src/HOL/Groups_Big.thy
 author haftmann Fri Jun 12 08:53:23 2015 +0200 (2015-06-12) changeset 60429 d3d1e185cd63 parent 60353 838025c6e278 child 60494 e726f88232d3 permissions -rw-r--r--
uniform _ div _ as infix syntax for ring division
```     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 {* Big sum and product over finite (non-empty) sets *}
```
```     7
```
```     8 theory Groups_Big
```
```     9 imports Finite_Set
```
```    10 begin
```
```    11
```
```    12 subsection {* Generic monoid operation over a set *}
```
```    13
```
```    14 no_notation times (infixl "*" 70)
```
```    15 no_notation Groups.one ("1")
```
```    16
```
```    17 locale comm_monoid_set = comm_monoid
```
```    18 begin
```
```    19
```
```    20 interpretation comp_fun_commute f
```
```    21   by default (simp add: fun_eq_iff left_commute)
```
```    22
```
```    23 interpretation comp?: comp_fun_commute "f \<circ> g"
```
```    24   by (fact comp_comp_fun_commute)
```
```    25
```
```    26 definition F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
```
```    27 where
```
```    28   eq_fold: "F g A = Finite_Set.fold (f \<circ> g) 1 A"
```
```    29
```
```    30 lemma infinite [simp]:
```
```    31   "\<not> finite A \<Longrightarrow> F g A = 1"
```
```    32   by (simp add: eq_fold)
```
```    33
```
```    34 lemma empty [simp]:
```
```    35   "F g {} = 1"
```
```    36   by (simp add: eq_fold)
```
```    37
```
```    38 lemma insert [simp]:
```
```    39   assumes "finite A" and "x \<notin> A"
```
```    40   shows "F g (insert x A) = g x * F g A"
```
```    41   using assms by (simp add: eq_fold)
```
```    42
```
```    43 lemma remove:
```
```    44   assumes "finite A" and "x \<in> A"
```
```    45   shows "F g A = g x * F g (A - {x})"
```
```    46 proof -
```
```    47   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
```
```    48     by (auto dest: mk_disjoint_insert)
```
```    49   moreover from `finite A` A have "finite B" by simp
```
```    50   ultimately show ?thesis by simp
```
```    51 qed
```
```    52
```
```    53 lemma insert_remove:
```
```    54   assumes "finite A"
```
```    55   shows "F g (insert x A) = g x * F g (A - {x})"
```
```    56   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
```
```    57
```
```    58 lemma neutral:
```
```    59   assumes "\<forall>x\<in>A. g x = 1"
```
```    60   shows "F g A = 1"
```
```    61   using assms by (induct A rule: infinite_finite_induct) simp_all
```
```    62
```
```    63 lemma neutral_const [simp]:
```
```    64   "F (\<lambda>_. 1) A = 1"
```
```    65   by (simp add: neutral)
```
```    66
```
```    67 lemma union_inter:
```
```    68   assumes "finite A" and "finite B"
```
```    69   shows "F g (A \<union> B) * F g (A \<inter> B) = F g A * F g B"
```
```    70   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
```
```    71 using assms proof (induct A)
```
```    72   case empty then show ?case by simp
```
```    73 next
```
```    74   case (insert x A) then show ?case
```
```    75     by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
```
```    76 qed
```
```    77
```
```    78 corollary union_inter_neutral:
```
```    79   assumes "finite A" and "finite B"
```
```    80   and I0: "\<forall>x \<in> A \<inter> B. g x = 1"
```
```    81   shows "F g (A \<union> B) = F g A * F g B"
```
```    82   using assms by (simp add: union_inter [symmetric] neutral)
```
```    83
```
```    84 corollary union_disjoint:
```
```    85   assumes "finite A" and "finite B"
```
```    86   assumes "A \<inter> B = {}"
```
```    87   shows "F g (A \<union> B) = F g A * F g B"
```
```    88   using assms by (simp add: union_inter_neutral)
```
```    89
```
```    90 lemma 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 `A = B`
```
```   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 (split 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 `S \<subseteq> T` by blast
```
```   211   have d: "S \<inter> (T - S) = {}" using `S \<subseteq> T` by blast
```
```   212   from `finite T` `S \<subseteq> T` 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 `finite S` 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 `finite A`, 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 (split g) (A <*> 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 `finite A`
```
```   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 `finite B` `A \<notin> B` 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 `finite C` 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 `finite A` `finite (C - A)` 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 `finite B` `finite (C - B)` 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 {* Generalized summation over a set *}
```
```   466
```
```   467 context comm_monoid_add
```
```   468 begin
```
```   469
```
```   470 definition setsum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
```
```   471 where
```
```   472   "setsum = comm_monoid_set.F plus 0"
```
```   473
```
```   474 sublocale setsum!: comm_monoid_set plus 0
```
```   475 where
```
```   476   "comm_monoid_set.F plus 0 = setsum"
```
```   477 proof -
```
```   478   show "comm_monoid_set plus 0" ..
```
```   479   then interpret setsum!: comm_monoid_set plus 0 .
```
```   480   from setsum_def show "comm_monoid_set.F plus 0 = setsum" by rule
```
```   481 qed
```
```   482
```
```   483 abbreviation
```
```   484   Setsum ("\<Sum>_"  999) where
```
```   485   "\<Sum>A \<equiv> setsum (%x. x) A"
```
```   486
```
```   487 end
```
```   488
```
```   489 text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
```
```   490 written @{text"\<Sum>x\<in>A. e"}. *}
```
```   491
```
```   492 syntax
```
```   493   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
```
```   494 syntax (xsymbols)
```
```   495   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```   496 syntax (HTML output)
```
```   497   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```   498
```
```   499 translations -- {* Beware of argument permutation! *}
```
```   500   "SUM i:A. b" == "CONST setsum (%i. b) A"
```
```   501   "\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A"
```
```   502
```
```   503 text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
```
```   504  @{text"\<Sum>x|P. e"}. *}
```
```   505
```
```   506 syntax
```
```   507   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
```
```   508 syntax (xsymbols)
```
```   509   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   510 syntax (HTML output)
```
```   511   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   512
```
```   513 translations
```
```   514   "SUM x|P. t" => "CONST setsum (%x. t) {x. P}"
```
```   515   "\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}"
```
```   516
```
```   517 print_translation {*
```
```   518 let
```
```   519   fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) \$ Abs (y, Ty, P)] =
```
```   520         if x <> y then raise Match
```
```   521         else
```
```   522           let
```
```   523             val x' = Syntax_Trans.mark_bound_body (x, Tx);
```
```   524             val t' = subst_bound (x', t);
```
```   525             val P' = subst_bound (x', P);
```
```   526           in
```
```   527             Syntax.const @{syntax_const "_qsetsum"} \$ Syntax_Trans.mark_bound_abs (x, Tx) \$ P' \$ t'
```
```   528           end
```
```   529     | setsum_tr' _ = raise Match;
```
```   530 in [(@{const_syntax setsum}, K setsum_tr')] end
```
```   531 *}
```
```   532
```
```   533 text {* TODO generalization candidates *}
```
```   534
```
```   535 lemma setsum_image_gen:
```
```   536   assumes fS: "finite S"
```
```   537   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
```
```   538 proof-
```
```   539   { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }
```
```   540   hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
```
```   541     by simp
```
```   542   also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
```
```   543     by (rule setsum.commute_restrict [OF fS finite_imageI [OF fS]])
```
```   544   finally show ?thesis .
```
```   545 qed
```
```   546
```
```   547
```
```   548 subsubsection {* Properties in more restricted classes of structures *}
```
```   549
```
```   550 lemma setsum_Un: "finite A ==> finite B ==>
```
```   551   (setsum f (A Un B) :: 'a :: ab_group_add) =
```
```   552    setsum f A + setsum f B - setsum f (A Int B)"
```
```   553 by (subst setsum.union_inter [symmetric], auto simp add: algebra_simps)
```
```   554
```
```   555 lemma setsum_Un2:
```
```   556   assumes "finite (A \<union> B)"
```
```   557   shows "setsum f (A \<union> B) = setsum f (A - B) + setsum f (B - A) + setsum f (A \<inter> B)"
```
```   558 proof -
```
```   559   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
```
```   560     by auto
```
```   561   with assms show ?thesis by simp (subst setsum.union_disjoint, auto)+
```
```   562 qed
```
```   563
```
```   564 lemma setsum_diff1: "finite A \<Longrightarrow>
```
```   565   (setsum f (A - {a}) :: ('a::ab_group_add)) =
```
```   566   (if a:A then setsum f A - f a else setsum f A)"
```
```   567 by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```   568
```
```   569 lemma setsum_diff:
```
```   570   assumes le: "finite A" "B \<subseteq> A"
```
```   571   shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
```
```   572 proof -
```
```   573   from le have finiteB: "finite B" using finite_subset by auto
```
```   574   show ?thesis using finiteB le
```
```   575   proof induct
```
```   576     case empty
```
```   577     thus ?case by auto
```
```   578   next
```
```   579     case (insert x F)
```
```   580     thus ?case using le finiteB
```
```   581       by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
```
```   582   qed
```
```   583 qed
```
```   584
```
```   585 lemma setsum_mono:
```
```   586   assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))"
```
```   587   shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
```
```   588 proof (cases "finite K")
```
```   589   case True
```
```   590   thus ?thesis using le
```
```   591   proof induct
```
```   592     case empty
```
```   593     thus ?case by simp
```
```   594   next
```
```   595     case insert
```
```   596     thus ?case using add_mono by fastforce
```
```   597   qed
```
```   598 next
```
```   599   case False then show ?thesis by simp
```
```   600 qed
```
```   601
```
```   602 lemma setsum_strict_mono:
```
```   603   fixes f :: "'a \<Rightarrow> 'b::{ordered_cancel_ab_semigroup_add,comm_monoid_add}"
```
```   604   assumes "finite A"  "A \<noteq> {}"
```
```   605     and "!!x. x:A \<Longrightarrow> f x < g x"
```
```   606   shows "setsum f A < setsum g A"
```
```   607   using assms
```
```   608 proof (induct rule: finite_ne_induct)
```
```   609   case singleton thus ?case by simp
```
```   610 next
```
```   611   case insert thus ?case by (auto simp: add_strict_mono)
```
```   612 qed
```
```   613
```
```   614 lemma setsum_strict_mono_ex1:
```
```   615 fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_cancel_ab_semigroup_add}"
```
```   616 assumes "finite A" and "ALL x:A. f x \<le> g x" and "EX a:A. f a < g a"
```
```   617 shows "setsum f A < setsum g A"
```
```   618 proof-
```
```   619   from assms(3) obtain a where a: "a:A" "f a < g a" by blast
```
```   620   have "setsum f A = setsum f ((A-{a}) \<union> {a})"
```
```   621     by(simp add:insert_absorb[OF `a:A`])
```
```   622   also have "\<dots> = setsum f (A-{a}) + setsum f {a}"
```
```   623     using `finite A` by(subst setsum.union_disjoint) auto
```
```   624   also have "setsum f (A-{a}) \<le> setsum g (A-{a})"
```
```   625     by(rule setsum_mono)(simp add: assms(2))
```
```   626   also have "setsum f {a} < setsum g {a}" using a by simp
```
```   627   also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"
```
```   628     using `finite A` by(subst setsum.union_disjoint[symmetric]) auto
```
```   629   also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF `a:A`])
```
```   630   finally show ?thesis by (auto simp add: add_right_mono add_strict_left_mono)
```
```   631 qed
```
```   632
```
```   633 lemma setsum_negf: "(\<Sum>x\<in>A. - f x::'a::ab_group_add) = - (\<Sum>x\<in>A. f x)"
```
```   634 proof (cases "finite A")
```
```   635   case True thus ?thesis by (induct set: finite) auto
```
```   636 next
```
```   637   case False thus ?thesis by simp
```
```   638 qed
```
```   639
```
```   640 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)"
```
```   641   using setsum.distrib [of f "- g" A] by (simp add: setsum_negf)
```
```   642
```
```   643 lemma setsum_subtractf_nat:
```
```   644   "(\<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)"
```
```   645   by (induction A rule: infinite_finite_induct) (auto simp: setsum_mono)
```
```   646
```
```   647 lemma setsum_nonneg:
```
```   648   assumes nn: "\<forall>x\<in>A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
```
```   649   shows "0 \<le> setsum f A"
```
```   650 proof (cases "finite A")
```
```   651   case True thus ?thesis using nn
```
```   652   proof induct
```
```   653     case empty then show ?case by simp
```
```   654   next
```
```   655     case (insert x F)
```
```   656     then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
```
```   657     with insert show ?case by simp
```
```   658   qed
```
```   659 next
```
```   660   case False thus ?thesis by simp
```
```   661 qed
```
```   662
```
```   663 lemma setsum_nonpos:
```
```   664   assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})"
```
```   665   shows "setsum f A \<le> 0"
```
```   666 proof (cases "finite A")
```
```   667   case True thus ?thesis using np
```
```   668   proof induct
```
```   669     case empty then show ?case by simp
```
```   670   next
```
```   671     case (insert x F)
```
```   672     then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
```
```   673     with insert show ?case by simp
```
```   674   qed
```
```   675 next
```
```   676   case False thus ?thesis by simp
```
```   677 qed
```
```   678
```
```   679 lemma setsum_nonneg_leq_bound:
```
```   680   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
```
```   681   assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"
```
```   682   shows "f i \<le> B"
```
```   683 proof -
```
```   684   have "0 \<le> (\<Sum> i \<in> s - {i}. f i)" and "0 \<le> f i"
```
```   685     using assms by (auto intro!: setsum_nonneg)
```
```   686   moreover
```
```   687   have "(\<Sum> i \<in> s - {i}. f i) + f i = B"
```
```   688     using assms by (simp add: setsum_diff1)
```
```   689   ultimately show ?thesis by auto
```
```   690 qed
```
```   691
```
```   692 lemma setsum_nonneg_0:
```
```   693   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
```
```   694   assumes "finite s" and pos: "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"
```
```   695   and "(\<Sum> i \<in> s. f i) = 0" and i: "i \<in> s"
```
```   696   shows "f i = 0"
```
```   697   using setsum_nonneg_leq_bound[OF assms] pos[OF i] by auto
```
```   698
```
```   699 lemma setsum_mono2:
```
```   700 fixes f :: "'a \<Rightarrow> 'b :: ordered_comm_monoid_add"
```
```   701 assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
```
```   702 shows "setsum f A \<le> setsum f B"
```
```   703 proof -
```
```   704   have "setsum f A \<le> setsum f A + setsum f (B-A)"
```
```   705     by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
```
```   706   also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
```
```   707     by (simp add: setsum.union_disjoint del:Un_Diff_cancel)
```
```   708   also have "A \<union> (B-A) = B" using sub by blast
```
```   709   finally show ?thesis .
```
```   710 qed
```
```   711
```
```   712 lemma setsum_le_included:
```
```   713   fixes f :: "'a \<Rightarrow> 'b::ordered_comm_monoid_add"
```
```   714   assumes "finite s" "finite t"
```
```   715   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)"
```
```   716   shows "setsum f s \<le> setsum g t"
```
```   717 proof -
```
```   718   have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"
```
```   719   proof (rule setsum_mono)
```
```   720     fix y assume "y \<in> s"
```
```   721     with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto
```
```   722     with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")
```
```   723       using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]
```
```   724       by (auto intro!: setsum_mono2)
```
```   725   qed
```
```   726   also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"
```
```   727     using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)
```
```   728   also have "... \<le> setsum g t"
```
```   729     using assms by (auto simp: setsum_image_gen[symmetric])
```
```   730   finally show ?thesis .
```
```   731 qed
```
```   732
```
```   733 lemma setsum_mono3: "finite B ==> A <= B ==>
```
```   734     ALL x: B - A.
```
```   735       0 <= ((f x)::'a::{comm_monoid_add,ordered_ab_semigroup_add}) ==>
```
```   736         setsum f A <= setsum f B"
```
```   737   apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
```
```   738   apply (erule ssubst)
```
```   739   apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
```
```   740   apply simp
```
```   741   apply (rule add_left_mono)
```
```   742   apply (erule setsum_nonneg)
```
```   743   apply (subst setsum.union_disjoint [THEN sym])
```
```   744   apply (erule finite_subset, assumption)
```
```   745   apply (rule finite_subset)
```
```   746   prefer 2
```
```   747   apply assumption
```
```   748   apply (auto simp add: sup_absorb2)
```
```   749 done
```
```   750
```
```   751 lemma setsum_right_distrib:
```
```   752   fixes f :: "'a => ('b::semiring_0)"
```
```   753   shows "r * setsum f A = setsum (%n. r * f n) A"
```
```   754 proof (cases "finite A")
```
```   755   case True
```
```   756   thus ?thesis
```
```   757   proof induct
```
```   758     case empty thus ?case by simp
```
```   759   next
```
```   760     case (insert x A) thus ?case by (simp add: distrib_left)
```
```   761   qed
```
```   762 next
```
```   763   case False thus ?thesis by simp
```
```   764 qed
```
```   765
```
```   766 lemma setsum_left_distrib:
```
```   767   "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
```
```   768 proof (cases "finite A")
```
```   769   case True
```
```   770   then show ?thesis
```
```   771   proof induct
```
```   772     case empty thus ?case by simp
```
```   773   next
```
```   774     case (insert x A) thus ?case by (simp add: distrib_right)
```
```   775   qed
```
```   776 next
```
```   777   case False thus ?thesis by simp
```
```   778 qed
```
```   779
```
```   780 lemma setsum_divide_distrib:
```
```   781   "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
```
```   782 proof (cases "finite A")
```
```   783   case True
```
```   784   then show ?thesis
```
```   785   proof induct
```
```   786     case empty thus ?case by simp
```
```   787   next
```
```   788     case (insert x A) thus ?case by (simp add: add_divide_distrib)
```
```   789   qed
```
```   790 next
```
```   791   case False thus ?thesis by simp
```
```   792 qed
```
```   793
```
```   794 lemma setsum_abs[iff]:
```
```   795   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   796   shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
```
```   797 proof (cases "finite A")
```
```   798   case True
```
```   799   thus ?thesis
```
```   800   proof induct
```
```   801     case empty thus ?case by simp
```
```   802   next
```
```   803     case (insert x A)
```
```   804     thus ?case by (auto intro: abs_triangle_ineq order_trans)
```
```   805   qed
```
```   806 next
```
```   807   case False thus ?thesis by simp
```
```   808 qed
```
```   809
```
```   810 lemma setsum_abs_ge_zero[iff]:
```
```   811   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   812   shows "0 \<le> setsum (%i. abs(f i)) A"
```
```   813 proof (cases "finite A")
```
```   814   case True
```
```   815   thus ?thesis
```
```   816   proof induct
```
```   817     case empty thus ?case by simp
```
```   818   next
```
```   819     case (insert x A) thus ?case by auto
```
```   820   qed
```
```   821 next
```
```   822   case False thus ?thesis by simp
```
```   823 qed
```
```   824
```
```   825 lemma abs_setsum_abs[simp]:
```
```   826   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   827   shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
```
```   828 proof (cases "finite A")
```
```   829   case True
```
```   830   thus ?thesis
```
```   831   proof induct
```
```   832     case empty thus ?case by simp
```
```   833   next
```
```   834     case (insert a A)
```
```   835     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
```
```   836     also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
```
```   837     also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
```
```   838       by (simp del: abs_of_nonneg)
```
```   839     also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
```
```   840     finally show ?case .
```
```   841   qed
```
```   842 next
```
```   843   case False thus ?thesis by simp
```
```   844 qed
```
```   845
```
```   846 lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
```
```   847   shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
```
```   848   unfolding setsum.remove [OF assms] by auto
```
```   849
```
```   850 lemma setsum_product:
```
```   851   fixes f :: "'a => ('b::semiring_0)"
```
```   852   shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
```
```   853   by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum.commute)
```
```   854
```
```   855 lemma setsum_mult_setsum_if_inj:
```
```   856 fixes f :: "'a => ('b::semiring_0)"
```
```   857 shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
```
```   858   setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
```
```   859 by(auto simp: setsum_product setsum.cartesian_product
```
```   860         intro!:  setsum.reindex_cong[symmetric])
```
```   861
```
```   862 lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
```
```   863 apply (case_tac "finite A")
```
```   864  prefer 2 apply simp
```
```   865 apply (erule rev_mp)
```
```   866 apply (erule finite_induct, auto)
```
```   867 done
```
```   868
```
```   869 lemma setsum_eq_0_iff [simp]:
```
```   870   "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
```
```   871   by (induct set: finite) auto
```
```   872
```
```   873 lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
```
```   874   setsum f A = Suc 0 \<longleftrightarrow> (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
```
```   875 apply(erule finite_induct)
```
```   876 apply (auto simp add:add_is_1)
```
```   877 done
```
```   878
```
```   879 lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
```
```   880
```
```   881 lemma setsum_Un_nat: "finite A ==> finite B ==>
```
```   882   (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
```
```   883   -- {* For the natural numbers, we have subtraction. *}
```
```   884 by (subst setsum.union_inter [symmetric], auto simp add: algebra_simps)
```
```   885
```
```   886 lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
```
```   887   (if a:A then setsum f A - f a else setsum f A)"
```
```   888 apply (case_tac "finite A")
```
```   889  prefer 2 apply simp
```
```   890 apply (erule finite_induct)
```
```   891  apply (auto simp add: insert_Diff_if)
```
```   892 apply (drule_tac a = a in mk_disjoint_insert, auto)
```
```   893 done
```
```   894
```
```   895 lemma setsum_diff_nat:
```
```   896 assumes "finite B" and "B \<subseteq> A"
```
```   897 shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
```
```   898 using assms
```
```   899 proof induct
```
```   900   show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
```
```   901 next
```
```   902   fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
```
```   903     and xFinA: "insert x F \<subseteq> A"
```
```   904     and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
```
```   905   from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
```
```   906   from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
```
```   907     by (simp add: setsum_diff1_nat)
```
```   908   from xFinA have "F \<subseteq> A" by simp
```
```   909   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
```
```   910   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
```
```   911     by simp
```
```   912   from xnotinF have "A - insert x F = (A - F) - {x}" by auto
```
```   913   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
```
```   914     by simp
```
```   915   from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
```
```   916   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
```
```   917     by simp
```
```   918   thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
```
```   919 qed
```
```   920
```
```   921 lemma setsum_comp_morphism:
```
```   922   assumes "h 0 = 0" and "\<And>x y. h (x + y) = h x + h y"
```
```   923   shows "setsum (h \<circ> g) A = h (setsum g A)"
```
```   924 proof (cases "finite A")
```
```   925   case False then show ?thesis by (simp add: assms)
```
```   926 next
```
```   927   case True then show ?thesis by (induct A) (simp_all add: assms)
```
```   928 qed
```
```   929
```
```   930 lemma (in comm_semiring_1) dvd_setsum:
```
```   931   "(\<And>a. a \<in> A \<Longrightarrow> d dvd f a) \<Longrightarrow> d dvd setsum f A"
```
```   932   by (induct A rule: infinite_finite_induct) simp_all
```
```   933
```
```   934
```
```   935 subsubsection {* Cardinality as special case of @{const setsum} *}
```
```   936
```
```   937 lemma card_eq_setsum:
```
```   938   "card A = setsum (\<lambda>x. 1) A"
```
```   939 proof -
```
```   940   have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)"
```
```   941     by (simp add: fun_eq_iff)
```
```   942   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) = Finite_Set.fold (\<lambda>_. Suc)"
```
```   943     by (rule arg_cong)
```
```   944   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
```
```   945     by (blast intro: fun_cong)
```
```   946   then show ?thesis by (simp add: card.eq_fold setsum.eq_fold)
```
```   947 qed
```
```   948
```
```   949 lemma setsum_constant [simp]:
```
```   950   "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
```
```   951 apply (cases "finite A")
```
```   952 apply (erule finite_induct)
```
```   953 apply (auto simp add: algebra_simps)
```
```   954 done
```
```   955
```
```   956 lemma setsum_Suc: "setsum (\<lambda>x. Suc(f x)) A = setsum f A + card A"
```
```   957   using setsum.distrib[of f "\<lambda>_. 1" A]
```
```   958   by simp
```
```   959
```
```   960 lemma setsum_bounded:
```
```   961   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"
```
```   962   shows "setsum f A \<le> of_nat (card A) * K"
```
```   963 proof (cases "finite A")
```
```   964   case True
```
```   965   thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
```
```   966 next
```
```   967   case False thus ?thesis by simp
```
```   968 qed
```
```   969
```
```   970 lemma card_UN_disjoint:
```
```   971   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
```
```   972     and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
```
```   973   shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
```
```   974 proof -
```
```   975   have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)" by simp
```
```   976   with assms show ?thesis by (simp add: card_eq_setsum setsum.UNION_disjoint del: setsum_constant)
```
```   977 qed
```
```   978
```
```   979 lemma card_Union_disjoint:
```
```   980   "finite C ==> (ALL A:C. finite A) ==>
```
```   981    (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {})
```
```   982    ==> card (Union C) = setsum card C"
```
```   983 apply (frule card_UN_disjoint [of C id])
```
```   984 apply simp_all
```
```   985 done
```
```   986
```
```   987 lemma setsum_multicount_gen:
```
```   988   assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
```
```   989   shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")
```
```   990 proof-
```
```   991   have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto
```
```   992   also have "\<dots> = ?r" 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" by (rule setsum_multicount_gen) (auto simp: assms)
```
```  1002   also have "\<dots> = ?r" by (simp add: mult.commute)
```
```  1003   finally show ?thesis by auto
```
```  1004 qed
```
```  1005
```
```  1006 lemma (in ordered_comm_monoid_add) setsum_pos:
```
```  1007   "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> 0 < f i) \<Longrightarrow> 0 < setsum f I"
```
```  1008   by (induct I rule: finite_ne_induct) (auto intro: add_pos_pos)
```
```  1009
```
```  1010
```
```  1011 subsubsection {* Cardinality of products *}
```
```  1012
```
```  1013 lemma card_SigmaI [simp]:
```
```  1014   "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
```
```  1015   \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
```
```  1016 by(simp add: card_eq_setsum setsum.Sigma del:setsum_constant)
```
```  1017
```
```  1018 (*
```
```  1019 lemma SigmaI_insert: "y \<notin> A ==>
```
```  1020   (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
```
```  1021   by auto
```
```  1022 *)
```
```  1023
```
```  1024 lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
```
```  1025   by (cases "finite A \<and> finite B")
```
```  1026     (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1027
```
```  1028 lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
```
```  1029 by (simp add: card_cartesian_product)
```
```  1030
```
```  1031
```
```  1032 subsection {* Generalized product over a set *}
```
```  1033
```
```  1034 context comm_monoid_mult
```
```  1035 begin
```
```  1036
```
```  1037 definition setprod :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
```
```  1038 where
```
```  1039   "setprod = comm_monoid_set.F times 1"
```
```  1040
```
```  1041 sublocale setprod!: comm_monoid_set times 1
```
```  1042 where
```
```  1043   "comm_monoid_set.F times 1 = setprod"
```
```  1044 proof -
```
```  1045   show "comm_monoid_set times 1" ..
```
```  1046   then interpret setprod!: comm_monoid_set times 1 .
```
```  1047   from setprod_def show "comm_monoid_set.F times 1 = setprod" by rule
```
```  1048 qed
```
```  1049
```
```  1050 abbreviation
```
```  1051   Setprod ("\<Prod>_"  999) where
```
```  1052   "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"
```
```  1053
```
```  1054 end
```
```  1055
```
```  1056 syntax
```
```  1057   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
```
```  1058 syntax (xsymbols)
```
```  1059   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```  1060 syntax (HTML output)
```
```  1061   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```  1062
```
```  1063 translations -- {* Beware of argument permutation! *}
```
```  1064   "PROD i:A. b" == "CONST setprod (%i. b) A"
```
```  1065   "\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A"
```
```  1066
```
```  1067 text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
```
```  1068  @{text"\<Prod>x|P. e"}. *}
```
```  1069
```
```  1070 syntax
```
```  1071   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
```
```  1072 syntax (xsymbols)
```
```  1073   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```  1074 syntax (HTML output)
```
```  1075   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```  1076
```
```  1077 translations
```
```  1078   "PROD x|P. t" => "CONST setprod (%x. t) {x. P}"
```
```  1079   "\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"
```
```  1080
```
```  1081 context comm_monoid_mult
```
```  1082 begin
```
```  1083
```
```  1084 lemma setprod_dvd_setprod:
```
```  1085   "(\<And>a. a \<in> A \<Longrightarrow> f a dvd g a) \<Longrightarrow> setprod f A dvd setprod g A"
```
```  1086 proof (induct A rule: infinite_finite_induct)
```
```  1087   case infinite then show ?case by (auto intro: dvdI)
```
```  1088 next
```
```  1089   case empty then show ?case by (auto intro: dvdI)
```
```  1090 next
```
```  1091   case (insert a A) then
```
```  1092   have "f a dvd g a" and "setprod f A dvd setprod g A" by simp_all
```
```  1093   then obtain r s where "g a = f a * r" and "setprod g A = setprod f A * s" by (auto elim!: dvdE)
```
```  1094   then have "g a * setprod g A = f a * setprod f A * (r * s)" by (simp add: ac_simps)
```
```  1095   with insert.hyps show ?case by (auto intro: dvdI)
```
```  1096 qed
```
```  1097
```
```  1098 lemma setprod_dvd_setprod_subset:
```
```  1099   "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> setprod f A dvd setprod f B"
```
```  1100   by (auto simp add: setprod.subset_diff ac_simps intro: dvdI)
```
```  1101
```
```  1102 end
```
```  1103
```
```  1104
```
```  1105 subsubsection {* Properties in more restricted classes of structures *}
```
```  1106
```
```  1107 context comm_semiring_1
```
```  1108 begin
```
```  1109
```
```  1110 lemma dvd_setprod_eqI [intro]:
```
```  1111   assumes "finite A" and "a \<in> A" and "b = f a"
```
```  1112   shows "b dvd setprod f A"
```
```  1113 proof -
```
```  1114   from `finite A` have "setprod f (insert a (A - {a})) = f a * setprod f (A - {a})"
```
```  1115     by (intro setprod.insert) auto
```
```  1116   also from `a \<in> A` have "insert a (A - {a}) = A" by blast
```
```  1117   finally have "setprod f A = f a * setprod f (A - {a})" .
```
```  1118   with `b = f a` show ?thesis by simp
```
```  1119 qed
```
```  1120
```
```  1121 lemma dvd_setprodI [intro]:
```
```  1122   assumes "finite A" and "a \<in> A"
```
```  1123   shows "f a dvd setprod f A"
```
```  1124   using assms by auto
```
```  1125
```
```  1126 lemma setprod_zero:
```
```  1127   assumes "finite A" and "\<exists>a\<in>A. f a = 0"
```
```  1128   shows "setprod f A = 0"
```
```  1129 using assms proof (induct A)
```
```  1130   case empty then show ?case by simp
```
```  1131 next
```
```  1132   case (insert a A)
```
```  1133   then have "f a = 0 \<or> (\<exists>a\<in>A. f a = 0)" by simp
```
```  1134   then have "f a * setprod f A = 0" by rule (simp_all add: insert)
```
```  1135   with insert show ?case by simp
```
```  1136 qed
```
```  1137
```
```  1138 lemma setprod_dvd_setprod_subset2:
```
```  1139   assumes "finite B" and "A \<subseteq> B" and "\<And>a. a \<in> A \<Longrightarrow> f a dvd g a"
```
```  1140   shows "setprod f A dvd setprod g B"
```
```  1141 proof -
```
```  1142   from assms have "setprod f A dvd setprod g A"
```
```  1143     by (auto intro: setprod_dvd_setprod)
```
```  1144   moreover from assms have "setprod g A dvd setprod g B"
```
```  1145     by (auto intro: setprod_dvd_setprod_subset)
```
```  1146   ultimately show ?thesis by (rule dvd_trans)
```
```  1147 qed
```
```  1148
```
```  1149 end
```
```  1150
```
```  1151 lemma setprod_zero_iff [simp]:
```
```  1152   assumes "finite A"
```
```  1153   shows "setprod f A = (0::'a::semidom) \<longleftrightarrow> (\<exists>a\<in>A. f a = 0)"
```
```  1154   using assms by (induct A) (auto simp: no_zero_divisors)
```
```  1155
```
```  1156 lemma (in semidom_divide) setprod_diff1:
```
```  1157   assumes "finite A" and "f a \<noteq> 0"
```
```  1158   shows "setprod f (A - {a}) = (if a \<in> A then setprod f A div f a else setprod f A)"
```
```  1159 proof (cases "a \<notin> A")
```
```  1160   case True then show ?thesis by simp
```
```  1161 next
```
```  1162   case False with assms show ?thesis
```
```  1163   proof (induct A rule: finite_induct)
```
```  1164     case empty then show ?case by simp
```
```  1165   next
```
```  1166     case (insert b B)
```
```  1167     then show ?case
```
```  1168     proof (cases "a = b")
```
```  1169       case True with insert show ?thesis by simp
```
```  1170     next
```
```  1171       case False with insert have "a \<in> B" by simp
```
```  1172       def C \<equiv> "B - {a}"
```
```  1173       with `finite B` `a \<in> B`
```
```  1174         have *: "B = insert a C" "finite C" "a \<notin> C" by auto
```
```  1175       with insert show ?thesis by (auto simp add: insert_commute ac_simps)
```
```  1176     qed
```
```  1177   qed
```
```  1178 qed
```
```  1179
```
```  1180 lemma (in field) setprod_inversef:
```
```  1181   "finite A \<Longrightarrow> setprod (inverse \<circ> f) A = inverse (setprod f A)"
```
```  1182   by (induct A rule: finite_induct) simp_all
```
```  1183
```
```  1184 lemma (in field) setprod_dividef:
```
```  1185   "finite A \<Longrightarrow> (\<Prod>x\<in>A. f x / g x) = setprod f A / setprod g A"
```
```  1186   using setprod_inversef [of A g] by (simp add: divide_inverse setprod.distrib)
```
```  1187
```
```  1188 lemma setprod_Un:
```
```  1189   fixes f :: "'b \<Rightarrow> 'a :: field"
```
```  1190   assumes "finite A" and "finite B"
```
```  1191   and "\<forall>x\<in>A \<inter> B. f x \<noteq> 0"
```
```  1192   shows "setprod f (A \<union> B) = setprod f A * setprod f B / setprod f (A \<inter> B)"
```
```  1193 proof -
```
```  1194   from assms have "setprod f A * setprod f B = setprod f (A \<union> B) * setprod f (A \<inter> B)"
```
```  1195     by (simp add: setprod.union_inter [symmetric, of A B])
```
```  1196   with assms show ?thesis by simp
```
```  1197 qed
```
```  1198
```
```  1199 lemma (in linordered_semidom) setprod_nonneg:
```
```  1200   "(\<forall>a\<in>A. 0 \<le> f a) \<Longrightarrow> 0 \<le> setprod f A"
```
```  1201   by (induct A rule: infinite_finite_induct) simp_all
```
```  1202
```
```  1203 lemma (in linordered_semidom) setprod_pos:
```
```  1204   "(\<forall>a\<in>A. 0 < f a) \<Longrightarrow> 0 < setprod f A"
```
```  1205   by (induct A rule: infinite_finite_induct) simp_all
```
```  1206
```
```  1207 lemma (in linordered_semidom) setprod_mono:
```
```  1208   assumes "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i"
```
```  1209   shows "setprod f A \<le> setprod g A"
```
```  1210   using assms by (induct A rule: infinite_finite_induct)
```
```  1211     (auto intro!: setprod_nonneg mult_mono)
```
```  1212
```
```  1213 lemma (in linordered_field) abs_setprod:
```
```  1214   "\<bar>setprod f A\<bar> = (\<Prod>x\<in>A. \<bar>f x\<bar>)"
```
```  1215   by (induct A rule: infinite_finite_induct) (simp_all add: abs_mult)
```
```  1216
```
```  1217 lemma setprod_eq_1_iff [simp]:
```
```  1218   "finite A \<Longrightarrow> setprod f A = 1 \<longleftrightarrow> (\<forall>a\<in>A. f a = (1::nat))"
```
```  1219   by (induct A rule: finite_induct) simp_all
```
```  1220
```
```  1221 lemma setprod_pos_nat:
```
```  1222   "finite A \<Longrightarrow> (\<forall>a\<in>A. f a > (0::nat)) \<Longrightarrow> setprod f A > 0"
```
```  1223   using setprod_zero_iff by (simp del: neq0_conv add: neq0_conv [symmetric])
```
```  1224
```
```  1225 lemma setprod_pos_nat_iff [simp]:
```
```  1226   "finite A \<Longrightarrow> setprod f A > 0 \<longleftrightarrow> (\<forall>a\<in>A. f a > (0::nat))"
```
```  1227   using setprod_zero_iff by (simp del:neq0_conv add:neq0_conv [symmetric])
```
```  1228
```
```  1229 end
```