src/HOL/Groups_Big.thy
author paulson <lp15@cam.ac.uk>
Mon Feb 22 14:37:56 2016 +0000 (2016-02-22)
changeset 62379 340738057c8c
parent 62378 85ed00c1fe7c
child 62481 b5d8e57826df
permissions -rw-r--r--
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
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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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section \<open>Big sum and product over finite (non-empty) sets\<close>
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theory Groups_Big
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imports Finite_Set
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begin
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subsection \<open>Generic monoid operation over a set\<close>
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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 standard (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 \<open>x \<in> A\<close> 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 \<open>finite A\<close> 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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  \<comment> \<open>The reversed orientation looks more natural, but LOOPS as a simprule!\<close>
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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 union_diff2:
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  assumes "finite A" and "finite B"
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  shows "F g (A \<union> B) = F g (A - B) * F g (B - A) * F g (A \<inter> B)"
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proof -
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  have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
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    by auto
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  with assms show ?thesis by simp (subst union_disjoint, auto)+
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qed
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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 \<open>A = B\<close>
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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 reindex_cong:
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  assumes "inj_on l B"
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  assumes "A = l ` B"
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  assumes "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x"
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  shows "F g A = F h B"
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  using assms by (simp add: reindex)
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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 \<circ> 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 (case_prod 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 \<open>S \<subseteq> T\<close> by blast
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  have d: "S \<inter> (T - S) = {}" using \<open>S \<subseteq> T\<close> by blast
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  from \<open>finite T\<close> \<open>S \<subseteq> T\<close> 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 \<open>finite S\<close> 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_nontrivial:
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  assumes "finite A"
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  and nz: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> h x = h y \<Longrightarrow> g (h x) = 1"
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  shows "F g (h ` A) = F (g \<circ> h) A"
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proof (subst reindex_bij_betw_not_neutral [symmetric])
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  show "bij_betw h (A - {x \<in> A. (g \<circ> h) x = 1}) (h ` A - h ` {x \<in> A. (g \<circ> h) x = 1})"
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    using nz by (auto intro!: inj_onI simp: bij_betw_def)
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qed (insert \<open>finite A\<close>, auto)
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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 -
hoelzl@57129
   299
  have bij: "bij_betw j (S - (S' \<inter> S)) (T - (T' \<inter> T))"
hoelzl@57129
   300
    using witness by (intro bij_betw_byWitness[where f'=i]) auto
hoelzl@57129
   301
  have F_eq: "F g S = F (\<lambda>x. h (j x)) S"
hoelzl@57129
   302
    by (intro cong) (auto simp: eq)
hoelzl@57129
   303
  show ?thesis
hoelzl@57129
   304
    unfolding F_eq using fin nn eq
hoelzl@57129
   305
    by (intro reindex_bij_betw_not_neutral[OF _ _ bij]) auto
hoelzl@57129
   306
qed
hoelzl@57129
   307
hoelzl@62376
   308
lemma delta:
haftmann@54744
   309
  assumes fS: "finite S"
haftmann@54744
   310
  shows "F (\<lambda>k. if k = a then b k else 1) S = (if a \<in> S then b a else 1)"
haftmann@54744
   311
proof-
haftmann@54744
   312
  let ?f = "(\<lambda>k. if k=a then b k else 1)"
haftmann@54744
   313
  { assume a: "a \<notin> S"
haftmann@54744
   314
    hence "\<forall>k\<in>S. ?f k = 1" by simp
haftmann@54744
   315
    hence ?thesis  using a by simp }
haftmann@54744
   316
  moreover
haftmann@54744
   317
  { assume a: "a \<in> S"
haftmann@54744
   318
    let ?A = "S - {a}"
haftmann@54744
   319
    let ?B = "{a}"
hoelzl@62376
   320
    have eq: "S = ?A \<union> ?B" using a by blast
haftmann@54744
   321
    have dj: "?A \<inter> ?B = {}" by simp
hoelzl@62376
   322
    from fS have fAB: "finite ?A" "finite ?B" by auto
haftmann@54744
   323
    have "F ?f S = F ?f ?A * F ?f ?B"
haftmann@54744
   324
      using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]]
haftmann@54744
   325
      by simp
haftmann@54744
   326
    then have ?thesis using a by simp }
haftmann@54744
   327
  ultimately show ?thesis by blast
haftmann@54744
   328
qed
haftmann@54744
   329
hoelzl@62376
   330
lemma delta':
haftmann@54744
   331
  assumes fS: "finite S"
haftmann@54744
   332
  shows "F (\<lambda>k. if a = k then b k else 1) S = (if a \<in> S then b a else 1)"
haftmann@54744
   333
  using delta [OF fS, of a b, symmetric] by (auto intro: cong)
haftmann@54744
   334
haftmann@54744
   335
lemma If_cases:
haftmann@54744
   336
  fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
haftmann@54744
   337
  assumes fA: "finite A"
haftmann@54744
   338
  shows "F (\<lambda>x. if P x then h x else g x) A =
haftmann@54744
   339
    F h (A \<inter> {x. P x}) * F g (A \<inter> - {x. P x})"
haftmann@54744
   340
proof -
hoelzl@62376
   341
  have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}"
hoelzl@62376
   342
          "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}"
haftmann@54744
   343
    by blast+
hoelzl@62376
   344
  from fA
haftmann@54744
   345
  have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
haftmann@54744
   346
  let ?g = "\<lambda>x. if P x then h x else g x"
haftmann@54744
   347
  from union_disjoint [OF f a(2), of ?g] a(1)
haftmann@54744
   348
  show ?thesis
haftmann@54744
   349
    by (subst (1 2) cong) simp_all
haftmann@54744
   350
qed
haftmann@54744
   351
haftmann@54744
   352
lemma cartesian_product:
wenzelm@61943
   353
   "F (\<lambda>x. F (g x) B) A = F (case_prod g) (A \<times> B)"
haftmann@54744
   354
apply (rule sym)
hoelzl@62376
   355
apply (cases "finite A")
hoelzl@62376
   356
 apply (cases "finite B")
haftmann@54744
   357
  apply (simp add: Sigma)
haftmann@54744
   358
 apply (cases "A={}", simp)
haftmann@54744
   359
 apply simp
haftmann@54744
   360
apply (auto intro: infinite dest: finite_cartesian_productD2)
haftmann@54744
   361
apply (cases "B = {}") apply (auto intro: infinite dest: finite_cartesian_productD1)
haftmann@54744
   362
done
haftmann@54744
   363
haftmann@57418
   364
lemma inter_restrict:
haftmann@57418
   365
  assumes "finite A"
haftmann@57418
   366
  shows "F g (A \<inter> B) = F (\<lambda>x. if x \<in> B then g x else 1) A"
haftmann@57418
   367
proof -
haftmann@57418
   368
  let ?g = "\<lambda>x. if x \<in> A \<inter> B then g x else 1"
haftmann@57418
   369
  have "\<forall>i\<in>A - A \<inter> B. (if i \<in> A \<inter> B then g i else 1) = 1"
haftmann@57418
   370
   by simp
haftmann@57418
   371
  moreover have "A \<inter> B \<subseteq> A" by blast
wenzelm@60758
   372
  ultimately have "F ?g (A \<inter> B) = F ?g A" using \<open>finite A\<close>
haftmann@57418
   373
    by (intro mono_neutral_left) auto
haftmann@57418
   374
  then show ?thesis by simp
haftmann@57418
   375
qed
haftmann@57418
   376
haftmann@57418
   377
lemma inter_filter:
haftmann@57418
   378
  "finite A \<Longrightarrow> F g {x \<in> A. P x} = F (\<lambda>x. if P x then g x else 1) A"
haftmann@57418
   379
  by (simp add: inter_restrict [symmetric, of A "{x. P x}" g, simplified mem_Collect_eq] Int_def)
haftmann@57418
   380
haftmann@57418
   381
lemma Union_comp:
haftmann@57418
   382
  assumes "\<forall>A \<in> B. finite A"
haftmann@57418
   383
    and "\<And>A1 A2 x. A1 \<in> B \<Longrightarrow> A2 \<in> B  \<Longrightarrow> A1 \<noteq> A2 \<Longrightarrow> x \<in> A1 \<Longrightarrow> x \<in> A2 \<Longrightarrow> g x = 1"
haftmann@57418
   384
  shows "F g (\<Union>B) = (F \<circ> F) g B"
haftmann@57418
   385
using assms proof (induct B rule: infinite_finite_induct)
haftmann@57418
   386
  case (infinite A)
haftmann@57418
   387
  then have "\<not> finite (\<Union>A)" by (blast dest: finite_UnionD)
haftmann@57418
   388
  with infinite show ?case by simp
haftmann@57418
   389
next
haftmann@57418
   390
  case empty then show ?case by simp
haftmann@57418
   391
next
haftmann@57418
   392
  case (insert A B)
haftmann@57418
   393
  then have "finite A" "finite B" "finite (\<Union>B)" "A \<notin> B"
haftmann@57418
   394
    and "\<forall>x\<in>A \<inter> \<Union>B. g x = 1"
haftmann@57418
   395
    and H: "F g (\<Union>B) = (F o F) g B" by auto
haftmann@57418
   396
  then have "F g (A \<union> \<Union>B) = F g A * F g (\<Union>B)"
haftmann@57418
   397
    by (simp add: union_inter_neutral)
wenzelm@60758
   398
  with \<open>finite B\<close> \<open>A \<notin> B\<close> show ?case
haftmann@57418
   399
    by (simp add: H)
haftmann@57418
   400
qed
haftmann@57418
   401
haftmann@57418
   402
lemma commute:
haftmann@57418
   403
  "F (\<lambda>i. F (g i) B) A = F (\<lambda>j. F (\<lambda>i. g i j) A) B"
haftmann@57418
   404
  unfolding cartesian_product
haftmann@57418
   405
  by (rule reindex_bij_witness [where i = "\<lambda>(i, j). (j, i)" and j = "\<lambda>(i, j). (j, i)"]) auto
haftmann@57418
   406
haftmann@57418
   407
lemma commute_restrict:
haftmann@57418
   408
  "finite A \<Longrightarrow> finite B \<Longrightarrow>
haftmann@57418
   409
    F (\<lambda>x. F (g x) {y. y \<in> B \<and> R x y}) A = F (\<lambda>y. F (\<lambda>x. g x y) {x. x \<in> A \<and> R x y}) B"
haftmann@57418
   410
  by (simp add: inter_filter) (rule commute)
haftmann@57418
   411
haftmann@57418
   412
lemma Plus:
haftmann@57418
   413
  fixes A :: "'b set" and B :: "'c set"
haftmann@57418
   414
  assumes fin: "finite A" "finite B"
haftmann@57418
   415
  shows "F g (A <+> B) = F (g \<circ> Inl) A * F (g \<circ> Inr) B"
haftmann@57418
   416
proof -
haftmann@57418
   417
  have "A <+> B = Inl ` A \<union> Inr ` B" by auto
haftmann@57418
   418
  moreover from fin have "finite (Inl ` A :: ('b + 'c) set)" "finite (Inr ` B :: ('b + 'c) set)"
haftmann@57418
   419
    by auto
haftmann@57418
   420
  moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('b + 'c) set)" by auto
haftmann@57418
   421
  moreover have "inj_on (Inl :: 'b \<Rightarrow> 'b + 'c) A" "inj_on (Inr :: 'c \<Rightarrow> 'b + 'c) B"
haftmann@57418
   422
    by (auto intro: inj_onI)
haftmann@57418
   423
  ultimately show ?thesis using fin
haftmann@57418
   424
    by (simp add: union_disjoint reindex)
haftmann@57418
   425
qed
haftmann@57418
   426
haftmann@58195
   427
lemma same_carrier:
haftmann@58195
   428
  assumes "finite C"
haftmann@58195
   429
  assumes subset: "A \<subseteq> C" "B \<subseteq> C"
haftmann@58195
   430
  assumes trivial: "\<And>a. a \<in> C - A \<Longrightarrow> g a = 1" "\<And>b. b \<in> C - B \<Longrightarrow> h b = 1"
haftmann@58195
   431
  shows "F g A = F h B \<longleftrightarrow> F g C = F h C"
haftmann@58195
   432
proof -
wenzelm@60758
   433
  from \<open>finite C\<close> subset have
haftmann@58195
   434
    "finite A" and "finite B" and "finite (C - A)" and "finite (C - B)"
haftmann@58195
   435
    by (auto elim: finite_subset)
haftmann@58195
   436
  from subset have [simp]: "A - (C - A) = A" by auto
haftmann@58195
   437
  from subset have [simp]: "B - (C - B) = B" by auto
haftmann@58195
   438
  from subset have "C = A \<union> (C - A)" by auto
haftmann@58195
   439
  then have "F g C = F g (A \<union> (C - A))" by simp
haftmann@58195
   440
  also have "\<dots> = F g (A - (C - A)) * F g (C - A - A) * F g (A \<inter> (C - A))"
wenzelm@60758
   441
    using \<open>finite A\<close> \<open>finite (C - A)\<close> by (simp only: union_diff2)
haftmann@58195
   442
  finally have P: "F g C = F g A" using trivial by simp
haftmann@58195
   443
  from subset have "C = B \<union> (C - B)" by auto
haftmann@58195
   444
  then have "F h C = F h (B \<union> (C - B))" by simp
haftmann@58195
   445
  also have "\<dots> = F h (B - (C - B)) * F h (C - B - B) * F h (B \<inter> (C - B))"
wenzelm@60758
   446
    using \<open>finite B\<close> \<open>finite (C - B)\<close> by (simp only: union_diff2)
haftmann@58195
   447
  finally have Q: "F h C = F h B" using trivial by simp
haftmann@58195
   448
  from P Q show ?thesis by simp
haftmann@58195
   449
qed
haftmann@58195
   450
haftmann@58195
   451
lemma same_carrierI:
haftmann@58195
   452
  assumes "finite C"
haftmann@58195
   453
  assumes subset: "A \<subseteq> C" "B \<subseteq> C"
haftmann@58195
   454
  assumes trivial: "\<And>a. a \<in> C - A \<Longrightarrow> g a = 1" "\<And>b. b \<in> C - B \<Longrightarrow> h b = 1"
haftmann@58195
   455
  assumes "F g C = F h C"
haftmann@58195
   456
  shows "F g A = F h B"
haftmann@58195
   457
  using assms same_carrier [of C A B] by simp
haftmann@58195
   458
haftmann@54744
   459
end
haftmann@54744
   460
haftmann@54744
   461
notation times (infixl "*" 70)
haftmann@54744
   462
notation Groups.one ("1")
haftmann@54744
   463
haftmann@54744
   464
wenzelm@60758
   465
subsection \<open>Generalized summation over a set\<close>
haftmann@54744
   466
haftmann@54744
   467
context comm_monoid_add
haftmann@54744
   468
begin
haftmann@54744
   469
wenzelm@61605
   470
sublocale setsum: comm_monoid_set plus 0
haftmann@61776
   471
defines
haftmann@61776
   472
  setsum = setsum.F ..
haftmann@54744
   473
wenzelm@61955
   474
abbreviation Setsum ("\<Sum>_" [1000] 999)
wenzelm@61955
   475
  where "\<Sum>A \<equiv> setsum (\<lambda>x. x) A"
haftmann@54744
   476
haftmann@54744
   477
end
haftmann@54744
   478
wenzelm@61955
   479
text \<open>Now: lot's of fancy syntax. First, @{term "setsum (\<lambda>x. e) A"} is written \<open>\<Sum>x\<in>A. e\<close>.\<close>
haftmann@54744
   480
wenzelm@61955
   481
syntax (ASCII)
wenzelm@61955
   482
  "_setsum" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b::comm_monoid_add"  ("(3SUM _:_./ _)" [0, 51, 10] 10)
haftmann@54744
   483
syntax
wenzelm@61955
   484
  "_setsum" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b::comm_monoid_add"  ("(2\<Sum>_\<in>_./ _)" [0, 51, 10] 10)
wenzelm@61799
   485
translations \<comment> \<open>Beware of argument permutation!\<close>
wenzelm@61955
   486
  "\<Sum>i\<in>A. b" \<rightleftharpoons> "CONST setsum (\<lambda>i. b) A"
haftmann@54744
   487
wenzelm@61955
   488
text \<open>Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter \<open>\<Sum>x|P. e\<close>.\<close>
haftmann@54744
   489
wenzelm@61955
   490
syntax (ASCII)
wenzelm@61955
   491
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(3SUM _ |/ _./ _)" [0, 0, 10] 10)
haftmann@54744
   492
syntax
wenzelm@61955
   493
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(2\<Sum>_ | (_)./ _)" [0, 0, 10] 10)
haftmann@54744
   494
translations
wenzelm@61955
   495
  "\<Sum>x|P. t" => "CONST setsum (\<lambda>x. t) {x. P}"
haftmann@54744
   496
wenzelm@60758
   497
print_translation \<open>
haftmann@54744
   498
let
haftmann@54744
   499
  fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) $ Abs (y, Ty, P)] =
haftmann@54744
   500
        if x <> y then raise Match
haftmann@54744
   501
        else
haftmann@54744
   502
          let
haftmann@54744
   503
            val x' = Syntax_Trans.mark_bound_body (x, Tx);
haftmann@54744
   504
            val t' = subst_bound (x', t);
haftmann@54744
   505
            val P' = subst_bound (x', P);
haftmann@54744
   506
          in
haftmann@54744
   507
            Syntax.const @{syntax_const "_qsetsum"} $ Syntax_Trans.mark_bound_abs (x, Tx) $ P' $ t'
haftmann@54744
   508
          end
haftmann@54744
   509
    | setsum_tr' _ = raise Match;
haftmann@54744
   510
in [(@{const_syntax setsum}, K setsum_tr')] end
wenzelm@60758
   511
\<close>
haftmann@54744
   512
wenzelm@60758
   513
text \<open>TODO generalization candidates\<close>
haftmann@54744
   514
hoelzl@62376
   515
lemma (in comm_monoid_add) setsum_image_gen:
haftmann@57418
   516
  assumes fS: "finite S"
haftmann@57418
   517
  shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
haftmann@57418
   518
proof-
haftmann@57418
   519
  { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }
haftmann@57418
   520
  hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
haftmann@57418
   521
    by simp
haftmann@57418
   522
  also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
haftmann@57418
   523
    by (rule setsum.commute_restrict [OF fS finite_imageI [OF fS]])
haftmann@57418
   524
  finally show ?thesis .
haftmann@54744
   525
qed
haftmann@54744
   526
haftmann@54744
   527
wenzelm@60758
   528
subsubsection \<open>Properties in more restricted classes of structures\<close>
haftmann@54744
   529
haftmann@54744
   530
lemma setsum_Un: "finite A ==> finite B ==>
haftmann@54744
   531
  (setsum f (A Un B) :: 'a :: ab_group_add) =
haftmann@54744
   532
   setsum f A + setsum f B - setsum f (A Int B)"
haftmann@57418
   533
by (subst setsum.union_inter [symmetric], auto simp add: algebra_simps)
haftmann@54744
   534
haftmann@54744
   535
lemma setsum_Un2:
haftmann@54744
   536
  assumes "finite (A \<union> B)"
haftmann@54744
   537
  shows "setsum f (A \<union> B) = setsum f (A - B) + setsum f (B - A) + setsum f (A \<inter> B)"
haftmann@54744
   538
proof -
haftmann@54744
   539
  have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
haftmann@54744
   540
    by auto
haftmann@57418
   541
  with assms show ?thesis by simp (subst setsum.union_disjoint, auto)+
haftmann@54744
   542
qed
haftmann@54744
   543
haftmann@54744
   544
lemma setsum_diff1: "finite A \<Longrightarrow>
haftmann@54744
   545
  (setsum f (A - {a}) :: ('a::ab_group_add)) =
haftmann@54744
   546
  (if a:A then setsum f A - f a else setsum f A)"
haftmann@54744
   547
by (erule finite_induct) (auto simp add: insert_Diff_if)
haftmann@54744
   548
haftmann@54744
   549
lemma setsum_diff:
haftmann@54744
   550
  assumes le: "finite A" "B \<subseteq> A"
haftmann@54744
   551
  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
haftmann@54744
   552
proof -
haftmann@54744
   553
  from le have finiteB: "finite B" using finite_subset by auto
haftmann@54744
   554
  show ?thesis using finiteB le
haftmann@54744
   555
  proof induct
haftmann@54744
   556
    case empty
haftmann@54744
   557
    thus ?case by auto
haftmann@54744
   558
  next
haftmann@54744
   559
    case (insert x F)
hoelzl@62376
   560
    thus ?case using le finiteB
haftmann@54744
   561
      by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
haftmann@54744
   562
  qed
haftmann@54744
   563
qed
haftmann@54744
   564
hoelzl@62376
   565
lemma (in ordered_comm_monoid_add) setsum_mono:
hoelzl@62376
   566
  assumes le: "\<And>i. i\<in>K \<Longrightarrow> f i \<le> g i"
haftmann@54744
   567
  shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
haftmann@54744
   568
proof (cases "finite K")
haftmann@54744
   569
  case True
haftmann@54744
   570
  thus ?thesis using le
haftmann@54744
   571
  proof induct
haftmann@54744
   572
    case empty
haftmann@54744
   573
    thus ?case by simp
haftmann@54744
   574
  next
haftmann@54744
   575
    case insert
haftmann@54744
   576
    thus ?case using add_mono by fastforce
haftmann@54744
   577
  qed
haftmann@54744
   578
next
haftmann@54744
   579
  case False then show ?thesis by simp
haftmann@54744
   580
qed
haftmann@54744
   581
hoelzl@62377
   582
lemma (in strict_ordered_comm_monoid_add) setsum_strict_mono:
hoelzl@62376
   583
  assumes "finite A"  "A \<noteq> {}" and "\<And>x. x \<in> A \<Longrightarrow> f x < g x"
haftmann@54744
   584
  shows "setsum f A < setsum g A"
haftmann@54744
   585
  using assms
haftmann@54744
   586
proof (induct rule: finite_ne_induct)
haftmann@54744
   587
  case singleton thus ?case by simp
haftmann@54744
   588
next
haftmann@54744
   589
  case insert thus ?case by (auto simp: add_strict_mono)
haftmann@54744
   590
qed
haftmann@54744
   591
haftmann@54744
   592
lemma setsum_strict_mono_ex1:
hoelzl@62376
   593
  fixes f g :: "'i \<Rightarrow> 'a::ordered_cancel_comm_monoid_add"
hoelzl@62377
   594
  assumes "finite A" and "\<forall>x\<in>A. f x \<le> g x" and "\<exists>a\<in>A. f a < g a"
hoelzl@62376
   595
  shows "setsum f A < setsum g A"
haftmann@54744
   596
proof-
haftmann@54744
   597
  from assms(3) obtain a where a: "a:A" "f a < g a" by blast
haftmann@54744
   598
  have "setsum f A = setsum f ((A-{a}) \<union> {a})"
wenzelm@60758
   599
    by(simp add:insert_absorb[OF \<open>a:A\<close>])
haftmann@54744
   600
  also have "\<dots> = setsum f (A-{a}) + setsum f {a}"
wenzelm@60758
   601
    using \<open>finite A\<close> by(subst setsum.union_disjoint) auto
haftmann@54744
   602
  also have "setsum f (A-{a}) \<le> setsum g (A-{a})"
haftmann@54744
   603
    by(rule setsum_mono)(simp add: assms(2))
haftmann@54744
   604
  also have "setsum f {a} < setsum g {a}" using a by simp
haftmann@54744
   605
  also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"
wenzelm@60758
   606
    using \<open>finite A\<close> by(subst setsum.union_disjoint[symmetric]) auto
wenzelm@60758
   607
  also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF \<open>a:A\<close>])
haftmann@54744
   608
  finally show ?thesis by (auto simp add: add_right_mono add_strict_left_mono)
haftmann@54744
   609
qed
haftmann@54744
   610
hoelzl@59416
   611
lemma setsum_negf: "(\<Sum>x\<in>A. - f x::'a::ab_group_add) = - (\<Sum>x\<in>A. f x)"
haftmann@54744
   612
proof (cases "finite A")
haftmann@54744
   613
  case True thus ?thesis by (induct set: finite) auto
haftmann@54744
   614
next
haftmann@54744
   615
  case False thus ?thesis by simp
haftmann@54744
   616
qed
haftmann@54744
   617
hoelzl@59416
   618
lemma setsum_subtractf: "(\<Sum>x\<in>A. f x - g x::'a::ab_group_add) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
haftmann@57418
   619
  using setsum.distrib [of f "- g" A] by (simp add: setsum_negf)
haftmann@54744
   620
hoelzl@59416
   621
lemma setsum_subtractf_nat:
hoelzl@59416
   622
  "(\<And>x. x \<in> A \<Longrightarrow> g x \<le> f x) \<Longrightarrow> (\<Sum>x\<in>A. f x - g x::nat) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
hoelzl@59416
   623
  by (induction A rule: infinite_finite_induct) (auto simp: setsum_mono)
hoelzl@59416
   624
hoelzl@62376
   625
lemma (in ordered_comm_monoid_add) setsum_nonneg:
hoelzl@62376
   626
  assumes nn: "\<forall>x\<in>A. 0 \<le> f x"
haftmann@54744
   627
  shows "0 \<le> setsum f A"
haftmann@54744
   628
proof (cases "finite A")
haftmann@54744
   629
  case True thus ?thesis using nn
haftmann@54744
   630
  proof induct
haftmann@54744
   631
    case empty then show ?case by simp
haftmann@54744
   632
  next
haftmann@54744
   633
    case (insert x F)
haftmann@54744
   634
    then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
haftmann@54744
   635
    with insert show ?case by simp
haftmann@54744
   636
  qed
haftmann@54744
   637
next
haftmann@54744
   638
  case False thus ?thesis by simp
haftmann@54744
   639
qed
haftmann@54744
   640
hoelzl@62376
   641
lemma (in ordered_comm_monoid_add) setsum_nonpos:
hoelzl@62376
   642
  assumes np: "\<forall>x\<in>A. f x \<le> 0"
haftmann@54744
   643
  shows "setsum f A \<le> 0"
haftmann@54744
   644
proof (cases "finite A")
haftmann@54744
   645
  case True thus ?thesis using np
haftmann@54744
   646
  proof induct
haftmann@54744
   647
    case empty then show ?case by simp
haftmann@54744
   648
  next
haftmann@54744
   649
    case (insert x F)
haftmann@54744
   650
    then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
haftmann@54744
   651
    with insert show ?case by simp
haftmann@54744
   652
  qed
haftmann@54744
   653
next
haftmann@54744
   654
  case False thus ?thesis by simp
haftmann@54744
   655
qed
haftmann@54744
   656
hoelzl@62376
   657
lemma (in ordered_comm_monoid_add) setsum_nonneg_eq_0_iff:
hoelzl@62376
   658
  "finite A \<Longrightarrow> \<forall>x\<in>A. 0 \<le> f x \<Longrightarrow> setsum f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
hoelzl@62376
   659
  by (induct set: finite, simp) (simp add: add_nonneg_eq_0_iff setsum_nonneg)
hoelzl@62376
   660
hoelzl@62376
   661
lemma (in ordered_comm_monoid_add) setsum_nonneg_0:
hoelzl@62376
   662
  "finite s \<Longrightarrow> (\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0) \<Longrightarrow> (\<Sum> i \<in> s. f i) = 0 \<Longrightarrow> i \<in> s \<Longrightarrow> f i = 0"
hoelzl@62376
   663
  by (simp add: setsum_nonneg_eq_0_iff)
hoelzl@62376
   664
hoelzl@62376
   665
lemma (in ordered_comm_monoid_add) setsum_nonneg_leq_bound:
haftmann@54744
   666
  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
   667
  shows "f i \<le> B"
haftmann@54744
   668
proof -
hoelzl@62376
   669
  have "f i \<le> f i + (\<Sum>i \<in> s - {i}. f i)"
hoelzl@62376
   670
    using assms by (intro add_increasing2 setsum_nonneg) auto
hoelzl@62376
   671
  also have "\<dots> = B"
hoelzl@62376
   672
    using setsum.remove[of s i f] assms by simp
hoelzl@62376
   673
  finally show ?thesis by auto
haftmann@54744
   674
qed
haftmann@54744
   675
hoelzl@62376
   676
lemma (in ordered_comm_monoid_add) setsum_mono2:
hoelzl@62376
   677
  assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
hoelzl@62376
   678
  shows "setsum f A \<le> setsum f B"
haftmann@54744
   679
proof -
haftmann@54744
   680
  have "setsum f A \<le> setsum f A + setsum f (B-A)"
haftmann@54744
   681
    by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
haftmann@54744
   682
  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
haftmann@57418
   683
    by (simp add: setsum.union_disjoint del:Un_Diff_cancel)
haftmann@54744
   684
  also have "A \<union> (B-A) = B" using sub by blast
haftmann@54744
   685
  finally show ?thesis .
haftmann@54744
   686
qed
haftmann@54744
   687
hoelzl@62376
   688
lemma (in ordered_comm_monoid_add) setsum_le_included:
haftmann@57418
   689
  assumes "finite s" "finite t"
haftmann@57418
   690
  and "\<forall>y\<in>t. 0 \<le> g y" "(\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<le> g y)"
haftmann@57418
   691
  shows "setsum f s \<le> setsum g t"
haftmann@57418
   692
proof -
haftmann@57418
   693
  have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"
haftmann@57418
   694
  proof (rule setsum_mono)
haftmann@57418
   695
    fix y assume "y \<in> s"
haftmann@57418
   696
    with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto
haftmann@57418
   697
    with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")
haftmann@57418
   698
      using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]
haftmann@57418
   699
      by (auto intro!: setsum_mono2)
haftmann@57418
   700
  qed
haftmann@57418
   701
  also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"
haftmann@57418
   702
    using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)
haftmann@57418
   703
  also have "... \<le> setsum g t"
haftmann@57418
   704
    using assms by (auto simp: setsum_image_gen[symmetric])
haftmann@57418
   705
  finally show ?thesis .
haftmann@57418
   706
qed
haftmann@57418
   707
hoelzl@62376
   708
lemma (in ordered_comm_monoid_add) setsum_mono3:
hoelzl@62376
   709
  "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> \<forall>x\<in>B - A. 0 \<le> f x \<Longrightarrow> setsum f A \<le> setsum f B"
hoelzl@62376
   710
  by (rule setsum_mono2) auto
haftmann@54744
   711
hoelzl@62376
   712
lemma (in canonically_ordered_monoid_add) setsum_eq_0_iff [simp]:
hoelzl@62376
   713
  "finite F \<Longrightarrow> (setsum f F = 0) = (\<forall>a\<in>F. f a = 0)"
hoelzl@62376
   714
  by (intro ballI setsum_nonneg_eq_0_iff zero_le)
hoelzl@62376
   715
hoelzl@62376
   716
lemma setsum_right_distrib:
haftmann@54744
   717
  fixes f :: "'a => ('b::semiring_0)"
haftmann@54744
   718
  shows "r * setsum f A = setsum (%n. r * f n) A"
haftmann@54744
   719
proof (cases "finite A")
haftmann@54744
   720
  case True
haftmann@54744
   721
  thus ?thesis
haftmann@54744
   722
  proof induct
haftmann@54744
   723
    case empty thus ?case by simp
haftmann@54744
   724
  next
haftmann@54744
   725
    case (insert x A) thus ?case by (simp add: distrib_left)
haftmann@54744
   726
  qed
haftmann@54744
   727
next
haftmann@54744
   728
  case False thus ?thesis by simp
haftmann@54744
   729
qed
haftmann@54744
   730
haftmann@54744
   731
lemma setsum_left_distrib:
haftmann@54744
   732
  "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
haftmann@54744
   733
proof (cases "finite A")
haftmann@54744
   734
  case True
haftmann@54744
   735
  then show ?thesis
haftmann@54744
   736
  proof induct
haftmann@54744
   737
    case empty thus ?case by simp
haftmann@54744
   738
  next
haftmann@54744
   739
    case (insert x A) thus ?case by (simp add: distrib_right)
haftmann@54744
   740
  qed
haftmann@54744
   741
next
haftmann@54744
   742
  case False thus ?thesis by simp
haftmann@54744
   743
qed
haftmann@54744
   744
haftmann@54744
   745
lemma setsum_divide_distrib:
haftmann@54744
   746
  "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
haftmann@54744
   747
proof (cases "finite A")
haftmann@54744
   748
  case True
haftmann@54744
   749
  then show ?thesis
haftmann@54744
   750
  proof induct
haftmann@54744
   751
    case empty thus ?case by simp
haftmann@54744
   752
  next
haftmann@54744
   753
    case (insert x A) thus ?case by (simp add: add_divide_distrib)
haftmann@54744
   754
  qed
haftmann@54744
   755
next
haftmann@54744
   756
  case False thus ?thesis by simp
haftmann@54744
   757
qed
haftmann@54744
   758
hoelzl@62376
   759
lemma setsum_abs[iff]:
haftmann@54744
   760
  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
wenzelm@61944
   761
  shows "\<bar>setsum f A\<bar> \<le> setsum (%i. \<bar>f i\<bar>) A"
haftmann@54744
   762
proof (cases "finite A")
haftmann@54744
   763
  case True
haftmann@54744
   764
  thus ?thesis
haftmann@54744
   765
  proof induct
haftmann@54744
   766
    case empty thus ?case by simp
haftmann@54744
   767
  next
haftmann@54744
   768
    case (insert x A)
haftmann@54744
   769
    thus ?case by (auto intro: abs_triangle_ineq order_trans)
haftmann@54744
   770
  qed
haftmann@54744
   771
next
haftmann@54744
   772
  case False thus ?thesis by simp
haftmann@54744
   773
qed
haftmann@54744
   774
lp15@60974
   775
lemma setsum_abs_ge_zero[iff]:
haftmann@54744
   776
  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
wenzelm@61944
   777
  shows "0 \<le> setsum (%i. \<bar>f i\<bar>) A"
lp15@60974
   778
  by (simp add: setsum_nonneg)
haftmann@54744
   779
hoelzl@62376
   780
lemma abs_setsum_abs[simp]:
haftmann@54744
   781
  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
wenzelm@61944
   782
  shows "\<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar> = (\<Sum>a\<in>A. \<bar>f a\<bar>)"
haftmann@54744
   783
proof (cases "finite A")
haftmann@54744
   784
  case True
haftmann@54744
   785
  thus ?thesis
haftmann@54744
   786
  proof induct
haftmann@54744
   787
    case empty thus ?case by simp
haftmann@54744
   788
  next
haftmann@54744
   789
    case (insert a A)
haftmann@54744
   790
    hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
haftmann@54744
   791
    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
   792
    also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
haftmann@54744
   793
      by (simp del: abs_of_nonneg)
haftmann@54744
   794
    also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
haftmann@54744
   795
    finally show ?case .
haftmann@54744
   796
  qed
haftmann@54744
   797
next
haftmann@54744
   798
  case False thus ?thesis by simp
haftmann@54744
   799
qed
haftmann@54744
   800
haftmann@54744
   801
lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
haftmann@54744
   802
  shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
haftmann@57418
   803
  unfolding setsum.remove [OF assms] by auto
haftmann@54744
   804
haftmann@54744
   805
lemma setsum_product:
haftmann@54744
   806
  fixes f :: "'a => ('b::semiring_0)"
haftmann@54744
   807
  shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
haftmann@57418
   808
  by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum.commute)
haftmann@54744
   809
haftmann@54744
   810
lemma setsum_mult_setsum_if_inj:
haftmann@54744
   811
fixes f :: "'a => ('b::semiring_0)"
haftmann@54744
   812
shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
haftmann@54744
   813
  setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
haftmann@57418
   814
by(auto simp: setsum_product setsum.cartesian_product
haftmann@57418
   815
        intro!:  setsum.reindex_cong[symmetric])
haftmann@54744
   816
haftmann@54744
   817
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
haftmann@54744
   818
apply (case_tac "finite A")
haftmann@54744
   819
 prefer 2 apply simp
haftmann@54744
   820
apply (erule rev_mp)
haftmann@54744
   821
apply (erule finite_induct, auto)
haftmann@54744
   822
done
haftmann@54744
   823
haftmann@54744
   824
lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
haftmann@54744
   825
  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
   826
apply(erule finite_induct)
haftmann@54744
   827
apply (auto simp add:add_is_1)
haftmann@54744
   828
done
haftmann@54744
   829
haftmann@54744
   830
lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
haftmann@54744
   831
haftmann@54744
   832
lemma setsum_Un_nat: "finite A ==> finite B ==>
haftmann@54744
   833
  (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
wenzelm@61799
   834
  \<comment> \<open>For the natural numbers, we have subtraction.\<close>
haftmann@57418
   835
by (subst setsum.union_inter [symmetric], auto simp add: algebra_simps)
haftmann@54744
   836
haftmann@54744
   837
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
haftmann@54744
   838
  (if a:A then setsum f A - f a else setsum f A)"
haftmann@54744
   839
apply (case_tac "finite A")
haftmann@54744
   840
 prefer 2 apply simp
haftmann@54744
   841
apply (erule finite_induct)
haftmann@54744
   842
 apply (auto simp add: insert_Diff_if)
haftmann@54744
   843
apply (drule_tac a = a in mk_disjoint_insert, auto)
haftmann@54744
   844
done
haftmann@54744
   845
hoelzl@62376
   846
lemma setsum_diff_nat:
haftmann@54744
   847
assumes "finite B" and "B \<subseteq> A"
haftmann@54744
   848
shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
haftmann@54744
   849
using assms
haftmann@54744
   850
proof induct
haftmann@54744
   851
  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
haftmann@54744
   852
next
haftmann@54744
   853
  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
haftmann@54744
   854
    and xFinA: "insert x F \<subseteq> A"
haftmann@54744
   855
    and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
haftmann@54744
   856
  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
haftmann@54744
   857
  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
haftmann@54744
   858
    by (simp add: setsum_diff1_nat)
haftmann@54744
   859
  from xFinA have "F \<subseteq> A" by simp
haftmann@54744
   860
  with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
haftmann@54744
   861
  with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
haftmann@54744
   862
    by simp
haftmann@54744
   863
  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
haftmann@54744
   864
  with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
haftmann@54744
   865
    by simp
haftmann@54744
   866
  from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
haftmann@54744
   867
  with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
haftmann@54744
   868
    by simp
haftmann@54744
   869
  thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
haftmann@54744
   870
qed
haftmann@54744
   871
haftmann@54744
   872
lemma setsum_comp_morphism:
haftmann@54744
   873
  assumes "h 0 = 0" and "\<And>x y. h (x + y) = h x + h y"
haftmann@54744
   874
  shows "setsum (h \<circ> g) A = h (setsum g A)"
haftmann@54744
   875
proof (cases "finite A")
haftmann@54744
   876
  case False then show ?thesis by (simp add: assms)
haftmann@54744
   877
next
haftmann@54744
   878
  case True then show ?thesis by (induct A) (simp_all add: assms)
haftmann@54744
   879
qed
haftmann@54744
   880
haftmann@59010
   881
lemma (in comm_semiring_1) dvd_setsum:
haftmann@59010
   882
  "(\<And>a. a \<in> A \<Longrightarrow> d dvd f a) \<Longrightarrow> d dvd setsum f A"
haftmann@59010
   883
  by (induct A rule: infinite_finite_induct) simp_all
haftmann@59010
   884
hoelzl@62377
   885
lemma (in ordered_comm_monoid_add) setsum_pos:
hoelzl@62377
   886
  "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> 0 < f i) \<Longrightarrow> 0 < setsum f I"
hoelzl@62377
   887
  by (induct I rule: finite_ne_induct) (auto intro: add_pos_pos)
hoelzl@62377
   888
hoelzl@62377
   889
lemma (in ordered_comm_monoid_add) setsum_pos2:
hoelzl@62377
   890
  assumes I: "finite I" "i \<in> I" "0 < f i" "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
hoelzl@62377
   891
  shows "0 < setsum f I"
lp15@60974
   892
proof -
hoelzl@62377
   893
  have "0 < f i + setsum f (I - {i})"
hoelzl@62377
   894
    using assms by (intro add_pos_nonneg setsum_nonneg) auto
hoelzl@62377
   895
  also have "\<dots> = setsum f I"
lp15@60974
   896
    using assms by (simp add: setsum.remove)
lp15@60974
   897
  finally show ?thesis .
lp15@60974
   898
qed
lp15@60974
   899
eberlm@61524
   900
lemma setsum_cong_Suc:
eberlm@61524
   901
  assumes "0 \<notin> A" "\<And>x. Suc x \<in> A \<Longrightarrow> f (Suc x) = g (Suc x)"
eberlm@61524
   902
  shows   "setsum f A = setsum g A"
eberlm@61524
   903
proof (rule setsum.cong)
eberlm@61524
   904
  fix x assume "x \<in> A"
eberlm@61524
   905
  with assms(1) show "f x = g x" by (cases x) (auto intro!: assms(2))
eberlm@61524
   906
qed simp_all
eberlm@61524
   907
haftmann@54744
   908
wenzelm@60758
   909
subsubsection \<open>Cardinality as special case of @{const setsum}\<close>
haftmann@54744
   910
haftmann@54744
   911
lemma card_eq_setsum:
haftmann@54744
   912
  "card A = setsum (\<lambda>x. 1) A"
haftmann@54744
   913
proof -
haftmann@54744
   914
  have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)"
haftmann@54744
   915
    by (simp add: fun_eq_iff)
haftmann@54744
   916
  then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) = Finite_Set.fold (\<lambda>_. Suc)"
haftmann@54744
   917
    by (rule arg_cong)
haftmann@54744
   918
  then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
haftmann@54744
   919
    by (blast intro: fun_cong)
haftmann@54744
   920
  then show ?thesis by (simp add: card.eq_fold setsum.eq_fold)
haftmann@54744
   921
qed
haftmann@54744
   922
haftmann@54744
   923
lemma setsum_constant [simp]:
haftmann@54744
   924
  "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
haftmann@54744
   925
apply (cases "finite A")
haftmann@54744
   926
apply (erule finite_induct)
haftmann@54744
   927
apply (auto simp add: algebra_simps)
haftmann@54744
   928
done
haftmann@54744
   929
lp15@59615
   930
lemma setsum_Suc: "setsum (\<lambda>x. Suc(f x)) A = setsum f A + card A"
hoelzl@62376
   931
  using setsum.distrib[of f "\<lambda>_. 1" A]
lp15@59615
   932
  by simp
nipkow@58349
   933
lp15@60974
   934
lemma setsum_bounded_above:
hoelzl@62376
   935
  assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_comm_monoid_add})"
haftmann@54744
   936
  shows "setsum f A \<le> of_nat (card A) * K"
haftmann@54744
   937
proof (cases "finite A")
haftmann@54744
   938
  case True
haftmann@54744
   939
  thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
haftmann@54744
   940
next
haftmann@54744
   941
  case False thus ?thesis by simp
haftmann@54744
   942
qed
haftmann@54744
   943
lp15@60974
   944
lemma setsum_bounded_above_strict:
hoelzl@62376
   945
  assumes "\<And>i. i\<in>A \<Longrightarrow> f i < (K::'a::{ordered_cancel_comm_monoid_add,semiring_1})"
lp15@60974
   946
          "card A > 0"
lp15@60974
   947
  shows "setsum f A < of_nat (card A) * K"
lp15@60974
   948
using assms setsum_strict_mono[where A=A and g = "%x. K"]
lp15@60974
   949
by (simp add: card_gt_0_iff)
lp15@60974
   950
lp15@60974
   951
lemma setsum_bounded_below:
hoelzl@62376
   952
  assumes le: "\<And>i. i\<in>A \<Longrightarrow> (K::'a::{semiring_1, ordered_comm_monoid_add}) \<le> f i"
lp15@60974
   953
  shows "of_nat (card A) * K \<le> setsum f A"
lp15@60974
   954
proof (cases "finite A")
lp15@60974
   955
  case True
lp15@60974
   956
  thus ?thesis using le setsum_mono[where K=A and f = "%x. K"] by simp
lp15@60974
   957
next
lp15@60974
   958
  case False thus ?thesis by simp
lp15@60974
   959
qed
lp15@60974
   960
haftmann@54744
   961
lemma card_UN_disjoint:
haftmann@54744
   962
  assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
haftmann@54744
   963
    and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
haftmann@54744
   964
  shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
haftmann@54744
   965
proof -
haftmann@54744
   966
  have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)" by simp
haftmann@57418
   967
  with assms show ?thesis by (simp add: card_eq_setsum setsum.UNION_disjoint del: setsum_constant)
haftmann@54744
   968
qed
haftmann@54744
   969
haftmann@54744
   970
lemma card_Union_disjoint:
haftmann@54744
   971
  "finite C ==> (ALL A:C. finite A) ==>
haftmann@54744
   972
   (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {})
wenzelm@61952
   973
   ==> card (\<Union>C) = setsum card C"
haftmann@54744
   974
apply (frule card_UN_disjoint [of C id])
haftmann@56166
   975
apply simp_all
haftmann@54744
   976
done
haftmann@54744
   977
haftmann@57418
   978
lemma setsum_multicount_gen:
haftmann@57418
   979
  assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
haftmann@57418
   980
  shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")
haftmann@57418
   981
proof-
haftmann@57418
   982
  have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto
haftmann@57418
   983
  also have "\<dots> = ?r" unfolding setsum.commute_restrict [OF assms(1-2)]
haftmann@57418
   984
    using assms(3) by auto
haftmann@57418
   985
  finally show ?thesis .
haftmann@57418
   986
qed
haftmann@57418
   987
haftmann@57418
   988
lemma setsum_multicount:
haftmann@57418
   989
  assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
haftmann@57418
   990
  shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
haftmann@57418
   991
proof-
haftmann@57418
   992
  have "?l = setsum (\<lambda>i. k) T" by (rule setsum_multicount_gen) (auto simp: assms)
haftmann@57512
   993
  also have "\<dots> = ?r" by (simp add: mult.commute)
haftmann@57418
   994
  finally show ?thesis by auto
haftmann@57418
   995
qed
haftmann@57418
   996
wenzelm@60758
   997
subsubsection \<open>Cardinality of products\<close>
haftmann@54744
   998
haftmann@54744
   999
lemma card_SigmaI [simp]:
haftmann@54744
  1000
  "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
haftmann@54744
  1001
  \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
haftmann@57418
  1002
by(simp add: card_eq_setsum setsum.Sigma del:setsum_constant)
haftmann@54744
  1003
haftmann@54744
  1004
(*
haftmann@54744
  1005
lemma SigmaI_insert: "y \<notin> A ==>
wenzelm@61943
  1006
  (SIGMA x:(insert y A). B x) = (({y} \<times> (B y)) \<union> (SIGMA x: A. B x))"
haftmann@54744
  1007
  by auto
haftmann@54744
  1008
*)
haftmann@54744
  1009
wenzelm@61943
  1010
lemma card_cartesian_product: "card (A \<times> B) = card(A) * card(B)"
haftmann@54744
  1011
  by (cases "finite A \<and> finite B")
haftmann@54744
  1012
    (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
haftmann@54744
  1013
wenzelm@61943
  1014
lemma card_cartesian_product_singleton:  "card({x} \<times> A) = card(A)"
haftmann@54744
  1015
by (simp add: card_cartesian_product)
haftmann@54744
  1016
haftmann@54744
  1017
wenzelm@60758
  1018
subsection \<open>Generalized product over a set\<close>
haftmann@54744
  1019
haftmann@54744
  1020
context comm_monoid_mult
haftmann@54744
  1021
begin
haftmann@54744
  1022
wenzelm@61605
  1023
sublocale setprod: comm_monoid_set times 1
haftmann@61776
  1024
defines
haftmann@61776
  1025
  setprod = setprod.F ..
haftmann@54744
  1026
haftmann@54744
  1027
abbreviation
haftmann@54744
  1028
  Setprod ("\<Prod>_" [1000] 999) where
haftmann@54744
  1029
  "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"
haftmann@54744
  1030
haftmann@54744
  1031
end
haftmann@54744
  1032
wenzelm@61955
  1033
syntax (ASCII)
lp15@60494
  1034
  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(4PROD _:_./ _)" [0, 51, 10] 10)
wenzelm@61955
  1035
syntax
lp15@60494
  1036
  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(2\<Prod>_\<in>_./ _)" [0, 51, 10] 10)
wenzelm@61799
  1037
translations \<comment> \<open>Beware of argument permutation!\<close>
hoelzl@62376
  1038
  "\<Prod>i\<in>A. b" == "CONST setprod (\<lambda>i. b) A"
haftmann@54744
  1039
wenzelm@61955
  1040
text \<open>Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter \<open>\<Prod>x|P. e\<close>.\<close>
haftmann@54744
  1041
wenzelm@61955
  1042
syntax (ASCII)
wenzelm@61955
  1043
  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(4PROD _ |/ _./ _)" [0, 0, 10] 10)
haftmann@54744
  1044
syntax
wenzelm@61955
  1045
  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(2\<Prod>_ | (_)./ _)" [0, 0, 10] 10)
haftmann@54744
  1046
translations
wenzelm@61955
  1047
  "\<Prod>x|P. t" => "CONST setprod (\<lambda>x. t) {x. P}"
haftmann@54744
  1048
haftmann@59010
  1049
context comm_monoid_mult
haftmann@59010
  1050
begin
haftmann@59010
  1051
hoelzl@62376
  1052
lemma setprod_dvd_setprod:
haftmann@59010
  1053
  "(\<And>a. a \<in> A \<Longrightarrow> f a dvd g a) \<Longrightarrow> setprod f A dvd setprod g A"
haftmann@59010
  1054
proof (induct A rule: infinite_finite_induct)
haftmann@59010
  1055
  case infinite then show ?case by (auto intro: dvdI)
haftmann@59010
  1056
next
haftmann@59010
  1057
  case empty then show ?case by (auto intro: dvdI)
haftmann@59010
  1058
next
haftmann@59010
  1059
  case (insert a A) then
haftmann@59010
  1060
  have "f a dvd g a" and "setprod f A dvd setprod g A" by simp_all
haftmann@59010
  1061
  then obtain r s where "g a = f a * r" and "setprod g A = setprod f A * s" by (auto elim!: dvdE)
haftmann@59010
  1062
  then have "g a * setprod g A = f a * setprod f A * (r * s)" by (simp add: ac_simps)
haftmann@59010
  1063
  with insert.hyps show ?case by (auto intro: dvdI)
haftmann@59010
  1064
qed
haftmann@59010
  1065
haftmann@59010
  1066
lemma setprod_dvd_setprod_subset:
haftmann@59010
  1067
  "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> setprod f A dvd setprod f B"
haftmann@59010
  1068
  by (auto simp add: setprod.subset_diff ac_simps intro: dvdI)
haftmann@59010
  1069
haftmann@59010
  1070
end
haftmann@59010
  1071
haftmann@54744
  1072
wenzelm@60758
  1073
subsubsection \<open>Properties in more restricted classes of structures\<close>
haftmann@54744
  1074
haftmann@59010
  1075
context comm_semiring_1
haftmann@59010
  1076
begin
haftmann@54744
  1077
haftmann@59010
  1078
lemma dvd_setprod_eqI [intro]:
haftmann@59010
  1079
  assumes "finite A" and "a \<in> A" and "b = f a"
haftmann@59010
  1080
  shows "b dvd setprod f A"
haftmann@59010
  1081
proof -
wenzelm@60758
  1082
  from \<open>finite A\<close> have "setprod f (insert a (A - {a})) = f a * setprod f (A - {a})"
haftmann@59010
  1083
    by (intro setprod.insert) auto
wenzelm@60758
  1084
  also from \<open>a \<in> A\<close> have "insert a (A - {a}) = A" by blast
haftmann@59010
  1085
  finally have "setprod f A = f a * setprod f (A - {a})" .
wenzelm@60758
  1086
  with \<open>b = f a\<close> show ?thesis by simp
haftmann@59010
  1087
qed
haftmann@54744
  1088
haftmann@59010
  1089
lemma dvd_setprodI [intro]:
haftmann@59010
  1090
  assumes "finite A" and "a \<in> A"
haftmann@59010
  1091
  shows "f a dvd setprod f A"
haftmann@59010
  1092
  using assms by auto
haftmann@54744
  1093
haftmann@59010
  1094
lemma setprod_zero:
haftmann@59010
  1095
  assumes "finite A" and "\<exists>a\<in>A. f a = 0"
haftmann@59010
  1096
  shows "setprod f A = 0"
haftmann@59010
  1097
using assms proof (induct A)
haftmann@59010
  1098
  case empty then show ?case by simp
haftmann@59010
  1099
next
haftmann@59010
  1100
  case (insert a A)
haftmann@59010
  1101
  then have "f a = 0 \<or> (\<exists>a\<in>A. f a = 0)" by simp
haftmann@59010
  1102
  then have "f a * setprod f A = 0" by rule (simp_all add: insert)
haftmann@59010
  1103
  with insert show ?case by simp
haftmann@59010
  1104
qed
haftmann@54744
  1105
haftmann@54744
  1106
lemma setprod_dvd_setprod_subset2:
haftmann@59010
  1107
  assumes "finite B" and "A \<subseteq> B" and "\<And>a. a \<in> A \<Longrightarrow> f a dvd g a"
haftmann@59010
  1108
  shows "setprod f A dvd setprod g B"
haftmann@59010
  1109
proof -
haftmann@59010
  1110
  from assms have "setprod f A dvd setprod g A"
haftmann@59010
  1111
    by (auto intro: setprod_dvd_setprod)
haftmann@59010
  1112
  moreover from assms have "setprod g A dvd setprod g B"
haftmann@59010
  1113
    by (auto intro: setprod_dvd_setprod_subset)
haftmann@59010
  1114
  ultimately show ?thesis by (rule dvd_trans)
haftmann@59010
  1115
qed
haftmann@59010
  1116
haftmann@59010
  1117
end
haftmann@59010
  1118
haftmann@59010
  1119
lemma setprod_zero_iff [simp]:
haftmann@59010
  1120
  assumes "finite A"
haftmann@59833
  1121
  shows "setprod f A = (0::'a::semidom) \<longleftrightarrow> (\<exists>a\<in>A. f a = 0)"
haftmann@59010
  1122
  using assms by (induct A) (auto simp: no_zero_divisors)
haftmann@59010
  1123
haftmann@60353
  1124
lemma (in semidom_divide) setprod_diff1:
haftmann@60353
  1125
  assumes "finite A" and "f a \<noteq> 0"
haftmann@60429
  1126
  shows "setprod f (A - {a}) = (if a \<in> A then setprod f A div f a else setprod f A)"
haftmann@60353
  1127
proof (cases "a \<notin> A")
haftmann@60353
  1128
  case True then show ?thesis by simp
haftmann@60353
  1129
next
haftmann@60353
  1130
  case False with assms show ?thesis
haftmann@60353
  1131
  proof (induct A rule: finite_induct)
haftmann@60353
  1132
    case empty then show ?case by simp
haftmann@60353
  1133
  next
haftmann@60353
  1134
    case (insert b B)
haftmann@60353
  1135
    then show ?case
haftmann@60353
  1136
    proof (cases "a = b")
haftmann@60353
  1137
      case True with insert show ?thesis by simp
haftmann@60353
  1138
    next
haftmann@60353
  1139
      case False with insert have "a \<in> B" by simp
haftmann@60353
  1140
      def C \<equiv> "B - {a}"
wenzelm@60758
  1141
      with \<open>finite B\<close> \<open>a \<in> B\<close>
haftmann@60353
  1142
        have *: "B = insert a C" "finite C" "a \<notin> C" by auto
haftmann@60353
  1143
      with insert show ?thesis by (auto simp add: insert_commute ac_simps)
haftmann@60353
  1144
    qed
haftmann@60353
  1145
  qed
haftmann@60353
  1146
qed
haftmann@54744
  1147
hoelzl@62376
  1148
lemma (in field) setprod_inversef:
haftmann@59010
  1149
  "finite A \<Longrightarrow> setprod (inverse \<circ> f) A = inverse (setprod f A)"
haftmann@59010
  1150
  by (induct A rule: finite_induct) simp_all
haftmann@59010
  1151
haftmann@59867
  1152
lemma (in field) setprod_dividef:
haftmann@59010
  1153
  "finite A \<Longrightarrow> (\<Prod>x\<in>A. f x / g x) = setprod f A / setprod g A"
haftmann@59010
  1154
  using setprod_inversef [of A g] by (simp add: divide_inverse setprod.distrib)
haftmann@54744
  1155
haftmann@59010
  1156
lemma setprod_Un:
haftmann@59010
  1157
  fixes f :: "'b \<Rightarrow> 'a :: field"
haftmann@59010
  1158
  assumes "finite A" and "finite B"
haftmann@59010
  1159
  and "\<forall>x\<in>A \<inter> B. f x \<noteq> 0"
haftmann@59010
  1160
  shows "setprod f (A \<union> B) = setprod f A * setprod f B / setprod f (A \<inter> B)"
haftmann@59010
  1161
proof -
haftmann@59010
  1162
  from assms have "setprod f A * setprod f B = setprod f (A \<union> B) * setprod f (A \<inter> B)"
haftmann@59010
  1163
    by (simp add: setprod.union_inter [symmetric, of A B])
haftmann@59010
  1164
  with assms show ?thesis by simp
haftmann@59010
  1165
qed
haftmann@54744
  1166
haftmann@59010
  1167
lemma (in linordered_semidom) setprod_nonneg:
haftmann@59010
  1168
  "(\<forall>a\<in>A. 0 \<le> f a) \<Longrightarrow> 0 \<le> setprod f A"
haftmann@59010
  1169
  by (induct A rule: infinite_finite_induct) simp_all
haftmann@59010
  1170
haftmann@59010
  1171
lemma (in linordered_semidom) setprod_pos:
haftmann@59010
  1172
  "(\<forall>a\<in>A. 0 < f a) \<Longrightarrow> 0 < setprod f A"
haftmann@59010
  1173
  by (induct A rule: infinite_finite_induct) simp_all
haftmann@59010
  1174
haftmann@59010
  1175
lemma (in linordered_semidom) setprod_mono:
hoelzl@62376
  1176
  "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i \<Longrightarrow> setprod f A \<le> setprod g A"
hoelzl@62376
  1177
  by (induct A rule: infinite_finite_induct) (auto intro!: setprod_nonneg mult_mono)
haftmann@54744
  1178
lp15@60974
  1179
lemma (in linordered_semidom) setprod_mono_strict:
lp15@60974
  1180
    assumes"finite A" "\<forall>i\<in>A. 0 \<le> f i \<and> f i < g i" "A \<noteq> {}"
lp15@60974
  1181
    shows "setprod f A < setprod g A"
hoelzl@62376
  1182
using assms
lp15@60974
  1183
apply (induct A rule: finite_induct)
lp15@60974
  1184
apply (simp add: )
lp15@60974
  1185
apply (force intro: mult_strict_mono' setprod_nonneg)
lp15@60974
  1186
done
lp15@60974
  1187
haftmann@59010
  1188
lemma (in linordered_field) abs_setprod:
haftmann@59010
  1189
  "\<bar>setprod f A\<bar> = (\<Prod>x\<in>A. \<bar>f x\<bar>)"
haftmann@59010
  1190
  by (induct A rule: infinite_finite_induct) (simp_all add: abs_mult)
haftmann@54744
  1191
haftmann@54744
  1192
lemma setprod_eq_1_iff [simp]:
haftmann@59010
  1193
  "finite A \<Longrightarrow> setprod f A = 1 \<longleftrightarrow> (\<forall>a\<in>A. f a = (1::nat))"
haftmann@59010
  1194
  by (induct A rule: finite_induct) simp_all
haftmann@54744
  1195
haftmann@59010
  1196
lemma setprod_pos_nat_iff [simp]:
haftmann@59010
  1197
  "finite A \<Longrightarrow> setprod f A > 0 \<longleftrightarrow> (\<forall>a\<in>A. f a > (0::nat))"
hoelzl@62378
  1198
  using setprod_zero_iff by (simp del: neq0_conv add: zero_less_iff_neq_zero)
haftmann@54744
  1199
haftmann@54744
  1200
end