src/HOL/Groups_Big.thy
author haftmann
Thu Aug 27 21:19:48 2015 +0200 (2015-08-27)
changeset 61032 b57df8eecad6
parent 60974 6a6f15d8fbc4
child 61169 4de9ff3ea29a
permissions -rw-r--r--
standardized some occurences of ancient "split" alias
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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 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 \<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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  -- \<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 -
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  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
haftmann@54744
   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}"
haftmann@54744
   320
    have eq: "S = ?A \<union> ?B" using a by blast 
haftmann@54744
   321
    have dj: "?A \<inter> ?B = {}" by simp
haftmann@54744
   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
haftmann@54744
   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 -
haftmann@54744
   341
  have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}" 
haftmann@54744
   342
          "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}" 
haftmann@54744
   343
    by blast+
haftmann@54744
   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:
haftmann@61032
   353
   "F (\<lambda>x. F (g x) B) A = F (case_prod g) (A <*> B)"
haftmann@54744
   354
apply (rule sym)
haftmann@54744
   355
apply (cases "finite A") 
haftmann@54744
   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
haftmann@54744
   470
definition setsum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
haftmann@54744
   471
where
haftmann@54744
   472
  "setsum = comm_monoid_set.F plus 0"
haftmann@54744
   473
haftmann@54744
   474
sublocale setsum!: comm_monoid_set plus 0
haftmann@54744
   475
where
haftmann@54744
   476
  "comm_monoid_set.F plus 0 = setsum"
haftmann@54744
   477
proof -
haftmann@54744
   478
  show "comm_monoid_set plus 0" ..
haftmann@54744
   479
  then interpret setsum!: comm_monoid_set plus 0 .
haftmann@54744
   480
  from setsum_def show "comm_monoid_set.F plus 0 = setsum" by rule
haftmann@54744
   481
qed
haftmann@54744
   482
haftmann@54744
   483
abbreviation
haftmann@54744
   484
  Setsum ("\<Sum>_" [1000] 999) where
haftmann@54744
   485
  "\<Sum>A \<equiv> setsum (%x. x) A"
haftmann@54744
   486
haftmann@54744
   487
end
haftmann@54744
   488
wenzelm@60758
   489
text\<open>Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
wenzelm@60758
   490
written @{text"\<Sum>x\<in>A. e"}.\<close>
haftmann@54744
   491
haftmann@54744
   492
syntax
lp15@60494
   493
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_./ _)" [0, 51, 10] 10)
haftmann@54744
   494
syntax (xsymbols)
lp15@60494
   495
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(2\<Sum>_\<in>_./ _)" [0, 51, 10] 10)
haftmann@54744
   496
syntax (HTML output)
lp15@60494
   497
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(2\<Sum>_\<in>_./ _)" [0, 51, 10] 10)
haftmann@54744
   498
wenzelm@60758
   499
translations -- \<open>Beware of argument permutation!\<close>
haftmann@54744
   500
  "SUM i:A. b" == "CONST setsum (%i. b) A"
haftmann@54744
   501
  "\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A"
haftmann@54744
   502
wenzelm@60758
   503
text\<open>Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
wenzelm@60758
   504
 @{text"\<Sum>x|P. e"}.\<close>
haftmann@54744
   505
haftmann@54744
   506
syntax
haftmann@54744
   507
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
haftmann@54744
   508
syntax (xsymbols)
lp15@60494
   509
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(2\<Sum>_ | (_)./ _)" [0,0,10] 10)
haftmann@54744
   510
syntax (HTML output)
lp15@60494
   511
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(2\<Sum>_ | (_)./ _)" [0,0,10] 10)
haftmann@54744
   512
haftmann@54744
   513
translations
haftmann@54744
   514
  "SUM x|P. t" => "CONST setsum (%x. t) {x. P}"
haftmann@54744
   515
  "\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}"
haftmann@54744
   516
wenzelm@60758
   517
print_translation \<open>
haftmann@54744
   518
let
haftmann@54744
   519
  fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) $ Abs (y, Ty, P)] =
haftmann@54744
   520
        if x <> y then raise Match
haftmann@54744
   521
        else
haftmann@54744
   522
          let
haftmann@54744
   523
            val x' = Syntax_Trans.mark_bound_body (x, Tx);
haftmann@54744
   524
            val t' = subst_bound (x', t);
haftmann@54744
   525
            val P' = subst_bound (x', P);
haftmann@54744
   526
          in
haftmann@54744
   527
            Syntax.const @{syntax_const "_qsetsum"} $ Syntax_Trans.mark_bound_abs (x, Tx) $ P' $ t'
haftmann@54744
   528
          end
haftmann@54744
   529
    | setsum_tr' _ = raise Match;
haftmann@54744
   530
in [(@{const_syntax setsum}, K setsum_tr')] end
wenzelm@60758
   531
\<close>
haftmann@54744
   532
wenzelm@60758
   533
text \<open>TODO generalization candidates\<close>
haftmann@54744
   534
haftmann@57418
   535
lemma setsum_image_gen:
haftmann@57418
   536
  assumes fS: "finite S"
haftmann@57418
   537
  shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
haftmann@57418
   538
proof-
haftmann@57418
   539
  { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }
haftmann@57418
   540
  hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
haftmann@57418
   541
    by simp
haftmann@57418
   542
  also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
haftmann@57418
   543
    by (rule setsum.commute_restrict [OF fS finite_imageI [OF fS]])
haftmann@57418
   544
  finally show ?thesis .
haftmann@54744
   545
qed
haftmann@54744
   546
haftmann@54744
   547
wenzelm@60758
   548
subsubsection \<open>Properties in more restricted classes of structures\<close>
haftmann@54744
   549
haftmann@54744
   550
lemma setsum_Un: "finite A ==> finite B ==>
haftmann@54744
   551
  (setsum f (A Un B) :: 'a :: ab_group_add) =
haftmann@54744
   552
   setsum f A + setsum f B - setsum f (A Int B)"
haftmann@57418
   553
by (subst setsum.union_inter [symmetric], auto simp add: algebra_simps)
haftmann@54744
   554
haftmann@54744
   555
lemma setsum_Un2:
haftmann@54744
   556
  assumes "finite (A \<union> B)"
haftmann@54744
   557
  shows "setsum f (A \<union> B) = setsum f (A - B) + setsum f (B - A) + setsum f (A \<inter> B)"
haftmann@54744
   558
proof -
haftmann@54744
   559
  have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
haftmann@54744
   560
    by auto
haftmann@57418
   561
  with assms show ?thesis by simp (subst setsum.union_disjoint, auto)+
haftmann@54744
   562
qed
haftmann@54744
   563
haftmann@54744
   564
lemma setsum_diff1: "finite A \<Longrightarrow>
haftmann@54744
   565
  (setsum f (A - {a}) :: ('a::ab_group_add)) =
haftmann@54744
   566
  (if a:A then setsum f A - f a else setsum f A)"
haftmann@54744
   567
by (erule finite_induct) (auto simp add: insert_Diff_if)
haftmann@54744
   568
haftmann@54744
   569
lemma setsum_diff:
haftmann@54744
   570
  assumes le: "finite A" "B \<subseteq> A"
haftmann@54744
   571
  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
haftmann@54744
   572
proof -
haftmann@54744
   573
  from le have finiteB: "finite B" using finite_subset by auto
haftmann@54744
   574
  show ?thesis using finiteB le
haftmann@54744
   575
  proof induct
haftmann@54744
   576
    case empty
haftmann@54744
   577
    thus ?case by auto
haftmann@54744
   578
  next
haftmann@54744
   579
    case (insert x F)
haftmann@54744
   580
    thus ?case using le finiteB 
haftmann@54744
   581
      by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
haftmann@54744
   582
  qed
haftmann@54744
   583
qed
haftmann@54744
   584
haftmann@54744
   585
lemma setsum_mono:
haftmann@54744
   586
  assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))"
haftmann@54744
   587
  shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
haftmann@54744
   588
proof (cases "finite K")
haftmann@54744
   589
  case True
haftmann@54744
   590
  thus ?thesis using le
haftmann@54744
   591
  proof induct
haftmann@54744
   592
    case empty
haftmann@54744
   593
    thus ?case by simp
haftmann@54744
   594
  next
haftmann@54744
   595
    case insert
haftmann@54744
   596
    thus ?case using add_mono by fastforce
haftmann@54744
   597
  qed
haftmann@54744
   598
next
haftmann@54744
   599
  case False then show ?thesis by simp
haftmann@54744
   600
qed
haftmann@54744
   601
haftmann@54744
   602
lemma setsum_strict_mono:
haftmann@54744
   603
  fixes f :: "'a \<Rightarrow> 'b::{ordered_cancel_ab_semigroup_add,comm_monoid_add}"
haftmann@54744
   604
  assumes "finite A"  "A \<noteq> {}"
haftmann@54744
   605
    and "!!x. x:A \<Longrightarrow> f x < g x"
haftmann@54744
   606
  shows "setsum f A < setsum g A"
haftmann@54744
   607
  using assms
haftmann@54744
   608
proof (induct rule: finite_ne_induct)
haftmann@54744
   609
  case singleton thus ?case by simp
haftmann@54744
   610
next
haftmann@54744
   611
  case insert thus ?case by (auto simp: add_strict_mono)
haftmann@54744
   612
qed
haftmann@54744
   613
haftmann@54744
   614
lemma setsum_strict_mono_ex1:
haftmann@54744
   615
fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_cancel_ab_semigroup_add}"
haftmann@54744
   616
assumes "finite A" and "ALL x:A. f x \<le> g x" and "EX a:A. f a < g a"
haftmann@54744
   617
shows "setsum f A < setsum g A"
haftmann@54744
   618
proof-
haftmann@54744
   619
  from assms(3) obtain a where a: "a:A" "f a < g a" by blast
haftmann@54744
   620
  have "setsum f A = setsum f ((A-{a}) \<union> {a})"
wenzelm@60758
   621
    by(simp add:insert_absorb[OF \<open>a:A\<close>])
haftmann@54744
   622
  also have "\<dots> = setsum f (A-{a}) + setsum f {a}"
wenzelm@60758
   623
    using \<open>finite A\<close> by(subst setsum.union_disjoint) auto
haftmann@54744
   624
  also have "setsum f (A-{a}) \<le> setsum g (A-{a})"
haftmann@54744
   625
    by(rule setsum_mono)(simp add: assms(2))
haftmann@54744
   626
  also have "setsum f {a} < setsum g {a}" using a by simp
haftmann@54744
   627
  also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"
wenzelm@60758
   628
    using \<open>finite A\<close> by(subst setsum.union_disjoint[symmetric]) auto
wenzelm@60758
   629
  also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF \<open>a:A\<close>])
haftmann@54744
   630
  finally show ?thesis by (auto simp add: add_right_mono add_strict_left_mono)
haftmann@54744
   631
qed
haftmann@54744
   632
hoelzl@59416
   633
lemma setsum_negf: "(\<Sum>x\<in>A. - f x::'a::ab_group_add) = - (\<Sum>x\<in>A. f x)"
haftmann@54744
   634
proof (cases "finite A")
haftmann@54744
   635
  case True thus ?thesis by (induct set: finite) auto
haftmann@54744
   636
next
haftmann@54744
   637
  case False thus ?thesis by simp
haftmann@54744
   638
qed
haftmann@54744
   639
hoelzl@59416
   640
lemma setsum_subtractf: "(\<Sum>x\<in>A. f x - g x::'a::ab_group_add) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
haftmann@57418
   641
  using setsum.distrib [of f "- g" A] by (simp add: setsum_negf)
haftmann@54744
   642
hoelzl@59416
   643
lemma setsum_subtractf_nat:
hoelzl@59416
   644
  "(\<And>x. x \<in> A \<Longrightarrow> g x \<le> f x) \<Longrightarrow> (\<Sum>x\<in>A. f x - g x::nat) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
hoelzl@59416
   645
  by (induction A rule: infinite_finite_induct) (auto simp: setsum_mono)
hoelzl@59416
   646
haftmann@54744
   647
lemma setsum_nonneg:
haftmann@54744
   648
  assumes nn: "\<forall>x\<in>A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
haftmann@54744
   649
  shows "0 \<le> setsum f A"
haftmann@54744
   650
proof (cases "finite A")
haftmann@54744
   651
  case True thus ?thesis using nn
haftmann@54744
   652
  proof induct
haftmann@54744
   653
    case empty then show ?case by simp
haftmann@54744
   654
  next
haftmann@54744
   655
    case (insert x F)
haftmann@54744
   656
    then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
haftmann@54744
   657
    with insert show ?case by simp
haftmann@54744
   658
  qed
haftmann@54744
   659
next
haftmann@54744
   660
  case False thus ?thesis by simp
haftmann@54744
   661
qed
haftmann@54744
   662
haftmann@54744
   663
lemma setsum_nonpos:
haftmann@54744
   664
  assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})"
haftmann@54744
   665
  shows "setsum f A \<le> 0"
haftmann@54744
   666
proof (cases "finite A")
haftmann@54744
   667
  case True thus ?thesis using np
haftmann@54744
   668
  proof induct
haftmann@54744
   669
    case empty then show ?case by simp
haftmann@54744
   670
  next
haftmann@54744
   671
    case (insert x F)
haftmann@54744
   672
    then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
haftmann@54744
   673
    with insert show ?case by simp
haftmann@54744
   674
  qed
haftmann@54744
   675
next
haftmann@54744
   676
  case False thus ?thesis by simp
haftmann@54744
   677
qed
haftmann@54744
   678
haftmann@54744
   679
lemma setsum_nonneg_leq_bound:
haftmann@54744
   680
  fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
haftmann@54744
   681
  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
   682
  shows "f i \<le> B"
haftmann@54744
   683
proof -
haftmann@54744
   684
  have "0 \<le> (\<Sum> i \<in> s - {i}. f i)" and "0 \<le> f i"
haftmann@54744
   685
    using assms by (auto intro!: setsum_nonneg)
haftmann@54744
   686
  moreover
haftmann@54744
   687
  have "(\<Sum> i \<in> s - {i}. f i) + f i = B"
haftmann@54744
   688
    using assms by (simp add: setsum_diff1)
haftmann@54744
   689
  ultimately show ?thesis by auto
haftmann@54744
   690
qed
haftmann@54744
   691
haftmann@54744
   692
lemma setsum_nonneg_0:
haftmann@54744
   693
  fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
haftmann@54744
   694
  assumes "finite s" and pos: "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"
haftmann@54744
   695
  and "(\<Sum> i \<in> s. f i) = 0" and i: "i \<in> s"
haftmann@54744
   696
  shows "f i = 0"
haftmann@54744
   697
  using setsum_nonneg_leq_bound[OF assms] pos[OF i] by auto
haftmann@54744
   698
haftmann@54744
   699
lemma setsum_mono2:
haftmann@54744
   700
fixes f :: "'a \<Rightarrow> 'b :: ordered_comm_monoid_add"
haftmann@54744
   701
assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
haftmann@54744
   702
shows "setsum f A \<le> setsum f B"
haftmann@54744
   703
proof -
haftmann@54744
   704
  have "setsum f A \<le> setsum f A + setsum f (B-A)"
haftmann@54744
   705
    by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
haftmann@54744
   706
  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
haftmann@57418
   707
    by (simp add: setsum.union_disjoint del:Un_Diff_cancel)
haftmann@54744
   708
  also have "A \<union> (B-A) = B" using sub by blast
haftmann@54744
   709
  finally show ?thesis .
haftmann@54744
   710
qed
haftmann@54744
   711
haftmann@57418
   712
lemma setsum_le_included:
haftmann@57418
   713
  fixes f :: "'a \<Rightarrow> 'b::ordered_comm_monoid_add"
haftmann@57418
   714
  assumes "finite s" "finite t"
haftmann@57418
   715
  and "\<forall>y\<in>t. 0 \<le> g y" "(\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<le> g y)"
haftmann@57418
   716
  shows "setsum f s \<le> setsum g t"
haftmann@57418
   717
proof -
haftmann@57418
   718
  have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"
haftmann@57418
   719
  proof (rule setsum_mono)
haftmann@57418
   720
    fix y assume "y \<in> s"
haftmann@57418
   721
    with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto
haftmann@57418
   722
    with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")
haftmann@57418
   723
      using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]
haftmann@57418
   724
      by (auto intro!: setsum_mono2)
haftmann@57418
   725
  qed
haftmann@57418
   726
  also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"
haftmann@57418
   727
    using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)
haftmann@57418
   728
  also have "... \<le> setsum g t"
haftmann@57418
   729
    using assms by (auto simp: setsum_image_gen[symmetric])
haftmann@57418
   730
  finally show ?thesis .
haftmann@57418
   731
qed
haftmann@57418
   732
haftmann@54744
   733
lemma setsum_mono3: "finite B ==> A <= B ==> 
haftmann@54744
   734
    ALL x: B - A. 
haftmann@54744
   735
      0 <= ((f x)::'a::{comm_monoid_add,ordered_ab_semigroup_add}) ==>
haftmann@54744
   736
        setsum f A <= setsum f B"
haftmann@54744
   737
  apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
haftmann@54744
   738
  apply (erule ssubst)
haftmann@54744
   739
  apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
haftmann@54744
   740
  apply simp
haftmann@54744
   741
  apply (rule add_left_mono)
haftmann@54744
   742
  apply (erule setsum_nonneg)
haftmann@57418
   743
  apply (subst setsum.union_disjoint [THEN sym])
haftmann@54744
   744
  apply (erule finite_subset, assumption)
haftmann@54744
   745
  apply (rule finite_subset)
haftmann@54744
   746
  prefer 2
haftmann@54744
   747
  apply assumption
haftmann@54744
   748
  apply (auto simp add: sup_absorb2)
haftmann@54744
   749
done
haftmann@54744
   750
haftmann@54744
   751
lemma setsum_right_distrib: 
haftmann@54744
   752
  fixes f :: "'a => ('b::semiring_0)"
haftmann@54744
   753
  shows "r * setsum f A = setsum (%n. r * f n) A"
haftmann@54744
   754
proof (cases "finite A")
haftmann@54744
   755
  case True
haftmann@54744
   756
  thus ?thesis
haftmann@54744
   757
  proof induct
haftmann@54744
   758
    case empty thus ?case by simp
haftmann@54744
   759
  next
haftmann@54744
   760
    case (insert x A) thus ?case by (simp add: distrib_left)
haftmann@54744
   761
  qed
haftmann@54744
   762
next
haftmann@54744
   763
  case False thus ?thesis by simp
haftmann@54744
   764
qed
haftmann@54744
   765
haftmann@54744
   766
lemma setsum_left_distrib:
haftmann@54744
   767
  "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
haftmann@54744
   768
proof (cases "finite A")
haftmann@54744
   769
  case True
haftmann@54744
   770
  then show ?thesis
haftmann@54744
   771
  proof induct
haftmann@54744
   772
    case empty thus ?case by simp
haftmann@54744
   773
  next
haftmann@54744
   774
    case (insert x A) thus ?case by (simp add: distrib_right)
haftmann@54744
   775
  qed
haftmann@54744
   776
next
haftmann@54744
   777
  case False thus ?thesis by simp
haftmann@54744
   778
qed
haftmann@54744
   779
haftmann@54744
   780
lemma setsum_divide_distrib:
haftmann@54744
   781
  "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
haftmann@54744
   782
proof (cases "finite A")
haftmann@54744
   783
  case True
haftmann@54744
   784
  then show ?thesis
haftmann@54744
   785
  proof induct
haftmann@54744
   786
    case empty thus ?case by simp
haftmann@54744
   787
  next
haftmann@54744
   788
    case (insert x A) thus ?case by (simp add: add_divide_distrib)
haftmann@54744
   789
  qed
haftmann@54744
   790
next
haftmann@54744
   791
  case False thus ?thesis by simp
haftmann@54744
   792
qed
haftmann@54744
   793
haftmann@54744
   794
lemma setsum_abs[iff]: 
haftmann@54744
   795
  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
haftmann@54744
   796
  shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
haftmann@54744
   797
proof (cases "finite A")
haftmann@54744
   798
  case True
haftmann@54744
   799
  thus ?thesis
haftmann@54744
   800
  proof induct
haftmann@54744
   801
    case empty thus ?case by simp
haftmann@54744
   802
  next
haftmann@54744
   803
    case (insert x A)
haftmann@54744
   804
    thus ?case by (auto intro: abs_triangle_ineq order_trans)
haftmann@54744
   805
  qed
haftmann@54744
   806
next
haftmann@54744
   807
  case False thus ?thesis by simp
haftmann@54744
   808
qed
haftmann@54744
   809
lp15@60974
   810
lemma setsum_abs_ge_zero[iff]:
haftmann@54744
   811
  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
haftmann@54744
   812
  shows "0 \<le> setsum (%i. abs(f i)) A"
lp15@60974
   813
  by (simp add: setsum_nonneg)
haftmann@54744
   814
haftmann@54744
   815
lemma abs_setsum_abs[simp]: 
haftmann@54744
   816
  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
haftmann@54744
   817
  shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
haftmann@54744
   818
proof (cases "finite A")
haftmann@54744
   819
  case True
haftmann@54744
   820
  thus ?thesis
haftmann@54744
   821
  proof induct
haftmann@54744
   822
    case empty thus ?case by simp
haftmann@54744
   823
  next
haftmann@54744
   824
    case (insert a A)
haftmann@54744
   825
    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
   826
    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
   827
    also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
haftmann@54744
   828
      by (simp del: abs_of_nonneg)
haftmann@54744
   829
    also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
haftmann@54744
   830
    finally show ?case .
haftmann@54744
   831
  qed
haftmann@54744
   832
next
haftmann@54744
   833
  case False thus ?thesis by simp
haftmann@54744
   834
qed
haftmann@54744
   835
haftmann@54744
   836
lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
haftmann@54744
   837
  shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
haftmann@57418
   838
  unfolding setsum.remove [OF assms] by auto
haftmann@54744
   839
haftmann@54744
   840
lemma setsum_product:
haftmann@54744
   841
  fixes f :: "'a => ('b::semiring_0)"
haftmann@54744
   842
  shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
haftmann@57418
   843
  by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum.commute)
haftmann@54744
   844
haftmann@54744
   845
lemma setsum_mult_setsum_if_inj:
haftmann@54744
   846
fixes f :: "'a => ('b::semiring_0)"
haftmann@54744
   847
shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
haftmann@54744
   848
  setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
haftmann@57418
   849
by(auto simp: setsum_product setsum.cartesian_product
haftmann@57418
   850
        intro!:  setsum.reindex_cong[symmetric])
haftmann@54744
   851
haftmann@54744
   852
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
haftmann@54744
   853
apply (case_tac "finite A")
haftmann@54744
   854
 prefer 2 apply simp
haftmann@54744
   855
apply (erule rev_mp)
haftmann@54744
   856
apply (erule finite_induct, auto)
haftmann@54744
   857
done
haftmann@54744
   858
haftmann@54744
   859
lemma setsum_eq_0_iff [simp]:
haftmann@54744
   860
  "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
haftmann@54744
   861
  by (induct set: finite) auto
haftmann@54744
   862
haftmann@54744
   863
lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
haftmann@54744
   864
  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
   865
apply(erule finite_induct)
haftmann@54744
   866
apply (auto simp add:add_is_1)
haftmann@54744
   867
done
haftmann@54744
   868
haftmann@54744
   869
lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
haftmann@54744
   870
haftmann@54744
   871
lemma setsum_Un_nat: "finite A ==> finite B ==>
haftmann@54744
   872
  (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
wenzelm@60758
   873
  -- \<open>For the natural numbers, we have subtraction.\<close>
haftmann@57418
   874
by (subst setsum.union_inter [symmetric], auto simp add: algebra_simps)
haftmann@54744
   875
haftmann@54744
   876
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
haftmann@54744
   877
  (if a:A then setsum f A - f a else setsum f A)"
haftmann@54744
   878
apply (case_tac "finite A")
haftmann@54744
   879
 prefer 2 apply simp
haftmann@54744
   880
apply (erule finite_induct)
haftmann@54744
   881
 apply (auto simp add: insert_Diff_if)
haftmann@54744
   882
apply (drule_tac a = a in mk_disjoint_insert, auto)
haftmann@54744
   883
done
haftmann@54744
   884
haftmann@54744
   885
lemma setsum_diff_nat: 
haftmann@54744
   886
assumes "finite B" and "B \<subseteq> A"
haftmann@54744
   887
shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
haftmann@54744
   888
using assms
haftmann@54744
   889
proof induct
haftmann@54744
   890
  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
haftmann@54744
   891
next
haftmann@54744
   892
  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
haftmann@54744
   893
    and xFinA: "insert x F \<subseteq> A"
haftmann@54744
   894
    and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
haftmann@54744
   895
  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
haftmann@54744
   896
  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
haftmann@54744
   897
    by (simp add: setsum_diff1_nat)
haftmann@54744
   898
  from xFinA have "F \<subseteq> A" by simp
haftmann@54744
   899
  with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
haftmann@54744
   900
  with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
haftmann@54744
   901
    by simp
haftmann@54744
   902
  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
haftmann@54744
   903
  with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
haftmann@54744
   904
    by simp
haftmann@54744
   905
  from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
haftmann@54744
   906
  with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
haftmann@54744
   907
    by simp
haftmann@54744
   908
  thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
haftmann@54744
   909
qed
haftmann@54744
   910
haftmann@54744
   911
lemma setsum_comp_morphism:
haftmann@54744
   912
  assumes "h 0 = 0" and "\<And>x y. h (x + y) = h x + h y"
haftmann@54744
   913
  shows "setsum (h \<circ> g) A = h (setsum g A)"
haftmann@54744
   914
proof (cases "finite A")
haftmann@54744
   915
  case False then show ?thesis by (simp add: assms)
haftmann@54744
   916
next
haftmann@54744
   917
  case True then show ?thesis by (induct A) (simp_all add: assms)
haftmann@54744
   918
qed
haftmann@54744
   919
haftmann@59010
   920
lemma (in comm_semiring_1) dvd_setsum:
haftmann@59010
   921
  "(\<And>a. a \<in> A \<Longrightarrow> d dvd f a) \<Longrightarrow> d dvd setsum f A"
haftmann@59010
   922
  by (induct A rule: infinite_finite_induct) simp_all
haftmann@59010
   923
lp15@60974
   924
lemma setsum_pos2:
lp15@60974
   925
    assumes "finite I" "i \<in> I" "0 < f i" "(\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i)"
lp15@60974
   926
      shows "(0::'a::{ordered_ab_group_add,comm_monoid_add}) < setsum f I"
lp15@60974
   927
proof -
lp15@60974
   928
  have "0 \<le> setsum f (I-{i})"
lp15@60974
   929
    using assms by (simp add: setsum_nonneg)
lp15@60974
   930
  also have "... < setsum f (I-{i}) + f i"
lp15@60974
   931
    using assms by auto
lp15@60974
   932
  also have "... = setsum f I"
lp15@60974
   933
    using assms by (simp add: setsum.remove)
lp15@60974
   934
  finally show ?thesis .
lp15@60974
   935
qed
lp15@60974
   936
haftmann@54744
   937
wenzelm@60758
   938
subsubsection \<open>Cardinality as special case of @{const setsum}\<close>
haftmann@54744
   939
haftmann@54744
   940
lemma card_eq_setsum:
haftmann@54744
   941
  "card A = setsum (\<lambda>x. 1) A"
haftmann@54744
   942
proof -
haftmann@54744
   943
  have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)"
haftmann@54744
   944
    by (simp add: fun_eq_iff)
haftmann@54744
   945
  then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) = Finite_Set.fold (\<lambda>_. Suc)"
haftmann@54744
   946
    by (rule arg_cong)
haftmann@54744
   947
  then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
haftmann@54744
   948
    by (blast intro: fun_cong)
haftmann@54744
   949
  then show ?thesis by (simp add: card.eq_fold setsum.eq_fold)
haftmann@54744
   950
qed
haftmann@54744
   951
haftmann@54744
   952
lemma setsum_constant [simp]:
haftmann@54744
   953
  "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
haftmann@54744
   954
apply (cases "finite A")
haftmann@54744
   955
apply (erule finite_induct)
haftmann@54744
   956
apply (auto simp add: algebra_simps)
haftmann@54744
   957
done
haftmann@54744
   958
lp15@59615
   959
lemma setsum_Suc: "setsum (\<lambda>x. Suc(f x)) A = setsum f A + card A"
lp15@59615
   960
  using setsum.distrib[of f "\<lambda>_. 1" A] 
lp15@59615
   961
  by simp
nipkow@58349
   962
lp15@60974
   963
lemma setsum_bounded_above:
haftmann@54744
   964
  assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"
haftmann@54744
   965
  shows "setsum f A \<le> of_nat (card A) * K"
haftmann@54744
   966
proof (cases "finite A")
haftmann@54744
   967
  case True
haftmann@54744
   968
  thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
haftmann@54744
   969
next
haftmann@54744
   970
  case False thus ?thesis by simp
haftmann@54744
   971
qed
haftmann@54744
   972
lp15@60974
   973
lemma setsum_bounded_above_strict:
lp15@60974
   974
  assumes "\<And>i. i\<in>A \<Longrightarrow> f i < (K::'a::{ordered_cancel_ab_semigroup_add,semiring_1})"
lp15@60974
   975
          "card A > 0"
lp15@60974
   976
  shows "setsum f A < of_nat (card A) * K"
lp15@60974
   977
using assms setsum_strict_mono[where A=A and g = "%x. K"]
lp15@60974
   978
by (simp add: card_gt_0_iff)
lp15@60974
   979
lp15@60974
   980
lemma setsum_bounded_below:
lp15@60974
   981
  assumes le: "\<And>i. i\<in>A \<Longrightarrow> (K::'a::{semiring_1, ordered_ab_semigroup_add}) \<le> f i"
lp15@60974
   982
  shows "of_nat (card A) * K \<le> setsum f A"
lp15@60974
   983
proof (cases "finite A")
lp15@60974
   984
  case True
lp15@60974
   985
  thus ?thesis using le setsum_mono[where K=A and f = "%x. K"] by simp
lp15@60974
   986
next
lp15@60974
   987
  case False thus ?thesis by simp
lp15@60974
   988
qed
lp15@60974
   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@57418
   996
  with assms show ?thesis by (simp add: card_eq_setsum setsum.UNION_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@57418
  1007
lemma setsum_multicount_gen:
haftmann@57418
  1008
  assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
haftmann@57418
  1009
  shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")
haftmann@57418
  1010
proof-
haftmann@57418
  1011
  have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto
haftmann@57418
  1012
  also have "\<dots> = ?r" unfolding setsum.commute_restrict [OF assms(1-2)]
haftmann@57418
  1013
    using assms(3) by auto
haftmann@57418
  1014
  finally show ?thesis .
haftmann@57418
  1015
qed
haftmann@57418
  1016
haftmann@57418
  1017
lemma setsum_multicount:
haftmann@57418
  1018
  assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
haftmann@57418
  1019
  shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
haftmann@57418
  1020
proof-
haftmann@57418
  1021
  have "?l = setsum (\<lambda>i. k) T" by (rule setsum_multicount_gen) (auto simp: assms)
haftmann@57512
  1022
  also have "\<dots> = ?r" by (simp add: mult.commute)
haftmann@57418
  1023
  finally show ?thesis by auto
haftmann@57418
  1024
qed
haftmann@57418
  1025
haftmann@58437
  1026
lemma (in ordered_comm_monoid_add) setsum_pos: 
haftmann@58437
  1027
  "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> 0 < f i) \<Longrightarrow> 0 < setsum f I"
haftmann@58437
  1028
  by (induct I rule: finite_ne_induct) (auto intro: add_pos_pos)
haftmann@58437
  1029
haftmann@54744
  1030
wenzelm@60758
  1031
subsubsection \<open>Cardinality of products\<close>
haftmann@54744
  1032
haftmann@54744
  1033
lemma card_SigmaI [simp]:
haftmann@54744
  1034
  "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
haftmann@54744
  1035
  \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
haftmann@57418
  1036
by(simp add: card_eq_setsum setsum.Sigma del:setsum_constant)
haftmann@54744
  1037
haftmann@54744
  1038
(*
haftmann@54744
  1039
lemma SigmaI_insert: "y \<notin> A ==>
haftmann@54744
  1040
  (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
haftmann@54744
  1041
  by auto
haftmann@54744
  1042
*)
haftmann@54744
  1043
haftmann@54744
  1044
lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
haftmann@54744
  1045
  by (cases "finite A \<and> finite B")
haftmann@54744
  1046
    (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
haftmann@54744
  1047
haftmann@54744
  1048
lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
haftmann@54744
  1049
by (simp add: card_cartesian_product)
haftmann@54744
  1050
haftmann@54744
  1051
wenzelm@60758
  1052
subsection \<open>Generalized product over a set\<close>
haftmann@54744
  1053
haftmann@54744
  1054
context comm_monoid_mult
haftmann@54744
  1055
begin
haftmann@54744
  1056
haftmann@54744
  1057
definition setprod :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
haftmann@54744
  1058
where
haftmann@54744
  1059
  "setprod = comm_monoid_set.F times 1"
haftmann@54744
  1060
haftmann@54744
  1061
sublocale setprod!: comm_monoid_set times 1
haftmann@54744
  1062
where
haftmann@54744
  1063
  "comm_monoid_set.F times 1 = setprod"
haftmann@54744
  1064
proof -
haftmann@54744
  1065
  show "comm_monoid_set times 1" ..
haftmann@54744
  1066
  then interpret setprod!: comm_monoid_set times 1 .
haftmann@54744
  1067
  from setprod_def show "comm_monoid_set.F times 1 = setprod" by rule
haftmann@54744
  1068
qed
haftmann@54744
  1069
haftmann@54744
  1070
abbreviation
haftmann@54744
  1071
  Setprod ("\<Prod>_" [1000] 999) where
haftmann@54744
  1072
  "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"
haftmann@54744
  1073
haftmann@54744
  1074
end
haftmann@54744
  1075
haftmann@54744
  1076
syntax
lp15@60494
  1077
  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(4PROD _:_./ _)" [0, 51, 10] 10)
haftmann@54744
  1078
syntax (xsymbols)
lp15@60494
  1079
  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(2\<Prod>_\<in>_./ _)" [0, 51, 10] 10)
haftmann@54744
  1080
syntax (HTML output)
lp15@60494
  1081
  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(2\<Prod>_\<in>_./ _)" [0, 51, 10] 10)
haftmann@54744
  1082
wenzelm@60758
  1083
translations -- \<open>Beware of argument permutation!\<close>
haftmann@54744
  1084
  "PROD i:A. b" == "CONST setprod (%i. b) A" 
haftmann@54744
  1085
  "\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A" 
haftmann@54744
  1086
wenzelm@60758
  1087
text\<open>Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
wenzelm@60758
  1088
 @{text"\<Prod>x|P. e"}.\<close>
haftmann@54744
  1089
haftmann@54744
  1090
syntax
lp15@60494
  1091
  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(4PROD _ |/ _./ _)" [0,0,10] 10)
haftmann@54744
  1092
syntax (xsymbols)
lp15@60494
  1093
  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(2\<Prod>_ | (_)./ _)" [0,0,10] 10)
haftmann@54744
  1094
syntax (HTML output)
lp15@60494
  1095
  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(2\<Prod>_ | (_)./ _)" [0,0,10] 10)
haftmann@54744
  1096
haftmann@54744
  1097
translations
haftmann@54744
  1098
  "PROD x|P. t" => "CONST setprod (%x. t) {x. P}"
haftmann@54744
  1099
  "\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"
haftmann@54744
  1100
haftmann@59010
  1101
context comm_monoid_mult
haftmann@59010
  1102
begin
haftmann@59010
  1103
haftmann@59010
  1104
lemma setprod_dvd_setprod: 
haftmann@59010
  1105
  "(\<And>a. a \<in> A \<Longrightarrow> f a dvd g a) \<Longrightarrow> setprod f A dvd setprod g A"
haftmann@59010
  1106
proof (induct A rule: infinite_finite_induct)
haftmann@59010
  1107
  case infinite then show ?case by (auto intro: dvdI)
haftmann@59010
  1108
next
haftmann@59010
  1109
  case empty then show ?case by (auto intro: dvdI)
haftmann@59010
  1110
next
haftmann@59010
  1111
  case (insert a A) then
haftmann@59010
  1112
  have "f a dvd g a" and "setprod f A dvd setprod g A" by simp_all
haftmann@59010
  1113
  then obtain r s where "g a = f a * r" and "setprod g A = setprod f A * s" by (auto elim!: dvdE)
haftmann@59010
  1114
  then have "g a * setprod g A = f a * setprod f A * (r * s)" by (simp add: ac_simps)
haftmann@59010
  1115
  with insert.hyps show ?case by (auto intro: dvdI)
haftmann@59010
  1116
qed
haftmann@59010
  1117
haftmann@59010
  1118
lemma setprod_dvd_setprod_subset:
haftmann@59010
  1119
  "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> setprod f A dvd setprod f B"
haftmann@59010
  1120
  by (auto simp add: setprod.subset_diff ac_simps intro: dvdI)
haftmann@59010
  1121
haftmann@59010
  1122
end
haftmann@59010
  1123
haftmann@54744
  1124
wenzelm@60758
  1125
subsubsection \<open>Properties in more restricted classes of structures\<close>
haftmann@54744
  1126
haftmann@59010
  1127
context comm_semiring_1
haftmann@59010
  1128
begin
haftmann@54744
  1129
haftmann@59010
  1130
lemma dvd_setprod_eqI [intro]:
haftmann@59010
  1131
  assumes "finite A" and "a \<in> A" and "b = f a"
haftmann@59010
  1132
  shows "b dvd setprod f A"
haftmann@59010
  1133
proof -
wenzelm@60758
  1134
  from \<open>finite A\<close> have "setprod f (insert a (A - {a})) = f a * setprod f (A - {a})"
haftmann@59010
  1135
    by (intro setprod.insert) auto
wenzelm@60758
  1136
  also from \<open>a \<in> A\<close> have "insert a (A - {a}) = A" by blast
haftmann@59010
  1137
  finally have "setprod f A = f a * setprod f (A - {a})" .
wenzelm@60758
  1138
  with \<open>b = f a\<close> show ?thesis by simp
haftmann@59010
  1139
qed
haftmann@54744
  1140
haftmann@59010
  1141
lemma dvd_setprodI [intro]:
haftmann@59010
  1142
  assumes "finite A" and "a \<in> A"
haftmann@59010
  1143
  shows "f a dvd setprod f A"
haftmann@59010
  1144
  using assms by auto
haftmann@54744
  1145
haftmann@59010
  1146
lemma setprod_zero:
haftmann@59010
  1147
  assumes "finite A" and "\<exists>a\<in>A. f a = 0"
haftmann@59010
  1148
  shows "setprod f A = 0"
haftmann@59010
  1149
using assms proof (induct A)
haftmann@59010
  1150
  case empty then show ?case by simp
haftmann@59010
  1151
next
haftmann@59010
  1152
  case (insert a A)
haftmann@59010
  1153
  then have "f a = 0 \<or> (\<exists>a\<in>A. f a = 0)" by simp
haftmann@59010
  1154
  then have "f a * setprod f A = 0" by rule (simp_all add: insert)
haftmann@59010
  1155
  with insert show ?case by simp
haftmann@59010
  1156
qed
haftmann@54744
  1157
haftmann@54744
  1158
lemma setprod_dvd_setprod_subset2:
haftmann@59010
  1159
  assumes "finite B" and "A \<subseteq> B" and "\<And>a. a \<in> A \<Longrightarrow> f a dvd g a"
haftmann@59010
  1160
  shows "setprod f A dvd setprod g B"
haftmann@59010
  1161
proof -
haftmann@59010
  1162
  from assms have "setprod f A dvd setprod g A"
haftmann@59010
  1163
    by (auto intro: setprod_dvd_setprod)
haftmann@59010
  1164
  moreover from assms have "setprod g A dvd setprod g B"
haftmann@59010
  1165
    by (auto intro: setprod_dvd_setprod_subset)
haftmann@59010
  1166
  ultimately show ?thesis by (rule dvd_trans)
haftmann@59010
  1167
qed
haftmann@59010
  1168
haftmann@59010
  1169
end
haftmann@59010
  1170
haftmann@59010
  1171
lemma setprod_zero_iff [simp]:
haftmann@59010
  1172
  assumes "finite A"
haftmann@59833
  1173
  shows "setprod f A = (0::'a::semidom) \<longleftrightarrow> (\<exists>a\<in>A. f a = 0)"
haftmann@59010
  1174
  using assms by (induct A) (auto simp: no_zero_divisors)
haftmann@59010
  1175
haftmann@60353
  1176
lemma (in semidom_divide) setprod_diff1:
haftmann@60353
  1177
  assumes "finite A" and "f a \<noteq> 0"
haftmann@60429
  1178
  shows "setprod f (A - {a}) = (if a \<in> A then setprod f A div f a else setprod f A)"
haftmann@60353
  1179
proof (cases "a \<notin> A")
haftmann@60353
  1180
  case True then show ?thesis by simp
haftmann@60353
  1181
next
haftmann@60353
  1182
  case False with assms show ?thesis
haftmann@60353
  1183
  proof (induct A rule: finite_induct)
haftmann@60353
  1184
    case empty then show ?case by simp
haftmann@60353
  1185
  next
haftmann@60353
  1186
    case (insert b B)
haftmann@60353
  1187
    then show ?case
haftmann@60353
  1188
    proof (cases "a = b")
haftmann@60353
  1189
      case True with insert show ?thesis by simp
haftmann@60353
  1190
    next
haftmann@60353
  1191
      case False with insert have "a \<in> B" by simp
haftmann@60353
  1192
      def C \<equiv> "B - {a}"
wenzelm@60758
  1193
      with \<open>finite B\<close> \<open>a \<in> B\<close>
haftmann@60353
  1194
        have *: "B = insert a C" "finite C" "a \<notin> C" by auto
haftmann@60353
  1195
      with insert show ?thesis by (auto simp add: insert_commute ac_simps)
haftmann@60353
  1196
    qed
haftmann@60353
  1197
  qed
haftmann@60353
  1198
qed
haftmann@54744
  1199
haftmann@59867
  1200
lemma (in field) setprod_inversef: 
haftmann@59010
  1201
  "finite A \<Longrightarrow> setprod (inverse \<circ> f) A = inverse (setprod f A)"
haftmann@59010
  1202
  by (induct A rule: finite_induct) simp_all
haftmann@59010
  1203
haftmann@59867
  1204
lemma (in field) setprod_dividef:
haftmann@59010
  1205
  "finite A \<Longrightarrow> (\<Prod>x\<in>A. f x / g x) = setprod f A / setprod g A"
haftmann@59010
  1206
  using setprod_inversef [of A g] by (simp add: divide_inverse setprod.distrib)
haftmann@54744
  1207
haftmann@59010
  1208
lemma setprod_Un:
haftmann@59010
  1209
  fixes f :: "'b \<Rightarrow> 'a :: field"
haftmann@59010
  1210
  assumes "finite A" and "finite B"
haftmann@59010
  1211
  and "\<forall>x\<in>A \<inter> B. f x \<noteq> 0"
haftmann@59010
  1212
  shows "setprod f (A \<union> B) = setprod f A * setprod f B / setprod f (A \<inter> B)"
haftmann@59010
  1213
proof -
haftmann@59010
  1214
  from assms have "setprod f A * setprod f B = setprod f (A \<union> B) * setprod f (A \<inter> B)"
haftmann@59010
  1215
    by (simp add: setprod.union_inter [symmetric, of A B])
haftmann@59010
  1216
  with assms show ?thesis by simp
haftmann@59010
  1217
qed
haftmann@54744
  1218
haftmann@59010
  1219
lemma (in linordered_semidom) setprod_nonneg:
haftmann@59010
  1220
  "(\<forall>a\<in>A. 0 \<le> f a) \<Longrightarrow> 0 \<le> setprod f A"
haftmann@59010
  1221
  by (induct A rule: infinite_finite_induct) simp_all
haftmann@59010
  1222
haftmann@59010
  1223
lemma (in linordered_semidom) setprod_pos:
haftmann@59010
  1224
  "(\<forall>a\<in>A. 0 < f a) \<Longrightarrow> 0 < setprod f A"
haftmann@59010
  1225
  by (induct A rule: infinite_finite_induct) simp_all
haftmann@59010
  1226
haftmann@59010
  1227
lemma (in linordered_semidom) setprod_mono:
haftmann@54744
  1228
  assumes "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i"
haftmann@54744
  1229
  shows "setprod f A \<le> setprod g A"
haftmann@59010
  1230
  using assms by (induct A rule: infinite_finite_induct)
haftmann@59010
  1231
    (auto intro!: setprod_nonneg mult_mono)
haftmann@54744
  1232
lp15@60974
  1233
lemma (in linordered_semidom) setprod_mono_strict:
lp15@60974
  1234
    assumes"finite A" "\<forall>i\<in>A. 0 \<le> f i \<and> f i < g i" "A \<noteq> {}"
lp15@60974
  1235
    shows "setprod f A < setprod g A"
lp15@60974
  1236
using assms 
lp15@60974
  1237
apply (induct A rule: finite_induct)
lp15@60974
  1238
apply (simp add: )
lp15@60974
  1239
apply (force intro: mult_strict_mono' setprod_nonneg)
lp15@60974
  1240
done
lp15@60974
  1241
haftmann@59010
  1242
lemma (in linordered_field) abs_setprod:
haftmann@59010
  1243
  "\<bar>setprod f A\<bar> = (\<Prod>x\<in>A. \<bar>f x\<bar>)"
haftmann@59010
  1244
  by (induct A rule: infinite_finite_induct) (simp_all add: abs_mult)
haftmann@54744
  1245
haftmann@54744
  1246
lemma setprod_eq_1_iff [simp]:
haftmann@59010
  1247
  "finite A \<Longrightarrow> setprod f A = 1 \<longleftrightarrow> (\<forall>a\<in>A. f a = (1::nat))"
haftmann@59010
  1248
  by (induct A rule: finite_induct) simp_all
haftmann@54744
  1249
haftmann@59010
  1250
lemma setprod_pos_nat_iff [simp]:
haftmann@59010
  1251
  "finite A \<Longrightarrow> setprod f A > 0 \<longleftrightarrow> (\<forall>a\<in>A. f a > (0::nat))"
haftmann@59010
  1252
  using setprod_zero_iff by (simp del:neq0_conv add:neq0_conv [symmetric])
haftmann@54744
  1253
lp15@60974
  1254
lemma setsum_nonneg_eq_0_iff:
lp15@60974
  1255
  fixes f :: "'a \<Rightarrow> 'b::ordered_ab_group_add"
lp15@60974
  1256
  shows "\<lbrakk>finite A; \<forall>x\<in>A. 0 \<le> f x\<rbrakk> \<Longrightarrow> setsum f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
lp15@60974
  1257
  apply (induct set: finite, simp)
lp15@60974
  1258
  apply (simp add: add_nonneg_eq_0_iff setsum_nonneg)
lp15@60974
  1259
  done
lp15@60974
  1260
haftmann@54744
  1261
end