src/HOL/Groups_Big.thy
author hoelzl
Wed Jun 18 07:31:12 2014 +0200 (2014-06-18)
changeset 57275 0ddb5b755cdc
parent 57129 7edb7550663e
child 57418 6ab1c7cb0b8d
permissions -rw-r--r--
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
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(*  Title:      HOL/Groups_Big.thy
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    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
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                with contributions by Jeremy Avigad
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*)
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header {* Big sum and product over finite (non-empty) sets *}
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theory Groups_Big
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imports Finite_Set
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begin
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subsection {* Generic monoid operation over a set *}
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no_notation times (infixl "*" 70)
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no_notation Groups.one ("1")
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locale comm_monoid_set = comm_monoid
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begin
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interpretation comp_fun_commute f
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  by default (simp add: fun_eq_iff left_commute)
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interpretation comp?: comp_fun_commute "f \<circ> g"
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  by (fact comp_comp_fun_commute)
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definition F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
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where
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  eq_fold: "F g A = Finite_Set.fold (f \<circ> g) 1 A"
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lemma infinite [simp]:
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  "\<not> finite A \<Longrightarrow> F g A = 1"
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  by (simp add: eq_fold)
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lemma empty [simp]:
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  "F g {} = 1"
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  by (simp add: eq_fold)
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lemma insert [simp]:
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  assumes "finite A" and "x \<notin> A"
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  shows "F g (insert x A) = g x * F g A"
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  using assms by (simp add: eq_fold)
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lemma remove:
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  assumes "finite A" and "x \<in> A"
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  shows "F g A = g x * F g (A - {x})"
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proof -
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  from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
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    by (auto dest: mk_disjoint_insert)
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  moreover from `finite A` A have "finite B" by simp
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  ultimately show ?thesis by simp
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qed
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lemma insert_remove:
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  assumes "finite A"
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  shows "F g (insert x A) = g x * F g (A - {x})"
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  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
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lemma neutral:
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  assumes "\<forall>x\<in>A. g x = 1"
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  shows "F g A = 1"
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  using assms by (induct A rule: infinite_finite_induct) simp_all
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lemma neutral_const [simp]:
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  "F (\<lambda>_. 1) A = 1"
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  by (simp add: neutral)
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lemma union_inter:
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  assumes "finite A" and "finite B"
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  shows "F g (A \<union> B) * F g (A \<inter> B) = F g A * F g B"
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  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
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using assms proof (induct A)
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  case empty then show ?case by simp
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next
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  case (insert x A) then show ?case
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    by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
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qed
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corollary union_inter_neutral:
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  assumes "finite A" and "finite B"
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  and I0: "\<forall>x \<in> A \<inter> B. g x = 1"
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  shows "F g (A \<union> B) = F g A * F g B"
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  using assms by (simp add: union_inter [symmetric] neutral)
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corollary union_disjoint:
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  assumes "finite A" and "finite B"
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  assumes "A \<inter> B = {}"
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  shows "F g (A \<union> B) = F g A * F g B"
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  using assms by (simp add: union_inter_neutral)
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lemma subset_diff:
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  assumes "B \<subseteq> A" and "finite A"
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  shows "F g A = F g (A - B) * F g B"
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proof -
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  from assms have "finite (A - B)" by auto
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  moreover from assms have "finite B" by (rule finite_subset)
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  moreover from assms have "(A - B) \<inter> B = {}" by auto
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  ultimately have "F g (A - B \<union> B) = F g (A - B) * F g B" by (rule union_disjoint)
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  moreover from assms have "A \<union> B = A" by auto
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  ultimately show ?thesis by simp
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qed
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lemma setdiff_irrelevant:
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  assumes "finite A"
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  shows "F g (A - {x. g x = z}) = F g A"
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  using assms by (induct A) (simp_all add: insert_Diff_if) 
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lemma not_neutral_contains_not_neutral:
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  assumes "F g A \<noteq> z"
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  obtains a where "a \<in> A" and "g a \<noteq> z"
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proof -
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  from assms have "\<exists>a\<in>A. g a \<noteq> z"
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  proof (induct A rule: infinite_finite_induct)
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    case (insert a A)
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    then show ?case by simp (rule, simp)
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  qed simp_all
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  with that show thesis by blast
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qed
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lemma reindex:
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  assumes "inj_on h A"
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  shows "F g (h ` A) = F (g \<circ> h) A"
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proof (cases "finite A")
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  case True
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  with assms show ?thesis by (simp add: eq_fold fold_image comp_assoc)
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next
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  case False with assms have "\<not> finite (h ` A)" by (blast dest: finite_imageD)
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  with False show ?thesis by simp
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qed
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lemma cong:
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  assumes "A = B"
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  assumes g_h: "\<And>x. x \<in> B \<Longrightarrow> g x = h x"
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  shows "F g A = F h B"
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  using g_h unfolding `A = B`
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  by (induct B rule: infinite_finite_induct) auto
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lemma strong_cong [cong]:
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  assumes "A = B" "\<And>x. x \<in> B =simp=> g x = h x"
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  shows "F (\<lambda>x. g x) A = F (\<lambda>x. h x) B"
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  by (rule cong) (insert assms, simp_all add: simp_implies_def)
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lemma UNION_disjoint:
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  assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
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  and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
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  shows "F g (UNION I A) = F (\<lambda>x. F g (A x)) I"
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apply (insert assms)
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apply (induct rule: finite_induct)
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apply simp
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apply atomize
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apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
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 prefer 2 apply blast
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apply (subgoal_tac "A x Int UNION Fa A = {}")
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 prefer 2 apply blast
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apply (simp add: union_disjoint)
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done
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lemma Union_disjoint:
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  assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
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  shows "F g (Union C) = F (F g) C"
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proof cases
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  assume "finite C"
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  from UNION_disjoint [OF this assms]
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  show ?thesis by simp
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qed (auto dest: finite_UnionD intro: infinite)
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lemma distrib:
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  "F (\<lambda>x. g x * h x) A = F g A * F h A"
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  using assms by (induct A rule: infinite_finite_induct) (simp_all add: assoc commute left_commute)
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lemma Sigma:
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  "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)"
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apply (subst Sigma_def)
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apply (subst UNION_disjoint, assumption, simp)
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 apply blast
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apply (rule cong)
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apply rule
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apply (simp add: fun_eq_iff)
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apply (subst UNION_disjoint, simp, simp)
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 apply blast
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apply (simp add: comp_def)
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done
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lemma related: 
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  assumes Re: "R 1 1" 
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  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
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  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
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  shows "R (F h S) (F g S)"
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  using fS by (rule finite_subset_induct) (insert assms, auto)
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lemma mono_neutral_cong_left:
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  assumes "finite T" and "S \<subseteq> T" and "\<forall>i \<in> T - S. h i = 1"
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  and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g S = F h T"
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proof-
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  have eq: "T = S \<union> (T - S)" using `S \<subseteq> T` by blast
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  have d: "S \<inter> (T - S) = {}" using `S \<subseteq> T` by blast
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  from `finite T` `S \<subseteq> T` have f: "finite S" "finite (T - S)"
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    by (auto intro: finite_subset)
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  show ?thesis using assms(4)
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    by (simp add: union_disjoint [OF f d, unfolded eq [symmetric]] neutral [OF assms(3)])
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qed
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lemma mono_neutral_cong_right:
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  "\<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>
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   \<Longrightarrow> F g T = F h S"
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  by (auto intro!: mono_neutral_cong_left [symmetric])
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lemma mono_neutral_left:
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  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g S = F g T"
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  by (blast intro: mono_neutral_cong_left)
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lemma mono_neutral_right:
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  "\<lbrakk> finite T;  S \<subseteq> T;  \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g T = F g S"
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  by (blast intro!: mono_neutral_left [symmetric])
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lemma reindex_bij_betw: "bij_betw h S T \<Longrightarrow> F (\<lambda>x. g (h x)) S = F g T"
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  by (auto simp: bij_betw_def reindex)
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lemma reindex_bij_witness:
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  assumes witness:
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    "\<And>a. a \<in> S \<Longrightarrow> i (j a) = a"
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    "\<And>a. a \<in> S \<Longrightarrow> j a \<in> T"
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    "\<And>b. b \<in> T \<Longrightarrow> j (i b) = b"
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    "\<And>b. b \<in> T \<Longrightarrow> i b \<in> S"
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  assumes eq:
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    "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a"
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  shows "F g S = F h T"
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proof -
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  have "bij_betw j S T"
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    using bij_betw_byWitness[where A=S and f=j and f'=i and A'=T] witness by auto
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  moreover have "F g S = F (\<lambda>x. h (j x)) S"
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    by (intro cong) (auto simp: eq)
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  ultimately show ?thesis
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    by (simp add: reindex_bij_betw)
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qed
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lemma reindex_bij_betw_not_neutral:
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  assumes fin: "finite S'" "finite T'"
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  assumes bij: "bij_betw h (S - S') (T - T')"
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  assumes nn:
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    "\<And>a. a \<in> S' \<Longrightarrow> g (h a) = z"
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    "\<And>b. b \<in> T' \<Longrightarrow> g b = z"
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  shows "F (\<lambda>x. g (h x)) S = F g T"
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proof -
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  have [simp]: "finite S \<longleftrightarrow> finite T"
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    using bij_betw_finite[OF bij] fin by auto
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  show ?thesis
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  proof cases
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    assume "finite S"
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    with nn have "F (\<lambda>x. g (h x)) S = F (\<lambda>x. g (h x)) (S - S')"
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      by (intro mono_neutral_cong_right) auto
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    also have "\<dots> = F g (T - T')"
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      using bij by (rule reindex_bij_betw)
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    also have "\<dots> = F g T"
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      using nn `finite S` by (intro mono_neutral_cong_left) auto
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    finally show ?thesis .
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  qed simp
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qed
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lemma reindex_bij_witness_not_neutral:
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  assumes fin: "finite S'" "finite T'"
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  assumes witness:
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    "\<And>a. a \<in> S - S' \<Longrightarrow> i (j a) = a"
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    "\<And>a. a \<in> S - S' \<Longrightarrow> j a \<in> T - T'"
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    "\<And>b. b \<in> T - T' \<Longrightarrow> j (i b) = b"
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    "\<And>b. b \<in> T - T' \<Longrightarrow> i b \<in> S - S'"
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  assumes nn:
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    "\<And>a. a \<in> S' \<Longrightarrow> g a = z"
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    "\<And>b. b \<in> T' \<Longrightarrow> h b = z"
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  assumes eq:
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    "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a"
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  shows "F g S = F h T"
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proof -
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  have bij: "bij_betw j (S - (S' \<inter> S)) (T - (T' \<inter> T))"
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    using witness by (intro bij_betw_byWitness[where f'=i]) auto
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  have F_eq: "F g S = F (\<lambda>x. h (j x)) S"
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    by (intro cong) (auto simp: eq)
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  show ?thesis
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    unfolding F_eq using fin nn eq
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    by (intro reindex_bij_betw_not_neutral[OF _ _ bij]) auto
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qed
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lemma delta: 
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  assumes fS: "finite S"
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  shows "F (\<lambda>k. if k = a then b k else 1) S = (if a \<in> S then b a else 1)"
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proof-
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  let ?f = "(\<lambda>k. if k=a then b k else 1)"
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  { assume a: "a \<notin> S"
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    hence "\<forall>k\<in>S. ?f k = 1" by simp
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    hence ?thesis  using a by simp }
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  moreover
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  { assume a: "a \<in> S"
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    let ?A = "S - {a}"
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    let ?B = "{a}"
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    have eq: "S = ?A \<union> ?B" using a by blast 
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    have dj: "?A \<inter> ?B = {}" by simp
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    from fS have fAB: "finite ?A" "finite ?B" by auto  
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    have "F ?f S = F ?f ?A * F ?f ?B"
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      using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]]
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      by simp
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    then have ?thesis using a by simp }
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  ultimately show ?thesis by blast
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   303
qed
haftmann@54744
   304
haftmann@54744
   305
lemma delta': 
haftmann@54744
   306
  assumes fS: "finite S"
haftmann@54744
   307
  shows "F (\<lambda>k. if a = k then b k else 1) S = (if a \<in> S then b a else 1)"
haftmann@54744
   308
  using delta [OF fS, of a b, symmetric] by (auto intro: cong)
haftmann@54744
   309
haftmann@54744
   310
lemma If_cases:
haftmann@54744
   311
  fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
haftmann@54744
   312
  assumes fA: "finite A"
haftmann@54744
   313
  shows "F (\<lambda>x. if P x then h x else g x) A =
haftmann@54744
   314
    F h (A \<inter> {x. P x}) * F g (A \<inter> - {x. P x})"
haftmann@54744
   315
proof -
haftmann@54744
   316
  have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}" 
haftmann@54744
   317
          "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}" 
haftmann@54744
   318
    by blast+
haftmann@54744
   319
  from fA 
haftmann@54744
   320
  have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
haftmann@54744
   321
  let ?g = "\<lambda>x. if P x then h x else g x"
haftmann@54744
   322
  from union_disjoint [OF f a(2), of ?g] a(1)
haftmann@54744
   323
  show ?thesis
haftmann@54744
   324
    by (subst (1 2) cong) simp_all
haftmann@54744
   325
qed
haftmann@54744
   326
haftmann@54744
   327
lemma cartesian_product:
haftmann@54744
   328
   "F (\<lambda>x. F (g x) B) A = F (split g) (A <*> B)"
haftmann@54744
   329
apply (rule sym)
haftmann@54744
   330
apply (cases "finite A") 
haftmann@54744
   331
 apply (cases "finite B") 
haftmann@54744
   332
  apply (simp add: Sigma)
haftmann@54744
   333
 apply (cases "A={}", simp)
haftmann@54744
   334
 apply simp
haftmann@54744
   335
apply (auto intro: infinite dest: finite_cartesian_productD2)
haftmann@54744
   336
apply (cases "B = {}") apply (auto intro: infinite dest: finite_cartesian_productD1)
haftmann@54744
   337
done
haftmann@54744
   338
haftmann@54744
   339
end
haftmann@54744
   340
haftmann@54744
   341
notation times (infixl "*" 70)
haftmann@54744
   342
notation Groups.one ("1")
haftmann@54744
   343
haftmann@54744
   344
haftmann@54744
   345
subsection {* Generalized summation over a set *}
haftmann@54744
   346
haftmann@54744
   347
context comm_monoid_add
haftmann@54744
   348
begin
haftmann@54744
   349
haftmann@54744
   350
definition setsum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
haftmann@54744
   351
where
haftmann@54744
   352
  "setsum = comm_monoid_set.F plus 0"
haftmann@54744
   353
haftmann@54744
   354
sublocale setsum!: comm_monoid_set plus 0
haftmann@54744
   355
where
haftmann@54744
   356
  "comm_monoid_set.F plus 0 = setsum"
haftmann@54744
   357
proof -
haftmann@54744
   358
  show "comm_monoid_set plus 0" ..
haftmann@54744
   359
  then interpret setsum!: comm_monoid_set plus 0 .
haftmann@54744
   360
  from setsum_def show "comm_monoid_set.F plus 0 = setsum" by rule
haftmann@54744
   361
qed
haftmann@54744
   362
haftmann@54744
   363
abbreviation
haftmann@54744
   364
  Setsum ("\<Sum>_" [1000] 999) where
haftmann@54744
   365
  "\<Sum>A \<equiv> setsum (%x. x) A"
haftmann@54744
   366
haftmann@54744
   367
end
haftmann@54744
   368
haftmann@54744
   369
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
haftmann@54744
   370
written @{text"\<Sum>x\<in>A. e"}. *}
haftmann@54744
   371
haftmann@54744
   372
syntax
haftmann@54744
   373
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
haftmann@54744
   374
syntax (xsymbols)
haftmann@54744
   375
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
haftmann@54744
   376
syntax (HTML output)
haftmann@54744
   377
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
haftmann@54744
   378
haftmann@54744
   379
translations -- {* Beware of argument permutation! *}
haftmann@54744
   380
  "SUM i:A. b" == "CONST setsum (%i. b) A"
haftmann@54744
   381
  "\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A"
haftmann@54744
   382
haftmann@54744
   383
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
haftmann@54744
   384
 @{text"\<Sum>x|P. e"}. *}
haftmann@54744
   385
haftmann@54744
   386
syntax
haftmann@54744
   387
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
haftmann@54744
   388
syntax (xsymbols)
haftmann@54744
   389
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
haftmann@54744
   390
syntax (HTML output)
haftmann@54744
   391
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
haftmann@54744
   392
haftmann@54744
   393
translations
haftmann@54744
   394
  "SUM x|P. t" => "CONST setsum (%x. t) {x. P}"
haftmann@54744
   395
  "\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}"
haftmann@54744
   396
haftmann@54744
   397
print_translation {*
haftmann@54744
   398
let
haftmann@54744
   399
  fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) $ Abs (y, Ty, P)] =
haftmann@54744
   400
        if x <> y then raise Match
haftmann@54744
   401
        else
haftmann@54744
   402
          let
haftmann@54744
   403
            val x' = Syntax_Trans.mark_bound_body (x, Tx);
haftmann@54744
   404
            val t' = subst_bound (x', t);
haftmann@54744
   405
            val P' = subst_bound (x', P);
haftmann@54744
   406
          in
haftmann@54744
   407
            Syntax.const @{syntax_const "_qsetsum"} $ Syntax_Trans.mark_bound_abs (x, Tx) $ P' $ t'
haftmann@54744
   408
          end
haftmann@54744
   409
    | setsum_tr' _ = raise Match;
haftmann@54744
   410
in [(@{const_syntax setsum}, K setsum_tr')] end
haftmann@54744
   411
*}
haftmann@54744
   412
haftmann@54744
   413
text {* TODO These are candidates for generalization *}
haftmann@54744
   414
haftmann@54744
   415
context comm_monoid_add
haftmann@54744
   416
begin
haftmann@54744
   417
haftmann@54744
   418
lemma setsum_reindex_id: 
hoelzl@57129
   419
  "inj_on f B \<Longrightarrow> setsum f B = setsum id (f ` B)"
haftmann@54744
   420
  by (simp add: setsum.reindex)
haftmann@54744
   421
haftmann@54744
   422
lemma setsum_reindex_nonzero:
haftmann@54744
   423
  assumes fS: "finite S"
haftmann@54744
   424
  and nz: "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x = f y \<Longrightarrow> h (f x) = 0"
haftmann@54744
   425
  shows "setsum h (f ` S) = setsum (h \<circ> f) S"
hoelzl@57129
   426
proof (subst setsum.reindex_bij_betw_not_neutral[symmetric])
hoelzl@57129
   427
  show "bij_betw f (S - {x\<in>S. h (f x) = 0}) (f`S - f`{x\<in>S. h (f x) = 0})"
hoelzl@57129
   428
    using nz by (auto intro!: inj_onI simp: bij_betw_def)
hoelzl@57129
   429
qed (insert fS, auto)
haftmann@54744
   430
haftmann@54744
   431
lemma setsum_cong2:
haftmann@54744
   432
  "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> setsum f A = setsum g A"
haftmann@54744
   433
  by (auto intro: setsum.cong)
haftmann@54744
   434
haftmann@54744
   435
lemma setsum_reindex_cong:
haftmann@54744
   436
   "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|] 
haftmann@54744
   437
    ==> setsum h B = setsum g A"
haftmann@54744
   438
  by (simp add: setsum.reindex)
haftmann@54744
   439
haftmann@54744
   440
lemma setsum_restrict_set:
haftmann@54744
   441
  assumes fA: "finite A"
haftmann@54744
   442
  shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"
haftmann@54744
   443
proof-
haftmann@54744
   444
  from fA have fab: "finite (A \<inter> B)" by auto
haftmann@54744
   445
  have aba: "A \<inter> B \<subseteq> A" by blast
haftmann@54744
   446
  let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"
haftmann@54744
   447
  from setsum.mono_neutral_left [OF fA aba, of ?g]
haftmann@54744
   448
  show ?thesis by simp
haftmann@54744
   449
qed
haftmann@54744
   450
haftmann@54744
   451
lemma setsum_Union_disjoint:
haftmann@54744
   452
  assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"
haftmann@54744
   453
  shows "setsum f (Union C) = setsum (setsum f) C"
haftmann@54744
   454
  using assms by (fact setsum.Union_disjoint)
haftmann@54744
   455
haftmann@54744
   456
lemma setsum_cartesian_product:
haftmann@54744
   457
  "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
haftmann@54744
   458
  by (fact setsum.cartesian_product)
haftmann@54744
   459
haftmann@54744
   460
lemma setsum_UNION_zero:
haftmann@54744
   461
  assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"
haftmann@54744
   462
  and f0: "\<And>T1 T2 x. T1\<in>S \<Longrightarrow> T2\<in>S \<Longrightarrow> T1 \<noteq> T2 \<Longrightarrow> x \<in> T1 \<Longrightarrow> x \<in> T2 \<Longrightarrow> f x = 0"
haftmann@54744
   463
  shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"
haftmann@54744
   464
  using fSS f0
haftmann@54744
   465
proof(induct rule: finite_induct[OF fS])
haftmann@54744
   466
  case 1 thus ?case by simp
haftmann@54744
   467
next
haftmann@54744
   468
  case (2 T F)
haftmann@54744
   469
  then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F" 
haftmann@54744
   470
    and H: "setsum f (\<Union> F) = setsum (setsum f) F" by auto
haftmann@54744
   471
  from fTF have fUF: "finite (\<Union>F)" by auto
haftmann@54744
   472
  from "2.prems" TF fTF
haftmann@54744
   473
  show ?case 
haftmann@54744
   474
    by (auto simp add: H [symmetric] intro: setsum.union_inter_neutral [OF fTF(1) fUF, of f])
haftmann@54744
   475
qed
haftmann@54744
   476
haftmann@54744
   477
text {* Commuting outer and inner summation *}
haftmann@54744
   478
haftmann@54744
   479
lemma setsum_commute:
haftmann@54744
   480
  "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
hoelzl@57129
   481
  unfolding setsum_cartesian_product
hoelzl@57129
   482
  by (rule setsum.reindex_bij_witness[where i="\<lambda>(i, j). (j, i)" and j="\<lambda>(i, j). (j, i)"]) auto
haftmann@54744
   483
haftmann@54744
   484
lemma setsum_Plus:
haftmann@54744
   485
  fixes A :: "'a set" and B :: "'b set"
haftmann@54744
   486
  assumes fin: "finite A" "finite B"
haftmann@54744
   487
  shows "setsum f (A <+> B) = setsum (f \<circ> Inl) A + setsum (f \<circ> Inr) B"
haftmann@54744
   488
proof -
haftmann@54744
   489
  have "A <+> B = Inl ` A \<union> Inr ` B" by auto
haftmann@54744
   490
  moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"
haftmann@54744
   491
    by auto
haftmann@54744
   492
  moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto
haftmann@54744
   493
  moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)
haftmann@54744
   494
  ultimately show ?thesis using fin by(simp add: setsum.union_disjoint setsum.reindex)
haftmann@54744
   495
qed
haftmann@54744
   496
haftmann@54744
   497
end
haftmann@54744
   498
haftmann@54744
   499
text {* TODO These are legacy *}
haftmann@54744
   500
haftmann@54744
   501
lemma setsum_empty:
haftmann@54744
   502
  "setsum f {} = 0"
haftmann@54744
   503
  by (fact setsum.empty)
haftmann@54744
   504
haftmann@54744
   505
lemma setsum_insert:
haftmann@54744
   506
  "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
haftmann@54744
   507
  by (fact setsum.insert)
haftmann@54744
   508
haftmann@54744
   509
lemma setsum_infinite:
haftmann@54744
   510
  "~ finite A ==> setsum f A = 0"
haftmann@54744
   511
  by (fact setsum.infinite)
haftmann@54744
   512
haftmann@54744
   513
lemma setsum_reindex:
haftmann@54744
   514
  "inj_on f B \<Longrightarrow> setsum h (f ` B) = setsum (h \<circ> f) B"
haftmann@54744
   515
  by (fact setsum.reindex)
haftmann@54744
   516
haftmann@54744
   517
lemma setsum_cong:
haftmann@54744
   518
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
haftmann@54744
   519
  by (fact setsum.cong)
haftmann@54744
   520
haftmann@54744
   521
lemma strong_setsum_cong:
haftmann@54744
   522
  "A = B ==> (!!x. x:B =simp=> f x = g x)
haftmann@54744
   523
   ==> setsum (%x. f x) A = setsum (%x. g x) B"
haftmann@54744
   524
  by (fact setsum.strong_cong)
haftmann@54744
   525
haftmann@54744
   526
lemmas setsum_0 = setsum.neutral_const
haftmann@54744
   527
lemmas setsum_0' = setsum.neutral
haftmann@54744
   528
haftmann@54744
   529
lemma setsum_Un_Int: "finite A ==> finite B ==>
haftmann@54744
   530
  setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
haftmann@54744
   531
  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
haftmann@54744
   532
  by (fact setsum.union_inter)
haftmann@54744
   533
haftmann@54744
   534
lemma setsum_Un_disjoint: "finite A ==> finite B
haftmann@54744
   535
  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
haftmann@54744
   536
  by (fact setsum.union_disjoint)
haftmann@54744
   537
haftmann@54744
   538
lemma setsum_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
haftmann@54744
   539
    setsum f A = setsum f (A - B) + setsum f B"
haftmann@54744
   540
  by (fact setsum.subset_diff)
haftmann@54744
   541
haftmann@54744
   542
lemma setsum_mono_zero_left: 
haftmann@54744
   543
  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 0 \<rbrakk> \<Longrightarrow> setsum f S = setsum f T"
haftmann@54744
   544
  by (fact setsum.mono_neutral_left)
haftmann@54744
   545
haftmann@54744
   546
lemmas setsum_mono_zero_right = setsum.mono_neutral_right
haftmann@54744
   547
haftmann@54744
   548
lemma setsum_mono_zero_cong_left: 
haftmann@54744
   549
  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 0; \<And>x. x \<in> S \<Longrightarrow> f x = g x \<rbrakk>
haftmann@54744
   550
  \<Longrightarrow> setsum f S = setsum g T"
haftmann@54744
   551
  by (fact setsum.mono_neutral_cong_left)
haftmann@54744
   552
haftmann@54744
   553
lemmas setsum_mono_zero_cong_right = setsum.mono_neutral_cong_right
haftmann@54744
   554
haftmann@54744
   555
lemma setsum_delta: "finite S \<Longrightarrow>
haftmann@54744
   556
  setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
haftmann@54744
   557
  by (fact setsum.delta)
haftmann@54744
   558
haftmann@54744
   559
lemma setsum_delta': "finite S \<Longrightarrow>
haftmann@54744
   560
  setsum (\<lambda>k. if a = k then b k else 0) S = (if a\<in> S then b a else 0)"
haftmann@54744
   561
  by (fact setsum.delta')
haftmann@54744
   562
haftmann@54744
   563
lemma setsum_cases:
haftmann@54744
   564
  assumes "finite A"
haftmann@54744
   565
  shows "setsum (\<lambda>x. if P x then f x else g x) A =
haftmann@54744
   566
         setsum f (A \<inter> {x. P x}) + setsum g (A \<inter> - {x. P x})"
haftmann@54744
   567
  using assms by (fact setsum.If_cases)
haftmann@54744
   568
haftmann@54744
   569
(*But we can't get rid of finite I. If infinite, although the rhs is 0, 
haftmann@54744
   570
  the lhs need not be, since UNION I A could still be finite.*)
haftmann@54744
   571
lemma setsum_UN_disjoint:
haftmann@54744
   572
  assumes "finite I" and "ALL i:I. finite (A i)"
haftmann@54744
   573
    and "ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}"
haftmann@54744
   574
  shows "setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
haftmann@54744
   575
  using assms by (fact setsum.UNION_disjoint)
haftmann@54744
   576
haftmann@54744
   577
(*But we can't get rid of finite A. If infinite, although the lhs is 0, 
haftmann@54744
   578
  the rhs need not be, since SIGMA A B could still be finite.*)
haftmann@54744
   579
lemma setsum_Sigma:
haftmann@54744
   580
  assumes "finite A" and  "ALL x:A. finite (B x)"
haftmann@54744
   581
  shows "(\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"
haftmann@54744
   582
  using assms by (fact setsum.Sigma)
haftmann@54744
   583
haftmann@54744
   584
lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
haftmann@54744
   585
  by (fact setsum.distrib)
haftmann@54744
   586
haftmann@54744
   587
lemma setsum_Un_zero:  
haftmann@54744
   588
  "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 0 \<rbrakk> \<Longrightarrow>
haftmann@54744
   589
  setsum f (S \<union> T) = setsum f S + setsum f T"
haftmann@54744
   590
  by (fact setsum.union_inter_neutral)
haftmann@54744
   591
haftmann@54744
   592
subsubsection {* Properties in more restricted classes of structures *}
haftmann@54744
   593
haftmann@54744
   594
lemma setsum_Un: "finite A ==> finite B ==>
haftmann@54744
   595
  (setsum f (A Un B) :: 'a :: ab_group_add) =
haftmann@54744
   596
   setsum f A + setsum f B - setsum f (A Int B)"
haftmann@54744
   597
by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
haftmann@54744
   598
haftmann@54744
   599
lemma setsum_Un2:
haftmann@54744
   600
  assumes "finite (A \<union> B)"
haftmann@54744
   601
  shows "setsum f (A \<union> B) = setsum f (A - B) + setsum f (B - A) + setsum f (A \<inter> B)"
haftmann@54744
   602
proof -
haftmann@54744
   603
  have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
haftmann@54744
   604
    by auto
haftmann@54744
   605
  with assms show ?thesis by simp (subst setsum_Un_disjoint, auto)+
haftmann@54744
   606
qed
haftmann@54744
   607
haftmann@54744
   608
lemma setsum_diff1: "finite A \<Longrightarrow>
haftmann@54744
   609
  (setsum f (A - {a}) :: ('a::ab_group_add)) =
haftmann@54744
   610
  (if a:A then setsum f A - f a else setsum f A)"
haftmann@54744
   611
by (erule finite_induct) (auto simp add: insert_Diff_if)
haftmann@54744
   612
haftmann@54744
   613
lemma setsum_diff:
haftmann@54744
   614
  assumes le: "finite A" "B \<subseteq> A"
haftmann@54744
   615
  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
haftmann@54744
   616
proof -
haftmann@54744
   617
  from le have finiteB: "finite B" using finite_subset by auto
haftmann@54744
   618
  show ?thesis using finiteB le
haftmann@54744
   619
  proof induct
haftmann@54744
   620
    case empty
haftmann@54744
   621
    thus ?case by auto
haftmann@54744
   622
  next
haftmann@54744
   623
    case (insert x F)
haftmann@54744
   624
    thus ?case using le finiteB 
haftmann@54744
   625
      by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
haftmann@54744
   626
  qed
haftmann@54744
   627
qed
haftmann@54744
   628
haftmann@54744
   629
lemma setsum_mono:
haftmann@54744
   630
  assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))"
haftmann@54744
   631
  shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
haftmann@54744
   632
proof (cases "finite K")
haftmann@54744
   633
  case True
haftmann@54744
   634
  thus ?thesis using le
haftmann@54744
   635
  proof induct
haftmann@54744
   636
    case empty
haftmann@54744
   637
    thus ?case by simp
haftmann@54744
   638
  next
haftmann@54744
   639
    case insert
haftmann@54744
   640
    thus ?case using add_mono by fastforce
haftmann@54744
   641
  qed
haftmann@54744
   642
next
haftmann@54744
   643
  case False then show ?thesis by simp
haftmann@54744
   644
qed
haftmann@54744
   645
haftmann@54744
   646
lemma setsum_strict_mono:
haftmann@54744
   647
  fixes f :: "'a \<Rightarrow> 'b::{ordered_cancel_ab_semigroup_add,comm_monoid_add}"
haftmann@54744
   648
  assumes "finite A"  "A \<noteq> {}"
haftmann@54744
   649
    and "!!x. x:A \<Longrightarrow> f x < g x"
haftmann@54744
   650
  shows "setsum f A < setsum g A"
haftmann@54744
   651
  using assms
haftmann@54744
   652
proof (induct rule: finite_ne_induct)
haftmann@54744
   653
  case singleton thus ?case by simp
haftmann@54744
   654
next
haftmann@54744
   655
  case insert thus ?case by (auto simp: add_strict_mono)
haftmann@54744
   656
qed
haftmann@54744
   657
haftmann@54744
   658
lemma setsum_strict_mono_ex1:
haftmann@54744
   659
fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_cancel_ab_semigroup_add}"
haftmann@54744
   660
assumes "finite A" and "ALL x:A. f x \<le> g x" and "EX a:A. f a < g a"
haftmann@54744
   661
shows "setsum f A < setsum g A"
haftmann@54744
   662
proof-
haftmann@54744
   663
  from assms(3) obtain a where a: "a:A" "f a < g a" by blast
haftmann@54744
   664
  have "setsum f A = setsum f ((A-{a}) \<union> {a})"
haftmann@54744
   665
    by(simp add:insert_absorb[OF `a:A`])
haftmann@54744
   666
  also have "\<dots> = setsum f (A-{a}) + setsum f {a}"
haftmann@54744
   667
    using `finite A` by(subst setsum_Un_disjoint) auto
haftmann@54744
   668
  also have "setsum f (A-{a}) \<le> setsum g (A-{a})"
haftmann@54744
   669
    by(rule setsum_mono)(simp add: assms(2))
haftmann@54744
   670
  also have "setsum f {a} < setsum g {a}" using a by simp
haftmann@54744
   671
  also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"
haftmann@54744
   672
    using `finite A` by(subst setsum_Un_disjoint[symmetric]) auto
haftmann@54744
   673
  also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF `a:A`])
haftmann@54744
   674
  finally show ?thesis by (auto simp add: add_right_mono add_strict_left_mono)
haftmann@54744
   675
qed
haftmann@54744
   676
haftmann@54744
   677
lemma setsum_negf:
haftmann@54744
   678
  "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
haftmann@54744
   679
proof (cases "finite A")
haftmann@54744
   680
  case True thus ?thesis by (induct set: finite) auto
haftmann@54744
   681
next
haftmann@54744
   682
  case False thus ?thesis by simp
haftmann@54744
   683
qed
haftmann@54744
   684
haftmann@54744
   685
lemma setsum_subtractf:
haftmann@54744
   686
  "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
haftmann@54744
   687
    setsum f A - setsum g A"
haftmann@54744
   688
  using setsum_addf [of f "- g" A] by (simp add: setsum_negf)
haftmann@54744
   689
haftmann@54744
   690
lemma setsum_nonneg:
haftmann@54744
   691
  assumes nn: "\<forall>x\<in>A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
haftmann@54744
   692
  shows "0 \<le> setsum f A"
haftmann@54744
   693
proof (cases "finite A")
haftmann@54744
   694
  case True thus ?thesis using nn
haftmann@54744
   695
  proof induct
haftmann@54744
   696
    case empty then show ?case by simp
haftmann@54744
   697
  next
haftmann@54744
   698
    case (insert x F)
haftmann@54744
   699
    then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
haftmann@54744
   700
    with insert show ?case by simp
haftmann@54744
   701
  qed
haftmann@54744
   702
next
haftmann@54744
   703
  case False thus ?thesis by simp
haftmann@54744
   704
qed
haftmann@54744
   705
haftmann@54744
   706
lemma setsum_nonpos:
haftmann@54744
   707
  assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})"
haftmann@54744
   708
  shows "setsum f A \<le> 0"
haftmann@54744
   709
proof (cases "finite A")
haftmann@54744
   710
  case True thus ?thesis using np
haftmann@54744
   711
  proof induct
haftmann@54744
   712
    case empty then show ?case by simp
haftmann@54744
   713
  next
haftmann@54744
   714
    case (insert x F)
haftmann@54744
   715
    then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
haftmann@54744
   716
    with insert show ?case by simp
haftmann@54744
   717
  qed
haftmann@54744
   718
next
haftmann@54744
   719
  case False thus ?thesis by simp
haftmann@54744
   720
qed
haftmann@54744
   721
haftmann@54744
   722
lemma setsum_nonneg_leq_bound:
haftmann@54744
   723
  fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
haftmann@54744
   724
  assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"
haftmann@54744
   725
  shows "f i \<le> B"
haftmann@54744
   726
proof -
haftmann@54744
   727
  have "0 \<le> (\<Sum> i \<in> s - {i}. f i)" and "0 \<le> f i"
haftmann@54744
   728
    using assms by (auto intro!: setsum_nonneg)
haftmann@54744
   729
  moreover
haftmann@54744
   730
  have "(\<Sum> i \<in> s - {i}. f i) + f i = B"
haftmann@54744
   731
    using assms by (simp add: setsum_diff1)
haftmann@54744
   732
  ultimately show ?thesis by auto
haftmann@54744
   733
qed
haftmann@54744
   734
haftmann@54744
   735
lemma setsum_nonneg_0:
haftmann@54744
   736
  fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
haftmann@54744
   737
  assumes "finite s" and pos: "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"
haftmann@54744
   738
  and "(\<Sum> i \<in> s. f i) = 0" and i: "i \<in> s"
haftmann@54744
   739
  shows "f i = 0"
haftmann@54744
   740
  using setsum_nonneg_leq_bound[OF assms] pos[OF i] by auto
haftmann@54744
   741
haftmann@54744
   742
lemma setsum_mono2:
haftmann@54744
   743
fixes f :: "'a \<Rightarrow> 'b :: ordered_comm_monoid_add"
haftmann@54744
   744
assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
haftmann@54744
   745
shows "setsum f A \<le> setsum f B"
haftmann@54744
   746
proof -
haftmann@54744
   747
  have "setsum f A \<le> setsum f A + setsum f (B-A)"
haftmann@54744
   748
    by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
haftmann@54744
   749
  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
haftmann@54744
   750
    by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
haftmann@54744
   751
  also have "A \<union> (B-A) = B" using sub by blast
haftmann@54744
   752
  finally show ?thesis .
haftmann@54744
   753
qed
haftmann@54744
   754
haftmann@54744
   755
lemma setsum_mono3: "finite B ==> A <= B ==> 
haftmann@54744
   756
    ALL x: B - A. 
haftmann@54744
   757
      0 <= ((f x)::'a::{comm_monoid_add,ordered_ab_semigroup_add}) ==>
haftmann@54744
   758
        setsum f A <= setsum f B"
haftmann@54744
   759
  apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
haftmann@54744
   760
  apply (erule ssubst)
haftmann@54744
   761
  apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
haftmann@54744
   762
  apply simp
haftmann@54744
   763
  apply (rule add_left_mono)
haftmann@54744
   764
  apply (erule setsum_nonneg)
haftmann@54744
   765
  apply (subst setsum_Un_disjoint [THEN sym])
haftmann@54744
   766
  apply (erule finite_subset, assumption)
haftmann@54744
   767
  apply (rule finite_subset)
haftmann@54744
   768
  prefer 2
haftmann@54744
   769
  apply assumption
haftmann@54744
   770
  apply (auto simp add: sup_absorb2)
haftmann@54744
   771
done
haftmann@54744
   772
haftmann@54744
   773
lemma setsum_right_distrib: 
haftmann@54744
   774
  fixes f :: "'a => ('b::semiring_0)"
haftmann@54744
   775
  shows "r * setsum f A = setsum (%n. r * f n) A"
haftmann@54744
   776
proof (cases "finite A")
haftmann@54744
   777
  case True
haftmann@54744
   778
  thus ?thesis
haftmann@54744
   779
  proof induct
haftmann@54744
   780
    case empty thus ?case by simp
haftmann@54744
   781
  next
haftmann@54744
   782
    case (insert x A) thus ?case by (simp add: distrib_left)
haftmann@54744
   783
  qed
haftmann@54744
   784
next
haftmann@54744
   785
  case False thus ?thesis by simp
haftmann@54744
   786
qed
haftmann@54744
   787
haftmann@54744
   788
lemma setsum_left_distrib:
haftmann@54744
   789
  "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
haftmann@54744
   790
proof (cases "finite A")
haftmann@54744
   791
  case True
haftmann@54744
   792
  then show ?thesis
haftmann@54744
   793
  proof induct
haftmann@54744
   794
    case empty thus ?case by simp
haftmann@54744
   795
  next
haftmann@54744
   796
    case (insert x A) thus ?case by (simp add: distrib_right)
haftmann@54744
   797
  qed
haftmann@54744
   798
next
haftmann@54744
   799
  case False thus ?thesis by simp
haftmann@54744
   800
qed
haftmann@54744
   801
haftmann@54744
   802
lemma setsum_divide_distrib:
haftmann@54744
   803
  "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
haftmann@54744
   804
proof (cases "finite A")
haftmann@54744
   805
  case True
haftmann@54744
   806
  then show ?thesis
haftmann@54744
   807
  proof induct
haftmann@54744
   808
    case empty thus ?case by simp
haftmann@54744
   809
  next
haftmann@54744
   810
    case (insert x A) thus ?case by (simp add: add_divide_distrib)
haftmann@54744
   811
  qed
haftmann@54744
   812
next
haftmann@54744
   813
  case False thus ?thesis by simp
haftmann@54744
   814
qed
haftmann@54744
   815
haftmann@54744
   816
lemma setsum_abs[iff]: 
haftmann@54744
   817
  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
haftmann@54744
   818
  shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
haftmann@54744
   819
proof (cases "finite A")
haftmann@54744
   820
  case True
haftmann@54744
   821
  thus ?thesis
haftmann@54744
   822
  proof induct
haftmann@54744
   823
    case empty thus ?case by simp
haftmann@54744
   824
  next
haftmann@54744
   825
    case (insert x A)
haftmann@54744
   826
    thus ?case by (auto intro: abs_triangle_ineq order_trans)
haftmann@54744
   827
  qed
haftmann@54744
   828
next
haftmann@54744
   829
  case False thus ?thesis by simp
haftmann@54744
   830
qed
haftmann@54744
   831
haftmann@54744
   832
lemma setsum_abs_ge_zero[iff]: 
haftmann@54744
   833
  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
haftmann@54744
   834
  shows "0 \<le> setsum (%i. abs(f i)) A"
haftmann@54744
   835
proof (cases "finite A")
haftmann@54744
   836
  case True
haftmann@54744
   837
  thus ?thesis
haftmann@54744
   838
  proof induct
haftmann@54744
   839
    case empty thus ?case by simp
haftmann@54744
   840
  next
haftmann@54744
   841
    case (insert x A) thus ?case by auto
haftmann@54744
   842
  qed
haftmann@54744
   843
next
haftmann@54744
   844
  case False thus ?thesis by simp
haftmann@54744
   845
qed
haftmann@54744
   846
haftmann@54744
   847
lemma abs_setsum_abs[simp]: 
haftmann@54744
   848
  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
haftmann@54744
   849
  shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
haftmann@54744
   850
proof (cases "finite A")
haftmann@54744
   851
  case True
haftmann@54744
   852
  thus ?thesis
haftmann@54744
   853
  proof induct
haftmann@54744
   854
    case empty thus ?case by simp
haftmann@54744
   855
  next
haftmann@54744
   856
    case (insert a A)
haftmann@54744
   857
    hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
haftmann@54744
   858
    also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
haftmann@54744
   859
    also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
haftmann@54744
   860
      by (simp del: abs_of_nonneg)
haftmann@54744
   861
    also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
haftmann@54744
   862
    finally show ?case .
haftmann@54744
   863
  qed
haftmann@54744
   864
next
haftmann@54744
   865
  case False thus ?thesis by simp
haftmann@54744
   866
qed
haftmann@54744
   867
haftmann@54744
   868
lemma setsum_diff1'[rule_format]:
haftmann@54744
   869
  "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
haftmann@54744
   870
apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"])
haftmann@54744
   871
apply (auto simp add: insert_Diff_if add_ac)
haftmann@54744
   872
done
haftmann@54744
   873
haftmann@54744
   874
lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
haftmann@54744
   875
  shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
haftmann@54744
   876
unfolding setsum_diff1'[OF assms] by auto
haftmann@54744
   877
haftmann@54744
   878
lemma setsum_product:
haftmann@54744
   879
  fixes f :: "'a => ('b::semiring_0)"
haftmann@54744
   880
  shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
haftmann@54744
   881
  by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)
haftmann@54744
   882
haftmann@54744
   883
lemma setsum_mult_setsum_if_inj:
haftmann@54744
   884
fixes f :: "'a => ('b::semiring_0)"
haftmann@54744
   885
shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
haftmann@54744
   886
  setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
haftmann@54744
   887
by(auto simp: setsum_product setsum_cartesian_product
haftmann@54744
   888
        intro!:  setsum_reindex_cong[symmetric])
haftmann@54744
   889
haftmann@54744
   890
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
haftmann@54744
   891
apply (case_tac "finite A")
haftmann@54744
   892
 prefer 2 apply simp
haftmann@54744
   893
apply (erule rev_mp)
haftmann@54744
   894
apply (erule finite_induct, auto)
haftmann@54744
   895
done
haftmann@54744
   896
haftmann@54744
   897
lemma setsum_eq_0_iff [simp]:
haftmann@54744
   898
  "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
haftmann@54744
   899
  by (induct set: finite) auto
haftmann@54744
   900
haftmann@54744
   901
lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
haftmann@54744
   902
  setsum f A = Suc 0 \<longleftrightarrow> (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
haftmann@54744
   903
apply(erule finite_induct)
haftmann@54744
   904
apply (auto simp add:add_is_1)
haftmann@54744
   905
done
haftmann@54744
   906
haftmann@54744
   907
lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
haftmann@54744
   908
haftmann@54744
   909
lemma setsum_Un_nat: "finite A ==> finite B ==>
haftmann@54744
   910
  (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
haftmann@54744
   911
  -- {* For the natural numbers, we have subtraction. *}
haftmann@54744
   912
by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
haftmann@54744
   913
haftmann@54744
   914
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
haftmann@54744
   915
  (if a:A then setsum f A - f a else setsum f A)"
haftmann@54744
   916
apply (case_tac "finite A")
haftmann@54744
   917
 prefer 2 apply simp
haftmann@54744
   918
apply (erule finite_induct)
haftmann@54744
   919
 apply (auto simp add: insert_Diff_if)
haftmann@54744
   920
apply (drule_tac a = a in mk_disjoint_insert, auto)
haftmann@54744
   921
done
haftmann@54744
   922
haftmann@54744
   923
lemma setsum_diff_nat: 
haftmann@54744
   924
assumes "finite B" and "B \<subseteq> A"
haftmann@54744
   925
shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
haftmann@54744
   926
using assms
haftmann@54744
   927
proof induct
haftmann@54744
   928
  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
haftmann@54744
   929
next
haftmann@54744
   930
  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
haftmann@54744
   931
    and xFinA: "insert x F \<subseteq> A"
haftmann@54744
   932
    and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
haftmann@54744
   933
  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
haftmann@54744
   934
  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
haftmann@54744
   935
    by (simp add: setsum_diff1_nat)
haftmann@54744
   936
  from xFinA have "F \<subseteq> A" by simp
haftmann@54744
   937
  with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
haftmann@54744
   938
  with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
haftmann@54744
   939
    by simp
haftmann@54744
   940
  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
haftmann@54744
   941
  with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
haftmann@54744
   942
    by simp
haftmann@54744
   943
  from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
haftmann@54744
   944
  with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
haftmann@54744
   945
    by simp
haftmann@54744
   946
  thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
haftmann@54744
   947
qed
haftmann@54744
   948
haftmann@54744
   949
lemma setsum_comp_morphism:
haftmann@54744
   950
  assumes "h 0 = 0" and "\<And>x y. h (x + y) = h x + h y"
haftmann@54744
   951
  shows "setsum (h \<circ> g) A = h (setsum g A)"
haftmann@54744
   952
proof (cases "finite A")
haftmann@54744
   953
  case False then show ?thesis by (simp add: assms)
haftmann@54744
   954
next
haftmann@54744
   955
  case True then show ?thesis by (induct A) (simp_all add: assms)
haftmann@54744
   956
qed
haftmann@54744
   957
haftmann@54744
   958
haftmann@54744
   959
subsubsection {* Cardinality as special case of @{const setsum} *}
haftmann@54744
   960
haftmann@54744
   961
lemma card_eq_setsum:
haftmann@54744
   962
  "card A = setsum (\<lambda>x. 1) A"
haftmann@54744
   963
proof -
haftmann@54744
   964
  have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)"
haftmann@54744
   965
    by (simp add: fun_eq_iff)
haftmann@54744
   966
  then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) = Finite_Set.fold (\<lambda>_. Suc)"
haftmann@54744
   967
    by (rule arg_cong)
haftmann@54744
   968
  then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
haftmann@54744
   969
    by (blast intro: fun_cong)
haftmann@54744
   970
  then show ?thesis by (simp add: card.eq_fold setsum.eq_fold)
haftmann@54744
   971
qed
haftmann@54744
   972
haftmann@54744
   973
lemma setsum_constant [simp]:
haftmann@54744
   974
  "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
haftmann@54744
   975
apply (cases "finite A")
haftmann@54744
   976
apply (erule finite_induct)
haftmann@54744
   977
apply (auto simp add: algebra_simps)
haftmann@54744
   978
done
haftmann@54744
   979
haftmann@54744
   980
lemma setsum_bounded:
haftmann@54744
   981
  assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"
haftmann@54744
   982
  shows "setsum f A \<le> of_nat (card A) * K"
haftmann@54744
   983
proof (cases "finite A")
haftmann@54744
   984
  case True
haftmann@54744
   985
  thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
haftmann@54744
   986
next
haftmann@54744
   987
  case False thus ?thesis by simp
haftmann@54744
   988
qed
haftmann@54744
   989
haftmann@54744
   990
lemma card_UN_disjoint:
haftmann@54744
   991
  assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
haftmann@54744
   992
    and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
haftmann@54744
   993
  shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
haftmann@54744
   994
proof -
haftmann@54744
   995
  have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)" by simp
haftmann@54744
   996
  with assms show ?thesis by (simp add: card_eq_setsum setsum_UN_disjoint del: setsum_constant)
haftmann@54744
   997
qed
haftmann@54744
   998
haftmann@54744
   999
lemma card_Union_disjoint:
haftmann@54744
  1000
  "finite C ==> (ALL A:C. finite A) ==>
haftmann@54744
  1001
   (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {})
haftmann@54744
  1002
   ==> card (Union C) = setsum card C"
haftmann@54744
  1003
apply (frule card_UN_disjoint [of C id])
haftmann@56166
  1004
apply simp_all
haftmann@54744
  1005
done
haftmann@54744
  1006
haftmann@54744
  1007
haftmann@54744
  1008
subsubsection {* Cardinality of products *}
haftmann@54744
  1009
haftmann@54744
  1010
lemma card_SigmaI [simp]:
haftmann@54744
  1011
  "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
haftmann@54744
  1012
  \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
haftmann@54744
  1013
by(simp add: card_eq_setsum setsum_Sigma del:setsum_constant)
haftmann@54744
  1014
haftmann@54744
  1015
(*
haftmann@54744
  1016
lemma SigmaI_insert: "y \<notin> A ==>
haftmann@54744
  1017
  (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
haftmann@54744
  1018
  by auto
haftmann@54744
  1019
*)
haftmann@54744
  1020
haftmann@54744
  1021
lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
haftmann@54744
  1022
  by (cases "finite A \<and> finite B")
haftmann@54744
  1023
    (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
haftmann@54744
  1024
haftmann@54744
  1025
lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
haftmann@54744
  1026
by (simp add: card_cartesian_product)
haftmann@54744
  1027
haftmann@54744
  1028
haftmann@54744
  1029
subsection {* Generalized product over a set *}
haftmann@54744
  1030
haftmann@54744
  1031
context comm_monoid_mult
haftmann@54744
  1032
begin
haftmann@54744
  1033
haftmann@54744
  1034
definition setprod :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
haftmann@54744
  1035
where
haftmann@54744
  1036
  "setprod = comm_monoid_set.F times 1"
haftmann@54744
  1037
haftmann@54744
  1038
sublocale setprod!: comm_monoid_set times 1
haftmann@54744
  1039
where
haftmann@54744
  1040
  "comm_monoid_set.F times 1 = setprod"
haftmann@54744
  1041
proof -
haftmann@54744
  1042
  show "comm_monoid_set times 1" ..
haftmann@54744
  1043
  then interpret setprod!: comm_monoid_set times 1 .
haftmann@54744
  1044
  from setprod_def show "comm_monoid_set.F times 1 = setprod" by rule
haftmann@54744
  1045
qed
haftmann@54744
  1046
haftmann@54744
  1047
abbreviation
haftmann@54744
  1048
  Setprod ("\<Prod>_" [1000] 999) where
haftmann@54744
  1049
  "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"
haftmann@54744
  1050
haftmann@54744
  1051
end
haftmann@54744
  1052
haftmann@54744
  1053
syntax
haftmann@54744
  1054
  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
haftmann@54744
  1055
syntax (xsymbols)
haftmann@54744
  1056
  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
haftmann@54744
  1057
syntax (HTML output)
haftmann@54744
  1058
  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
haftmann@54744
  1059
haftmann@54744
  1060
translations -- {* Beware of argument permutation! *}
haftmann@54744
  1061
  "PROD i:A. b" == "CONST setprod (%i. b) A" 
haftmann@54744
  1062
  "\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A" 
haftmann@54744
  1063
haftmann@54744
  1064
text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
haftmann@54744
  1065
 @{text"\<Prod>x|P. e"}. *}
haftmann@54744
  1066
haftmann@54744
  1067
syntax
haftmann@54744
  1068
  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
haftmann@54744
  1069
syntax (xsymbols)
haftmann@54744
  1070
  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
haftmann@54744
  1071
syntax (HTML output)
haftmann@54744
  1072
  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
haftmann@54744
  1073
haftmann@54744
  1074
translations
haftmann@54744
  1075
  "PROD x|P. t" => "CONST setprod (%x. t) {x. P}"
haftmann@54744
  1076
  "\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"
haftmann@54744
  1077
haftmann@54744
  1078
text {* TODO These are candidates for generalization *}
haftmann@54744
  1079
haftmann@54744
  1080
context comm_monoid_mult
haftmann@54744
  1081
begin
haftmann@54744
  1082
haftmann@54744
  1083
lemma setprod_reindex_id:
haftmann@54744
  1084
  "inj_on f B ==> setprod f B = setprod id (f ` B)"
haftmann@54744
  1085
  by (auto simp add: setprod.reindex)
haftmann@54744
  1086
haftmann@54744
  1087
lemma setprod_reindex_cong:
haftmann@54744
  1088
  "inj_on f A ==> B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
haftmann@54744
  1089
  by (frule setprod.reindex, simp)
haftmann@54744
  1090
haftmann@54744
  1091
lemma strong_setprod_reindex_cong:
hoelzl@57129
  1092
  "inj_on f A \<Longrightarrow> B = f ` A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x) \<Longrightarrow> setprod h B = setprod g A"
hoelzl@57129
  1093
  by (subst setprod.reindex_bij_betw[symmetric, where h=f])
hoelzl@57129
  1094
     (auto simp: bij_betw_def)
haftmann@54744
  1095
haftmann@54744
  1096
lemma setprod_Union_disjoint:
haftmann@54744
  1097
  assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}" 
haftmann@54744
  1098
  shows "setprod f (Union C) = setprod (setprod f) C"
haftmann@54744
  1099
  using assms by (fact setprod.Union_disjoint)
haftmann@54744
  1100
haftmann@54744
  1101
text{*Here we can eliminate the finiteness assumptions, by cases.*}
haftmann@54744
  1102
lemma setprod_cartesian_product:
haftmann@54744
  1103
  "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
haftmann@54744
  1104
  by (fact setprod.cartesian_product)
haftmann@54744
  1105
haftmann@54744
  1106
lemma setprod_Un2:
haftmann@54744
  1107
  assumes "finite (A \<union> B)"
haftmann@54744
  1108
  shows "setprod f (A \<union> B) = setprod f (A - B) * setprod f (B - A) * setprod f (A \<inter> B)"
haftmann@54744
  1109
proof -
haftmann@54744
  1110
  have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
haftmann@54744
  1111
    by auto
haftmann@54744
  1112
  with assms show ?thesis by simp (subst setprod.union_disjoint, auto)+
haftmann@54744
  1113
qed
haftmann@54744
  1114
haftmann@54744
  1115
end
haftmann@54744
  1116
haftmann@54744
  1117
text {* TODO These are legacy *}
haftmann@54744
  1118
haftmann@54744
  1119
lemma setprod_empty: "setprod f {} = 1"
haftmann@54744
  1120
  by (fact setprod.empty)
haftmann@54744
  1121
haftmann@54744
  1122
lemma setprod_insert: "[| finite A; a \<notin> A |] ==>
haftmann@54744
  1123
    setprod f (insert a A) = f a * setprod f A"
haftmann@54744
  1124
  by (fact setprod.insert)
haftmann@54744
  1125
haftmann@54744
  1126
lemma setprod_infinite: "~ finite A ==> setprod f A = 1"
haftmann@54744
  1127
  by (fact setprod.infinite)
haftmann@54744
  1128
haftmann@54744
  1129
lemma setprod_reindex:
haftmann@54744
  1130
  "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
haftmann@54744
  1131
  by (fact setprod.reindex)
haftmann@54744
  1132
haftmann@54744
  1133
lemma setprod_cong:
haftmann@54744
  1134
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
haftmann@54744
  1135
  by (fact setprod.cong)
haftmann@54744
  1136
haftmann@54744
  1137
lemma strong_setprod_cong:
haftmann@54744
  1138
  "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
haftmann@54744
  1139
  by (fact setprod.strong_cong)
haftmann@54744
  1140
haftmann@54744
  1141
lemma setprod_Un_one:
haftmann@54744
  1142
  "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 1 \<rbrakk>
haftmann@54744
  1143
  \<Longrightarrow> setprod f (S \<union> T) = setprod f S  * setprod f T"
haftmann@54744
  1144
  by (fact setprod.union_inter_neutral)
haftmann@54744
  1145
haftmann@54744
  1146
lemmas setprod_1 = setprod.neutral_const
haftmann@54744
  1147
lemmas setprod_1' = setprod.neutral
haftmann@54744
  1148
haftmann@54744
  1149
lemma setprod_Un_Int: "finite A ==> finite B
haftmann@54744
  1150
    ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
haftmann@54744
  1151
  by (fact setprod.union_inter)
haftmann@54744
  1152
haftmann@54744
  1153
lemma setprod_Un_disjoint: "finite A ==> finite B
haftmann@54744
  1154
  ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
haftmann@54744
  1155
  by (fact setprod.union_disjoint)
haftmann@54744
  1156
haftmann@54744
  1157
lemma setprod_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
haftmann@54744
  1158
    setprod f A = setprod f (A - B) * setprod f B"
haftmann@54744
  1159
  by (fact setprod.subset_diff)
haftmann@54744
  1160
haftmann@54744
  1161
lemma setprod_mono_one_left:
haftmann@54744
  1162
  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 1 \<rbrakk> \<Longrightarrow> setprod f S = setprod f T"
haftmann@54744
  1163
  by (fact setprod.mono_neutral_left)
haftmann@54744
  1164
haftmann@54744
  1165
lemmas setprod_mono_one_right = setprod.mono_neutral_right
haftmann@54744
  1166
haftmann@54744
  1167
lemma setprod_mono_one_cong_left: 
haftmann@54744
  1168
  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1; \<And>x. x \<in> S \<Longrightarrow> f x = g x \<rbrakk>
haftmann@54744
  1169
  \<Longrightarrow> setprod f S = setprod g T"
haftmann@54744
  1170
  by (fact setprod.mono_neutral_cong_left)
haftmann@54744
  1171
haftmann@54744
  1172
lemmas setprod_mono_one_cong_right = setprod.mono_neutral_cong_right
haftmann@54744
  1173
haftmann@54744
  1174
lemma setprod_delta: "finite S \<Longrightarrow>
haftmann@54744
  1175
  setprod (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"
haftmann@54744
  1176
  by (fact setprod.delta)
haftmann@54744
  1177
haftmann@54744
  1178
lemma setprod_delta': "finite S \<Longrightarrow>
haftmann@54744
  1179
  setprod (\<lambda>k. if a = k then b k else 1) S = (if a\<in> S then b a else 1)"
haftmann@54744
  1180
  by (fact setprod.delta')
haftmann@54744
  1181
haftmann@54744
  1182
lemma setprod_UN_disjoint:
haftmann@54744
  1183
    "finite I ==> (ALL i:I. finite (A i)) ==>
haftmann@54744
  1184
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
haftmann@54744
  1185
      setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
haftmann@54744
  1186
  by (fact setprod.UNION_disjoint)
haftmann@54744
  1187
haftmann@54744
  1188
lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
haftmann@54744
  1189
    (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
haftmann@54744
  1190
    (\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
haftmann@54744
  1191
  by (fact setprod.Sigma)
haftmann@54744
  1192
haftmann@54744
  1193
lemma setprod_timesf: "setprod (\<lambda>x. f x * g x) A = setprod f A * setprod g A"
haftmann@54744
  1194
  by (fact setprod.distrib)
haftmann@54744
  1195
haftmann@54744
  1196
haftmann@54744
  1197
subsubsection {* Properties in more restricted classes of structures *}
haftmann@54744
  1198
haftmann@54744
  1199
lemma setprod_zero:
haftmann@54744
  1200
     "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0"
haftmann@54744
  1201
apply (induct set: finite, force, clarsimp)
haftmann@54744
  1202
apply (erule disjE, auto)
haftmann@54744
  1203
done
haftmann@54744
  1204
haftmann@54744
  1205
lemma setprod_zero_iff[simp]: "finite A ==> 
haftmann@54744
  1206
  (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) =
haftmann@54744
  1207
  (EX x: A. f x = 0)"
haftmann@54744
  1208
by (erule finite_induct, auto simp:no_zero_divisors)
haftmann@54744
  1209
haftmann@54744
  1210
lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
haftmann@54744
  1211
  (setprod f (A Un B) :: 'a ::{field})
haftmann@54744
  1212
   = setprod f A * setprod f B / setprod f (A Int B)"
haftmann@54744
  1213
by (subst setprod_Un_Int [symmetric], auto)
haftmann@54744
  1214
haftmann@54744
  1215
lemma setprod_nonneg [rule_format]:
haftmann@54744
  1216
   "(ALL x: A. (0::'a::linordered_semidom) \<le> f x) --> 0 \<le> setprod f A"
nipkow@56536
  1217
by (cases "finite A", induct set: finite, simp_all)
haftmann@54744
  1218
haftmann@54744
  1219
lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::linordered_semidom) < f x)
haftmann@54744
  1220
  --> 0 < setprod f A"
nipkow@56544
  1221
by (cases "finite A", induct set: finite, simp_all)
haftmann@54744
  1222
haftmann@54744
  1223
lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
haftmann@54744
  1224
  (setprod f (A - {a}) :: 'a :: {field}) =
haftmann@54744
  1225
  (if a:A then setprod f A / f a else setprod f A)"
haftmann@54744
  1226
  by (erule finite_induct) (auto simp add: insert_Diff_if)
haftmann@54744
  1227
haftmann@54744
  1228
lemma setprod_inversef: 
haftmann@54744
  1229
  fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
haftmann@54744
  1230
  shows "finite A ==> setprod (inverse \<circ> f) A = inverse (setprod f A)"
haftmann@54744
  1231
by (erule finite_induct) auto
haftmann@54744
  1232
haftmann@54744
  1233
lemma setprod_dividef:
haftmann@54744
  1234
  fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
haftmann@54744
  1235
  shows "finite A
haftmann@54744
  1236
    ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
haftmann@54744
  1237
apply (subgoal_tac
haftmann@54744
  1238
         "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
haftmann@54744
  1239
apply (erule ssubst)
haftmann@54744
  1240
apply (subst divide_inverse)
haftmann@54744
  1241
apply (subst setprod_timesf)
haftmann@54744
  1242
apply (subst setprod_inversef, assumption+, rule refl)
haftmann@54744
  1243
apply (rule setprod_cong, rule refl)
haftmann@54744
  1244
apply (subst divide_inverse, auto)
haftmann@54744
  1245
done
haftmann@54744
  1246
haftmann@54744
  1247
lemma setprod_dvd_setprod [rule_format]: 
haftmann@54744
  1248
    "(ALL x : A. f x dvd g x) \<longrightarrow> setprod f A dvd setprod g A"
haftmann@54744
  1249
  apply (cases "finite A")
haftmann@54744
  1250
  apply (induct set: finite)
haftmann@54744
  1251
  apply (auto simp add: dvd_def)
haftmann@54744
  1252
  apply (rule_tac x = "k * ka" in exI)
haftmann@54744
  1253
  apply (simp add: algebra_simps)
haftmann@54744
  1254
done
haftmann@54744
  1255
haftmann@54744
  1256
lemma setprod_dvd_setprod_subset:
haftmann@54744
  1257
  "finite B \<Longrightarrow> A <= B \<Longrightarrow> setprod f A dvd setprod f B"
haftmann@54744
  1258
  apply (subgoal_tac "setprod f B = setprod f A * setprod f (B - A)")
haftmann@54744
  1259
  apply (unfold dvd_def, blast)
haftmann@54744
  1260
  apply (subst setprod_Un_disjoint [symmetric])
haftmann@54744
  1261
  apply (auto elim: finite_subset intro: setprod_cong)
haftmann@54744
  1262
done
haftmann@54744
  1263
haftmann@54744
  1264
lemma setprod_dvd_setprod_subset2:
haftmann@54744
  1265
  "finite B \<Longrightarrow> A <= B \<Longrightarrow> ALL x : A. (f x::'a::comm_semiring_1) dvd g x \<Longrightarrow> 
haftmann@54744
  1266
      setprod f A dvd setprod g B"
haftmann@54744
  1267
  apply (rule dvd_trans)
haftmann@54744
  1268
  apply (rule setprod_dvd_setprod, erule (1) bspec)
haftmann@54744
  1269
  apply (erule (1) setprod_dvd_setprod_subset)
haftmann@54744
  1270
done
haftmann@54744
  1271
haftmann@54744
  1272
lemma dvd_setprod: "finite A \<Longrightarrow> i:A \<Longrightarrow> 
haftmann@54744
  1273
    (f i ::'a::comm_semiring_1) dvd setprod f A"
haftmann@54744
  1274
by (induct set: finite) (auto intro: dvd_mult)
haftmann@54744
  1275
haftmann@54744
  1276
lemma dvd_setsum [rule_format]: "(ALL i : A. d dvd f i) \<longrightarrow> 
haftmann@54744
  1277
    (d::'a::comm_semiring_1) dvd (SUM x : A. f x)"
haftmann@54744
  1278
  apply (cases "finite A")
haftmann@54744
  1279
  apply (induct set: finite)
haftmann@54744
  1280
  apply auto
haftmann@54744
  1281
done
haftmann@54744
  1282
haftmann@54744
  1283
lemma setprod_mono:
haftmann@54744
  1284
  fixes f :: "'a \<Rightarrow> 'b\<Colon>linordered_semidom"
haftmann@54744
  1285
  assumes "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i"
haftmann@54744
  1286
  shows "setprod f A \<le> setprod g A"
haftmann@54744
  1287
proof (cases "finite A")
haftmann@54744
  1288
  case True
haftmann@54744
  1289
  hence ?thesis "setprod f A \<ge> 0" using subset_refl[of A]
haftmann@54744
  1290
  proof (induct A rule: finite_subset_induct)
haftmann@54744
  1291
    case (insert a F)
haftmann@54744
  1292
    thus "setprod f (insert a F) \<le> setprod g (insert a F)" "0 \<le> setprod f (insert a F)"
haftmann@54744
  1293
      unfolding setprod_insert[OF insert(1,3)]
haftmann@54744
  1294
      using assms[rule_format,OF insert(2)] insert
nipkow@56536
  1295
      by (auto intro: mult_mono)
haftmann@54744
  1296
  qed auto
haftmann@54744
  1297
  thus ?thesis by simp
haftmann@54744
  1298
qed auto
haftmann@54744
  1299
haftmann@54744
  1300
lemma abs_setprod:
haftmann@54744
  1301
  fixes f :: "'a \<Rightarrow> 'b\<Colon>{linordered_field,abs}"
haftmann@54744
  1302
  shows "abs (setprod f A) = setprod (\<lambda>x. abs (f x)) A"
haftmann@54744
  1303
proof (cases "finite A")
haftmann@54744
  1304
  case True thus ?thesis
haftmann@54744
  1305
    by induct (auto simp add: field_simps abs_mult)
haftmann@54744
  1306
qed auto
haftmann@54744
  1307
haftmann@54744
  1308
lemma setprod_eq_1_iff [simp]:
haftmann@54744
  1309
  "finite F ==> setprod f F = 1 \<longleftrightarrow> (ALL a:F. f a = (1::nat))"
haftmann@54744
  1310
  by (induct set: finite) auto
haftmann@54744
  1311
haftmann@54744
  1312
lemma setprod_pos_nat:
haftmann@54744
  1313
  "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0"
haftmann@54744
  1314
using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
haftmann@54744
  1315
haftmann@54744
  1316
lemma setprod_pos_nat_iff[simp]:
haftmann@54744
  1317
  "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))"
haftmann@54744
  1318
using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
haftmann@54744
  1319
hoelzl@57275
  1320
lemma (in ordered_comm_monoid_add) setsum_pos: 
hoelzl@57275
  1321
  "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> 0 < f i) \<Longrightarrow> 0 < setsum f I"
hoelzl@57275
  1322
  by (induct I rule: finite_ne_induct) (auto intro: add_pos_pos)
hoelzl@57275
  1323
haftmann@54744
  1324
end