))" by simp
```
```   450   also have "\<dots> = F h (B - (C - B)) \<^bold>* F h (C - B - B) \<^bold>* F h (B \<inter> (C - B))"
```
```   451     using \<open>finite B\<close> \<open>finite (C - B)\<close> by (simp only: union_diff2)
```
```   452   finally have "F h C = F h B"
```
```   453     using trivial by simp
```
```   454   with * show ?thesis by simp
```
```   455 qed
```
```   456
```
```   457 lemma same_carrierI:
```
```   458   assumes "finite C"
```
```   459   assumes subset: "A \<subseteq> C" "B \<subseteq> C"
```
```   460   assumes trivial: "\<And>a. a \<in> C - A \<Longrightarrow> g a = \<^bold>1" "\<And>b. b \<in> C - B \<Longrightarrow> h b = \<^bold>1"
```
```   461   assumes "F g C = F h C"
```
```   462   shows "F g A = F h B"
```
```   463   using assms same_carrier [of C A B] by simp
```
```   464
```
```   465 end
```
```   466
```
```   467
```
```   468 subsection \<open>Generalized summation over a set\<close>
```
```   469
```
```   470 context comm_monoid_add
```
```   471 begin
```
```   472
```
```   473 sublocale setsum: comm_monoid_set plus 0
```
```   474   defines setsum = setsum.F ..
```
```   475
```
```   476 abbreviation Setsum ("\<Sum>_"  999)
```
```   477   where "\<Sum>A \<equiv> setsum (\<lambda>x. x) A"
```
```   478
```
```   479 end
```
```   480
```
```   481 text \<open>Now: lot's of fancy syntax. First, @{term "setsum (\<lambda>x. e) A"} is written \<open>\<Sum>x\<in>A. e\<close>.\<close>
```
```   482
```
```   483 syntax (ASCII)
```
```   484   "_setsum" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b::comm_monoid_add"  ("(3SUM _:_./ _)" [0, 51, 10] 10)
```
```   485 syntax
```
```   486   "_setsum" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b::comm_monoid_add"  ("(2\<Sum>_\<in>_./ _)" [0, 51, 10] 10)
```
```   487 translations \<comment> \<open>Beware of argument permutation!\<close>
```
```   488   "\<Sum>i\<in>A. b" \<rightleftharpoons> "CONST setsum (\<lambda>i. b) A"
```
```   489
```
```   490 text \<open>Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter \<open>\<Sum>x|P. e\<close>.\<close>
```
```   491
```
```   492 syntax (ASCII)
```
```   493   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(3SUM _ |/ _./ _)" [0, 0, 10] 10)
```
```   494 syntax
```
```   495   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(2\<Sum>_ | (_)./ _)" [0, 0, 10] 10)
```
```   496 translations
```
```   497   "\<Sum>x|P. t" => "CONST setsum (\<lambda>x. t) {x. P}"
```
```   498
```
```   499 print_translation \<open>
```
```   500 let
```
```   501   fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) \$ Abs (y, Ty, P)] =
```
```   502         if x <> y then raise Match
```
```   503         else
```
```   504           let
```
```   505             val x' = Syntax_Trans.mark_bound_body (x, Tx);
```
```   506             val t' = subst_bound (x', t);
```
```   507             val P' = subst_bound (x', P);
```
```   508           in
```
```   509             Syntax.const @{syntax_const "_qsetsum"} \$ Syntax_Trans.mark_bound_abs (x, Tx) \$ P' \$ t'
```
```   510           end
```
```   511     | setsum_tr' _ = raise Match;
```
```   512 in [(@{const_syntax setsum}, K setsum_tr')] end
```
```   513 \<close>
```
```   514
```
```   515 (* TODO generalization candidates *)
```
```   516
```
```   517 lemma (in comm_monoid_add) setsum_image_gen:
```
```   518   assumes fin: "finite S"
```
```   519   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
```
```   520 proof -
```
```   521   have "{y. y\<in> f`S \<and> f x = y} = {f x}" if "x \<in> S" for x
```
```   522     using that by auto
```
```   523   then have "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
```
```   524     by simp
```
```   525   also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
```
```   526     by (rule setsum.commute_restrict [OF fin finite_imageI [OF fin]])
```
```   527   finally show ?thesis .
```
```   528 qed
```
```   529
```
```   530
```
```   531 subsubsection \<open>Properties in more restricted classes of structures\<close>
```
```   532
```
```   533 lemma setsum_Un:
```
```   534   "finite A \<Longrightarrow> finite B \<Longrightarrow> setsum f (A \<union> B) = setsum f A + setsum f B - setsum f (A \<inter> B)"
```
```   535   for f :: "'b \<Rightarrow> 'a::ab_group_add"
```
```   536   by (subst setsum.union_inter [symmetric]) (auto simp add: algebra_simps)
```
```   537
```
```   538 lemma setsum_Un2:
```
```   539   assumes "finite (A \<union> B)"
```
```   540   shows "setsum f (A \<union> B) = setsum f (A - B) + setsum f (B - A) + setsum f (A \<inter> B)"
```
```   541 proof -
```
```   542   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
```
```   543     by auto
```
```   544   with assms show ?thesis
```
```   545     by simp (subst setsum.union_disjoint, auto)+
```
```   546 qed
```
```   547
```
```   548 lemma setsum_diff1:
```
```   549   fixes f :: "'b \<Rightarrow> 'a::ab_group_add"
```
```   550   assumes "finite A"
```
```   551   shows "setsum f (A - {a}) = (if a \<in> A then setsum f A - f a else setsum f A)"
```
```   552   using assms by induct (auto simp: insert_Diff_if)
```
```   553
```
```   554 lemma setsum_diff:
```
```   555   fixes f :: "'b \<Rightarrow> 'a::ab_group_add"
```
```   556   assumes "finite A" "B \<subseteq> A"
```
```   557   shows "setsum f (A - B) = setsum f A - setsum f B"
```
```   558 proof -
```
```   559   from assms(2,1) have "finite B" by (rule finite_subset)
```
```   560   from this \<open>B \<subseteq> A\<close>
```
```   561   show ?thesis
```
```   562   proof induct
```
```   563     case empty
```
```   564     thus ?case by simp
```
```   565   next
```
```   566     case (insert x F)
```
```   567     with \<open>finite A\<close> \<open>finite B\<close> show ?case
```
```   568       by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
```
```   569   qed
```
```   570 qed
```
```   571
```
```   572 lemma (in ordered_comm_monoid_add) setsum_mono:
```
```   573   "(\<And>i. i\<in>K \<Longrightarrow> f i \<le> g i) \<Longrightarrow> (\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
```
```   574   by (induct K rule: infinite_finite_induct) (use add_mono in auto)
```
```   575
```
```   576 lemma (in strict_ordered_comm_monoid_add) setsum_strict_mono:
```
```   577   assumes "finite A" "A \<noteq> {}"
```
```   578     and "\<And>x. x \<in> A \<Longrightarrow> f x < g x"
```
```   579   shows "setsum f A < setsum g A"
```
```   580   using assms
```
```   581 proof (induct rule: finite_ne_induct)
```
```   582   case singleton
```
```   583   then show ?case by simp
```
```   584 next
```
```   585   case insert
```
```   586   then show ?case by (auto simp: add_strict_mono)
```
```   587 qed
```
```   588
```
```   589 lemma setsum_strict_mono_ex1:
```
```   590   fixes f g :: "'i \<Rightarrow> 'a::ordered_cancel_comm_monoid_add"
```
```   591   assumes "finite A"
```
```   592     and "\<forall>x\<in>A. f x \<le> g x"
```
```   593     and "\<exists>a\<in>A. f a < g a"
```
```   594   shows "setsum f A < setsum g A"
```
```   595 proof-
```
```   596   from assms(3) obtain a where a: "a \<in> A" "f a < g a" by blast
```
```   597   have "setsum f A = setsum f ((A - {a}) \<union> {a})"
```
```   598     by(simp add: insert_absorb[OF \<open>a \<in> A\<close>])
```
```   599   also have "\<dots> = setsum f (A - {a}) + setsum f {a}"
```
```   600     using \<open>finite A\<close> by(subst setsum.union_disjoint) auto
```
```   601   also have "setsum f (A - {a}) \<le> setsum g (A - {a})"
```
```   602     by (rule setsum_mono) (simp add: assms(2))
```
```   603   also from a have "setsum f {a} < setsum g {a}" by simp
```
```   604   also have "setsum g (A - {a}) + setsum g {a} = setsum g((A - {a}) \<union> {a})"
```
```   605     using \<open>finite A\<close> by (subst setsum.union_disjoint[symmetric]) auto
```
```   606   also have "\<dots> = setsum g A" by (simp add: insert_absorb[OF \<open>a \<in> A\<close>])
```
```   607   finally show ?thesis
```
```   608     by (auto simp add: add_right_mono add_strict_left_mono)
```
```   609 qed
```
```   610
```
```   611 lemma setsum_mono_inv:
```
```   612   fixes f g :: "'i \<Rightarrow> 'a :: ordered_cancel_comm_monoid_add"
```
```   613   assumes eq: "setsum f I = setsum g I"
```
```   614   assumes le: "\<And>i. i \<in> I \<Longrightarrow> f i \<le> g i"
```
```   615   assumes i: "i \<in> I"
```
```   616   assumes I: "finite I"
```
```   617   shows "f i = g i"
```
```   618 proof (rule ccontr)
```
```   619   assume "\<not> ?thesis"
```
```   620   with le[OF i] have "f i < g i" by simp
```
```   621   with i have "\<exists>i\<in>I. f i < g i" ..
```
```   622   from setsum_strict_mono_ex1[OF I _ this] le have "setsum f I < setsum g I"
```
```   623     by blast
```
```   624   with eq show False by simp
```
```   625 qed
```
```   626
```
```   627 lemma setsum_negf: "(\<Sum>x\<in>A. - f x) = - (\<Sum>x\<in>A. f x)"
```
```   628   for f :: "'b \<Rightarrow> 'a::ab_group_add"
```
```   629   by (induct A rule: infinite_finite_induct) auto
```
```   630
```
```   631 lemma setsum_subtractf: "(\<Sum>x\<in>A. f x - g x) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
```
```   632   for f g :: "'b \<Rightarrow>'a::ab_group_add"
```
```   633   using setsum.distrib [of f "- g" A] by (simp add: setsum_negf)
```
```   634
```
```   635 lemma setsum_subtractf_nat:
```
```   636   "(\<And>x. x \<in> A \<Longrightarrow> g x \<le> f x) \<Longrightarrow> (\<Sum>x\<in>A. f x - g x) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
```
```   637   for f g :: "'a \<Rightarrow> nat"
```
```   638   by (induct A rule: infinite_finite_induct) (auto simp: setsum_mono)
```
```   639
```
```   640 context ordered_comm_monoid_add
```
```   641 begin
```
```   642
```
```   643 lemma setsum_nonneg: "\<forall>x\<in>A. 0 \<le> f x \<Longrightarrow> 0 \<le> setsum f A"
```
```   644 proof (induct A rule: infinite_finite_induct)
```
```   645   case infinite
```
```   646   then show ?case by simp
```
```   647 next
```
```   648   case empty
```
```   649   then show ?case by simp
```
```   650 next
```
```   651   case (insert x F)
```
```   652   then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
```
```   653   with insert show ?case by simp
```
```   654 qed
```
```   655
```
```   656 lemma setsum_nonpos: "\<forall>x\<in>A. f x \<le> 0 \<Longrightarrow> setsum f A \<le> 0"
```
```   657 proof (induct A rule: infinite_finite_induct)
```
```   658   case infinite
```
```   659   then show ?case by simp
```
```   660 next
```
```   661   case empty
```
```   662   then show ?case by simp
```
```   663 next
```
```   664   case (insert x F)
```
```   665   then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
```
```   666   with insert show ?case by simp
```
```   667 qed
```
```   668
```
```   669 lemma setsum_nonneg_eq_0_iff:
```
```   670   "finite A \<Longrightarrow> \<forall>x\<in>A. 0 \<le> f x \<Longrightarrow> setsum f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
```
```   671   by (induct set: finite) (simp_all add: add_nonneg_eq_0_iff setsum_nonneg)
```
```   672
```
```   673 lemma setsum_nonneg_0:
```
```   674   "finite s \<Longrightarrow> (\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0) \<Longrightarrow> (\<Sum> i \<in> s. f i) = 0 \<Longrightarrow> i \<in> s \<Longrightarrow> f i = 0"
```
```   675   by (simp add: setsum_nonneg_eq_0_iff)
```
```   676
```
```   677 lemma setsum_nonneg_leq_bound:
```
```   678   assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"
```
```   679   shows "f i \<le> B"
```
```   680 proof -
```
```   681   from assms have "f i \<le> f i + (\<Sum>i \<in> s - {i}. f i)"
```
```   682     by (intro add_increasing2 setsum_nonneg) auto
```
```   683   also have "\<dots> = B"
```
```   684     using setsum.remove[of s i f] assms by simp
```
```   685   finally show ?thesis by auto
```
```   686 qed
```
```   687
```
```   688 lemma setsum_mono2:
```
```   689   assumes fin: "finite B"
```
```   690     and sub: "A \<subseteq> B"
```
```   691     and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
```
```   692   shows "setsum f A \<le> setsum f B"
```
```   693 proof -
```
```   694   have "setsum f A \<le> setsum f A + setsum f (B-A)"
```
```   695     by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
```
```   696   also from fin finite_subset[OF sub fin] have "\<dots> = setsum f (A \<union> (B-A))"
```
```   697     by (simp add: setsum.union_disjoint del: Un_Diff_cancel)
```
```   698   also from sub have "A \<union> (B-A) = B" by blast
```
```   699   finally show ?thesis .
```
```   700 qed
```
```   701
```
```   702 lemma setsum_le_included:
```
```   703   assumes "finite s" "finite t"
```
```   704   and "\<forall>y\<in>t. 0 \<le> g y" "(\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<le> g y)"
```
```   705   shows "setsum f s \<le> setsum g t"
```
```   706 proof -
```
```   707   have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"
```
```   708   proof (rule setsum_mono)
```
```   709     fix y
```
```   710     assume "y \<in> s"
```
```   711     with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto
```
```   712     with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")
```
```   713       using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]
```
```   714       by (auto intro!: setsum_mono2)
```
```   715   qed
```
```   716   also have "\<dots> \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"
```
```   717     using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)
```
```   718   also have "\<dots> \<le> setsum g t"
```
```   719     using assms by (auto simp: setsum_image_gen[symmetric])
```
```   720   finally show ?thesis .
```
```   721 qed
```
```   722
```
```   723 lemma setsum_mono3: "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> \<forall>x\<in>B - A. 0 \<le> f x \<Longrightarrow> setsum f A \<le> setsum f B"
```
```   724   by (rule setsum_mono2) auto
```
```   725
```
```   726 end
```
```   727
```
```   728 lemma (in canonically_ordered_monoid_add) setsum_eq_0_iff [simp]:
```
```   729   "finite F \<Longrightarrow> (setsum f F = 0) = (\<forall>a\<in>F. f a = 0)"
```
```   730   by (intro ballI setsum_nonneg_eq_0_iff zero_le)
```
```   731
```
```   732 lemma setsum_right_distrib: "r * setsum f A = setsum (\<lambda>n. r * f n) A"
```
```   733   for f :: "'a \<Rightarrow> 'b::semiring_0"
```
```   734 proof (induct A rule: infinite_finite_induct)
```
```   735   case infinite
```
```   736   then show ?case by simp
```
```   737 next
```
```   738   case empty
```
```   739   then show ?case by simp
```
```   740 next
```
```   741   case insert
```
```   742   then show ?case by (simp add: distrib_left)
```
```   743 qed
```
```   744
```
```   745 lemma setsum_left_distrib: "setsum f A * r = (\<Sum>n\<in>A. f n * r)"
```
```   746   for r :: "'a::semiring_0"
```
```   747 proof (induct A rule: infinite_finite_induct)
```
```   748   case infinite
```
```   749   then show ?case by simp
```
```   750 next
```
```   751   case empty
```
```   752   then show ?case by simp
```
```   753 next
```
```   754   case insert
```
```   755   then show ?case by (simp add: distrib_right)
```
```   756 qed
```
```   757
```
```   758 lemma setsum_divide_distrib: "setsum f A / r = (\<Sum>n\<in>A. f n / r)"
```
```   759   for r :: "'a::field"
```
```   760 proof (induct A rule: infinite_finite_induct)
```
```   761   case infinite
```
```   762   then show ?case by simp
```
```   763 next
```
```   764   case empty
```
```   765   then show ?case by simp
```
```   766 next
```
```   767   case insert
```
```   768   then show ?case by (simp add: add_divide_distrib)
```
```   769 qed
```
```   770
```
```   771 lemma setsum_abs[iff]: "\<bar>setsum f A\<bar> \<le> setsum (\<lambda>i. \<bar>f i\<bar>) A"
```
```   772   for f :: "'a \<Rightarrow> 'b::ordered_ab_group_add_abs"
```
```   773 proof (induct A rule: infinite_finite_induct)
```
```   774   case infinite
```
```   775   then show ?case by simp
```
```   776 next
```
```   777   case empty
```
```   778   then show ?case by simp
```
```   779 next
```
```   780   case insert
```
```   781   then show ?case by (auto intro: abs_triangle_ineq order_trans)
```
```   782 qed
```
```   783
```
```   784 lemma setsum_abs_ge_zero[iff]: "0 \<le> setsum (\<lambda>i. \<bar>f i\<bar>) A"
```
```   785   for f :: "'a \<Rightarrow> 'b::ordered_ab_group_add_abs"
```
```   786   by (simp add: setsum_nonneg)
```
```   787
```
```   788 lemma abs_setsum_abs[simp]: "\<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar> = (\<Sum>a\<in>A. \<bar>f a\<bar>)"
```
```   789   for f :: "'a \<Rightarrow> 'b::ordered_ab_group_add_abs"
```
```   790 proof (induct A rule: infinite_finite_induct)
```
```   791   case infinite
```
```   792   then show ?case by simp
```
```   793 next
```
```   794   case empty
```
```   795   then show ?case by simp
```
```   796 next
```
```   797   case (insert a A)
```
```   798   then have "\<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
```
```   799   also from insert have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>" by simp
```
```   800   also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>" by (simp del: abs_of_nonneg)
```
```   801   also from insert have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" by simp
```
```   802   finally show ?case .
```
```   803 qed
```
```   804
```
```   805 lemma setsum_diff1_ring:
```
```   806   fixes f :: "'b \<Rightarrow> 'a::ring"
```
```   807   assumes "finite A" "a \<in> A"
```
```   808   shows "setsum f (A - {a}) = setsum f A - (f a)"
```
```   809   unfolding setsum.remove [OF assms] by auto
```
```   810
```
```   811 lemma setsum_product:
```
```   812   fixes f :: "'a \<Rightarrow> 'b::semiring_0"
```
```   813   shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
```
```   814   by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum.commute)
```
```   815
```
```   816 lemma setsum_mult_setsum_if_inj:
```
```   817   fixes f :: "'a \<Rightarrow> 'b::semiring_0"
```
```   818   shows "inj_on (\<lambda>(a, b). f a * g b) (A \<times> B) \<Longrightarrow>
```
```   819     setsum f A * setsum g B = setsum id {f a * g b |a b. a \<in> A \<and> b \<in> B}"
```
```   820   by(auto simp: setsum_product setsum.cartesian_product intro!: setsum.reindex_cong[symmetric])
```
```   821
```
```   822 lemma setsum_SucD: "setsum f A = Suc n \<Longrightarrow> \<exists>a\<in>A. 0 < f a"
```
```   823   by (induct A rule: infinite_finite_induct) auto
```
```   824
```
```   825 lemma setsum_eq_Suc0_iff:
```
```   826   "finite A \<Longrightarrow> setsum f A = Suc 0 \<longleftrightarrow> (\<exists>a\<in>A. f a = Suc 0 \<and> (\<forall>b\<in>A. a \<noteq> b \<longrightarrow> f b = 0))"
```
```   827   by (induct A rule: finite_induct) (auto simp add: add_is_1)
```
```   828
```
```   829 lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
```
```   830
```
```   831 lemma setsum_Un_nat:
```
```   832   "finite A \<Longrightarrow> finite B \<Longrightarrow> setsum f (A \<union> B) = setsum f A + setsum f B - setsum f (A \<inter> B)"
```
```   833   for f :: "'a \<Rightarrow> nat"
```
```   834   \<comment> \<open>For the natural numbers, we have subtraction.\<close>
```
```   835   by (subst setsum.union_inter [symmetric]) (auto simp: algebra_simps)
```
```   836
```
```   837 lemma setsum_diff1_nat: "setsum f (A - {a}) = (if a \<in> A then setsum f A - f a else setsum f A)"
```
```   838   for f :: "'a \<Rightarrow> nat"
```
```   839 proof (induct A rule: infinite_finite_induct)
```
```   840   case infinite
```
```   841   then show ?case by simp
```
```   842 next
```
```   843   case empty
```
```   844   then show ?case by simp
```
```   845 next
```
```   846   case insert
```
```   847   then show ?case
```
```   848     apply (auto simp: insert_Diff_if)
```
```   849     apply (drule mk_disjoint_insert)
```
```   850     apply auto
```
```   851     done
```
```   852 qed
```
```   853
```
```   854 lemma setsum_diff_nat:
```
```   855   fixes f :: "'a \<Rightarrow> nat"
```
```   856   assumes "finite B" and "B \<subseteq> A"
```
```   857   shows "setsum f (A - B) = setsum f A - setsum f B"
```
```   858   using assms
```
```   859 proof induct
```
```   860   case empty
```
```   861   then show ?case by simp
```
```   862 next
```
```   863   case (insert x F)
```
```   864   note IH = \<open>F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F\<close>
```
```   865   from \<open>x \<notin> F\<close> \<open>insert x F \<subseteq> A\<close> have "x \<in> A - F" by simp
```
```   866   then have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
```
```   867     by (simp add: setsum_diff1_nat)
```
```   868   from \<open>insert x F \<subseteq> A\<close> have "F \<subseteq> A" by simp
```
```   869   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
```
```   870   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
```
```   871     by simp
```
```   872   from \<open>x \<notin> F\<close> have "A - insert x F = (A - F) - {x}" by auto
```
```   873   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
```
```   874     by simp
```
```   875   from \<open>finite F\<close> \<open>x \<notin> F\<close> have "setsum f (insert x F) = setsum f F + f x"
```
```   876     by simp
```
```   877   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
```
```   878     by simp
```
```   879   then show ?case by simp
```
```   880 qed
```
```   881
```
```   882 lemma setsum_comp_morphism:
```
```   883   "h 0 = 0 \<Longrightarrow> (\<And>x y. h (x + y) = h x + h y) \<Longrightarrow> setsum (h \<circ> g) A = h (setsum g A)"
```
```   884   by (induct A rule: infinite_finite_induct) simp_all
```
```   885
```
```   886 lemma (in comm_semiring_1) dvd_setsum: "(\<And>a. a \<in> A \<Longrightarrow> d dvd f a) \<Longrightarrow> d dvd setsum f A"
```
```   887   by (induct A rule: infinite_finite_induct) simp_all
```
```   888
```
```   889 lemma (in ordered_comm_monoid_add) setsum_pos:
```
```   890   "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> 0 < f i) \<Longrightarrow> 0 < setsum f I"
```
```   891   by (induct I rule: finite_ne_induct) (auto intro: add_pos_pos)
```
```   892
```
```   893 lemma (in ordered_comm_monoid_add) setsum_pos2:
```
```   894   assumes I: "finite I" "i \<in> I" "0 < f i" "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
```
```   895   shows "0 < setsum f I"
```
```   896 proof -
```
```   897   have "0 < f i + setsum f (I - {i})"
```
```   898     using assms by (intro add_pos_nonneg setsum_nonneg) auto
```
```   899   also have "\<dots> = setsum f I"
```
```   900     using assms by (simp add: setsum.remove)
```
```   901   finally show ?thesis .
```
```   902 qed
```
```   903
```
```   904 lemma setsum_cong_Suc:
```
```   905   assumes "0 \<notin> A" "\<And>x. Suc x \<in> A \<Longrightarrow> f (Suc x) = g (Suc x)"
```
```   906   shows "setsum f A = setsum g A"
```
```   907 proof (rule setsum.cong)
```
```   908   fix x
```
```   909   assume "x \<in> A"
```
```   910   with assms(1) show "f x = g x"
```
```   911     by (cases x) (auto intro!: assms(2))
```
```   912 qed simp_all
```
```   913
```
```   914
```
```   915 subsubsection \<open>Cardinality as special case of @{const setsum}\<close>
```
```   916
```
```   917 lemma card_eq_setsum: "card A = setsum (\<lambda>x. 1) A"
```
```   918 proof -
```
```   919   have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)"
```
```   920     by (simp add: fun_eq_iff)
```
```   921   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) = Finite_Set.fold (\<lambda>_. Suc)"
```
```   922     by (rule arg_cong)
```
```   923   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
```
```   924     by (blast intro: fun_cong)
```
```   925   then show ?thesis
```
```   926     by (simp add: card.eq_fold setsum.eq_fold)
```
```   927 qed
```
```   928
```
```   929 lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
```
```   930   by (induct A rule: infinite_finite_induct) (auto simp: algebra_simps)
```
```   931
```
```   932 lemma setsum_Suc: "setsum (\<lambda>x. Suc(f x)) A = setsum f A + card A"
```
```   933   using setsum.distrib[of f "\<lambda>_. 1" A] by simp
```
```   934
```
```   935 lemma setsum_bounded_above:
```
```   936   fixes K :: "'a::{semiring_1,ordered_comm_monoid_add}"
```
```   937   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> K"
```
```   938   shows "setsum f A \<le> of_nat (card A) * K"
```
```   939 proof (cases "finite A")
```
```   940   case True
```
```   941   then show ?thesis
```
```   942     using le setsum_mono[where K=A and g = "\<lambda>x. K"] by simp
```
```   943 next
```
```   944   case False
```
```   945   then show ?thesis by simp
```
```   946 qed
```
```   947
```
```   948 lemma setsum_bounded_above_strict:
```
```   949   fixes K :: "'a::{ordered_cancel_comm_monoid_add,semiring_1}"
```
```   950   assumes "\<And>i. i\<in>A \<Longrightarrow> f i < K" "card A > 0"
```
```   951   shows "setsum f A < of_nat (card A) * K"
```
```   952   using assms setsum_strict_mono[where A=A and g = "\<lambda>x. K"]
```
```   953   by (simp add: card_gt_0_iff)
```
```   954
```
```   955 lemma setsum_bounded_below:
```
```   956   fixes K :: "'a::{semiring_1,ordered_comm_monoid_add}"
```
```   957   assumes le: "\<And>i. i\<in>A \<Longrightarrow> K \<le> f i"
```
```   958   shows "of_nat (card A) * K \<le> setsum f A"
```
```   959 proof (cases "finite A")
```
```   960   case True
```
```   961   then show ?thesis
```
```   962     using le setsum_mono[where K=A and f = "\<lambda>x. K"] by simp
```
```   963 next
```
```   964   case False
```
```   965   then show ?thesis by simp
```
```   966 qed
```
```   967
```
```   968 lemma card_UN_disjoint:
```
```   969   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
```
```   970     and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
```
```   971   shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
```
```   972 proof -
```
```   973   have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)"
```
```   974     by simp
```
```   975   with assms show ?thesis
```
```   976     by (simp add: card_eq_setsum setsum.UNION_disjoint del: setsum_constant)
```
```   977 qed
```
```   978
```
```   979 lemma card_Union_disjoint:
```
```   980   "finite C \<Longrightarrow> \<forall>A\<in>C. finite A \<Longrightarrow> \<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {} \<Longrightarrow>
```
```   981     card (\<Union>C) = setsum card C"
```
```   982   by (frule card_UN_disjoint [of C id]) simp_all
```
```   983
```
```   984 lemma setsum_multicount_gen:
```
```   985   assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
```
```   986   shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t"
```
```   987     (is "?l = ?r")
```
```   988 proof-
```
```   989   have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s"
```
```   990     by auto
```
```   991   also have "\<dots> = ?r"
```
```   992     unfolding setsum.commute_restrict [OF assms(1-2)]
```
```   993     using assms(3) by auto
```
```   994   finally show ?thesis .
```
```   995 qed
```
```   996
```
```   997 lemma setsum_multicount:
```
```   998   assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
```
```   999   shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
```
```  1000 proof-
```
```  1001   have "?l = setsum (\<lambda>i. k) T"
```
```  1002     by (rule setsum_multicount_gen) (auto simp: assms)
```
```  1003   also have "\<dots> = ?r" by (simp add: mult.commute)
```
```  1004   finally show ?thesis by auto
```
```  1005 qed
```
```  1006
```
```  1007
```
```  1008 subsubsection \<open>Cardinality of products\<close>
```
```  1009
```
```  1010 lemma card_SigmaI [simp]:
```
```  1011   "finite A \<Longrightarrow> \<forall>a\<in>A. finite (B a) \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
```
```  1012   by (simp add: card_eq_setsum setsum.Sigma del: setsum_constant)
```
```  1013
```
```  1014 (*
```
```  1015 lemma SigmaI_insert: "y \<notin> A ==>
```
```  1016   (SIGMA x:(insert y A). B x) = (({y} \<times> (B y)) \<union> (SIGMA x: A. B x))"
```
```  1017   by auto
```
```  1018 *)
```
```  1019
```
```  1020 lemma card_cartesian_product: "card (A \<times> B) = card A * card B"
```
```  1021   by (cases "finite A \<and> finite B")
```
```  1022     (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1023
```
```  1024 lemma card_cartesian_product_singleton:  "card ({x} \<times> A) = card A"
```
```  1025   by (simp add: card_cartesian_product)
```
```  1026
```
```  1027
```
```  1028 subsection \<open>Generalized product over a set\<close>
```
```  1029
```
```  1030 context comm_monoid_mult
```
```  1031 begin
```
```  1032
```
```  1033 sublocale setprod: comm_monoid_set times 1
```
```  1034   defines setprod = setprod.F ..
```
```  1035
```
```  1036 abbreviation Setprod ("\<Prod>_"  999)
```
```  1037   where "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"
```
```  1038
```
```  1039 end
```
```  1040
```
```  1041 syntax (ASCII)
```
```  1042   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(4PROD _:_./ _)" [0, 51, 10] 10)
```
```  1043 syntax
```
```  1044   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(2\<Prod>_\<in>_./ _)" [0, 51, 10] 10)
```
```  1045 translations \<comment> \<open>Beware of argument permutation!\<close>
```
```  1046   "\<Prod>i\<in>A. b" == "CONST setprod (\<lambda>i. b) A"
```
```  1047
```
```  1048 text \<open>Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter \<open>\<Prod>x|P. e\<close>.\<close>
```
```  1049
```
```  1050 syntax (ASCII)
```
```  1051   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(4PROD _ |/ _./ _)" [0, 0, 10] 10)
```
```  1052 syntax
```
```  1053   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(2\<Prod>_ | (_)./ _)" [0, 0, 10] 10)
```
```  1054 translations
```
```  1055   "\<Prod>x|P. t" => "CONST setprod (\<lambda>x. t) {x. P}"
```
```  1056
```
```  1057 context comm_monoid_mult
```
```  1058 begin
```
```  1059
```
```  1060 lemma setprod_dvd_setprod: "(\<And>a. a \<in> A \<Longrightarrow> f a dvd g a) \<Longrightarrow> setprod f A dvd setprod g A"
```
```  1061 proof (induct A rule: infinite_finite_induct)
```
```  1062   case infinite
```
```  1063   then show ?case by (auto intro: dvdI)
```
```  1064 next
```
```  1065   case empty
```
```  1066   then show ?case by (auto intro: dvdI)
```
```  1067 next
```
```  1068   case (insert a A)
```
```  1069   then have "f a dvd g a" and "setprod f A dvd setprod g A"
```
```  1070     by simp_all
```
```  1071   then obtain r s where "g a = f a * r" and "setprod g A = setprod f A * s"
```
```  1072     by (auto elim!: dvdE)
```
```  1073   then have "g a * setprod g A = f a * setprod f A * (r * s)"
```
```  1074     by (simp add: ac_simps)
```
```  1075   with insert.hyps show ?case
```
```  1076     by (auto intro: dvdI)
```
```  1077 qed
```
```  1078
```
```  1079 lemma setprod_dvd_setprod_subset: "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> setprod f A dvd setprod f B"
```
```  1080   by (auto simp add: setprod.subset_diff ac_simps intro: dvdI)
```
```  1081
```
```  1082 end
```
```  1083
```
```  1084
```
```  1085 subsubsection \<open>Properties in more restricted classes of structures\<close>
```
```  1086
```
```  1087 context comm_semiring_1
```
```  1088 begin
```
```  1089
```
```  1090 lemma dvd_setprod_eqI [intro]:
```
```  1091   assumes "finite A" and "a \<in> A" and "b = f a"
```
```  1092   shows "b dvd setprod f A"
```
```  1093 proof -
```
```  1094   from \<open>finite A\<close> have "setprod f (insert a (A - {a})) = f a * setprod f (A - {a})"
```
```  1095     by (intro setprod.insert) auto
```
```  1096   also from \<open>a \<in> A\<close> have "insert a (A - {a}) = A"
```
```  1097     by blast
```
```  1098   finally have "setprod f A = f a * setprod f (A - {a})" .
```
```  1099   with \<open>b = f a\<close> show ?thesis
```
```  1100     by simp
```
```  1101 qed
```
```  1102
```
```  1103 lemma dvd_setprodI [intro]: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> f a dvd setprod f A"
```
```  1104   by auto
```
```  1105
```
```  1106 lemma setprod_zero:
```
```  1107   assumes "finite A" and "\<exists>a\<in>A. f a = 0"
```
```  1108   shows "setprod f A = 0"
```
```  1109   using assms
```
```  1110 proof (induct A)
```
```  1111   case empty
```
```  1112   then show ?case by simp
```
```  1113 next
```
```  1114   case (insert a A)
```
```  1115   then have "f a = 0 \<or> (\<exists>a\<in>A. f a = 0)" by simp
```
```  1116   then have "f a * setprod f A = 0" by rule (simp_all add: insert)
```
```  1117   with insert show ?case by simp
```
```  1118 qed
```
```  1119
```
```  1120 lemma setprod_dvd_setprod_subset2:
```
```  1121   assumes "finite B" and "A \<subseteq> B" and "\<And>a. a \<in> A \<Longrightarrow> f a dvd g a"
```
```  1122   shows "setprod f A dvd setprod g B"
```
```  1123 proof -
```
```  1124   from assms have "setprod f A dvd setprod g A"
```
```  1125     by (auto intro: setprod_dvd_setprod)
```
```  1126   moreover from assms have "setprod g A dvd setprod g B"
```
```  1127     by (auto intro: setprod_dvd_setprod_subset)
```
```  1128   ultimately show ?thesis by (rule dvd_trans)
```
```  1129 qed
```
```  1130
```
```  1131 end
```
```  1132
```
```  1133 lemma setprod_zero_iff [simp]:
```
```  1134   fixes f :: "'b \<Rightarrow> 'a::semidom"
```
```  1135   assumes "finite A"
```
```  1136   shows "setprod f A = 0 \<longleftrightarrow> (\<exists>a\<in>A. f a = 0)"
```
```  1137   using assms by (induct A) (auto simp: no_zero_divisors)
```
```  1138
```
```  1139 lemma (in semidom_divide) setprod_diff1:
```
```  1140   assumes "finite A" and "f a \<noteq> 0"
```
```  1141   shows "setprod f (A - {a}) = (if a \<in> A then setprod f A div f a else setprod f A)"
```
```  1142 proof (cases "a \<notin> A")
```
```  1143   case True
```
```  1144   then show ?thesis by simp
```
```  1145 next
```
```  1146   case False
```
```  1147   with assms show ?thesis
```
```  1148   proof induct
```
```  1149     case empty
```
```  1150     then show ?case by simp
```
```  1151   next
```
```  1152     case (insert b B)
```
```  1153     then show ?case
```
```  1154     proof (cases "a = b")
```
```  1155       case True
```
```  1156       with insert show ?thesis by simp
```
```  1157     next
```
```  1158       case False
```
```  1159       with insert have "a \<in> B" by simp
```
```  1160       define C where "C = B - {a}"
```
```  1161       with \<open>finite B\<close> \<open>a \<in> B\<close> have "B = insert a C" "finite C" "a \<notin> C"
```
```  1162         by auto
```
```  1163       with insert show ?thesis
```
```  1164         by (auto simp add: insert_commute ac_simps)
```
```  1165     qed
```
```  1166   qed
```
```  1167 qed
```
```  1168
```
```  1169 lemma setsum_zero_power [simp]: "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
```
```  1170   for c :: "nat \<Rightarrow> 'a::division_ring"
```
```  1171   by (induct A rule: infinite_finite_induct) auto
```
```  1172
```
```  1173 lemma setsum_zero_power' [simp]:
```
```  1174   "(\<Sum>i\<in>A. c i * 0^i / d i) = (if finite A \<and> 0 \<in> A then c 0 / d 0 else 0)"
```
```  1175   for c :: "nat \<Rightarrow> 'a::field"
```
```  1176   using setsum_zero_power [of "\<lambda>i. c i / d i" A] by auto
```
```  1177
```
```  1178 lemma (in field) setprod_inversef:
```
```  1179   "finite A \<Longrightarrow> setprod (inverse \<circ> f) A = inverse (setprod f A)"
```
```  1180   by (induct A rule: finite_induct) simp_all
```
```  1181
```
```  1182 lemma (in field) setprod_dividef: "finite A \<Longrightarrow> (\<Prod>x\<in>A. f x / g x) = setprod f A / setprod g A"
```
```  1183   using setprod_inversef [of A g] by (simp add: divide_inverse setprod.distrib)
```
```  1184
```
```  1185 lemma setprod_Un:
```
```  1186   fixes f :: "'b \<Rightarrow> 'a :: field"
```
```  1187   assumes "finite A" and "finite B"
```
```  1188     and "\<forall>x\<in>A \<inter> B. f x \<noteq> 0"
```
```  1189   shows "setprod f (A \<union> B) = setprod f A * setprod f B / setprod f (A \<inter> B)"
```
```  1190 proof -
```
```  1191   from assms have "setprod f A * setprod f B = setprod f (A \<union> B) * setprod f (A \<inter> B)"
```
```  1192     by (simp add: setprod.union_inter [symmetric, of A B])
```
```  1193   with assms show ?thesis
```
```  1194     by simp
```
```  1195 qed
```
```  1196
```
```  1197 lemma (in linordered_semidom) setprod_nonneg: "(\<forall>a\<in>A. 0 \<le> f a) \<Longrightarrow> 0 \<le> setprod f A"
```
```  1198   by (induct A rule: infinite_finite_induct) simp_all
```
```  1199
```
```  1200 lemma (in linordered_semidom) setprod_pos: "(\<forall>a\<in>A. 0 < f a) \<Longrightarrow> 0 < setprod f A"
```
```  1201   by (induct A rule: infinite_finite_induct) simp_all
```
```  1202
```
```  1203 lemma (in linordered_semidom) setprod_mono:
```
```  1204   "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i \<Longrightarrow> setprod f A \<le> setprod g A"
```
```  1205   by (induct A rule: infinite_finite_induct) (auto intro!: setprod_nonneg mult_mono)
```
```  1206
```
```  1207 lemma (in linordered_semidom) setprod_mono_strict:
```
```  1208   assumes "finite A" "\<forall>i\<in>A. 0 \<le> f i \<and> f i < g i" "A \<noteq> {}"
```
```  1209   shows "setprod f A < setprod g A"
```
```  1210   using assms
```
```  1211 proof (induct A rule: finite_induct)
```
```  1212   case empty
```
```  1213   then show ?case by simp
```
```  1214 next
```
```  1215   case insert
```
```  1216   then show ?case by (force intro: mult_strict_mono' setprod_nonneg)
```
```  1217 qed
```
```  1218
```
```  1219 lemma (in linordered_field) abs_setprod: "\<bar>setprod f A\<bar> = (\<Prod>x\<in>A. \<bar>f x\<bar>)"
```
```  1220   by (induct A rule: infinite_finite_induct) (simp_all add: abs_mult)
```
```  1221
```
```  1222 lemma setprod_eq_1_iff [simp]: "finite A \<Longrightarrow> setprod f A = 1 \<longleftrightarrow> (\<forall>a\<in>A. f a = 1)"
```
```  1223   for f :: "'a \<Rightarrow> nat"
```
```  1224   by (induct A rule: finite_induct) simp_all
```
```  1225
```
```  1226 lemma setprod_pos_nat_iff [simp]: "finite A \<Longrightarrow> setprod f A > 0 \<longleftrightarrow> (\<forall>a\<in>A. f a > 0)"
```
```  1227   for f :: "'a \<Rightarrow> nat"
```
```  1228   using setprod_zero_iff by (simp del: neq0_conv add: zero_less_iff_neq_zero)
```
```  1229
```
```  1230 lemma setprod_constant: "(\<Prod>x\<in> A. y) = y ^ card A"
```
```  1231   for y :: "'a::comm_monoid_mult"
```
```  1232   by (induct A rule: infinite_finite_induct) simp_all
```
```  1233
```
```  1234 lemma setprod_power_distrib: "setprod f A ^ n = setprod (\<lambda>x. (f x) ^ n) A"
```
```  1235   for f :: "'a \<Rightarrow> 'b::comm_semiring_1"
```
```  1236   by (induct A rule: infinite_finite_induct) (auto simp add: power_mult_distrib)
```
```  1237
```
```  1238 lemma power_setsum: "c ^ (\<Sum>a\<in>A. f a) = (\<Prod>a\<in>A. c ^ f a)"
```
```  1239   by (induct A rule: infinite_finite_induct) (simp_all add: power_add)
```
```  1240
```
```  1241 lemma setprod_gen_delta:
```
```  1242   fixes b :: "'b \<Rightarrow> 'a::comm_monoid_mult"
```
```  1243   assumes fin: "finite S"
```
```  1244   shows "setprod (\<lambda>k. if k = a then b k else c) S =
```
```  1245     (if a \<in> S then b a * c ^ (card S - 1) else c ^ card S)"
```
```  1246 proof -
```
```  1247   let ?f = "(\<lambda>k. if k=a then b k else c)"
```
```  1248   show ?thesis
```
```  1249   proof (cases "a \<in> S")
```
```  1250     case False
```
```  1251     then have "\<forall> k\<in> S. ?f k = c" by simp
```
```  1252     with False show ?thesis by (simp add: setprod_constant)
```
```  1253   next
```
```  1254     case True
```
```  1255     let ?A = "S - {a}"
```
```  1256     let ?B = "{a}"
```
```  1257     from True have eq: "S = ?A \<union> ?B" by blast
```
```  1258     have disjoint: "?A \<inter> ?B = {}" by simp
```
```  1259     from fin have fin': "finite ?A" "finite ?B" by auto
```
```  1260     have f_A0: "setprod ?f ?A = setprod (\<lambda>i. c) ?A"
```
```  1261       by (rule setprod.cong) auto
```
```  1262     from fin True have card_A: "card ?A = card S - 1" by auto
```
```  1263     have f_A1: "setprod ?f ?A = c ^ card ?A"
```
```  1264       unfolding f_A0 by (rule setprod_constant)
```
```  1265     have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
```
```  1266       using setprod.union_disjoint[OF fin' disjoint, of ?f, unfolded eq[symmetric]]
```
```  1267       by simp
```
```  1268     with True card_A show ?thesis
```
```  1269       by (simp add: f_A1 field_simps cong add: setprod.cong cong del: if_weak_cong)
```
```  1270   qed
```
```  1271 qed
```
```  1272
```
```  1273 end
```