src/HOL/Groups_Big.thy
 author paulson Mon Feb 22 14:37:56 2016 +0000 (2016-02-22) changeset 62379 340738057c8c parent 62378 85ed00c1fe7c child 62481 b5d8e57826df permissions -rw-r--r--
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
```     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 section \<open>Big sum and product over finite (non-empty) sets\<close>
```
```     7
```
```     8 theory Groups_Big
```
```     9 imports Finite_Set
```
```    10 begin
```
```    11
```
```    12 subsection \<open>Generic monoid operation over a set\<close>
```
```    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 standard (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 \<open>x \<in> A\<close> obtain B where A: "A = insert x B" and "x \<notin> B"
```
```    48     by (auto dest: mk_disjoint_insert)
```
```    49   moreover from \<open>finite A\<close> 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   \<comment> \<open>The reversed orientation looks more natural, but LOOPS as a simprule!\<close>
```
```    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 union_diff2:
```
```    91   assumes "finite A" and "finite B"
```
```    92   shows "F g (A \<union> B) = F g (A - B) * F g (B - A) * F g (A \<inter> B)"
```
```    93 proof -
```
```    94   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
```
```    95     by auto
```
```    96   with assms show ?thesis by simp (subst union_disjoint, auto)+
```
```    97 qed
```
```    98
```
```    99 lemma subset_diff:
```
```   100   assumes "B \<subseteq> A" and "finite A"
```
```   101   shows "F g A = F g (A - B) * F g B"
```
```   102 proof -
```
```   103   from assms have "finite (A - B)" by auto
```
```   104   moreover from assms have "finite B" by (rule finite_subset)
```
```   105   moreover from assms have "(A - B) \<inter> B = {}" by auto
```
```   106   ultimately have "F g (A - B \<union> B) = F g (A - B) * F g B" by (rule union_disjoint)
```
```   107   moreover from assms have "A \<union> B = A" by auto
```
```   108   ultimately show ?thesis by simp
```
```   109 qed
```
```   110
```
```   111 lemma setdiff_irrelevant:
```
```   112   assumes "finite A"
```
```   113   shows "F g (A - {x. g x = z}) = F g A"
```
```   114   using assms by (induct A) (simp_all add: insert_Diff_if)
```
```   115
```
```   116 lemma not_neutral_contains_not_neutral:
```
```   117   assumes "F g A \<noteq> z"
```
```   118   obtains a where "a \<in> A" and "g a \<noteq> z"
```
```   119 proof -
```
```   120   from assms have "\<exists>a\<in>A. g a \<noteq> z"
```
```   121   proof (induct A rule: infinite_finite_induct)
```
```   122     case (insert a A)
```
```   123     then show ?case by simp (rule, simp)
```
```   124   qed simp_all
```
```   125   with that show thesis by blast
```
```   126 qed
```
```   127
```
```   128 lemma reindex:
```
```   129   assumes "inj_on h A"
```
```   130   shows "F g (h ` A) = F (g \<circ> h) A"
```
```   131 proof (cases "finite A")
```
```   132   case True
```
```   133   with assms show ?thesis by (simp add: eq_fold fold_image comp_assoc)
```
```   134 next
```
```   135   case False with assms have "\<not> finite (h ` A)" by (blast dest: finite_imageD)
```
```   136   with False show ?thesis by simp
```
```   137 qed
```
```   138
```
```   139 lemma cong:
```
```   140   assumes "A = B"
```
```   141   assumes g_h: "\<And>x. x \<in> B \<Longrightarrow> g x = h x"
```
```   142   shows "F g A = F h B"
```
```   143   using g_h unfolding \<open>A = B\<close>
```
```   144   by (induct B rule: infinite_finite_induct) auto
```
```   145
```
```   146 lemma strong_cong [cong]:
```
```   147   assumes "A = B" "\<And>x. x \<in> B =simp=> g x = h x"
```
```   148   shows "F (\<lambda>x. g x) A = F (\<lambda>x. h x) B"
```
```   149   by (rule cong) (insert assms, simp_all add: simp_implies_def)
```
```   150
```
```   151 lemma reindex_cong:
```
```   152   assumes "inj_on l B"
```
```   153   assumes "A = l ` B"
```
```   154   assumes "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x"
```
```   155   shows "F g A = F h B"
```
```   156   using assms by (simp add: reindex)
```
```   157
```
```   158 lemma UNION_disjoint:
```
```   159   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
```
```   160   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
```
```   161   shows "F g (UNION I A) = F (\<lambda>x. F g (A x)) I"
```
```   162 apply (insert assms)
```
```   163 apply (induct rule: finite_induct)
```
```   164 apply simp
```
```   165 apply atomize
```
```   166 apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
```
```   167  prefer 2 apply blast
```
```   168 apply (subgoal_tac "A x Int UNION Fa A = {}")
```
```   169  prefer 2 apply blast
```
```   170 apply (simp add: union_disjoint)
```
```   171 done
```
```   172
```
```   173 lemma Union_disjoint:
```
```   174   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
```
```   175   shows "F g (\<Union>C) = (F \<circ> F) g C"
```
```   176 proof cases
```
```   177   assume "finite C"
```
```   178   from UNION_disjoint [OF this assms]
```
```   179   show ?thesis by simp
```
```   180 qed (auto dest: finite_UnionD intro: infinite)
```
```   181
```
```   182 lemma distrib:
```
```   183   "F (\<lambda>x. g x * h x) A = F g A * F h A"
```
```   184   using assms by (induct A rule: infinite_finite_induct) (simp_all add: assoc commute left_commute)
```
```   185
```
```   186 lemma Sigma:
```
```   187   "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)"
```
```   188 apply (subst Sigma_def)
```
```   189 apply (subst UNION_disjoint, assumption, simp)
```
```   190  apply blast
```
```   191 apply (rule cong)
```
```   192 apply rule
```
```   193 apply (simp add: fun_eq_iff)
```
```   194 apply (subst UNION_disjoint, simp, simp)
```
```   195  apply blast
```
```   196 apply (simp add: comp_def)
```
```   197 done
```
```   198
```
```   199 lemma related:
```
```   200   assumes Re: "R 1 1"
```
```   201   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
```
```   202   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
```
```   203   shows "R (F h S) (F g S)"
```
```   204   using fS by (rule finite_subset_induct) (insert assms, auto)
```
```   205
```
```   206 lemma mono_neutral_cong_left:
```
```   207   assumes "finite T" and "S \<subseteq> T" and "\<forall>i \<in> T - S. h i = 1"
```
```   208   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g S = F h T"
```
```   209 proof-
```
```   210   have eq: "T = S \<union> (T - S)" using \<open>S \<subseteq> T\<close> by blast
```
```   211   have d: "S \<inter> (T - S) = {}" using \<open>S \<subseteq> T\<close> by blast
```
```   212   from \<open>finite T\<close> \<open>S \<subseteq> T\<close> have f: "finite S" "finite (T - S)"
```
```   213     by (auto intro: finite_subset)
```
```   214   show ?thesis using assms(4)
```
```   215     by (simp add: union_disjoint [OF f d, unfolded eq [symmetric]] neutral [OF assms(3)])
```
```   216 qed
```
```   217
```
```   218 lemma mono_neutral_cong_right:
```
```   219   "\<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>
```
```   220    \<Longrightarrow> F g T = F h S"
```
```   221   by (auto intro!: mono_neutral_cong_left [symmetric])
```
```   222
```
```   223 lemma mono_neutral_left:
```
```   224   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g S = F g T"
```
```   225   by (blast intro: mono_neutral_cong_left)
```
```   226
```
```   227 lemma mono_neutral_right:
```
```   228   "\<lbrakk> finite T;  S \<subseteq> T;  \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g T = F g S"
```
```   229   by (blast intro!: mono_neutral_left [symmetric])
```
```   230
```
```   231 lemma reindex_bij_betw: "bij_betw h S T \<Longrightarrow> F (\<lambda>x. g (h x)) S = F g T"
```
```   232   by (auto simp: bij_betw_def reindex)
```
```   233
```
```   234 lemma reindex_bij_witness:
```
```   235   assumes witness:
```
```   236     "\<And>a. a \<in> S \<Longrightarrow> i (j a) = a"
```
```   237     "\<And>a. a \<in> S \<Longrightarrow> j a \<in> T"
```
```   238     "\<And>b. b \<in> T \<Longrightarrow> j (i b) = b"
```
```   239     "\<And>b. b \<in> T \<Longrightarrow> i b \<in> S"
```
```   240   assumes eq:
```
```   241     "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a"
```
```   242   shows "F g S = F h T"
```
```   243 proof -
```
```   244   have "bij_betw j S T"
```
```   245     using bij_betw_byWitness[where A=S and f=j and f'=i and A'=T] witness by auto
```
```   246   moreover have "F g S = F (\<lambda>x. h (j x)) S"
```
```   247     by (intro cong) (auto simp: eq)
```
```   248   ultimately show ?thesis
```
```   249     by (simp add: reindex_bij_betw)
```
```   250 qed
```
```   251
```
```   252 lemma reindex_bij_betw_not_neutral:
```
```   253   assumes fin: "finite S'" "finite T'"
```
```   254   assumes bij: "bij_betw h (S - S') (T - T')"
```
```   255   assumes nn:
```
```   256     "\<And>a. a \<in> S' \<Longrightarrow> g (h a) = z"
```
```   257     "\<And>b. b \<in> T' \<Longrightarrow> g b = z"
```
```   258   shows "F (\<lambda>x. g (h x)) S = F g T"
```
```   259 proof -
```
```   260   have [simp]: "finite S \<longleftrightarrow> finite T"
```
```   261     using bij_betw_finite[OF bij] fin by auto
```
```   262
```
```   263   show ?thesis
```
```   264   proof cases
```
```   265     assume "finite S"
```
```   266     with nn have "F (\<lambda>x. g (h x)) S = F (\<lambda>x. g (h x)) (S - S')"
```
```   267       by (intro mono_neutral_cong_right) auto
```
```   268     also have "\<dots> = F g (T - T')"
```
```   269       using bij by (rule reindex_bij_betw)
```
```   270     also have "\<dots> = F g T"
```
```   271       using nn \<open>finite S\<close> by (intro mono_neutral_cong_left) auto
```
```   272     finally show ?thesis .
```
```   273   qed simp
```
```   274 qed
```
```   275
```
```   276 lemma reindex_nontrivial:
```
```   277   assumes "finite A"
```
```   278   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) = 1"
```
```   279   shows "F g (h ` A) = F (g \<circ> h) A"
```
```   280 proof (subst reindex_bij_betw_not_neutral [symmetric])
```
```   281   show "bij_betw h (A - {x \<in> A. (g \<circ> h) x = 1}) (h ` A - h ` {x \<in> A. (g \<circ> h) x = 1})"
```
```   282     using nz by (auto intro!: inj_onI simp: bij_betw_def)
```
```   283 qed (insert \<open>finite A\<close>, auto)
```
```   284
```
```   285 lemma reindex_bij_witness_not_neutral:
```
```   286   assumes fin: "finite S'" "finite T'"
```
```   287   assumes witness:
```
```   288     "\<And>a. a \<in> S - S' \<Longrightarrow> i (j a) = a"
```
```   289     "\<And>a. a \<in> S - S' \<Longrightarrow> j a \<in> T - T'"
```
```   290     "\<And>b. b \<in> T - T' \<Longrightarrow> j (i b) = b"
```
```   291     "\<And>b. b \<in> T - T' \<Longrightarrow> i b \<in> S - S'"
```
```   292   assumes nn:
```
```   293     "\<And>a. a \<in> S' \<Longrightarrow> g a = z"
```
```   294     "\<And>b. b \<in> T' \<Longrightarrow> h b = z"
```
```   295   assumes eq:
```
```   296     "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a"
```
```   297   shows "F g S = F h T"
```
```   298 proof -
```
```   299   have bij: "bij_betw j (S - (S' \<inter> S)) (T - (T' \<inter> T))"
```
```   300     using witness by (intro bij_betw_byWitness[where f'=i]) auto
```
```   301   have F_eq: "F g S = F (\<lambda>x. h (j x)) S"
```
```   302     by (intro cong) (auto simp: eq)
```
```   303   show ?thesis
```
```   304     unfolding F_eq using fin nn eq
```
```   305     by (intro reindex_bij_betw_not_neutral[OF _ _ bij]) auto
```
```   306 qed
```
```   307
```
```   308 lemma delta:
```
```   309   assumes fS: "finite S"
```
```   310   shows "F (\<lambda>k. if k = a then b k else 1) S = (if a \<in> S then b a else 1)"
```
```   311 proof-
```
```   312   let ?f = "(\<lambda>k. if k=a then b k else 1)"
```
```   313   { assume a: "a \<notin> S"
```
```   314     hence "\<forall>k\<in>S. ?f k = 1" by simp
```
```   315     hence ?thesis  using a by simp }
```
```   316   moreover
```
```   317   { assume a: "a \<in> S"
```
```   318     let ?A = "S - {a}"
```
```   319     let ?B = "{a}"
```
```   320     have eq: "S = ?A \<union> ?B" using a by blast
```
```   321     have dj: "?A \<inter> ?B = {}" by simp
```
```   322     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```   323     have "F ?f S = F ?f ?A * F ?f ?B"
```
```   324       using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]]
```
```   325       by simp
```
```   326     then have ?thesis using a by simp }
```
```   327   ultimately show ?thesis by blast
```
```   328 qed
```
```   329
```
```   330 lemma delta':
```
```   331   assumes fS: "finite S"
```
```   332   shows "F (\<lambda>k. if a = k then b k else 1) S = (if a \<in> S then b a else 1)"
```
```   333   using delta [OF fS, of a b, symmetric] by (auto intro: cong)
```
```   334
```
```   335 lemma If_cases:
```
```   336   fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
```
```   337   assumes fA: "finite A"
```
```   338   shows "F (\<lambda>x. if P x then h x else g x) A =
```
```   339     F h (A \<inter> {x. P x}) * F g (A \<inter> - {x. P x})"
```
```   340 proof -
```
```   341   have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}"
```
```   342           "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}"
```
```   343     by blast+
```
```   344   from fA
```
```   345   have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
```
```   346   let ?g = "\<lambda>x. if P x then h x else g x"
```
```   347   from union_disjoint [OF f a(2), of ?g] a(1)
```
```   348   show ?thesis
```
```   349     by (subst (1 2) cong) simp_all
```
```   350 qed
```
```   351
```
```   352 lemma cartesian_product:
```
```   353    "F (\<lambda>x. F (g x) B) A = F (case_prod g) (A \<times> B)"
```
```   354 apply (rule sym)
```
```   355 apply (cases "finite A")
```
```   356  apply (cases "finite B")
```
```   357   apply (simp add: Sigma)
```
```   358  apply (cases "A={}", simp)
```
```   359  apply simp
```
```   360 apply (auto intro: infinite dest: finite_cartesian_productD2)
```
```   361 apply (cases "B = {}") apply (auto intro: infinite dest: finite_cartesian_productD1)
```
```   362 done
```
```   363
```
```   364 lemma inter_restrict:
```
```   365   assumes "finite A"
```
```   366   shows "F g (A \<inter> B) = F (\<lambda>x. if x \<in> B then g x else 1) A"
```
```   367 proof -
```
```   368   let ?g = "\<lambda>x. if x \<in> A \<inter> B then g x else 1"
```
```   369   have "\<forall>i\<in>A - A \<inter> B. (if i \<in> A \<inter> B then g i else 1) = 1"
```
```   370    by simp
```
```   371   moreover have "A \<inter> B \<subseteq> A" by blast
```
```   372   ultimately have "F ?g (A \<inter> B) = F ?g A" using \<open>finite A\<close>
```
```   373     by (intro mono_neutral_left) auto
```
```   374   then show ?thesis by simp
```
```   375 qed
```
```   376
```
```   377 lemma inter_filter:
```
```   378   "finite A \<Longrightarrow> F g {x \<in> A. P x} = F (\<lambda>x. if P x then g x else 1) A"
```
```   379   by (simp add: inter_restrict [symmetric, of A "{x. P x}" g, simplified mem_Collect_eq] Int_def)
```
```   380
```
```   381 lemma Union_comp:
```
```   382   assumes "\<forall>A \<in> B. finite A"
```
```   383     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 = 1"
```
```   384   shows "F g (\<Union>B) = (F \<circ> F) g B"
```
```   385 using assms proof (induct B rule: infinite_finite_induct)
```
```   386   case (infinite A)
```
```   387   then have "\<not> finite (\<Union>A)" by (blast dest: finite_UnionD)
```
```   388   with infinite show ?case by simp
```
```   389 next
```
```   390   case empty then show ?case by simp
```
```   391 next
```
```   392   case (insert A B)
```
```   393   then have "finite A" "finite B" "finite (\<Union>B)" "A \<notin> B"
```
```   394     and "\<forall>x\<in>A \<inter> \<Union>B. g x = 1"
```
```   395     and H: "F g (\<Union>B) = (F o F) g B" by auto
```
```   396   then have "F g (A \<union> \<Union>B) = F g A * F g (\<Union>B)"
```
```   397     by (simp add: union_inter_neutral)
```
```   398   with \<open>finite B\<close> \<open>A \<notin> B\<close> show ?case
```
```   399     by (simp add: H)
```
```   400 qed
```
```   401
```
```   402 lemma commute:
```
```   403   "F (\<lambda>i. F (g i) B) A = F (\<lambda>j. F (\<lambda>i. g i j) A) B"
```
```   404   unfolding cartesian_product
```
```   405   by (rule reindex_bij_witness [where i = "\<lambda>(i, j). (j, i)" and j = "\<lambda>(i, j). (j, i)"]) auto
```
```   406
```
```   407 lemma commute_restrict:
```
```   408   "finite A \<Longrightarrow> finite B \<Longrightarrow>
```
```   409     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"
```
```   410   by (simp add: inter_filter) (rule commute)
```
```   411
```
```   412 lemma Plus:
```
```   413   fixes A :: "'b set" and B :: "'c set"
```
```   414   assumes fin: "finite A" "finite B"
```
```   415   shows "F g (A <+> B) = F (g \<circ> Inl) A * F (g \<circ> Inr) B"
```
```   416 proof -
```
```   417   have "A <+> B = Inl ` A \<union> Inr ` B" by auto
```
```   418   moreover from fin have "finite (Inl ` A :: ('b + 'c) set)" "finite (Inr ` B :: ('b + 'c) set)"
```
```   419     by auto
```
```   420   moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('b + 'c) set)" by auto
```
```   421   moreover have "inj_on (Inl :: 'b \<Rightarrow> 'b + 'c) A" "inj_on (Inr :: 'c \<Rightarrow> 'b + 'c) B"
```
```   422     by (auto intro: inj_onI)
```
```   423   ultimately show ?thesis using fin
```
```   424     by (simp add: union_disjoint reindex)
```
```   425 qed
```
```   426
```
```   427 lemma same_carrier:
```
```   428   assumes "finite C"
```
```   429   assumes subset: "A \<subseteq> C" "B \<subseteq> C"
```
```   430   assumes trivial: "\<And>a. a \<in> C - A \<Longrightarrow> g a = 1" "\<And>b. b \<in> C - B \<Longrightarrow> h b = 1"
```
```   431   shows "F g A = F h B \<longleftrightarrow> F g C = F h C"
```
```   432 proof -
```
```   433   from \<open>finite C\<close> subset have
```
```   434     "finite A" and "finite B" and "finite (C - A)" and "finite (C - B)"
```
```   435     by (auto elim: finite_subset)
```
```   436   from subset have [simp]: "A - (C - A) = A" by auto
```
```   437   from subset have [simp]: "B - (C - B) = B" by auto
```
```   438   from subset have "C = A \<union> (C - A)" by auto
```
```   439   then have "F g C = F g (A \<union> (C - A))" by simp
```
```   440   also have "\<dots> = F g (A - (C - A)) * F g (C - A - A) * F g (A \<inter> (C - A))"
```
```   441     using \<open>finite A\<close> \<open>finite (C - A)\<close> by (simp only: union_diff2)
```
```   442   finally have P: "F g C = F g A" using trivial by simp
```
```   443   from subset have "C = B \<union> (C - B)" by auto
```
```   444   then have "F h C = F h (B \<union> (C - B))" by simp
```
```   445   also have "\<dots> = F h (B - (C - B)) * F h (C - B - B) * F h (B \<inter> (C - B))"
```
```   446     using \<open>finite B\<close> \<open>finite (C - B)\<close> by (simp only: union_diff2)
```
```   447   finally have Q: "F h C = F h B" using trivial by simp
```
```   448   from P Q show ?thesis by simp
```
```   449 qed
```
```   450
```
```   451 lemma same_carrierI:
```
```   452   assumes "finite C"
```
```   453   assumes subset: "A \<subseteq> C" "B \<subseteq> C"
```
```   454   assumes trivial: "\<And>a. a \<in> C - A \<Longrightarrow> g a = 1" "\<And>b. b \<in> C - B \<Longrightarrow> h b = 1"
```
```   455   assumes "F g C = F h C"
```
```   456   shows "F g A = F h B"
```
```   457   using assms same_carrier [of C A B] by simp
```
```   458
```
```   459 end
```
```   460
```
```   461 notation times (infixl "*" 70)
```
```   462 notation Groups.one ("1")
```
```   463
```
```   464
```
```   465 subsection \<open>Generalized summation over a set\<close>
```
```   466
```
```   467 context comm_monoid_add
```
```   468 begin
```
```   469
```
```   470 sublocale setsum: comm_monoid_set plus 0
```
```   471 defines
```
```   472   setsum = setsum.F ..
```
```   473
```
```   474 abbreviation Setsum ("\<Sum>_" [1000] 999)
```
```   475   where "\<Sum>A \<equiv> setsum (\<lambda>x. x) A"
```
```   476
```
```   477 end
```
```   478
```
```   479 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>
```
```   480
```
```   481 syntax (ASCII)
```
```   482   "_setsum" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b::comm_monoid_add"  ("(3SUM _:_./ _)" [0, 51, 10] 10)
```
```   483 syntax
```
```   484   "_setsum" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b::comm_monoid_add"  ("(2\<Sum>_\<in>_./ _)" [0, 51, 10] 10)
```
```   485 translations \<comment> \<open>Beware of argument permutation!\<close>
```
```   486   "\<Sum>i\<in>A. b" \<rightleftharpoons> "CONST setsum (\<lambda>i. b) A"
```
```   487
```
```   488 text \<open>Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter \<open>\<Sum>x|P. e\<close>.\<close>
```
```   489
```
```   490 syntax (ASCII)
```
```   491   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(3SUM _ |/ _./ _)" [0, 0, 10] 10)
```
```   492 syntax
```
```   493   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(2\<Sum>_ | (_)./ _)" [0, 0, 10] 10)
```
```   494 translations
```
```   495   "\<Sum>x|P. t" => "CONST setsum (\<lambda>x. t) {x. P}"
```
```   496
```
```   497 print_translation \<open>
```
```   498 let
```
```   499   fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) \$ Abs (y, Ty, P)] =
```
```   500         if x <> y then raise Match
```
```   501         else
```
```   502           let
```
```   503             val x' = Syntax_Trans.mark_bound_body (x, Tx);
```
```   504             val t' = subst_bound (x', t);
```
```   505             val P' = subst_bound (x', P);
```
```   506           in
```
```   507             Syntax.const @{syntax_const "_qsetsum"} \$ Syntax_Trans.mark_bound_abs (x, Tx) \$ P' \$ t'
```
```   508           end
```
```   509     | setsum_tr' _ = raise Match;
```
```   510 in [(@{const_syntax setsum}, K setsum_tr')] end
```
```   511 \<close>
```
```   512
```
```   513 text \<open>TODO generalization candidates\<close>
```
```   514
```
```   515 lemma (in comm_monoid_add) setsum_image_gen:
```
```   516   assumes fS: "finite S"
```
```   517   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
```
```   518 proof-
```
```   519   { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }
```
```   520   hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
```
```   521     by simp
```
```   522   also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
```
```   523     by (rule setsum.commute_restrict [OF fS finite_imageI [OF fS]])
```
```   524   finally show ?thesis .
```
```   525 qed
```
```   526
```
```   527
```
```   528 subsubsection \<open>Properties in more restricted classes of structures\<close>
```
```   529
```
```   530 lemma setsum_Un: "finite A ==> finite B ==>
```
```   531   (setsum f (A Un B) :: 'a :: ab_group_add) =
```
```   532    setsum f A + setsum f B - setsum f (A Int B)"
```
```   533 by (subst setsum.union_inter [symmetric], auto simp add: algebra_simps)
```
```   534
```
```   535 lemma setsum_Un2:
```
```   536   assumes "finite (A \<union> B)"
```
```   537   shows "setsum f (A \<union> B) = setsum f (A - B) + setsum f (B - A) + setsum f (A \<inter> B)"
```
```   538 proof -
```
```   539   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
```
```   540     by auto
```
```   541   with assms show ?thesis by simp (subst setsum.union_disjoint, auto)+
```
```   542 qed
```
```   543
```
```   544 lemma setsum_diff1: "finite A \<Longrightarrow>
```
```   545   (setsum f (A - {a}) :: ('a::ab_group_add)) =
```
```   546   (if a:A then setsum f A - f a else setsum f A)"
```
```   547 by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```   548
```
```   549 lemma setsum_diff:
```
```   550   assumes le: "finite A" "B \<subseteq> A"
```
```   551   shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
```
```   552 proof -
```
```   553   from le have finiteB: "finite B" using finite_subset by auto
```
```   554   show ?thesis using finiteB le
```
```   555   proof induct
```
```   556     case empty
```
```   557     thus ?case by auto
```
```   558   next
```
```   559     case (insert x F)
```
```   560     thus ?case using le finiteB
```
```   561       by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
```
```   562   qed
```
```   563 qed
```
```   564
```
```   565 lemma (in ordered_comm_monoid_add) setsum_mono:
```
```   566   assumes le: "\<And>i. i\<in>K \<Longrightarrow> f i \<le> g i"
```
```   567   shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
```
```   568 proof (cases "finite K")
```
```   569   case True
```
```   570   thus ?thesis using le
```
```   571   proof induct
```
```   572     case empty
```
```   573     thus ?case by simp
```
```   574   next
```
```   575     case insert
```
```   576     thus ?case using add_mono by fastforce
```
```   577   qed
```
```   578 next
```
```   579   case False then show ?thesis by simp
```
```   580 qed
```
```   581
```
```   582 lemma (in strict_ordered_comm_monoid_add) setsum_strict_mono:
```
```   583   assumes "finite A"  "A \<noteq> {}" and "\<And>x. x \<in> A \<Longrightarrow> f x < g x"
```
```   584   shows "setsum f A < setsum g A"
```
```   585   using assms
```
```   586 proof (induct rule: finite_ne_induct)
```
```   587   case singleton thus ?case by simp
```
```   588 next
```
```   589   case insert thus ?case by (auto simp: add_strict_mono)
```
```   590 qed
```
```   591
```
```   592 lemma setsum_strict_mono_ex1:
```
```   593   fixes f g :: "'i \<Rightarrow> 'a::ordered_cancel_comm_monoid_add"
```
```   594   assumes "finite A" and "\<forall>x\<in>A. f x \<le> g x" and "\<exists>a\<in>A. f a < g a"
```
```   595   shows "setsum f A < setsum g A"
```
```   596 proof-
```
```   597   from assms(3) obtain a where a: "a:A" "f a < g a" by blast
```
```   598   have "setsum f A = setsum f ((A-{a}) \<union> {a})"
```
```   599     by(simp add:insert_absorb[OF \<open>a:A\<close>])
```
```   600   also have "\<dots> = setsum f (A-{a}) + setsum f {a}"
```
```   601     using \<open>finite A\<close> by(subst setsum.union_disjoint) auto
```
```   602   also have "setsum f (A-{a}) \<le> setsum g (A-{a})"
```
```   603     by(rule setsum_mono)(simp add: assms(2))
```
```   604   also have "setsum f {a} < setsum g {a}" using a by simp
```
```   605   also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"
```
```   606     using \<open>finite A\<close> by(subst setsum.union_disjoint[symmetric]) auto
```
```   607   also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF \<open>a:A\<close>])
```
```   608   finally show ?thesis by (auto simp add: add_right_mono add_strict_left_mono)
```
```   609 qed
```
```   610
```
```   611 lemma setsum_negf: "(\<Sum>x\<in>A. - f x::'a::ab_group_add) = - (\<Sum>x\<in>A. f x)"
```
```   612 proof (cases "finite A")
```
```   613   case True thus ?thesis by (induct set: finite) auto
```
```   614 next
```
```   615   case False thus ?thesis by simp
```
```   616 qed
```
```   617
```
```   618 lemma setsum_subtractf: "(\<Sum>x\<in>A. f x - g x::'a::ab_group_add) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
```
```   619   using setsum.distrib [of f "- g" A] by (simp add: setsum_negf)
```
```   620
```
```   621 lemma setsum_subtractf_nat:
```
```   622   "(\<And>x. x \<in> A \<Longrightarrow> g x \<le> f x) \<Longrightarrow> (\<Sum>x\<in>A. f x - g x::nat) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
```
```   623   by (induction A rule: infinite_finite_induct) (auto simp: setsum_mono)
```
```   624
```
```   625 lemma (in ordered_comm_monoid_add) setsum_nonneg:
```
```   626   assumes nn: "\<forall>x\<in>A. 0 \<le> f x"
```
```   627   shows "0 \<le> setsum f A"
```
```   628 proof (cases "finite A")
```
```   629   case True thus ?thesis using nn
```
```   630   proof induct
```
```   631     case empty then show ?case by simp
```
```   632   next
```
```   633     case (insert x F)
```
```   634     then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
```
```   635     with insert show ?case by simp
```
```   636   qed
```
```   637 next
```
```   638   case False thus ?thesis by simp
```
```   639 qed
```
```   640
```
```   641 lemma (in ordered_comm_monoid_add) setsum_nonpos:
```
```   642   assumes np: "\<forall>x\<in>A. f x \<le> 0"
```
```   643   shows "setsum f A \<le> 0"
```
```   644 proof (cases "finite A")
```
```   645   case True thus ?thesis using np
```
```   646   proof induct
```
```   647     case empty then show ?case by simp
```
```   648   next
```
```   649     case (insert x F)
```
```   650     then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
```
```   651     with insert show ?case by simp
```
```   652   qed
```
```   653 next
```
```   654   case False thus ?thesis by simp
```
```   655 qed
```
```   656
```
```   657 lemma (in ordered_comm_monoid_add) setsum_nonneg_eq_0_iff:
```
```   658   "finite A \<Longrightarrow> \<forall>x\<in>A. 0 \<le> f x \<Longrightarrow> setsum f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
```
```   659   by (induct set: finite, simp) (simp add: add_nonneg_eq_0_iff setsum_nonneg)
```
```   660
```
```   661 lemma (in ordered_comm_monoid_add) setsum_nonneg_0:
```
```   662   "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"
```
```   663   by (simp add: setsum_nonneg_eq_0_iff)
```
```   664
```
```   665 lemma (in ordered_comm_monoid_add) setsum_nonneg_leq_bound:
```
```   666   assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"
```
```   667   shows "f i \<le> B"
```
```   668 proof -
```
```   669   have "f i \<le> f i + (\<Sum>i \<in> s - {i}. f i)"
```
```   670     using assms by (intro add_increasing2 setsum_nonneg) auto
```
```   671   also have "\<dots> = B"
```
```   672     using setsum.remove[of s i f] assms by simp
```
```   673   finally show ?thesis by auto
```
```   674 qed
```
```   675
```
```   676 lemma (in ordered_comm_monoid_add) setsum_mono2:
```
```   677   assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
```
```   678   shows "setsum f A \<le> setsum f B"
```
```   679 proof -
```
```   680   have "setsum f A \<le> setsum f A + setsum f (B-A)"
```
```   681     by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
```
```   682   also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
```
```   683     by (simp add: setsum.union_disjoint del:Un_Diff_cancel)
```
```   684   also have "A \<union> (B-A) = B" using sub by blast
```
```   685   finally show ?thesis .
```
```   686 qed
```
```   687
```
```   688 lemma (in ordered_comm_monoid_add) setsum_le_included:
```
```   689   assumes "finite s" "finite t"
```
```   690   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)"
```
```   691   shows "setsum f s \<le> setsum g t"
```
```   692 proof -
```
```   693   have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"
```
```   694   proof (rule setsum_mono)
```
```   695     fix y assume "y \<in> s"
```
```   696     with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto
```
```   697     with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")
```
```   698       using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]
```
```   699       by (auto intro!: setsum_mono2)
```
```   700   qed
```
```   701   also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"
```
```   702     using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)
```
```   703   also have "... \<le> setsum g t"
```
```   704     using assms by (auto simp: setsum_image_gen[symmetric])
```
```   705   finally show ?thesis .
```
```   706 qed
```
```   707
```
```   708 lemma (in ordered_comm_monoid_add) setsum_mono3:
```
```   709   "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> \<forall>x\<in>B - A. 0 \<le> f x \<Longrightarrow> setsum f A \<le> setsum f B"
```
```   710   by (rule setsum_mono2) auto
```
```   711
```
```   712 lemma (in canonically_ordered_monoid_add) setsum_eq_0_iff [simp]:
```
```   713   "finite F \<Longrightarrow> (setsum f F = 0) = (\<forall>a\<in>F. f a = 0)"
```
```   714   by (intro ballI setsum_nonneg_eq_0_iff zero_le)
```
```   715
```
```   716 lemma setsum_right_distrib:
```
```   717   fixes f :: "'a => ('b::semiring_0)"
```
```   718   shows "r * setsum f A = setsum (%n. r * f n) A"
```
```   719 proof (cases "finite A")
```
```   720   case True
```
```   721   thus ?thesis
```
```   722   proof induct
```
```   723     case empty thus ?case by simp
```
```   724   next
```
```   725     case (insert x A) thus ?case by (simp add: distrib_left)
```
```   726   qed
```
```   727 next
```
```   728   case False thus ?thesis by simp
```
```   729 qed
```
```   730
```
```   731 lemma setsum_left_distrib:
```
```   732   "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
```
```   733 proof (cases "finite A")
```
```   734   case True
```
```   735   then show ?thesis
```
```   736   proof induct
```
```   737     case empty thus ?case by simp
```
```   738   next
```
```   739     case (insert x A) thus ?case by (simp add: distrib_right)
```
```   740   qed
```
```   741 next
```
```   742   case False thus ?thesis by simp
```
```   743 qed
```
```   744
```
```   745 lemma setsum_divide_distrib:
```
```   746   "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
```
```   747 proof (cases "finite A")
```
```   748   case True
```
```   749   then show ?thesis
```
```   750   proof induct
```
```   751     case empty thus ?case by simp
```
```   752   next
```
```   753     case (insert x A) thus ?case by (simp add: add_divide_distrib)
```
```   754   qed
```
```   755 next
```
```   756   case False thus ?thesis by simp
```
```   757 qed
```
```   758
```
```   759 lemma setsum_abs[iff]:
```
```   760   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   761   shows "\<bar>setsum f A\<bar> \<le> setsum (%i. \<bar>f i\<bar>) A"
```
```   762 proof (cases "finite A")
```
```   763   case True
```
```   764   thus ?thesis
```
```   765   proof induct
```
```   766     case empty thus ?case by simp
```
```   767   next
```
```   768     case (insert x A)
```
```   769     thus ?case by (auto intro: abs_triangle_ineq order_trans)
```
```   770   qed
```
```   771 next
```
```   772   case False thus ?thesis by simp
```
```   773 qed
```
```   774
```
```   775 lemma setsum_abs_ge_zero[iff]:
```
```   776   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   777   shows "0 \<le> setsum (%i. \<bar>f i\<bar>) A"
```
```   778   by (simp add: setsum_nonneg)
```
```   779
```
```   780 lemma abs_setsum_abs[simp]:
```
```   781   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   782   shows "\<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar> = (\<Sum>a\<in>A. \<bar>f a\<bar>)"
```
```   783 proof (cases "finite A")
```
```   784   case True
```
```   785   thus ?thesis
```
```   786   proof induct
```
```   787     case empty thus ?case by simp
```
```   788   next
```
```   789     case (insert a A)
```
```   790     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
```
```   791     also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
```
```   792     also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
```
```   793       by (simp del: abs_of_nonneg)
```
```   794     also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
```
```   795     finally show ?case .
```
```   796   qed
```
```   797 next
```
```   798   case False thus ?thesis by simp
```
```   799 qed
```
```   800
```
```   801 lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
```
```   802   shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
```
```   803   unfolding setsum.remove [OF assms] by auto
```
```   804
```
```   805 lemma setsum_product:
```
```   806   fixes f :: "'a => ('b::semiring_0)"
```
```   807   shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
```
```   808   by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum.commute)
```
```   809
```
```   810 lemma setsum_mult_setsum_if_inj:
```
```   811 fixes f :: "'a => ('b::semiring_0)"
```
```   812 shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
```
```   813   setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
```
```   814 by(auto simp: setsum_product setsum.cartesian_product
```
```   815         intro!:  setsum.reindex_cong[symmetric])
```
```   816
```
```   817 lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
```
```   818 apply (case_tac "finite A")
```
```   819  prefer 2 apply simp
```
```   820 apply (erule rev_mp)
```
```   821 apply (erule finite_induct, auto)
```
```   822 done
```
```   823
```
```   824 lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
```
```   825   setsum f A = Suc 0 \<longleftrightarrow> (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
```
```   826 apply(erule finite_induct)
```
```   827 apply (auto simp add:add_is_1)
```
```   828 done
```
```   829
```
```   830 lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
```
```   831
```
```   832 lemma setsum_Un_nat: "finite A ==> finite B ==>
```
```   833   (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
```
```   834   \<comment> \<open>For the natural numbers, we have subtraction.\<close>
```
```   835 by (subst setsum.union_inter [symmetric], auto simp add: algebra_simps)
```
```   836
```
```   837 lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
```
```   838   (if a:A then setsum f A - f a else setsum f A)"
```
```   839 apply (case_tac "finite A")
```
```   840  prefer 2 apply simp
```
```   841 apply (erule finite_induct)
```
```   842  apply (auto simp add: insert_Diff_if)
```
```   843 apply (drule_tac a = a in mk_disjoint_insert, auto)
```
```   844 done
```
```   845
```
```   846 lemma setsum_diff_nat:
```
```   847 assumes "finite B" and "B \<subseteq> A"
```
```   848 shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
```
```   849 using assms
```
```   850 proof induct
```
```   851   show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
```
```   852 next
```
```   853   fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
```
```   854     and xFinA: "insert x F \<subseteq> A"
```
```   855     and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
```
```   856   from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
```
```   857   from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
```
```   858     by (simp add: setsum_diff1_nat)
```
```   859   from xFinA have "F \<subseteq> A" by simp
```
```   860   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
```
```   861   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
```
```   862     by simp
```
```   863   from xnotinF have "A - insert x F = (A - F) - {x}" by auto
```
```   864   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
```
```   865     by simp
```
```   866   from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
```
```   867   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
```
```   868     by simp
```
```   869   thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
```
```   870 qed
```
```   871
```
```   872 lemma setsum_comp_morphism:
```
```   873   assumes "h 0 = 0" and "\<And>x y. h (x + y) = h x + h y"
```
```   874   shows "setsum (h \<circ> g) A = h (setsum g A)"
```
```   875 proof (cases "finite A")
```
```   876   case False then show ?thesis by (simp add: assms)
```
```   877 next
```
```   878   case True then show ?thesis by (induct A) (simp_all add: assms)
```
```   879 qed
```
```   880
```
```   881 lemma (in comm_semiring_1) dvd_setsum:
```
```   882   "(\<And>a. a \<in> A \<Longrightarrow> d dvd f a) \<Longrightarrow> d dvd setsum f A"
```
```   883   by (induct A rule: infinite_finite_induct) simp_all
```
```   884
```
```   885 lemma (in ordered_comm_monoid_add) setsum_pos:
```
```   886   "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> 0 < f i) \<Longrightarrow> 0 < setsum f I"
```
```   887   by (induct I rule: finite_ne_induct) (auto intro: add_pos_pos)
```
```   888
```
```   889 lemma (in ordered_comm_monoid_add) setsum_pos2:
```
```   890   assumes I: "finite I" "i \<in> I" "0 < f i" "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
```
```   891   shows "0 < setsum f I"
```
```   892 proof -
```
```   893   have "0 < f i + setsum f (I - {i})"
```
```   894     using assms by (intro add_pos_nonneg setsum_nonneg) auto
```
```   895   also have "\<dots> = setsum f I"
```
```   896     using assms by (simp add: setsum.remove)
```
```   897   finally show ?thesis .
```
```   898 qed
```
```   899
```
```   900 lemma setsum_cong_Suc:
```
```   901   assumes "0 \<notin> A" "\<And>x. Suc x \<in> A \<Longrightarrow> f (Suc x) = g (Suc x)"
```
```   902   shows   "setsum f A = setsum g A"
```
```   903 proof (rule setsum.cong)
```
```   904   fix x assume "x \<in> A"
```
```   905   with assms(1) show "f x = g x" by (cases x) (auto intro!: assms(2))
```
```   906 qed simp_all
```
```   907
```
```   908
```
```   909 subsubsection \<open>Cardinality as special case of @{const setsum}\<close>
```
```   910
```
```   911 lemma card_eq_setsum:
```
```   912   "card A = setsum (\<lambda>x. 1) A"
```
```   913 proof -
```
```   914   have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)"
```
```   915     by (simp add: fun_eq_iff)
```
```   916   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) = Finite_Set.fold (\<lambda>_. Suc)"
```
```   917     by (rule arg_cong)
```
```   918   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
```
```   919     by (blast intro: fun_cong)
```
```   920   then show ?thesis by (simp add: card.eq_fold setsum.eq_fold)
```
```   921 qed
```
```   922
```
```   923 lemma setsum_constant [simp]:
```
```   924   "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
```
```   925 apply (cases "finite A")
```
```   926 apply (erule finite_induct)
```
```   927 apply (auto simp add: algebra_simps)
```
```   928 done
```
```   929
```
```   930 lemma setsum_Suc: "setsum (\<lambda>x. Suc(f x)) A = setsum f A + card A"
```
```   931   using setsum.distrib[of f "\<lambda>_. 1" A]
```
```   932   by simp
```
```   933
```
```   934 lemma setsum_bounded_above:
```
```   935   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_comm_monoid_add})"
```
```   936   shows "setsum f A \<le> of_nat (card A) * K"
```
```   937 proof (cases "finite A")
```
```   938   case True
```
```   939   thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
```
```   940 next
```
```   941   case False thus ?thesis by simp
```
```   942 qed
```
```   943
```
```   944 lemma setsum_bounded_above_strict:
```
```   945   assumes "\<And>i. i\<in>A \<Longrightarrow> f i < (K::'a::{ordered_cancel_comm_monoid_add,semiring_1})"
```
```   946           "card A > 0"
```
```   947   shows "setsum f A < of_nat (card A) * K"
```
```   948 using assms setsum_strict_mono[where A=A and g = "%x. K"]
```
```   949 by (simp add: card_gt_0_iff)
```
```   950
```
```   951 lemma setsum_bounded_below:
```
```   952   assumes le: "\<And>i. i\<in>A \<Longrightarrow> (K::'a::{semiring_1, ordered_comm_monoid_add}) \<le> f i"
```
```   953   shows "of_nat (card A) * K \<le> setsum f A"
```
```   954 proof (cases "finite A")
```
```   955   case True
```
```   956   thus ?thesis using le setsum_mono[where K=A and f = "%x. K"] by simp
```
```   957 next
```
```   958   case False thus ?thesis by simp
```
```   959 qed
```
```   960
```
```   961 lemma card_UN_disjoint:
```
```   962   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
```
```   963     and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
```
```   964   shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
```
```   965 proof -
```
```   966   have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)" by simp
```
```   967   with assms show ?thesis by (simp add: card_eq_setsum setsum.UNION_disjoint del: setsum_constant)
```
```   968 qed
```
```   969
```
```   970 lemma card_Union_disjoint:
```
```   971   "finite C ==> (ALL A:C. finite A) ==>
```
```   972    (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {})
```
```   973    ==> card (\<Union>C) = setsum card C"
```
```   974 apply (frule card_UN_disjoint [of C id])
```
```   975 apply simp_all
```
```   976 done
```
```   977
```
```   978 lemma setsum_multicount_gen:
```
```   979   assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
```
```   980   shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")
```
```   981 proof-
```
```   982   have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto
```
```   983   also have "\<dots> = ?r" unfolding setsum.commute_restrict [OF assms(1-2)]
```
```   984     using assms(3) by auto
```
```   985   finally show ?thesis .
```
```   986 qed
```
```   987
```
```   988 lemma setsum_multicount:
```
```   989   assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
```
```   990   shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
```
```   991 proof-
```
```   992   have "?l = setsum (\<lambda>i. k) T" by (rule setsum_multicount_gen) (auto simp: assms)
```
```   993   also have "\<dots> = ?r" by (simp add: mult.commute)
```
```   994   finally show ?thesis by auto
```
```   995 qed
```
```   996
```
```   997 subsubsection \<open>Cardinality of products\<close>
```
```   998
```
```   999 lemma card_SigmaI [simp]:
```
```  1000   "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
```
```  1001   \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
```
```  1002 by(simp add: card_eq_setsum setsum.Sigma del:setsum_constant)
```
```  1003
```
```  1004 (*
```
```  1005 lemma SigmaI_insert: "y \<notin> A ==>
```
```  1006   (SIGMA x:(insert y A). B x) = (({y} \<times> (B y)) \<union> (SIGMA x: A. B x))"
```
```  1007   by auto
```
```  1008 *)
```
```  1009
```
```  1010 lemma card_cartesian_product: "card (A \<times> B) = card(A) * card(B)"
```
```  1011   by (cases "finite A \<and> finite B")
```
```  1012     (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1013
```
```  1014 lemma card_cartesian_product_singleton:  "card({x} \<times> A) = card(A)"
```
```  1015 by (simp add: card_cartesian_product)
```
```  1016
```
```  1017
```
```  1018 subsection \<open>Generalized product over a set\<close>
```
```  1019
```
```  1020 context comm_monoid_mult
```
```  1021 begin
```
```  1022
```
```  1023 sublocale setprod: comm_monoid_set times 1
```
```  1024 defines
```
```  1025   setprod = setprod.F ..
```
```  1026
```
```  1027 abbreviation
```
```  1028   Setprod ("\<Prod>_" [1000] 999) where
```
```  1029   "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"
```
```  1030
```
```  1031 end
```
```  1032
```
```  1033 syntax (ASCII)
```
```  1034   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(4PROD _:_./ _)" [0, 51, 10] 10)
```
```  1035 syntax
```
```  1036   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(2\<Prod>_\<in>_./ _)" [0, 51, 10] 10)
```
```  1037 translations \<comment> \<open>Beware of argument permutation!\<close>
```
```  1038   "\<Prod>i\<in>A. b" == "CONST setprod (\<lambda>i. b) A"
```
```  1039
```
```  1040 text \<open>Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter \<open>\<Prod>x|P. e\<close>.\<close>
```
```  1041
```
```  1042 syntax (ASCII)
```
```  1043   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(4PROD _ |/ _./ _)" [0, 0, 10] 10)
```
```  1044 syntax
```
```  1045   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(2\<Prod>_ | (_)./ _)" [0, 0, 10] 10)
```
```  1046 translations
```
```  1047   "\<Prod>x|P. t" => "CONST setprod (\<lambda>x. t) {x. P}"
```
```  1048
```
```  1049 context comm_monoid_mult
```
```  1050 begin
```
```  1051
```
```  1052 lemma setprod_dvd_setprod:
```
```  1053   "(\<And>a. a \<in> A \<Longrightarrow> f a dvd g a) \<Longrightarrow> setprod f A dvd setprod g A"
```
```  1054 proof (induct A rule: infinite_finite_induct)
```
```  1055   case infinite then show ?case by (auto intro: dvdI)
```
```  1056 next
```
```  1057   case empty then show ?case by (auto intro: dvdI)
```
```  1058 next
```
```  1059   case (insert a A) then
```
```  1060   have "f a dvd g a" and "setprod f A dvd setprod g A" by simp_all
```
```  1061   then obtain r s where "g a = f a * r" and "setprod g A = setprod f A * s" by (auto elim!: dvdE)
```
```  1062   then have "g a * setprod g A = f a * setprod f A * (r * s)" by (simp add: ac_simps)
```
```  1063   with insert.hyps show ?case by (auto intro: dvdI)
```
```  1064 qed
```
```  1065
```
```  1066 lemma setprod_dvd_setprod_subset:
```
```  1067   "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> setprod f A dvd setprod f B"
```
```  1068   by (auto simp add: setprod.subset_diff ac_simps intro: dvdI)
```
```  1069
```
```  1070 end
```
```  1071
```
```  1072
```
```  1073 subsubsection \<open>Properties in more restricted classes of structures\<close>
```
```  1074
```
```  1075 context comm_semiring_1
```
```  1076 begin
```
```  1077
```
```  1078 lemma dvd_setprod_eqI [intro]:
```
```  1079   assumes "finite A" and "a \<in> A" and "b = f a"
```
```  1080   shows "b dvd setprod f A"
```
```  1081 proof -
```
```  1082   from \<open>finite A\<close> have "setprod f (insert a (A - {a})) = f a * setprod f (A - {a})"
```
```  1083     by (intro setprod.insert) auto
```
```  1084   also from \<open>a \<in> A\<close> have "insert a (A - {a}) = A" by blast
```
```  1085   finally have "setprod f A = f a * setprod f (A - {a})" .
```
```  1086   with \<open>b = f a\<close> show ?thesis by simp
```
```  1087 qed
```
```  1088
```
```  1089 lemma dvd_setprodI [intro]:
```
```  1090   assumes "finite A" and "a \<in> A"
```
```  1091   shows "f a dvd setprod f A"
```
```  1092   using assms by auto
```
```  1093
```
```  1094 lemma setprod_zero:
```
```  1095   assumes "finite A" and "\<exists>a\<in>A. f a = 0"
```
```  1096   shows "setprod f A = 0"
```
```  1097 using assms proof (induct A)
```
```  1098   case empty then show ?case by simp
```
```  1099 next
```
```  1100   case (insert a A)
```
```  1101   then have "f a = 0 \<or> (\<exists>a\<in>A. f a = 0)" by simp
```
```  1102   then have "f a * setprod f A = 0" by rule (simp_all add: insert)
```
```  1103   with insert show ?case by simp
```
```  1104 qed
```
```  1105
```
```  1106 lemma setprod_dvd_setprod_subset2:
```
```  1107   assumes "finite B" and "A \<subseteq> B" and "\<And>a. a \<in> A \<Longrightarrow> f a dvd g a"
```
```  1108   shows "setprod f A dvd setprod g B"
```
```  1109 proof -
```
```  1110   from assms have "setprod f A dvd setprod g A"
```
```  1111     by (auto intro: setprod_dvd_setprod)
```
```  1112   moreover from assms have "setprod g A dvd setprod g B"
```
```  1113     by (auto intro: setprod_dvd_setprod_subset)
```
```  1114   ultimately show ?thesis by (rule dvd_trans)
```
```  1115 qed
```
```  1116
```
```  1117 end
```
```  1118
```
```  1119 lemma setprod_zero_iff [simp]:
```
```  1120   assumes "finite A"
```
```  1121   shows "setprod f A = (0::'a::semidom) \<longleftrightarrow> (\<exists>a\<in>A. f a = 0)"
```
```  1122   using assms by (induct A) (auto simp: no_zero_divisors)
```
```  1123
```
```  1124 lemma (in semidom_divide) setprod_diff1:
```
```  1125   assumes "finite A" and "f a \<noteq> 0"
```
```  1126   shows "setprod f (A - {a}) = (if a \<in> A then setprod f A div f a else setprod f A)"
```
```  1127 proof (cases "a \<notin> A")
```
```  1128   case True then show ?thesis by simp
```
```  1129 next
```
```  1130   case False with assms show ?thesis
```
```  1131   proof (induct A rule: finite_induct)
```
```  1132     case empty then show ?case by simp
```
```  1133   next
```
```  1134     case (insert b B)
```
```  1135     then show ?case
```
```  1136     proof (cases "a = b")
```
```  1137       case True with insert show ?thesis by simp
```
```  1138     next
```
```  1139       case False with insert have "a \<in> B" by simp
```
```  1140       def C \<equiv> "B - {a}"
```
```  1141       with \<open>finite B\<close> \<open>a \<in> B\<close>
```
```  1142         have *: "B = insert a C" "finite C" "a \<notin> C" by auto
```
```  1143       with insert show ?thesis by (auto simp add: insert_commute ac_simps)
```
```  1144     qed
```
```  1145   qed
```
```  1146 qed
```
```  1147
```
```  1148 lemma (in field) setprod_inversef:
```
```  1149   "finite A \<Longrightarrow> setprod (inverse \<circ> f) A = inverse (setprod f A)"
```
```  1150   by (induct A rule: finite_induct) simp_all
```
```  1151
```
```  1152 lemma (in field) setprod_dividef:
```
```  1153   "finite A \<Longrightarrow> (\<Prod>x\<in>A. f x / g x) = setprod f A / setprod g A"
```
```  1154   using setprod_inversef [of A g] by (simp add: divide_inverse setprod.distrib)
```
```  1155
```
```  1156 lemma setprod_Un:
```
```  1157   fixes f :: "'b \<Rightarrow> 'a :: field"
```
```  1158   assumes "finite A" and "finite B"
```
```  1159   and "\<forall>x\<in>A \<inter> B. f x \<noteq> 0"
```
```  1160   shows "setprod f (A \<union> B) = setprod f A * setprod f B / setprod f (A \<inter> B)"
```
```  1161 proof -
```
```  1162   from assms have "setprod f A * setprod f B = setprod f (A \<union> B) * setprod f (A \<inter> B)"
```
```  1163     by (simp add: setprod.union_inter [symmetric, of A B])
```
```  1164   with assms show ?thesis by simp
```
```  1165 qed
```
```  1166
```
```  1167 lemma (in linordered_semidom) setprod_nonneg:
```
```  1168   "(\<forall>a\<in>A. 0 \<le> f a) \<Longrightarrow> 0 \<le> setprod f A"
```
```  1169   by (induct A rule: infinite_finite_induct) simp_all
```
```  1170
```
```  1171 lemma (in linordered_semidom) setprod_pos:
```
```  1172   "(\<forall>a\<in>A. 0 < f a) \<Longrightarrow> 0 < setprod f A"
```
```  1173   by (induct A rule: infinite_finite_induct) simp_all
```
```  1174
```
```  1175 lemma (in linordered_semidom) setprod_mono:
```
```  1176   "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i \<Longrightarrow> setprod f A \<le> setprod g A"
```
```  1177   by (induct A rule: infinite_finite_induct) (auto intro!: setprod_nonneg mult_mono)
```
```  1178
```
```  1179 lemma (in linordered_semidom) setprod_mono_strict:
```
```  1180     assumes"finite A" "\<forall>i\<in>A. 0 \<le> f i \<and> f i < g i" "A \<noteq> {}"
```
```  1181     shows "setprod f A < setprod g A"
```
```  1182 using assms
```
```  1183 apply (induct A rule: finite_induct)
```
```  1184 apply (simp add: )
```
```  1185 apply (force intro: mult_strict_mono' setprod_nonneg)
```
```  1186 done
```
```  1187
```
```  1188 lemma (in linordered_field) abs_setprod:
```
```  1189   "\<bar>setprod f A\<bar> = (\<Prod>x\<in>A. \<bar>f x\<bar>)"
```
```  1190   by (induct A rule: infinite_finite_induct) (simp_all add: abs_mult)
```
```  1191
```
```  1192 lemma setprod_eq_1_iff [simp]:
```
```  1193   "finite A \<Longrightarrow> setprod f A = 1 \<longleftrightarrow> (\<forall>a\<in>A. f a = (1::nat))"
```
```  1194   by (induct A rule: finite_induct) simp_all
```
```  1195
```
```  1196 lemma setprod_pos_nat_iff [simp]:
```
```  1197   "finite A \<Longrightarrow> setprod f A > 0 \<longleftrightarrow> (\<forall>a\<in>A. f a > (0::nat))"
```
```  1198   using setprod_zero_iff by (simp del: neq0_conv add: zero_less_iff_neq_zero)
```
```  1199
```
```  1200 end
```