src/HOL/Groups_Big.thy
 author wenzelm Sat Nov 04 15:24:40 2017 +0100 (19 months ago) changeset 67003 49850a679c2c parent 66936 cf8d8fc23891 child 67268 bdf25939a550 permissions -rw-r--r--
more robust sorted_entries;
```     1 (*  Title:      HOL/Groups_Big.thy
```
```     2     Author:     Tobias Nipkow
```
```     3     Author:     Lawrence C Paulson
```
```     4     Author:     Markus Wenzel
```
```     5     Author:     Jeremy Avigad
```
```     6 *)
```
```     7
```
```     8 section \<open>Big sum and product over finite (non-empty) sets\<close>
```
```     9
```
```    10 theory Groups_Big
```
```    11   imports Power
```
```    12 begin
```
```    13
```
```    14 subsection \<open>Generic monoid operation over a set\<close>
```
```    15
```
```    16 locale comm_monoid_set = comm_monoid
```
```    17 begin
```
```    18
```
```    19 interpretation comp_fun_commute f
```
```    20   by standard (simp add: fun_eq_iff left_commute)
```
```    21
```
```    22 interpretation comp?: comp_fun_commute "f \<circ> g"
```
```    23   by (fact comp_comp_fun_commute)
```
```    24
```
```    25 definition F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
```
```    26   where eq_fold: "F g A = Finite_Set.fold (f \<circ> g) \<^bold>1 A"
```
```    27
```
```    28 lemma infinite [simp]: "\<not> finite A \<Longrightarrow> F g A = \<^bold>1"
```
```    29   by (simp add: eq_fold)
```
```    30
```
```    31 lemma empty [simp]: "F g {} = \<^bold>1"
```
```    32   by (simp add: eq_fold)
```
```    33
```
```    34 lemma insert [simp]: "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> F g (insert x A) = g x \<^bold>* F g A"
```
```    35   by (simp add: eq_fold)
```
```    36
```
```    37 lemma remove:
```
```    38   assumes "finite A" and "x \<in> A"
```
```    39   shows "F g A = g x \<^bold>* F g (A - {x})"
```
```    40 proof -
```
```    41   from \<open>x \<in> A\<close> obtain B where B: "A = insert x B" and "x \<notin> B"
```
```    42     by (auto dest: mk_disjoint_insert)
```
```    43   moreover from \<open>finite A\<close> B have "finite B" by simp
```
```    44   ultimately show ?thesis by simp
```
```    45 qed
```
```    46
```
```    47 lemma insert_remove: "finite A \<Longrightarrow> F g (insert x A) = g x \<^bold>* F g (A - {x})"
```
```    48   by (cases "x \<in> A") (simp_all add: remove insert_absorb)
```
```    49
```
```    50 lemma insert_if: "finite A \<Longrightarrow> F g (insert x A) = (if x \<in> A then F g A else g x \<^bold>* F g A)"
```
```    51   by (cases "x \<in> A") (simp_all add: insert_absorb)
```
```    52
```
```    53 lemma neutral: "\<forall>x\<in>A. g x = \<^bold>1 \<Longrightarrow> F g A = \<^bold>1"
```
```    54   by (induct A rule: infinite_finite_induct) simp_all
```
```    55
```
```    56 lemma neutral_const [simp]: "F (\<lambda>_. \<^bold>1) A = \<^bold>1"
```
```    57   by (simp add: neutral)
```
```    58
```
```    59 lemma union_inter:
```
```    60   assumes "finite A" and "finite B"
```
```    61   shows "F g (A \<union> B) \<^bold>* F g (A \<inter> B) = F g A \<^bold>* F g B"
```
```    62   \<comment> \<open>The reversed orientation looks more natural, but LOOPS as a simprule!\<close>
```
```    63   using assms
```
```    64 proof (induct A)
```
```    65   case empty
```
```    66   then show ?case by simp
```
```    67 next
```
```    68   case (insert x A)
```
```    69   then show ?case
```
```    70     by (auto simp: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
```
```    71 qed
```
```    72
```
```    73 corollary union_inter_neutral:
```
```    74   assumes "finite A" and "finite B"
```
```    75     and "\<forall>x \<in> A \<inter> B. g x = \<^bold>1"
```
```    76   shows "F g (A \<union> B) = F g A \<^bold>* F g B"
```
```    77   using assms by (simp add: union_inter [symmetric] neutral)
```
```    78
```
```    79 corollary union_disjoint:
```
```    80   assumes "finite A" and "finite B"
```
```    81   assumes "A \<inter> B = {}"
```
```    82   shows "F g (A \<union> B) = F g A \<^bold>* F g B"
```
```    83   using assms by (simp add: union_inter_neutral)
```
```    84
```
```    85 lemma union_diff2:
```
```    86   assumes "finite A" and "finite B"
```
```    87   shows "F g (A \<union> B) = F g (A - B) \<^bold>* F g (B - A) \<^bold>* F g (A \<inter> B)"
```
```    88 proof -
```
```    89   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
```
```    90     by auto
```
```    91   with assms show ?thesis
```
```    92     by simp (subst union_disjoint, auto)+
```
```    93 qed
```
```    94
```
```    95 lemma subset_diff:
```
```    96   assumes "B \<subseteq> A" and "finite A"
```
```    97   shows "F g A = F g (A - B) \<^bold>* F g B"
```
```    98 proof -
```
```    99   from assms have "finite (A - B)" by auto
```
```   100   moreover from assms have "finite B" by (rule finite_subset)
```
```   101   moreover from assms have "(A - B) \<inter> B = {}" by auto
```
```   102   ultimately have "F g (A - B \<union> B) = F g (A - B) \<^bold>* F g B" by (rule union_disjoint)
```
```   103   moreover from assms have "A \<union> B = A" by auto
```
```   104   ultimately show ?thesis by simp
```
```   105 qed
```
```   106
```
```   107 lemma setdiff_irrelevant:
```
```   108   assumes "finite A"
```
```   109   shows "F g (A - {x. g x = z}) = F g A"
```
```   110   using assms by (induct A) (simp_all add: insert_Diff_if)
```
```   111
```
```   112 lemma not_neutral_contains_not_neutral:
```
```   113   assumes "F g A \<noteq> \<^bold>1"
```
```   114   obtains a where "a \<in> A" and "g a \<noteq> \<^bold>1"
```
```   115 proof -
```
```   116   from assms have "\<exists>a\<in>A. g a \<noteq> \<^bold>1"
```
```   117   proof (induct A rule: infinite_finite_induct)
```
```   118     case infinite
```
```   119     then show ?case by simp
```
```   120   next
```
```   121     case empty
```
```   122     then show ?case by simp
```
```   123   next
```
```   124     case (insert a A)
```
```   125     then show ?case by fastforce
```
```   126   qed
```
```   127   with that show thesis by blast
```
```   128 qed
```
```   129
```
```   130 lemma reindex:
```
```   131   assumes "inj_on h A"
```
```   132   shows "F g (h ` A) = F (g \<circ> h) A"
```
```   133 proof (cases "finite A")
```
```   134   case True
```
```   135   with assms show ?thesis
```
```   136     by (simp add: eq_fold fold_image comp_assoc)
```
```   137 next
```
```   138   case False
```
```   139   with assms have "\<not> finite (h ` A)" by (blast dest: finite_imageD)
```
```   140   with False show ?thesis by simp
```
```   141 qed
```
```   142
```
```   143 lemma cong [fundef_cong]:
```
```   144   assumes "A = B"
```
```   145   assumes g_h: "\<And>x. x \<in> B \<Longrightarrow> g x = h x"
```
```   146   shows "F g A = F h B"
```
```   147   using g_h unfolding \<open>A = B\<close>
```
```   148   by (induct B rule: infinite_finite_induct) auto
```
```   149
```
```   150 lemma strong_cong [cong]:
```
```   151   assumes "A = B" "\<And>x. x \<in> B =simp=> g x = h x"
```
```   152   shows "F (\<lambda>x. g x) A = F (\<lambda>x. h x) B"
```
```   153   by (rule cong) (use assms in \<open>simp_all add: simp_implies_def\<close>)
```
```   154
```
```   155 lemma reindex_cong:
```
```   156   assumes "inj_on l B"
```
```   157   assumes "A = l ` B"
```
```   158   assumes "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x"
```
```   159   shows "F g A = F h B"
```
```   160   using assms by (simp add: reindex)
```
```   161
```
```   162 lemma UNION_disjoint:
```
```   163   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
```
```   164     and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
```
```   165   shows "F g (UNION I A) = F (\<lambda>x. F g (A x)) I"
```
```   166   apply (insert assms)
```
```   167   apply (induct rule: finite_induct)
```
```   168    apply simp
```
```   169   apply atomize
```
```   170   apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
```
```   171    prefer 2 apply blast
```
```   172   apply (subgoal_tac "A x \<inter> UNION Fa A = {}")
```
```   173    prefer 2 apply blast
```
```   174   apply (simp add: union_disjoint)
```
```   175   done
```
```   176
```
```   177 lemma Union_disjoint:
```
```   178   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
```
```   179   shows "F g (\<Union>C) = (F \<circ> F) g C"
```
```   180 proof (cases "finite C")
```
```   181   case True
```
```   182   from UNION_disjoint [OF this assms] show ?thesis by simp
```
```   183 next
```
```   184   case False
```
```   185   then show ?thesis by (auto dest: finite_UnionD intro: infinite)
```
```   186 qed
```
```   187
```
```   188 lemma distrib: "F (\<lambda>x. g x \<^bold>* h x) A = F g A \<^bold>* F h A"
```
```   189   by (induct A rule: infinite_finite_induct) (simp_all add: assoc commute left_commute)
```
```   190
```
```   191 lemma Sigma:
```
```   192   "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)"
```
```   193   apply (subst Sigma_def)
```
```   194   apply (subst UNION_disjoint)
```
```   195      apply assumption
```
```   196     apply simp
```
```   197    apply blast
```
```   198   apply (rule cong)
```
```   199    apply rule
```
```   200   apply (simp add: fun_eq_iff)
```
```   201   apply (subst UNION_disjoint)
```
```   202      apply simp
```
```   203     apply simp
```
```   204    apply blast
```
```   205   apply (simp add: comp_def)
```
```   206   done
```
```   207
```
```   208 lemma related:
```
```   209   assumes Re: "R \<^bold>1 \<^bold>1"
```
```   210     and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 \<^bold>* y1) (x2 \<^bold>* y2)"
```
```   211     and fin: "finite S"
```
```   212     and R_h_g: "\<forall>x\<in>S. R (h x) (g x)"
```
```   213   shows "R (F h S) (F g S)"
```
```   214   using fin by (rule finite_subset_induct) (use assms in auto)
```
```   215
```
```   216 lemma mono_neutral_cong_left:
```
```   217   assumes "finite T"
```
```   218     and "S \<subseteq> T"
```
```   219     and "\<forall>i \<in> T - S. h i = \<^bold>1"
```
```   220     and "\<And>x. x \<in> S \<Longrightarrow> g x = h x"
```
```   221   shows "F g S = F h T"
```
```   222 proof-
```
```   223   have eq: "T = S \<union> (T - S)" using \<open>S \<subseteq> T\<close> by blast
```
```   224   have d: "S \<inter> (T - S) = {}" using \<open>S \<subseteq> T\<close> by blast
```
```   225   from \<open>finite T\<close> \<open>S \<subseteq> T\<close> have f: "finite S" "finite (T - S)"
```
```   226     by (auto intro: finite_subset)
```
```   227   show ?thesis using assms(4)
```
```   228     by (simp add: union_disjoint [OF f d, unfolded eq [symmetric]] neutral [OF assms(3)])
```
```   229 qed
```
```   230
```
```   231 lemma mono_neutral_cong_right:
```
```   232   "finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. g i = \<^bold>1 \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> g x = h x) \<Longrightarrow>
```
```   233     F g T = F h S"
```
```   234   by (auto intro!: mono_neutral_cong_left [symmetric])
```
```   235
```
```   236 lemma mono_neutral_left: "finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. g i = \<^bold>1 \<Longrightarrow> F g S = F g T"
```
```   237   by (blast intro: mono_neutral_cong_left)
```
```   238
```
```   239 lemma mono_neutral_right: "finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. g i = \<^bold>1 \<Longrightarrow> F g T = F g S"
```
```   240   by (blast intro!: mono_neutral_left [symmetric])
```
```   241
```
```   242 lemma mono_neutral_cong:
```
```   243   assumes [simp]: "finite T" "finite S"
```
```   244     and *: "\<And>i. i \<in> T - S \<Longrightarrow> h i = \<^bold>1" "\<And>i. i \<in> S - T \<Longrightarrow> g i = \<^bold>1"
```
```   245     and gh: "\<And>x. x \<in> S \<inter> T \<Longrightarrow> g x = h x"
```
```   246  shows "F g S = F h T"
```
```   247 proof-
```
```   248   have "F g S = F g (S \<inter> T)"
```
```   249     by(rule mono_neutral_right)(auto intro: *)
```
```   250   also have "\<dots> = F h (S \<inter> T)" using refl gh by(rule cong)
```
```   251   also have "\<dots> = F h T"
```
```   252     by(rule mono_neutral_left)(auto intro: *)
```
```   253   finally show ?thesis .
```
```   254 qed
```
```   255
```
```   256 lemma reindex_bij_betw: "bij_betw h S T \<Longrightarrow> F (\<lambda>x. g (h x)) S = F g T"
```
```   257   by (auto simp: bij_betw_def reindex)
```
```   258
```
```   259 lemma reindex_bij_witness:
```
```   260   assumes witness:
```
```   261     "\<And>a. a \<in> S \<Longrightarrow> i (j a) = a"
```
```   262     "\<And>a. a \<in> S \<Longrightarrow> j a \<in> T"
```
```   263     "\<And>b. b \<in> T \<Longrightarrow> j (i b) = b"
```
```   264     "\<And>b. b \<in> T \<Longrightarrow> i b \<in> S"
```
```   265   assumes eq:
```
```   266     "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a"
```
```   267   shows "F g S = F h T"
```
```   268 proof -
```
```   269   have "bij_betw j S T"
```
```   270     using bij_betw_byWitness[where A=S and f=j and f'=i and A'=T] witness by auto
```
```   271   moreover have "F g S = F (\<lambda>x. h (j x)) S"
```
```   272     by (intro cong) (auto simp: eq)
```
```   273   ultimately show ?thesis
```
```   274     by (simp add: reindex_bij_betw)
```
```   275 qed
```
```   276
```
```   277 lemma reindex_bij_betw_not_neutral:
```
```   278   assumes fin: "finite S'" "finite T'"
```
```   279   assumes bij: "bij_betw h (S - S') (T - T')"
```
```   280   assumes nn:
```
```   281     "\<And>a. a \<in> S' \<Longrightarrow> g (h a) = z"
```
```   282     "\<And>b. b \<in> T' \<Longrightarrow> g b = z"
```
```   283   shows "F (\<lambda>x. g (h x)) S = F g T"
```
```   284 proof -
```
```   285   have [simp]: "finite S \<longleftrightarrow> finite T"
```
```   286     using bij_betw_finite[OF bij] fin by auto
```
```   287   show ?thesis
```
```   288   proof (cases "finite S")
```
```   289     case True
```
```   290     with nn have "F (\<lambda>x. g (h x)) S = F (\<lambda>x. g (h x)) (S - S')"
```
```   291       by (intro mono_neutral_cong_right) auto
```
```   292     also have "\<dots> = F g (T - T')"
```
```   293       using bij by (rule reindex_bij_betw)
```
```   294     also have "\<dots> = F g T"
```
```   295       using nn \<open>finite S\<close> by (intro mono_neutral_cong_left) auto
```
```   296     finally show ?thesis .
```
```   297   next
```
```   298     case False
```
```   299     then show ?thesis by simp
```
```   300   qed
```
```   301 qed
```
```   302
```
```   303 lemma reindex_nontrivial:
```
```   304   assumes "finite A"
```
```   305     and nz: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> h x = h y \<Longrightarrow> g (h x) = \<^bold>1"
```
```   306   shows "F g (h ` A) = F (g \<circ> h) A"
```
```   307 proof (subst reindex_bij_betw_not_neutral [symmetric])
```
```   308   show "bij_betw h (A - {x \<in> A. (g \<circ> h) x = \<^bold>1}) (h ` A - h ` {x \<in> A. (g \<circ> h) x = \<^bold>1})"
```
```   309     using nz by (auto intro!: inj_onI simp: bij_betw_def)
```
```   310 qed (use \<open>finite A\<close> in auto)
```
```   311
```
```   312 lemma reindex_bij_witness_not_neutral:
```
```   313   assumes fin: "finite S'" "finite T'"
```
```   314   assumes witness:
```
```   315     "\<And>a. a \<in> S - S' \<Longrightarrow> i (j a) = a"
```
```   316     "\<And>a. a \<in> S - S' \<Longrightarrow> j a \<in> T - T'"
```
```   317     "\<And>b. b \<in> T - T' \<Longrightarrow> j (i b) = b"
```
```   318     "\<And>b. b \<in> T - T' \<Longrightarrow> i b \<in> S - S'"
```
```   319   assumes nn:
```
```   320     "\<And>a. a \<in> S' \<Longrightarrow> g a = z"
```
```   321     "\<And>b. b \<in> T' \<Longrightarrow> h b = z"
```
```   322   assumes eq:
```
```   323     "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a"
```
```   324   shows "F g S = F h T"
```
```   325 proof -
```
```   326   have bij: "bij_betw j (S - (S' \<inter> S)) (T - (T' \<inter> T))"
```
```   327     using witness by (intro bij_betw_byWitness[where f'=i]) auto
```
```   328   have F_eq: "F g S = F (\<lambda>x. h (j x)) S"
```
```   329     by (intro cong) (auto simp: eq)
```
```   330   show ?thesis
```
```   331     unfolding F_eq using fin nn eq
```
```   332     by (intro reindex_bij_betw_not_neutral[OF _ _ bij]) auto
```
```   333 qed
```
```   334
```
```   335 lemma delta [simp]:
```
```   336   assumes fS: "finite S"
```
```   337   shows "F (\<lambda>k. if k = a then b k else \<^bold>1) S = (if a \<in> S then b a else \<^bold>1)"
```
```   338 proof -
```
```   339   let ?f = "(\<lambda>k. if k = a then b k else \<^bold>1)"
```
```   340   show ?thesis
```
```   341   proof (cases "a \<in> S")
```
```   342     case False
```
```   343     then have "\<forall>k\<in>S. ?f k = \<^bold>1" by simp
```
```   344     with False show ?thesis by simp
```
```   345   next
```
```   346     case True
```
```   347     let ?A = "S - {a}"
```
```   348     let ?B = "{a}"
```
```   349     from True have eq: "S = ?A \<union> ?B" by blast
```
```   350     have dj: "?A \<inter> ?B = {}" by simp
```
```   351     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```   352     have "F ?f S = F ?f ?A \<^bold>* F ?f ?B"
```
```   353       using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]] by simp
```
```   354     with True show ?thesis by simp
```
```   355   qed
```
```   356 qed
```
```   357
```
```   358 lemma delta' [simp]:
```
```   359   assumes fin: "finite S"
```
```   360   shows "F (\<lambda>k. if a = k then b k else \<^bold>1) S = (if a \<in> S then b a else \<^bold>1)"
```
```   361   using delta [OF fin, of a b, symmetric] by (auto intro: cong)
```
```   362
```
```   363 lemma If_cases:
```
```   364   fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
```
```   365   assumes fin: "finite A"
```
```   366   shows "F (\<lambda>x. if P x then h x else g x) A = F h (A \<inter> {x. P x}) \<^bold>* F g (A \<inter> - {x. P x})"
```
```   367 proof -
```
```   368   have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}" "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}"
```
```   369     by blast+
```
```   370   from fin have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
```
```   371   let ?g = "\<lambda>x. if P x then h x else g x"
```
```   372   from union_disjoint [OF f a(2), of ?g] a(1) show ?thesis
```
```   373     by (subst (1 2) cong) simp_all
```
```   374 qed
```
```   375
```
```   376 lemma cartesian_product: "F (\<lambda>x. F (g x) B) A = F (case_prod g) (A \<times> B)"
```
```   377   apply (rule sym)
```
```   378   apply (cases "finite A")
```
```   379    apply (cases "finite B")
```
```   380     apply (simp add: Sigma)
```
```   381    apply (cases "A = {}")
```
```   382     apply simp
```
```   383    apply simp
```
```   384    apply (auto intro: infinite dest: finite_cartesian_productD2)
```
```   385   apply (cases "B = {}")
```
```   386    apply (auto intro: infinite dest: finite_cartesian_productD1)
```
```   387   done
```
```   388
```
```   389 lemma inter_restrict:
```
```   390   assumes "finite A"
```
```   391   shows "F g (A \<inter> B) = F (\<lambda>x. if x \<in> B then g x else \<^bold>1) A"
```
```   392 proof -
```
```   393   let ?g = "\<lambda>x. if x \<in> A \<inter> B then g x else \<^bold>1"
```
```   394   have "\<forall>i\<in>A - A \<inter> B. (if i \<in> A \<inter> B then g i else \<^bold>1) = \<^bold>1" by simp
```
```   395   moreover have "A \<inter> B \<subseteq> A" by blast
```
```   396   ultimately have "F ?g (A \<inter> B) = F ?g A"
```
```   397     using \<open>finite A\<close> by (intro mono_neutral_left) auto
```
```   398   then show ?thesis by simp
```
```   399 qed
```
```   400
```
```   401 lemma inter_filter:
```
```   402   "finite A \<Longrightarrow> F g {x \<in> A. P x} = F (\<lambda>x. if P x then g x else \<^bold>1) A"
```
```   403   by (simp add: inter_restrict [symmetric, of A "{x. P x}" g, simplified mem_Collect_eq] Int_def)
```
```   404
```
```   405 lemma Union_comp:
```
```   406   assumes "\<forall>A \<in> B. finite A"
```
```   407     and "\<And>A1 A2 x. A1 \<in> B \<Longrightarrow> A2 \<in> B \<Longrightarrow> A1 \<noteq> A2 \<Longrightarrow> x \<in> A1 \<Longrightarrow> x \<in> A2 \<Longrightarrow> g x = \<^bold>1"
```
```   408   shows "F g (\<Union>B) = (F \<circ> F) g B"
```
```   409   using assms
```
```   410 proof (induct B rule: infinite_finite_induct)
```
```   411   case (infinite A)
```
```   412   then have "\<not> finite (\<Union>A)" by (blast dest: finite_UnionD)
```
```   413   with infinite show ?case by simp
```
```   414 next
```
```   415   case empty
```
```   416   then show ?case by simp
```
```   417 next
```
```   418   case (insert A B)
```
```   419   then have "finite A" "finite B" "finite (\<Union>B)" "A \<notin> B"
```
```   420     and "\<forall>x\<in>A \<inter> \<Union>B. g x = \<^bold>1"
```
```   421     and H: "F g (\<Union>B) = (F \<circ> F) g B" by auto
```
```   422   then have "F g (A \<union> \<Union>B) = F g A \<^bold>* F g (\<Union>B)"
```
```   423     by (simp add: union_inter_neutral)
```
```   424   with \<open>finite B\<close> \<open>A \<notin> B\<close> show ?case
```
```   425     by (simp add: H)
```
```   426 qed
```
```   427
```
```   428 lemma swap: "F (\<lambda>i. F (g i) B) A = F (\<lambda>j. F (\<lambda>i. g i j) A) B"
```
```   429   unfolding cartesian_product
```
```   430   by (rule reindex_bij_witness [where i = "\<lambda>(i, j). (j, i)" and j = "\<lambda>(i, j). (j, i)"]) auto
```
```   431
```
```   432 lemma swap_restrict:
```
```   433   "finite A \<Longrightarrow> finite B \<Longrightarrow>
```
```   434     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"
```
```   435   by (simp add: inter_filter) (rule swap)
```
```   436
```
```   437 lemma Plus:
```
```   438   fixes A :: "'b set" and B :: "'c set"
```
```   439   assumes fin: "finite A" "finite B"
```
```   440   shows "F g (A <+> B) = F (g \<circ> Inl) A \<^bold>* F (g \<circ> Inr) B"
```
```   441 proof -
```
```   442   have "A <+> B = Inl ` A \<union> Inr ` B" by auto
```
```   443   moreover from fin have "finite (Inl ` A)" "finite (Inr ` B)" by auto
```
```   444   moreover have "Inl ` A \<inter> Inr ` B = {}" by auto
```
```   445   moreover have "inj_on Inl A" "inj_on Inr B" by (auto intro: inj_onI)
```
```   446   ultimately show ?thesis
```
```   447     using fin by (simp add: union_disjoint reindex)
```
```   448 qed
```
```   449
```
```   450 lemma same_carrier:
```
```   451   assumes "finite C"
```
```   452   assumes subset: "A \<subseteq> C" "B \<subseteq> C"
```
```   453   assumes trivial: "\<And>a. a \<in> C - A \<Longrightarrow> g a = \<^bold>1" "\<And>b. b \<in> C - B \<Longrightarrow> h b = \<^bold>1"
```
```   454   shows "F g A = F h B \<longleftrightarrow> F g C = F h C"
```
```   455 proof -
```
```   456   have "finite A" and "finite B" and "finite (C - A)" and "finite (C - B)"
```
```   457     using \<open>finite C\<close> subset by (auto elim: finite_subset)
```
```   458   from subset have [simp]: "A - (C - A) = A" by auto
```
```   459   from subset have [simp]: "B - (C - B) = B" by auto
```
```   460   from subset have "C = A \<union> (C - A)" by auto
```
```   461   then have "F g C = F g (A \<union> (C - A))" by simp
```
```   462   also have "\<dots> = F g (A - (C - A)) \<^bold>* F g (C - A - A) \<^bold>* F g (A \<inter> (C - A))"
```
```   463     using \<open>finite A\<close> \<open>finite (C - A)\<close> by (simp only: union_diff2)
```
```   464   finally have *: "F g C = F g A" using trivial by simp
```
```   465   from subset have "C = B \<union> (C - B)" by auto
```
```   466   then have "F h C = F h (B \<union> (C - B))" by simp
```
```   467   also have "\<dots> = F h (B - (C - B)) \<^bold>* F h (C - B - B) \<^bold>* F h (B \<inter> (C - B))"
```
```   468     using \<open>finite B\<close> \<open>finite (C - B)\<close> by (simp only: union_diff2)
```
```   469   finally have "F h C = F h B"
```
```   470     using trivial by simp
```
```   471   with * show ?thesis by simp
```
```   472 qed
```
```   473
```
```   474 lemma same_carrierI:
```
```   475   assumes "finite C"
```
```   476   assumes subset: "A \<subseteq> C" "B \<subseteq> C"
```
```   477   assumes trivial: "\<And>a. a \<in> C - A \<Longrightarrow> g a = \<^bold>1" "\<And>b. b \<in> C - B \<Longrightarrow> h b = \<^bold>1"
```
```   478   assumes "F g C = F h C"
```
```   479   shows "F g A = F h B"
```
```   480   using assms same_carrier [of C A B] by simp
```
```   481
```
```   482 end
```
```   483
```
```   484
```
```   485 subsection \<open>Generalized summation over a set\<close>
```
```   486
```
```   487 context comm_monoid_add
```
```   488 begin
```
```   489
```
```   490 sublocale sum: comm_monoid_set plus 0
```
```   491   defines sum = sum.F ..
```
```   492
```
```   493 abbreviation Sum ("\<Sum>_"  999)
```
```   494   where "\<Sum>A \<equiv> sum (\<lambda>x. x) A"
```
```   495
```
```   496 end
```
```   497
```
```   498 text \<open>Now: lot's of fancy syntax. First, @{term "sum (\<lambda>x. e) A"} is written \<open>\<Sum>x\<in>A. e\<close>.\<close>
```
```   499
```
```   500 syntax (ASCII)
```
```   501   "_sum" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b::comm_monoid_add"  ("(3SUM _:_./ _)" [0, 51, 10] 10)
```
```   502 syntax
```
```   503   "_sum" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b::comm_monoid_add"  ("(2\<Sum>_\<in>_./ _)" [0, 51, 10] 10)
```
```   504 translations \<comment> \<open>Beware of argument permutation!\<close>
```
```   505   "\<Sum>i\<in>A. b" \<rightleftharpoons> "CONST sum (\<lambda>i. b) A"
```
```   506
```
```   507 text \<open>Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter \<open>\<Sum>x|P. e\<close>.\<close>
```
```   508
```
```   509 syntax (ASCII)
```
```   510   "_qsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(3SUM _ |/ _./ _)" [0, 0, 10] 10)
```
```   511 syntax
```
```   512   "_qsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(2\<Sum>_ | (_)./ _)" [0, 0, 10] 10)
```
```   513 translations
```
```   514   "\<Sum>x|P. t" => "CONST sum (\<lambda>x. t) {x. P}"
```
```   515
```
```   516 print_translation \<open>
```
```   517 let
```
```   518   fun sum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) \$ Abs (y, Ty, P)] =
```
```   519         if x <> y then raise Match
```
```   520         else
```
```   521           let
```
```   522             val x' = Syntax_Trans.mark_bound_body (x, Tx);
```
```   523             val t' = subst_bound (x', t);
```
```   524             val P' = subst_bound (x', P);
```
```   525           in
```
```   526             Syntax.const @{syntax_const "_qsum"} \$ Syntax_Trans.mark_bound_abs (x, Tx) \$ P' \$ t'
```
```   527           end
```
```   528     | sum_tr' _ = raise Match;
```
```   529 in [(@{const_syntax sum}, K sum_tr')] end
```
```   530 \<close>
```
```   531
```
```   532 (* TODO generalization candidates *)
```
```   533
```
```   534 lemma (in comm_monoid_add) sum_image_gen:
```
```   535   assumes fin: "finite S"
```
```   536   shows "sum g S = sum (\<lambda>y. sum g {x. x \<in> S \<and> f x = y}) (f ` S)"
```
```   537 proof -
```
```   538   have "{y. y\<in> f`S \<and> f x = y} = {f x}" if "x \<in> S" for x
```
```   539     using that by auto
```
```   540   then have "sum g S = sum (\<lambda>x. sum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
```
```   541     by simp
```
```   542   also have "\<dots> = sum (\<lambda>y. sum g {x. x \<in> S \<and> f x = y}) (f ` S)"
```
```   543     by (rule sum.swap_restrict [OF fin finite_imageI [OF fin]])
```
```   544   finally show ?thesis .
```
```   545 qed
```
```   546
```
```   547
```
```   548 subsubsection \<open>Properties in more restricted classes of structures\<close>
```
```   549
```
```   550 lemma sum_Un:
```
```   551   "finite A \<Longrightarrow> finite B \<Longrightarrow> sum f (A \<union> B) = sum f A + sum f B - sum f (A \<inter> B)"
```
```   552   for f :: "'b \<Rightarrow> 'a::ab_group_add"
```
```   553   by (subst sum.union_inter [symmetric]) (auto simp add: algebra_simps)
```
```   554
```
```   555 lemma sum_Un2:
```
```   556   assumes "finite (A \<union> B)"
```
```   557   shows "sum f (A \<union> B) = sum f (A - B) + sum f (B - A) + sum 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
```
```   562     by simp (subst sum.union_disjoint, auto)+
```
```   563 qed
```
```   564
```
```   565 lemma sum_diff1:
```
```   566   fixes f :: "'b \<Rightarrow> 'a::ab_group_add"
```
```   567   assumes "finite A"
```
```   568   shows "sum f (A - {a}) = (if a \<in> A then sum f A - f a else sum f A)"
```
```   569   using assms by induct (auto simp: insert_Diff_if)
```
```   570
```
```   571 lemma sum_diff:
```
```   572   fixes f :: "'b \<Rightarrow> 'a::ab_group_add"
```
```   573   assumes "finite A" "B \<subseteq> A"
```
```   574   shows "sum f (A - B) = sum f A - sum f B"
```
```   575 proof -
```
```   576   from assms(2,1) have "finite B" by (rule finite_subset)
```
```   577   from this \<open>B \<subseteq> A\<close>
```
```   578   show ?thesis
```
```   579   proof induct
```
```   580     case empty
```
```   581     thus ?case by simp
```
```   582   next
```
```   583     case (insert x F)
```
```   584     with \<open>finite A\<close> \<open>finite B\<close> show ?case
```
```   585       by (simp add: Diff_insert[where a=x and B=F] sum_diff1 insert_absorb)
```
```   586   qed
```
```   587 qed
```
```   588
```
```   589 lemma (in ordered_comm_monoid_add) sum_mono:
```
```   590   "(\<And>i. i\<in>K \<Longrightarrow> f i \<le> g i) \<Longrightarrow> (\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
```
```   591   by (induct K rule: infinite_finite_induct) (use add_mono in auto)
```
```   592
```
```   593 lemma (in strict_ordered_comm_monoid_add) sum_strict_mono:
```
```   594   assumes "finite A" "A \<noteq> {}"
```
```   595     and "\<And>x. x \<in> A \<Longrightarrow> f x < g x"
```
```   596   shows "sum f A < sum g A"
```
```   597   using assms
```
```   598 proof (induct rule: finite_ne_induct)
```
```   599   case singleton
```
```   600   then show ?case by simp
```
```   601 next
```
```   602   case insert
```
```   603   then show ?case by (auto simp: add_strict_mono)
```
```   604 qed
```
```   605
```
```   606 lemma sum_strict_mono_ex1:
```
```   607   fixes f g :: "'i \<Rightarrow> 'a::ordered_cancel_comm_monoid_add"
```
```   608   assumes "finite A"
```
```   609     and "\<forall>x\<in>A. f x \<le> g x"
```
```   610     and "\<exists>a\<in>A. f a < g a"
```
```   611   shows "sum f A < sum g A"
```
```   612 proof-
```
```   613   from assms(3) obtain a where a: "a \<in> A" "f a < g a" by blast
```
```   614   have "sum f A = sum f ((A - {a}) \<union> {a})"
```
```   615     by(simp add: insert_absorb[OF \<open>a \<in> A\<close>])
```
```   616   also have "\<dots> = sum f (A - {a}) + sum f {a}"
```
```   617     using \<open>finite A\<close> by(subst sum.union_disjoint) auto
```
```   618   also have "sum f (A - {a}) \<le> sum g (A - {a})"
```
```   619     by (rule sum_mono) (simp add: assms(2))
```
```   620   also from a have "sum f {a} < sum g {a}" by simp
```
```   621   also have "sum g (A - {a}) + sum g {a} = sum g((A - {a}) \<union> {a})"
```
```   622     using \<open>finite A\<close> by (subst sum.union_disjoint[symmetric]) auto
```
```   623   also have "\<dots> = sum g A" by (simp add: insert_absorb[OF \<open>a \<in> A\<close>])
```
```   624   finally show ?thesis
```
```   625     by (auto simp add: add_right_mono add_strict_left_mono)
```
```   626 qed
```
```   627
```
```   628 lemma sum_mono_inv:
```
```   629   fixes f g :: "'i \<Rightarrow> 'a :: ordered_cancel_comm_monoid_add"
```
```   630   assumes eq: "sum f I = sum g I"
```
```   631   assumes le: "\<And>i. i \<in> I \<Longrightarrow> f i \<le> g i"
```
```   632   assumes i: "i \<in> I"
```
```   633   assumes I: "finite I"
```
```   634   shows "f i = g i"
```
```   635 proof (rule ccontr)
```
```   636   assume "\<not> ?thesis"
```
```   637   with le[OF i] have "f i < g i" by simp
```
```   638   with i have "\<exists>i\<in>I. f i < g i" ..
```
```   639   from sum_strict_mono_ex1[OF I _ this] le have "sum f I < sum g I"
```
```   640     by blast
```
```   641   with eq show False by simp
```
```   642 qed
```
```   643
```
```   644 lemma member_le_sum:
```
```   645   fixes f :: "_ \<Rightarrow> 'b::{semiring_1, ordered_comm_monoid_add}"
```
```   646   assumes "i \<in> A"
```
```   647     and le: "\<And>x. x \<in> A - {i} \<Longrightarrow> 0 \<le> f x"
```
```   648     and "finite A"
```
```   649   shows "f i \<le> sum f A"
```
```   650 proof -
```
```   651   have "f i \<le> sum f (A \<inter> {i})"
```
```   652     by (simp add: assms)
```
```   653   also have "... = (\<Sum>x\<in>A. if x \<in> {i} then f x else 0)"
```
```   654     using assms sum.inter_restrict by blast
```
```   655   also have "... \<le> sum f A"
```
```   656     apply (rule sum_mono)
```
```   657     apply (auto simp: le)
```
```   658     done
```
```   659   finally show ?thesis .
```
```   660 qed
```
```   661
```
```   662 lemma sum_negf: "(\<Sum>x\<in>A. - f x) = - (\<Sum>x\<in>A. f x)"
```
```   663   for f :: "'b \<Rightarrow> 'a::ab_group_add"
```
```   664   by (induct A rule: infinite_finite_induct) auto
```
```   665
```
```   666 lemma sum_subtractf: "(\<Sum>x\<in>A. f x - g x) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
```
```   667   for f g :: "'b \<Rightarrow>'a::ab_group_add"
```
```   668   using sum.distrib [of f "- g" A] by (simp add: sum_negf)
```
```   669
```
```   670 lemma sum_subtractf_nat:
```
```   671   "(\<And>x. x \<in> A \<Longrightarrow> g x \<le> f x) \<Longrightarrow> (\<Sum>x\<in>A. f x - g x) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
```
```   672   for f g :: "'a \<Rightarrow> nat"
```
```   673   by (induct A rule: infinite_finite_induct) (auto simp: sum_mono)
```
```   674
```
```   675 context ordered_comm_monoid_add
```
```   676 begin
```
```   677
```
```   678 lemma sum_nonneg: "(\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x) \<Longrightarrow> 0 \<le> sum f A"
```
```   679 proof (induct A rule: infinite_finite_induct)
```
```   680   case infinite
```
```   681   then show ?case by simp
```
```   682 next
```
```   683   case empty
```
```   684   then show ?case by simp
```
```   685 next
```
```   686   case (insert x F)
```
```   687   then have "0 + 0 \<le> f x + sum f F" by (blast intro: add_mono)
```
```   688   with insert show ?case by simp
```
```   689 qed
```
```   690
```
```   691 lemma sum_nonpos: "(\<And>x. x \<in> A \<Longrightarrow> f x \<le> 0) \<Longrightarrow> sum f A \<le> 0"
```
```   692 proof (induct A rule: infinite_finite_induct)
```
```   693   case infinite
```
```   694   then show ?case by simp
```
```   695 next
```
```   696   case empty
```
```   697   then show ?case by simp
```
```   698 next
```
```   699   case (insert x F)
```
```   700   then have "f x + sum f F \<le> 0 + 0" by (blast intro: add_mono)
```
```   701   with insert show ?case by simp
```
```   702 qed
```
```   703
```
```   704 lemma sum_nonneg_eq_0_iff:
```
```   705   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x) \<Longrightarrow> sum f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
```
```   706   by (induct set: finite) (simp_all add: add_nonneg_eq_0_iff sum_nonneg)
```
```   707
```
```   708 lemma sum_nonneg_0:
```
```   709   "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"
```
```   710   by (simp add: sum_nonneg_eq_0_iff)
```
```   711
```
```   712 lemma sum_nonneg_leq_bound:
```
```   713   assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"
```
```   714   shows "f i \<le> B"
```
```   715 proof -
```
```   716   from assms have "f i \<le> f i + (\<Sum>i \<in> s - {i}. f i)"
```
```   717     by (intro add_increasing2 sum_nonneg) auto
```
```   718   also have "\<dots> = B"
```
```   719     using sum.remove[of s i f] assms by simp
```
```   720   finally show ?thesis by auto
```
```   721 qed
```
```   722
```
```   723 lemma sum_mono2:
```
```   724   assumes fin: "finite B"
```
```   725     and sub: "A \<subseteq> B"
```
```   726     and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
```
```   727   shows "sum f A \<le> sum f B"
```
```   728 proof -
```
```   729   have "sum f A \<le> sum f A + sum f (B-A)"
```
```   730     by (auto intro: add_increasing2 [OF sum_nonneg] nn)
```
```   731   also from fin finite_subset[OF sub fin] have "\<dots> = sum f (A \<union> (B-A))"
```
```   732     by (simp add: sum.union_disjoint del: Un_Diff_cancel)
```
```   733   also from sub have "A \<union> (B-A) = B" by blast
```
```   734   finally show ?thesis .
```
```   735 qed
```
```   736
```
```   737 lemma sum_le_included:
```
```   738   assumes "finite s" "finite t"
```
```   739   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)"
```
```   740   shows "sum f s \<le> sum g t"
```
```   741 proof -
```
```   742   have "sum f s \<le> sum (\<lambda>y. sum g {x. x\<in>t \<and> i x = y}) s"
```
```   743   proof (rule sum_mono)
```
```   744     fix y
```
```   745     assume "y \<in> s"
```
```   746     with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto
```
```   747     with assms show "f y \<le> sum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")
```
```   748       using order_trans[of "?A (i z)" "sum g {z}" "?B (i z)", intro]
```
```   749       by (auto intro!: sum_mono2)
```
```   750   qed
```
```   751   also have "\<dots> \<le> sum (\<lambda>y. sum g {x. x\<in>t \<and> i x = y}) (i ` t)"
```
```   752     using assms(2-4) by (auto intro!: sum_mono2 sum_nonneg)
```
```   753   also have "\<dots> \<le> sum g t"
```
```   754     using assms by (auto simp: sum_image_gen[symmetric])
```
```   755   finally show ?thesis .
```
```   756 qed
```
```   757
```
```   758 end
```
```   759
```
```   760 lemma (in canonically_ordered_monoid_add) sum_eq_0_iff [simp]:
```
```   761   "finite F \<Longrightarrow> (sum f F = 0) = (\<forall>a\<in>F. f a = 0)"
```
```   762   by (intro ballI sum_nonneg_eq_0_iff zero_le)
```
```   763
```
```   764 context semiring_0
```
```   765 begin
```
```   766
```
```   767 lemma sum_distrib_left: "r * sum f A = (\<Sum>n\<in>A. r * f n)"
```
```   768   by (induct A rule: infinite_finite_induct) (simp_all add: algebra_simps)
```
```   769
```
```   770 lemma sum_distrib_right: "sum f A * r = (\<Sum>n\<in>A. f n * r)"
```
```   771   by (induct A rule: infinite_finite_induct) (simp_all add: algebra_simps)
```
```   772
```
```   773 end
```
```   774
```
```   775 lemma sum_divide_distrib: "sum f A / r = (\<Sum>n\<in>A. f n / r)"
```
```   776   for r :: "'a::field"
```
```   777 proof (induct A rule: infinite_finite_induct)
```
```   778   case infinite
```
```   779   then show ?case by simp
```
```   780 next
```
```   781   case empty
```
```   782   then show ?case by simp
```
```   783 next
```
```   784   case insert
```
```   785   then show ?case by (simp add: add_divide_distrib)
```
```   786 qed
```
```   787
```
```   788 lemma sum_abs[iff]: "\<bar>sum f A\<bar> \<le> sum (\<lambda>i. \<bar>f i\<bar>) A"
```
```   789   for f :: "'a \<Rightarrow> 'b::ordered_ab_group_add_abs"
```
```   790 proof (induct A rule: infinite_finite_induct)
```
```   791   case infinite
```
```   792   then show ?case by simp
```
```   793 next
```
```   794   case empty
```
```   795   then show ?case by simp
```
```   796 next
```
```   797   case insert
```
```   798   then show ?case by (auto intro: abs_triangle_ineq order_trans)
```
```   799 qed
```
```   800
```
```   801 lemma sum_abs_ge_zero[iff]: "0 \<le> sum (\<lambda>i. \<bar>f i\<bar>) A"
```
```   802   for f :: "'a \<Rightarrow> 'b::ordered_ab_group_add_abs"
```
```   803   by (simp add: sum_nonneg)
```
```   804
```
```   805 lemma abs_sum_abs[simp]: "\<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar> = (\<Sum>a\<in>A. \<bar>f a\<bar>)"
```
```   806   for f :: "'a \<Rightarrow> 'b::ordered_ab_group_add_abs"
```
```   807 proof (induct A rule: infinite_finite_induct)
```
```   808   case infinite
```
```   809   then show ?case by simp
```
```   810 next
```
```   811   case empty
```
```   812   then show ?case by simp
```
```   813 next
```
```   814   case (insert a A)
```
```   815   then have "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
```
```   816   also from insert have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>" by simp
```
```   817   also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>" by (simp del: abs_of_nonneg)
```
```   818   also from insert have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" by simp
```
```   819   finally show ?case .
```
```   820 qed
```
```   821
```
```   822 lemma sum_diff1_ring:
```
```   823   fixes f :: "'b \<Rightarrow> 'a::ring"
```
```   824   assumes "finite A" "a \<in> A"
```
```   825   shows "sum f (A - {a}) = sum f A - (f a)"
```
```   826   unfolding sum.remove [OF assms] by auto
```
```   827
```
```   828 lemma sum_product:
```
```   829   fixes f :: "'a \<Rightarrow> 'b::semiring_0"
```
```   830   shows "sum f A * sum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
```
```   831   by (simp add: sum_distrib_left sum_distrib_right) (rule sum.swap)
```
```   832
```
```   833 lemma sum_mult_sum_if_inj:
```
```   834   fixes f :: "'a \<Rightarrow> 'b::semiring_0"
```
```   835   shows "inj_on (\<lambda>(a, b). f a * g b) (A \<times> B) \<Longrightarrow>
```
```   836     sum f A * sum g B = sum id {f a * g b |a b. a \<in> A \<and> b \<in> B}"
```
```   837   by(auto simp: sum_product sum.cartesian_product intro!: sum.reindex_cong[symmetric])
```
```   838
```
```   839 lemma sum_SucD: "sum f A = Suc n \<Longrightarrow> \<exists>a\<in>A. 0 < f a"
```
```   840   by (induct A rule: infinite_finite_induct) auto
```
```   841
```
```   842 lemma sum_eq_Suc0_iff:
```
```   843   "finite A \<Longrightarrow> sum f A = Suc 0 \<longleftrightarrow> (\<exists>a\<in>A. f a = Suc 0 \<and> (\<forall>b\<in>A. a \<noteq> b \<longrightarrow> f b = 0))"
```
```   844   by (induct A rule: finite_induct) (auto simp add: add_is_1)
```
```   845
```
```   846 lemmas sum_eq_1_iff = sum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
```
```   847
```
```   848 lemma sum_Un_nat:
```
```   849   "finite A \<Longrightarrow> finite B \<Longrightarrow> sum f (A \<union> B) = sum f A + sum f B - sum f (A \<inter> B)"
```
```   850   for f :: "'a \<Rightarrow> nat"
```
```   851   \<comment> \<open>For the natural numbers, we have subtraction.\<close>
```
```   852   by (subst sum.union_inter [symmetric]) (auto simp: algebra_simps)
```
```   853
```
```   854 lemma sum_diff1_nat: "sum f (A - {a}) = (if a \<in> A then sum f A - f a else sum f A)"
```
```   855   for f :: "'a \<Rightarrow> nat"
```
```   856 proof (induct A rule: infinite_finite_induct)
```
```   857   case infinite
```
```   858   then show ?case by simp
```
```   859 next
```
```   860   case empty
```
```   861   then show ?case by simp
```
```   862 next
```
```   863   case insert
```
```   864   then show ?case
```
```   865     apply (auto simp: insert_Diff_if)
```
```   866     apply (drule mk_disjoint_insert)
```
```   867     apply auto
```
```   868     done
```
```   869 qed
```
```   870
```
```   871 lemma sum_diff_nat:
```
```   872   fixes f :: "'a \<Rightarrow> nat"
```
```   873   assumes "finite B" and "B \<subseteq> A"
```
```   874   shows "sum f (A - B) = sum f A - sum f B"
```
```   875   using assms
```
```   876 proof induct
```
```   877   case empty
```
```   878   then show ?case by simp
```
```   879 next
```
```   880   case (insert x F)
```
```   881   note IH = \<open>F \<subseteq> A \<Longrightarrow> sum f (A - F) = sum f A - sum f F\<close>
```
```   882   from \<open>x \<notin> F\<close> \<open>insert x F \<subseteq> A\<close> have "x \<in> A - F" by simp
```
```   883   then have A: "sum f ((A - F) - {x}) = sum f (A - F) - f x"
```
```   884     by (simp add: sum_diff1_nat)
```
```   885   from \<open>insert x F \<subseteq> A\<close> have "F \<subseteq> A" by simp
```
```   886   with IH have "sum f (A - F) = sum f A - sum f F" by simp
```
```   887   with A have B: "sum f ((A - F) - {x}) = sum f A - sum f F - f x"
```
```   888     by simp
```
```   889   from \<open>x \<notin> F\<close> have "A - insert x F = (A - F) - {x}" by auto
```
```   890   with B have C: "sum f (A - insert x F) = sum f A - sum f F - f x"
```
```   891     by simp
```
```   892   from \<open>finite F\<close> \<open>x \<notin> F\<close> have "sum f (insert x F) = sum f F + f x"
```
```   893     by simp
```
```   894   with C have "sum f (A - insert x F) = sum f A - sum f (insert x F)"
```
```   895     by simp
```
```   896   then show ?case by simp
```
```   897 qed
```
```   898
```
```   899 lemma sum_comp_morphism:
```
```   900   "h 0 = 0 \<Longrightarrow> (\<And>x y. h (x + y) = h x + h y) \<Longrightarrow> sum (h \<circ> g) A = h (sum g A)"
```
```   901   by (induct A rule: infinite_finite_induct) simp_all
```
```   902
```
```   903 lemma (in comm_semiring_1) dvd_sum: "(\<And>a. a \<in> A \<Longrightarrow> d dvd f a) \<Longrightarrow> d dvd sum f A"
```
```   904   by (induct A rule: infinite_finite_induct) simp_all
```
```   905
```
```   906 lemma (in ordered_comm_monoid_add) sum_pos:
```
```   907   "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> 0 < f i) \<Longrightarrow> 0 < sum f I"
```
```   908   by (induct I rule: finite_ne_induct) (auto intro: add_pos_pos)
```
```   909
```
```   910 lemma (in ordered_comm_monoid_add) sum_pos2:
```
```   911   assumes I: "finite I" "i \<in> I" "0 < f i" "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
```
```   912   shows "0 < sum f I"
```
```   913 proof -
```
```   914   have "0 < f i + sum f (I - {i})"
```
```   915     using assms by (intro add_pos_nonneg sum_nonneg) auto
```
```   916   also have "\<dots> = sum f I"
```
```   917     using assms by (simp add: sum.remove)
```
```   918   finally show ?thesis .
```
```   919 qed
```
```   920
```
```   921 lemma sum_cong_Suc:
```
```   922   assumes "0 \<notin> A" "\<And>x. Suc x \<in> A \<Longrightarrow> f (Suc x) = g (Suc x)"
```
```   923   shows "sum f A = sum g A"
```
```   924 proof (rule sum.cong)
```
```   925   fix x
```
```   926   assume "x \<in> A"
```
```   927   with assms(1) show "f x = g x"
```
```   928     by (cases x) (auto intro!: assms(2))
```
```   929 qed simp_all
```
```   930
```
```   931
```
```   932 subsubsection \<open>Cardinality as special case of @{const sum}\<close>
```
```   933
```
```   934 lemma card_eq_sum: "card A = sum (\<lambda>x. 1) A"
```
```   935 proof -
```
```   936   have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)"
```
```   937     by (simp add: fun_eq_iff)
```
```   938   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) = Finite_Set.fold (\<lambda>_. Suc)"
```
```   939     by (rule arg_cong)
```
```   940   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
```
```   941     by (blast intro: fun_cong)
```
```   942   then show ?thesis
```
```   943     by (simp add: card.eq_fold sum.eq_fold)
```
```   944 qed
```
```   945
```
```   946 context semiring_1
```
```   947 begin
```
```   948
```
```   949 lemma sum_constant [simp]:
```
```   950   "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
```
```   951   by (induct A rule: infinite_finite_induct) (simp_all add: algebra_simps)
```
```   952
```
```   953 end
```
```   954
```
```   955 lemma sum_Suc: "sum (\<lambda>x. Suc(f x)) A = sum f A + card A"
```
```   956   using sum.distrib[of f "\<lambda>_. 1" A] by simp
```
```   957
```
```   958 lemma sum_bounded_above:
```
```   959   fixes K :: "'a::{semiring_1,ordered_comm_monoid_add}"
```
```   960   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> K"
```
```   961   shows "sum f A \<le> of_nat (card A) * K"
```
```   962 proof (cases "finite A")
```
```   963   case True
```
```   964   then show ?thesis
```
```   965     using le sum_mono[where K=A and g = "\<lambda>x. K"] by simp
```
```   966 next
```
```   967   case False
```
```   968   then show ?thesis by simp
```
```   969 qed
```
```   970
```
```   971 lemma sum_bounded_above_strict:
```
```   972   fixes K :: "'a::{ordered_cancel_comm_monoid_add,semiring_1}"
```
```   973   assumes "\<And>i. i\<in>A \<Longrightarrow> f i < K" "card A > 0"
```
```   974   shows "sum f A < of_nat (card A) * K"
```
```   975   using assms sum_strict_mono[where A=A and g = "\<lambda>x. K"]
```
```   976   by (simp add: card_gt_0_iff)
```
```   977
```
```   978 lemma sum_bounded_below:
```
```   979   fixes K :: "'a::{semiring_1,ordered_comm_monoid_add}"
```
```   980   assumes le: "\<And>i. i\<in>A \<Longrightarrow> K \<le> f i"
```
```   981   shows "of_nat (card A) * K \<le> sum f A"
```
```   982 proof (cases "finite A")
```
```   983   case True
```
```   984   then show ?thesis
```
```   985     using le sum_mono[where K=A and f = "\<lambda>x. K"] by simp
```
```   986 next
```
```   987   case False
```
```   988   then show ?thesis by simp
```
```   989 qed
```
```   990
```
```   991 lemma card_UN_disjoint:
```
```   992   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
```
```   993     and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
```
```   994   shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
```
```   995 proof -
```
```   996   have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)"
```
```   997     by simp
```
```   998   with assms show ?thesis
```
```   999     by (simp add: card_eq_sum sum.UNION_disjoint del: sum_constant)
```
```  1000 qed
```
```  1001
```
```  1002 lemma card_Union_disjoint:
```
```  1003   "finite C \<Longrightarrow> \<forall>A\<in>C. finite A \<Longrightarrow> \<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {} \<Longrightarrow>
```
```  1004     card (\<Union>C) = sum card C"
```
```  1005   by (frule card_UN_disjoint [of C id]) simp_all
```
```  1006
```
```  1007 lemma sum_multicount_gen:
```
```  1008   assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
```
```  1009   shows "sum (\<lambda>i. (card {j\<in>t. R i j})) s = sum k t"
```
```  1010     (is "?l = ?r")
```
```  1011 proof-
```
```  1012   have "?l = sum (\<lambda>i. sum (\<lambda>x.1) {j\<in>t. R i j}) s"
```
```  1013     by auto
```
```  1014   also have "\<dots> = ?r"
```
```  1015     unfolding sum.swap_restrict [OF assms(1-2)]
```
```  1016     using assms(3) by auto
```
```  1017   finally show ?thesis .
```
```  1018 qed
```
```  1019
```
```  1020 lemma sum_multicount:
```
```  1021   assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
```
```  1022   shows "sum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
```
```  1023 proof-
```
```  1024   have "?l = sum (\<lambda>i. k) T"
```
```  1025     by (rule sum_multicount_gen) (auto simp: assms)
```
```  1026   also have "\<dots> = ?r" by (simp add: mult.commute)
```
```  1027   finally show ?thesis by auto
```
```  1028 qed
```
```  1029
```
```  1030
```
```  1031 subsubsection \<open>Cardinality of products\<close>
```
```  1032
```
```  1033 lemma card_SigmaI [simp]:
```
```  1034   "finite A \<Longrightarrow> \<forall>a\<in>A. finite (B a) \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
```
```  1035   by (simp add: card_eq_sum sum.Sigma del: sum_constant)
```
```  1036
```
```  1037 (*
```
```  1038 lemma SigmaI_insert: "y \<notin> A ==>
```
```  1039   (SIGMA x:(insert y A). B x) = (({y} \<times> (B y)) \<union> (SIGMA x: A. B x))"
```
```  1040   by auto
```
```  1041 *)
```
```  1042
```
```  1043 lemma card_cartesian_product: "card (A \<times> B) = card A * card B"
```
```  1044   by (cases "finite A \<and> finite B")
```
```  1045     (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1046
```
```  1047 lemma card_cartesian_product_singleton:  "card ({x} \<times> A) = card A"
```
```  1048   by (simp add: card_cartesian_product)
```
```  1049
```
```  1050
```
```  1051 subsection \<open>Generalized product over a set\<close>
```
```  1052
```
```  1053 context comm_monoid_mult
```
```  1054 begin
```
```  1055
```
```  1056 sublocale prod: comm_monoid_set times 1
```
```  1057   defines prod = prod.F ..
```
```  1058
```
```  1059 abbreviation Prod ("\<Prod>_"  999)
```
```  1060   where "\<Prod>A \<equiv> prod (\<lambda>x. x) A"
```
```  1061
```
```  1062 end
```
```  1063
```
```  1064 syntax (ASCII)
```
```  1065   "_prod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(4PROD _:_./ _)" [0, 51, 10] 10)
```
```  1066 syntax
```
```  1067   "_prod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(2\<Prod>_\<in>_./ _)" [0, 51, 10] 10)
```
```  1068 translations \<comment> \<open>Beware of argument permutation!\<close>
```
```  1069   "\<Prod>i\<in>A. b" == "CONST prod (\<lambda>i. b) A"
```
```  1070
```
```  1071 text \<open>Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter \<open>\<Prod>x|P. e\<close>.\<close>
```
```  1072
```
```  1073 syntax (ASCII)
```
```  1074   "_qprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(4PROD _ |/ _./ _)" [0, 0, 10] 10)
```
```  1075 syntax
```
```  1076   "_qprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(2\<Prod>_ | (_)./ _)" [0, 0, 10] 10)
```
```  1077 translations
```
```  1078   "\<Prod>x|P. t" => "CONST prod (\<lambda>x. t) {x. P}"
```
```  1079
```
```  1080 context comm_monoid_mult
```
```  1081 begin
```
```  1082
```
```  1083 lemma prod_dvd_prod: "(\<And>a. a \<in> A \<Longrightarrow> f a dvd g a) \<Longrightarrow> prod f A dvd prod g A"
```
```  1084 proof (induct A rule: infinite_finite_induct)
```
```  1085   case infinite
```
```  1086   then show ?case by (auto intro: dvdI)
```
```  1087 next
```
```  1088   case empty
```
```  1089   then show ?case by (auto intro: dvdI)
```
```  1090 next
```
```  1091   case (insert a A)
```
```  1092   then have "f a dvd g a" and "prod f A dvd prod g A"
```
```  1093     by simp_all
```
```  1094   then obtain r s where "g a = f a * r" and "prod g A = prod f A * s"
```
```  1095     by (auto elim!: dvdE)
```
```  1096   then have "g a * prod g A = f a * prod f A * (r * s)"
```
```  1097     by (simp add: ac_simps)
```
```  1098   with insert.hyps show ?case
```
```  1099     by (auto intro: dvdI)
```
```  1100 qed
```
```  1101
```
```  1102 lemma prod_dvd_prod_subset: "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> prod f A dvd prod f B"
```
```  1103   by (auto simp add: prod.subset_diff ac_simps intro: dvdI)
```
```  1104
```
```  1105 end
```
```  1106
```
```  1107
```
```  1108 subsubsection \<open>Properties in more restricted classes of structures\<close>
```
```  1109
```
```  1110 context linordered_nonzero_semiring
```
```  1111 begin
```
```  1112
```
```  1113 lemma prod_ge_1: "(\<And>x. x \<in> A \<Longrightarrow> 1 \<le> f x) \<Longrightarrow> 1 \<le> prod f A"
```
```  1114 proof (induct A rule: infinite_finite_induct)
```
```  1115   case infinite
```
```  1116   then show ?case by simp
```
```  1117 next
```
```  1118   case empty
```
```  1119   then show ?case by simp
```
```  1120 next
```
```  1121   case (insert x F)
```
```  1122   have "1 * 1 \<le> f x * prod f F"
```
```  1123     by (rule mult_mono') (use insert in auto)
```
```  1124   with insert show ?case by simp
```
```  1125 qed
```
```  1126
```
```  1127 lemma prod_le_1:
```
```  1128   fixes f :: "'b \<Rightarrow> 'a"
```
```  1129   assumes "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x \<and> f x \<le> 1"
```
```  1130   shows "prod f A \<le> 1"
```
```  1131     using assms
```
```  1132 proof (induct A rule: infinite_finite_induct)
```
```  1133   case infinite
```
```  1134   then show ?case by simp
```
```  1135 next
```
```  1136   case empty
```
```  1137   then show ?case by simp
```
```  1138 next
```
```  1139   case (insert x F)
```
```  1140   then show ?case by (force simp: mult.commute intro: dest: mult_le_one)
```
```  1141 qed
```
```  1142
```
```  1143 end
```
```  1144
```
```  1145 context comm_semiring_1
```
```  1146 begin
```
```  1147
```
```  1148 lemma dvd_prod_eqI [intro]:
```
```  1149   assumes "finite A" and "a \<in> A" and "b = f a"
```
```  1150   shows "b dvd prod f A"
```
```  1151 proof -
```
```  1152   from \<open>finite A\<close> have "prod f (insert a (A - {a})) = f a * prod f (A - {a})"
```
```  1153     by (intro prod.insert) auto
```
```  1154   also from \<open>a \<in> A\<close> have "insert a (A - {a}) = A"
```
```  1155     by blast
```
```  1156   finally have "prod f A = f a * prod f (A - {a})" .
```
```  1157   with \<open>b = f a\<close> show ?thesis
```
```  1158     by simp
```
```  1159 qed
```
```  1160
```
```  1161 lemma dvd_prodI [intro]: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> f a dvd prod f A"
```
```  1162   by auto
```
```  1163
```
```  1164 lemma prod_zero:
```
```  1165   assumes "finite A" and "\<exists>a\<in>A. f a = 0"
```
```  1166   shows "prod f A = 0"
```
```  1167   using assms
```
```  1168 proof (induct A)
```
```  1169   case empty
```
```  1170   then show ?case by simp
```
```  1171 next
```
```  1172   case (insert a A)
```
```  1173   then have "f a = 0 \<or> (\<exists>a\<in>A. f a = 0)" by simp
```
```  1174   then have "f a * prod f A = 0" by rule (simp_all add: insert)
```
```  1175   with insert show ?case by simp
```
```  1176 qed
```
```  1177
```
```  1178 lemma prod_dvd_prod_subset2:
```
```  1179   assumes "finite B" and "A \<subseteq> B" and "\<And>a. a \<in> A \<Longrightarrow> f a dvd g a"
```
```  1180   shows "prod f A dvd prod g B"
```
```  1181 proof -
```
```  1182   from assms have "prod f A dvd prod g A"
```
```  1183     by (auto intro: prod_dvd_prod)
```
```  1184   moreover from assms have "prod g A dvd prod g B"
```
```  1185     by (auto intro: prod_dvd_prod_subset)
```
```  1186   ultimately show ?thesis by (rule dvd_trans)
```
```  1187 qed
```
```  1188
```
```  1189 end
```
```  1190
```
```  1191 lemma (in semidom) prod_zero_iff [simp]:
```
```  1192   fixes f :: "'b \<Rightarrow> 'a"
```
```  1193   assumes "finite A"
```
```  1194   shows "prod f A = 0 \<longleftrightarrow> (\<exists>a\<in>A. f a = 0)"
```
```  1195   using assms by (induct A) (auto simp: no_zero_divisors)
```
```  1196
```
```  1197 lemma (in semidom_divide) prod_diff1:
```
```  1198   assumes "finite A" and "f a \<noteq> 0"
```
```  1199   shows "prod f (A - {a}) = (if a \<in> A then prod f A div f a else prod f A)"
```
```  1200 proof (cases "a \<notin> A")
```
```  1201   case True
```
```  1202   then show ?thesis by simp
```
```  1203 next
```
```  1204   case False
```
```  1205   with assms show ?thesis
```
```  1206   proof induct
```
```  1207     case empty
```
```  1208     then show ?case by simp
```
```  1209   next
```
```  1210     case (insert b B)
```
```  1211     then show ?case
```
```  1212     proof (cases "a = b")
```
```  1213       case True
```
```  1214       with insert show ?thesis by simp
```
```  1215     next
```
```  1216       case False
```
```  1217       with insert have "a \<in> B" by simp
```
```  1218       define C where "C = B - {a}"
```
```  1219       with \<open>finite B\<close> \<open>a \<in> B\<close> have "B = insert a C" "finite C" "a \<notin> C"
```
```  1220         by auto
```
```  1221       with insert show ?thesis
```
```  1222         by (auto simp add: insert_commute ac_simps)
```
```  1223     qed
```
```  1224   qed
```
```  1225 qed
```
```  1226
```
```  1227 lemma sum_zero_power [simp]: "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
```
```  1228   for c :: "nat \<Rightarrow> 'a::division_ring"
```
```  1229   by (induct A rule: infinite_finite_induct) auto
```
```  1230
```
```  1231 lemma sum_zero_power' [simp]:
```
```  1232   "(\<Sum>i\<in>A. c i * 0^i / d i) = (if finite A \<and> 0 \<in> A then c 0 / d 0 else 0)"
```
```  1233   for c :: "nat \<Rightarrow> 'a::field"
```
```  1234   using sum_zero_power [of "\<lambda>i. c i / d i" A] by auto
```
```  1235
```
```  1236 lemma (in field) prod_inversef: "prod (inverse \<circ> f) A = inverse (prod f A)"
```
```  1237  proof (cases "finite A")
```
```  1238    case True
```
```  1239    then show ?thesis
```
```  1240      by (induct A rule: finite_induct) simp_all
```
```  1241  next
```
```  1242    case False
```
```  1243    then show ?thesis
```
```  1244      by auto
```
```  1245  qed
```
```  1246
```
```  1247 lemma (in field) prod_dividef: "(\<Prod>x\<in>A. f x / g x) = prod f A / prod g A"
```
```  1248   using prod_inversef [of g A] by (simp add: divide_inverse prod.distrib)
```
```  1249
```
```  1250 lemma prod_Un:
```
```  1251   fixes f :: "'b \<Rightarrow> 'a :: field"
```
```  1252   assumes "finite A" and "finite B"
```
```  1253     and "\<forall>x\<in>A \<inter> B. f x \<noteq> 0"
```
```  1254   shows "prod f (A \<union> B) = prod f A * prod f B / prod f (A \<inter> B)"
```
```  1255 proof -
```
```  1256   from assms have "prod f A * prod f B = prod f (A \<union> B) * prod f (A \<inter> B)"
```
```  1257     by (simp add: prod.union_inter [symmetric, of A B])
```
```  1258   with assms show ?thesis
```
```  1259     by simp
```
```  1260 qed
```
```  1261
```
```  1262 lemma (in linordered_semidom) prod_nonneg: "(\<forall>a\<in>A. 0 \<le> f a) \<Longrightarrow> 0 \<le> prod f A"
```
```  1263   by (induct A rule: infinite_finite_induct) simp_all
```
```  1264
```
```  1265 lemma (in linordered_semidom) prod_pos: "(\<forall>a\<in>A. 0 < f a) \<Longrightarrow> 0 < prod f A"
```
```  1266   by (induct A rule: infinite_finite_induct) simp_all
```
```  1267
```
```  1268 lemma (in linordered_semidom) prod_mono:
```
```  1269   "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i \<Longrightarrow> prod f A \<le> prod g A"
```
```  1270   by (induct A rule: infinite_finite_induct) (auto intro!: prod_nonneg mult_mono)
```
```  1271
```
```  1272 lemma (in linordered_semidom) prod_mono_strict:
```
```  1273   assumes "finite A" "\<forall>i\<in>A. 0 \<le> f i \<and> f i < g i" "A \<noteq> {}"
```
```  1274   shows "prod f A < prod g A"
```
```  1275   using assms
```
```  1276 proof (induct A rule: finite_induct)
```
```  1277   case empty
```
```  1278   then show ?case by simp
```
```  1279 next
```
```  1280   case insert
```
```  1281   then show ?case by (force intro: mult_strict_mono' prod_nonneg)
```
```  1282 qed
```
```  1283
```
```  1284 lemma (in linordered_field) abs_prod: "\<bar>prod f A\<bar> = (\<Prod>x\<in>A. \<bar>f x\<bar>)"
```
```  1285   by (induct A rule: infinite_finite_induct) (simp_all add: abs_mult)
```
```  1286
```
```  1287 lemma prod_eq_1_iff [simp]: "finite A \<Longrightarrow> prod f A = 1 \<longleftrightarrow> (\<forall>a\<in>A. f a = 1)"
```
```  1288   for f :: "'a \<Rightarrow> nat"
```
```  1289   by (induct A rule: finite_induct) simp_all
```
```  1290
```
```  1291 lemma prod_pos_nat_iff [simp]: "finite A \<Longrightarrow> prod f A > 0 \<longleftrightarrow> (\<forall>a\<in>A. f a > 0)"
```
```  1292   for f :: "'a \<Rightarrow> nat"
```
```  1293   using prod_zero_iff by (simp del: neq0_conv add: zero_less_iff_neq_zero)
```
```  1294
```
```  1295 lemma prod_constant: "(\<Prod>x\<in> A. y) = y ^ card A"
```
```  1296   for y :: "'a::comm_monoid_mult"
```
```  1297   by (induct A rule: infinite_finite_induct) simp_all
```
```  1298
```
```  1299 lemma prod_power_distrib: "prod f A ^ n = prod (\<lambda>x. (f x) ^ n) A"
```
```  1300   for f :: "'a \<Rightarrow> 'b::comm_semiring_1"
```
```  1301   by (induct A rule: infinite_finite_induct) (auto simp add: power_mult_distrib)
```
```  1302
```
```  1303 lemma power_sum: "c ^ (\<Sum>a\<in>A. f a) = (\<Prod>a\<in>A. c ^ f a)"
```
```  1304   by (induct A rule: infinite_finite_induct) (simp_all add: power_add)
```
```  1305
```
```  1306 lemma prod_gen_delta:
```
```  1307   fixes b :: "'b \<Rightarrow> 'a::comm_monoid_mult"
```
```  1308   assumes fin: "finite S"
```
```  1309   shows "prod (\<lambda>k. if k = a then b k else c) S =
```
```  1310     (if a \<in> S then b a * c ^ (card S - 1) else c ^ card S)"
```
```  1311 proof -
```
```  1312   let ?f = "(\<lambda>k. if k=a then b k else c)"
```
```  1313   show ?thesis
```
```  1314   proof (cases "a \<in> S")
```
```  1315     case False
```
```  1316     then have "\<forall> k\<in> S. ?f k = c" by simp
```
```  1317     with False show ?thesis by (simp add: prod_constant)
```
```  1318   next
```
```  1319     case True
```
```  1320     let ?A = "S - {a}"
```
```  1321     let ?B = "{a}"
```
```  1322     from True have eq: "S = ?A \<union> ?B" by blast
```
```  1323     have disjoint: "?A \<inter> ?B = {}" by simp
```
```  1324     from fin have fin': "finite ?A" "finite ?B" by auto
```
```  1325     have f_A0: "prod ?f ?A = prod (\<lambda>i. c) ?A"
```
```  1326       by (rule prod.cong) auto
```
```  1327     from fin True have card_A: "card ?A = card S - 1" by auto
```
```  1328     have f_A1: "prod ?f ?A = c ^ card ?A"
```
```  1329       unfolding f_A0 by (rule prod_constant)
```
```  1330     have "prod ?f ?A * prod ?f ?B = prod ?f S"
```
```  1331       using prod.union_disjoint[OF fin' disjoint, of ?f, unfolded eq[symmetric]]
```
```  1332       by simp
```
```  1333     with True card_A show ?thesis
```
```  1334       by (simp add: f_A1 field_simps cong add: prod.cong cong del: if_weak_cong)
```
```  1335   qed
```
```  1336 qed
```
```  1337
```
```  1338 lemma sum_image_le:
```
```  1339   fixes g :: "'a \<Rightarrow> 'b::ordered_ab_group_add"
```
```  1340   assumes "finite I" "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> g(f i)"
```
```  1341     shows "sum g (f ` I) \<le> sum (g \<circ> f) I"
```
```  1342   using assms
```
```  1343 proof induction
```
```  1344   case empty
```
```  1345   then show ?case by auto
```
```  1346 next
```
```  1347   case (insert x F) then
```
```  1348   have "sum g (f ` insert x F) = sum g (insert (f x) (f ` F))" by simp
```
```  1349   also have "\<dots> \<le> g (f x) + sum g (f ` F)"
```
```  1350     by (simp add: insert sum.insert_if)
```
```  1351   also have "\<dots>  \<le> sum (g \<circ> f) (insert x F)"
```
```  1352     using insert by auto
```
```  1353   finally show ?case .
```
```  1354 qed
```
```  1355
```
```  1356 end
```