src/HOL/Groups_Big.thy
author wenzelm
Tue Sep 26 20:54:40 2017 +0200 (21 months ago)
changeset 66695 91500c024c7f
parent 66364 fa3247e6ee4b
child 66804 3f9bb52082c4
permissions -rw-r--r--
tuned;
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(*  Title:      HOL/Groups_Big.thy
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    Author:     Tobias Nipkow
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    Author:     Lawrence C Paulson
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    Author:     Markus Wenzel
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    Author:     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 Power
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begin
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subsection \<open>Generic monoid operation over a set\<close>
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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 eq_fold: "F g A = Finite_Set.fold (f \<circ> g) \<^bold>1 A"
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lemma infinite [simp]: "\<not> finite A \<Longrightarrow> F g A = \<^bold>1"
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  by (simp add: eq_fold)
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lemma empty [simp]: "F g {} = \<^bold>1"
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  by (simp add: eq_fold)
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lemma insert [simp]: "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> F g (insert x A) = g x \<^bold>* F g A"
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  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 \<^bold>* F g (A - {x})"
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proof -
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  from \<open>x \<in> A\<close> obtain B where B: "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> B 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: "finite A \<Longrightarrow> F g (insert x A) = g x \<^bold>* F g (A - {x})"
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  by (cases "x \<in> A") (simp_all add: remove insert_absorb)
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lemma insert_if: "finite A \<Longrightarrow> F g (insert x A) = (if x \<in> A then F g A else g x \<^bold>* F g A)"
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  by (cases "x \<in> A") (simp_all add: insert_absorb)
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lemma neutral: "\<forall>x\<in>A. g x = \<^bold>1 \<Longrightarrow> F g A = \<^bold>1"
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  by (induct A rule: infinite_finite_induct) simp_all
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lemma neutral_const [simp]: "F (\<lambda>_. \<^bold>1) A = \<^bold>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) \<^bold>* F g (A \<inter> B) = F g A \<^bold>* 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
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proof (induct A)
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  case empty
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  then show ?case by simp
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next
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  case (insert x A)
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  then show ?case
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    by (auto simp: 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 "\<forall>x \<in> A \<inter> B. g x = \<^bold>1"
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  shows "F g (A \<union> B) = F g A \<^bold>* 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 \<^bold>* 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) \<^bold>* F g (B - A) \<^bold>* 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
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    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) \<^bold>* 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) \<^bold>* 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> \<^bold>1"
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  obtains a where "a \<in> A" and "g a \<noteq> \<^bold>1"
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proof -
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  from assms have "\<exists>a\<in>A. g a \<noteq> \<^bold>1"
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  proof (induct A rule: infinite_finite_induct)
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    case infinite
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    then show ?case by simp
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  next
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    case empty
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    then show ?case by simp
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  next
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    case (insert a A)
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    then show ?case by fastforce
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  qed
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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
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    by (simp add: eq_fold fold_image comp_assoc)
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next
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  case False
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  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 [fundef_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) (use assms in \<open>simp_all add: simp_implies_def\<close>)
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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 \<inter> 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 "finite C")
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  case True
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  from UNION_disjoint [OF this assms] show ?thesis by simp
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next
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  case False
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  then show ?thesis by (auto dest: finite_UnionD intro: infinite)
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qed
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lemma distrib: "F (\<lambda>x. g x \<^bold>* h x) A = F g A \<^bold>* F h A"
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  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)
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     apply assumption
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    apply 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)
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     apply simp
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    apply 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 \<^bold>1 \<^bold>1"
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    and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 \<^bold>* y1) (x2 \<^bold>* y2)"
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    and fin: "finite S"
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    and R_h_g: "\<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 fin by (rule finite_subset_induct) (use assms in auto)
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lemma mono_neutral_cong_left:
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  assumes "finite T"
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    and "S \<subseteq> T"
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    and "\<forall>i \<in> T - S. h i = \<^bold>1"
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    and "\<And>x. x \<in> S \<Longrightarrow> g x = h x"
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  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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  "finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. g i = \<^bold>1 \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> g x = h x) \<Longrightarrow>
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    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: "finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. g i = \<^bold>1 \<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: "finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. g i = \<^bold>1 \<Longrightarrow> F g T = F g S"
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  by (blast intro!: mono_neutral_left [symmetric])
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lemma mono_neutral_cong:
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  assumes [simp]: "finite T" "finite S"
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    and *: "\<And>i. i \<in> T - S \<Longrightarrow> h i = \<^bold>1" "\<And>i. i \<in> S - T \<Longrightarrow> g i = \<^bold>1"
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    and gh: "\<And>x. x \<in> S \<inter> T \<Longrightarrow> g x = h x"
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 shows "F g S = F h T"
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proof-
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  have "F g S = F g (S \<inter> T)"
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    by(rule mono_neutral_right)(auto intro: *)
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  also have "\<dots> = F h (S \<inter> T)" using refl gh by(rule cong)
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  also have "\<dots> = F h T"
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    by(rule mono_neutral_left)(auto intro: *)
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  finally show ?thesis .
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qed
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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 "finite S")
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    case True
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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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  next
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    case False
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    then show ?thesis by simp
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  qed
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qed
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   303
lemma reindex_nontrivial:
haftmann@57418
   304
  assumes "finite A"
wenzelm@63654
   305
    and nz: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> h x = h y \<Longrightarrow> g (h x) = \<^bold>1"
haftmann@57418
   306
  shows "F g (h ` A) = F (g \<circ> h) A"
haftmann@57418
   307
proof (subst reindex_bij_betw_not_neutral [symmetric])
haftmann@63290
   308
  show "bij_betw h (A - {x \<in> A. (g \<circ> h) x = \<^bold>1}) (h ` A - h ` {x \<in> A. (g \<circ> h) x = \<^bold>1})"
haftmann@57418
   309
    using nz by (auto intro!: inj_onI simp: bij_betw_def)
wenzelm@63654
   310
qed (use \<open>finite A\<close> in auto)
haftmann@57418
   311
hoelzl@57129
   312
lemma reindex_bij_witness_not_neutral:
hoelzl@57129
   313
  assumes fin: "finite S'" "finite T'"
hoelzl@57129
   314
  assumes witness:
hoelzl@57129
   315
    "\<And>a. a \<in> S - S' \<Longrightarrow> i (j a) = a"
hoelzl@57129
   316
    "\<And>a. a \<in> S - S' \<Longrightarrow> j a \<in> T - T'"
hoelzl@57129
   317
    "\<And>b. b \<in> T - T' \<Longrightarrow> j (i b) = b"
hoelzl@57129
   318
    "\<And>b. b \<in> T - T' \<Longrightarrow> i b \<in> S - S'"
hoelzl@57129
   319
  assumes nn:
hoelzl@57129
   320
    "\<And>a. a \<in> S' \<Longrightarrow> g a = z"
hoelzl@57129
   321
    "\<And>b. b \<in> T' \<Longrightarrow> h b = z"
hoelzl@57129
   322
  assumes eq:
hoelzl@57129
   323
    "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a"
hoelzl@57129
   324
  shows "F g S = F h T"
hoelzl@57129
   325
proof -
hoelzl@57129
   326
  have bij: "bij_betw j (S - (S' \<inter> S)) (T - (T' \<inter> T))"
hoelzl@57129
   327
    using witness by (intro bij_betw_byWitness[where f'=i]) auto
hoelzl@57129
   328
  have F_eq: "F g S = F (\<lambda>x. h (j x)) S"
hoelzl@57129
   329
    by (intro cong) (auto simp: eq)
hoelzl@57129
   330
  show ?thesis
hoelzl@57129
   331
    unfolding F_eq using fin nn eq
hoelzl@57129
   332
    by (intro reindex_bij_betw_not_neutral[OF _ _ bij]) auto
hoelzl@57129
   333
qed
hoelzl@57129
   334
lp15@66089
   335
lemma delta [simp]:
haftmann@54744
   336
  assumes fS: "finite S"
haftmann@63290
   337
  shows "F (\<lambda>k. if k = a then b k else \<^bold>1) S = (if a \<in> S then b a else \<^bold>1)"
wenzelm@63654
   338
proof -
wenzelm@63654
   339
  let ?f = "(\<lambda>k. if k = a then b k else \<^bold>1)"
wenzelm@63654
   340
  show ?thesis
wenzelm@63654
   341
  proof (cases "a \<in> S")
wenzelm@63654
   342
    case False
wenzelm@63654
   343
    then have "\<forall>k\<in>S. ?f k = \<^bold>1" by simp
wenzelm@63654
   344
    with False show ?thesis by simp
wenzelm@63654
   345
  next
wenzelm@63654
   346
    case True
haftmann@54744
   347
    let ?A = "S - {a}"
haftmann@54744
   348
    let ?B = "{a}"
wenzelm@63654
   349
    from True have eq: "S = ?A \<union> ?B" by blast
haftmann@54744
   350
    have dj: "?A \<inter> ?B = {}" by simp
hoelzl@62376
   351
    from fS have fAB: "finite ?A" "finite ?B" by auto
haftmann@63290
   352
    have "F ?f S = F ?f ?A \<^bold>* F ?f ?B"
wenzelm@63654
   353
      using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]] by simp
wenzelm@63654
   354
    with True show ?thesis by simp
wenzelm@63654
   355
  qed
haftmann@54744
   356
qed
haftmann@54744
   357
lp15@66089
   358
lemma delta' [simp]:
wenzelm@63654
   359
  assumes fin: "finite S"
haftmann@63290
   360
  shows "F (\<lambda>k. if a = k then b k else \<^bold>1) S = (if a \<in> S then b a else \<^bold>1)"
wenzelm@63654
   361
  using delta [OF fin, of a b, symmetric] by (auto intro: cong)
haftmann@54744
   362
haftmann@54744
   363
lemma If_cases:
haftmann@54744
   364
  fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
wenzelm@63654
   365
  assumes fin: "finite A"
wenzelm@63654
   366
  shows "F (\<lambda>x. if P x then h x else g x) A = F h (A \<inter> {x. P x}) \<^bold>* F g (A \<inter> - {x. P x})"
haftmann@54744
   367
proof -
wenzelm@63654
   368
  have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}" "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}"
haftmann@54744
   369
    by blast+
wenzelm@63654
   370
  from fin have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
haftmann@54744
   371
  let ?g = "\<lambda>x. if P x then h x else g x"
wenzelm@63654
   372
  from union_disjoint [OF f a(2), of ?g] a(1) show ?thesis
haftmann@54744
   373
    by (subst (1 2) cong) simp_all
haftmann@54744
   374
qed
haftmann@54744
   375
wenzelm@63654
   376
lemma cartesian_product: "F (\<lambda>x. F (g x) B) A = F (case_prod g) (A \<times> B)"
wenzelm@63654
   377
  apply (rule sym)
wenzelm@63654
   378
  apply (cases "finite A")
wenzelm@63654
   379
   apply (cases "finite B")
wenzelm@63654
   380
    apply (simp add: Sigma)
wenzelm@63654
   381
   apply (cases "A = {}")
wenzelm@63654
   382
    apply simp
wenzelm@63654
   383
   apply simp
wenzelm@63654
   384
   apply (auto intro: infinite dest: finite_cartesian_productD2)
wenzelm@63654
   385
  apply (cases "B = {}")
wenzelm@63654
   386
   apply (auto intro: infinite dest: finite_cartesian_productD1)
wenzelm@63654
   387
  done
haftmann@54744
   388
haftmann@57418
   389
lemma inter_restrict:
haftmann@57418
   390
  assumes "finite A"
haftmann@63290
   391
  shows "F g (A \<inter> B) = F (\<lambda>x. if x \<in> B then g x else \<^bold>1) A"
haftmann@57418
   392
proof -
haftmann@63290
   393
  let ?g = "\<lambda>x. if x \<in> A \<inter> B then g x else \<^bold>1"
wenzelm@63654
   394
  have "\<forall>i\<in>A - A \<inter> B. (if i \<in> A \<inter> B then g i else \<^bold>1) = \<^bold>1" by simp
haftmann@57418
   395
  moreover have "A \<inter> B \<subseteq> A" by blast
wenzelm@63654
   396
  ultimately have "F ?g (A \<inter> B) = F ?g A"
wenzelm@63654
   397
    using \<open>finite A\<close> by (intro mono_neutral_left) auto
haftmann@57418
   398
  then show ?thesis by simp
haftmann@57418
   399
qed
haftmann@57418
   400
haftmann@57418
   401
lemma inter_filter:
haftmann@63290
   402
  "finite A \<Longrightarrow> F g {x \<in> A. P x} = F (\<lambda>x. if P x then g x else \<^bold>1) A"
haftmann@57418
   403
  by (simp add: inter_restrict [symmetric, of A "{x. P x}" g, simplified mem_Collect_eq] Int_def)
haftmann@57418
   404
haftmann@57418
   405
lemma Union_comp:
haftmann@57418
   406
  assumes "\<forall>A \<in> B. finite A"
wenzelm@63654
   407
    and "\<And>A1 A2 x. A1 \<in> B \<Longrightarrow> A2 \<in> B \<Longrightarrow> A1 \<noteq> A2 \<Longrightarrow> x \<in> A1 \<Longrightarrow> x \<in> A2 \<Longrightarrow> g x = \<^bold>1"
haftmann@57418
   408
  shows "F g (\<Union>B) = (F \<circ> F) g B"
wenzelm@63654
   409
  using assms
wenzelm@63654
   410
proof (induct B rule: infinite_finite_induct)
haftmann@57418
   411
  case (infinite A)
haftmann@57418
   412
  then have "\<not> finite (\<Union>A)" by (blast dest: finite_UnionD)
haftmann@57418
   413
  with infinite show ?case by simp
haftmann@57418
   414
next
wenzelm@63654
   415
  case empty
wenzelm@63654
   416
  then show ?case by simp
haftmann@57418
   417
next
haftmann@57418
   418
  case (insert A B)
haftmann@57418
   419
  then have "finite A" "finite B" "finite (\<Union>B)" "A \<notin> B"
haftmann@63290
   420
    and "\<forall>x\<in>A \<inter> \<Union>B. g x = \<^bold>1"
wenzelm@63654
   421
    and H: "F g (\<Union>B) = (F \<circ> F) g B" by auto
haftmann@63290
   422
  then have "F g (A \<union> \<Union>B) = F g A \<^bold>* F g (\<Union>B)"
haftmann@57418
   423
    by (simp add: union_inter_neutral)
wenzelm@60758
   424
  with \<open>finite B\<close> \<open>A \<notin> B\<close> show ?case
haftmann@57418
   425
    by (simp add: H)
haftmann@57418
   426
qed
haftmann@57418
   427
wenzelm@63654
   428
lemma commute: "F (\<lambda>i. F (g i) B) A = F (\<lambda>j. F (\<lambda>i. g i j) A) B"
haftmann@57418
   429
  unfolding cartesian_product
haftmann@57418
   430
  by (rule reindex_bij_witness [where i = "\<lambda>(i, j). (j, i)" and j = "\<lambda>(i, j). (j, i)"]) auto
haftmann@57418
   431
haftmann@57418
   432
lemma commute_restrict:
haftmann@57418
   433
  "finite A \<Longrightarrow> finite B \<Longrightarrow>
haftmann@57418
   434
    F (\<lambda>x. F (g x) {y. y \<in> B \<and> R x y}) A = F (\<lambda>y. F (\<lambda>x. g x y) {x. x \<in> A \<and> R x y}) B"
haftmann@57418
   435
  by (simp add: inter_filter) (rule commute)
haftmann@57418
   436
haftmann@57418
   437
lemma Plus:
haftmann@57418
   438
  fixes A :: "'b set" and B :: "'c set"
haftmann@57418
   439
  assumes fin: "finite A" "finite B"
haftmann@63290
   440
  shows "F g (A <+> B) = F (g \<circ> Inl) A \<^bold>* F (g \<circ> Inr) B"
haftmann@57418
   441
proof -
haftmann@57418
   442
  have "A <+> B = Inl ` A \<union> Inr ` B" by auto
wenzelm@63654
   443
  moreover from fin have "finite (Inl ` A)" "finite (Inr ` B)" by auto
wenzelm@63654
   444
  moreover have "Inl ` A \<inter> Inr ` B = {}" by auto
wenzelm@63654
   445
  moreover have "inj_on Inl A" "inj_on Inr B" by (auto intro: inj_onI)
wenzelm@63654
   446
  ultimately show ?thesis
wenzelm@63654
   447
    using fin by (simp add: union_disjoint reindex)
haftmann@57418
   448
qed
haftmann@57418
   449
haftmann@58195
   450
lemma same_carrier:
haftmann@58195
   451
  assumes "finite C"
haftmann@58195
   452
  assumes subset: "A \<subseteq> C" "B \<subseteq> C"
haftmann@63290
   453
  assumes trivial: "\<And>a. a \<in> C - A \<Longrightarrow> g a = \<^bold>1" "\<And>b. b \<in> C - B \<Longrightarrow> h b = \<^bold>1"
haftmann@58195
   454
  shows "F g A = F h B \<longleftrightarrow> F g C = F h C"
haftmann@58195
   455
proof -
wenzelm@63654
   456
  have "finite A" and "finite B" and "finite (C - A)" and "finite (C - B)"
wenzelm@63654
   457
    using \<open>finite C\<close> subset by (auto elim: finite_subset)
haftmann@58195
   458
  from subset have [simp]: "A - (C - A) = A" by auto
haftmann@58195
   459
  from subset have [simp]: "B - (C - B) = B" by auto
haftmann@58195
   460
  from subset have "C = A \<union> (C - A)" by auto
haftmann@58195
   461
  then have "F g C = F g (A \<union> (C - A))" by simp
haftmann@63290
   462
  also have "\<dots> = F g (A - (C - A)) \<^bold>* F g (C - A - A) \<^bold>* F g (A \<inter> (C - A))"
wenzelm@60758
   463
    using \<open>finite A\<close> \<open>finite (C - A)\<close> by (simp only: union_diff2)
wenzelm@63654
   464
  finally have *: "F g C = F g A" using trivial by simp
haftmann@58195
   465
  from subset have "C = B \<union> (C - B)" by auto
haftmann@58195
   466
  then have "F h C = F h (B \<union> (C - B))" by simp
haftmann@63290
   467
  also have "\<dots> = F h (B - (C - B)) \<^bold>* F h (C - B - B) \<^bold>* F h (B \<inter> (C - B))"
wenzelm@60758
   468
    using \<open>finite B\<close> \<open>finite (C - B)\<close> by (simp only: union_diff2)
wenzelm@63654
   469
  finally have "F h C = F h B"
wenzelm@63654
   470
    using trivial by simp
wenzelm@63654
   471
  with * show ?thesis by simp
haftmann@58195
   472
qed
haftmann@58195
   473
haftmann@58195
   474
lemma same_carrierI:
haftmann@58195
   475
  assumes "finite C"
haftmann@58195
   476
  assumes subset: "A \<subseteq> C" "B \<subseteq> C"
haftmann@63290
   477
  assumes trivial: "\<And>a. a \<in> C - A \<Longrightarrow> g a = \<^bold>1" "\<And>b. b \<in> C - B \<Longrightarrow> h b = \<^bold>1"
haftmann@58195
   478
  assumes "F g C = F h C"
haftmann@58195
   479
  shows "F g A = F h B"
haftmann@58195
   480
  using assms same_carrier [of C A B] by simp
haftmann@58195
   481
haftmann@54744
   482
end
haftmann@54744
   483
haftmann@54744
   484
wenzelm@60758
   485
subsection \<open>Generalized summation over a set\<close>
haftmann@54744
   486
haftmann@54744
   487
context comm_monoid_add
haftmann@54744
   488
begin
haftmann@54744
   489
nipkow@64267
   490
sublocale sum: comm_monoid_set plus 0
nipkow@64267
   491
  defines sum = sum.F ..
haftmann@54744
   492
nipkow@64267
   493
abbreviation Sum ("\<Sum>_" [1000] 999)
nipkow@64267
   494
  where "\<Sum>A \<equiv> sum (\<lambda>x. x) A"
haftmann@54744
   495
haftmann@54744
   496
end
haftmann@54744
   497
nipkow@64267
   498
text \<open>Now: lot's of fancy syntax. First, @{term "sum (\<lambda>x. e) A"} is written \<open>\<Sum>x\<in>A. e\<close>.\<close>
haftmann@54744
   499
wenzelm@61955
   500
syntax (ASCII)
nipkow@64267
   501
  "_sum" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b::comm_monoid_add"  ("(3SUM _:_./ _)" [0, 51, 10] 10)
haftmann@54744
   502
syntax
nipkow@64267
   503
  "_sum" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b::comm_monoid_add"  ("(2\<Sum>_\<in>_./ _)" [0, 51, 10] 10)
wenzelm@61799
   504
translations \<comment> \<open>Beware of argument permutation!\<close>
nipkow@64267
   505
  "\<Sum>i\<in>A. b" \<rightleftharpoons> "CONST sum (\<lambda>i. b) A"
haftmann@54744
   506
wenzelm@61955
   507
text \<open>Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter \<open>\<Sum>x|P. e\<close>.\<close>
haftmann@54744
   508
wenzelm@61955
   509
syntax (ASCII)
nipkow@64267
   510
  "_qsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(3SUM _ |/ _./ _)" [0, 0, 10] 10)
haftmann@54744
   511
syntax
nipkow@64267
   512
  "_qsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(2\<Sum>_ | (_)./ _)" [0, 0, 10] 10)
haftmann@54744
   513
translations
nipkow@64267
   514
  "\<Sum>x|P. t" => "CONST sum (\<lambda>x. t) {x. P}"
haftmann@54744
   515
wenzelm@60758
   516
print_translation \<open>
haftmann@54744
   517
let
nipkow@64267
   518
  fun sum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) $ Abs (y, Ty, P)] =
haftmann@54744
   519
        if x <> y then raise Match
haftmann@54744
   520
        else
haftmann@54744
   521
          let
haftmann@54744
   522
            val x' = Syntax_Trans.mark_bound_body (x, Tx);
haftmann@54744
   523
            val t' = subst_bound (x', t);
haftmann@54744
   524
            val P' = subst_bound (x', P);
haftmann@54744
   525
          in
nipkow@64267
   526
            Syntax.const @{syntax_const "_qsum"} $ Syntax_Trans.mark_bound_abs (x, Tx) $ P' $ t'
haftmann@54744
   527
          end
nipkow@64267
   528
    | sum_tr' _ = raise Match;
nipkow@64267
   529
in [(@{const_syntax sum}, K sum_tr')] end
wenzelm@60758
   530
\<close>
haftmann@54744
   531
wenzelm@63654
   532
(* TODO generalization candidates *)
haftmann@54744
   533
nipkow@64267
   534
lemma (in comm_monoid_add) sum_image_gen:
wenzelm@63654
   535
  assumes fin: "finite S"
nipkow@64267
   536
  shows "sum g S = sum (\<lambda>y. sum g {x. x \<in> S \<and> f x = y}) (f ` S)"
wenzelm@63654
   537
proof -
wenzelm@63654
   538
  have "{y. y\<in> f`S \<and> f x = y} = {f x}" if "x \<in> S" for x
wenzelm@63654
   539
    using that by auto
nipkow@64267
   540
  then have "sum g S = sum (\<lambda>x. sum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
haftmann@57418
   541
    by simp
nipkow@64267
   542
  also have "\<dots> = sum (\<lambda>y. sum g {x. x \<in> S \<and> f x = y}) (f ` S)"
nipkow@64267
   543
    by (rule sum.commute_restrict [OF fin finite_imageI [OF fin]])
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
nipkow@64267
   550
lemma sum_Un:
nipkow@64267
   551
  "finite A \<Longrightarrow> finite B \<Longrightarrow> sum f (A \<union> B) = sum f A + sum f B - sum f (A \<inter> B)"
wenzelm@63654
   552
  for f :: "'b \<Rightarrow> 'a::ab_group_add"
nipkow@64267
   553
  by (subst sum.union_inter [symmetric]) (auto simp add: algebra_simps)
haftmann@54744
   554
nipkow@64267
   555
lemma sum_Un2:
haftmann@54744
   556
  assumes "finite (A \<union> B)"
nipkow@64267
   557
  shows "sum f (A \<union> B) = sum f (A - B) + sum f (B - A) + sum 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
wenzelm@63654
   561
  with assms show ?thesis
nipkow@64267
   562
    by simp (subst sum.union_disjoint, auto)+
haftmann@54744
   563
qed
haftmann@54744
   564
nipkow@64267
   565
lemma sum_diff1:
wenzelm@63654
   566
  fixes f :: "'b \<Rightarrow> 'a::ab_group_add"
wenzelm@63654
   567
  assumes "finite A"
nipkow@64267
   568
  shows "sum f (A - {a}) = (if a \<in> A then sum f A - f a else sum f A)"
wenzelm@63654
   569
  using assms by induct (auto simp: insert_Diff_if)
haftmann@54744
   570
nipkow@64267
   571
lemma sum_diff:
wenzelm@63654
   572
  fixes f :: "'b \<Rightarrow> 'a::ab_group_add"
wenzelm@63654
   573
  assumes "finite A" "B \<subseteq> A"
nipkow@64267
   574
  shows "sum f (A - B) = sum f A - sum f B"
haftmann@54744
   575
proof -
wenzelm@63654
   576
  from assms(2,1) have "finite B" by (rule finite_subset)
wenzelm@63654
   577
  from this \<open>B \<subseteq> A\<close>
wenzelm@63654
   578
  show ?thesis
haftmann@54744
   579
  proof induct
haftmann@54744
   580
    case empty
wenzelm@63654
   581
    thus ?case by simp
haftmann@54744
   582
  next
haftmann@54744
   583
    case (insert x F)
wenzelm@63654
   584
    with \<open>finite A\<close> \<open>finite B\<close> show ?case
nipkow@64267
   585
      by (simp add: Diff_insert[where a=x and B=F] sum_diff1 insert_absorb)
haftmann@54744
   586
  qed
haftmann@54744
   587
qed
haftmann@54744
   588
nipkow@64267
   589
lemma (in ordered_comm_monoid_add) sum_mono:
wenzelm@63915
   590
  "(\<And>i. i\<in>K \<Longrightarrow> f i \<le> g i) \<Longrightarrow> (\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
wenzelm@63915
   591
  by (induct K rule: infinite_finite_induct) (use add_mono in auto)
haftmann@54744
   592
nipkow@64267
   593
lemma (in strict_ordered_comm_monoid_add) sum_strict_mono:
wenzelm@63654
   594
  assumes "finite A" "A \<noteq> {}"
wenzelm@63654
   595
    and "\<And>x. x \<in> A \<Longrightarrow> f x < g x"
nipkow@64267
   596
  shows "sum f A < sum g A"
haftmann@54744
   597
  using assms
haftmann@54744
   598
proof (induct rule: finite_ne_induct)
wenzelm@63654
   599
  case singleton
wenzelm@63654
   600
  then show ?case by simp
haftmann@54744
   601
next
wenzelm@63654
   602
  case insert
wenzelm@63654
   603
  then show ?case by (auto simp: add_strict_mono)
haftmann@54744
   604
qed
haftmann@54744
   605
nipkow@64267
   606
lemma sum_strict_mono_ex1:
hoelzl@62376
   607
  fixes f g :: "'i \<Rightarrow> 'a::ordered_cancel_comm_monoid_add"
wenzelm@63654
   608
  assumes "finite A"
wenzelm@63654
   609
    and "\<forall>x\<in>A. f x \<le> g x"
wenzelm@63654
   610
    and "\<exists>a\<in>A. f a < g a"
nipkow@64267
   611
  shows "sum f A < sum g A"
haftmann@54744
   612
proof-
wenzelm@63654
   613
  from assms(3) obtain a where a: "a \<in> A" "f a < g a" by blast
nipkow@64267
   614
  have "sum f A = sum f ((A - {a}) \<union> {a})"
wenzelm@63654
   615
    by(simp add: insert_absorb[OF \<open>a \<in> A\<close>])
nipkow@64267
   616
  also have "\<dots> = sum f (A - {a}) + sum f {a}"
nipkow@64267
   617
    using \<open>finite A\<close> by(subst sum.union_disjoint) auto
nipkow@64267
   618
  also have "sum f (A - {a}) \<le> sum g (A - {a})"
nipkow@64267
   619
    by (rule sum_mono) (simp add: assms(2))
nipkow@64267
   620
  also from a have "sum f {a} < sum g {a}" by simp
nipkow@64267
   621
  also have "sum g (A - {a}) + sum g {a} = sum g((A - {a}) \<union> {a})"
nipkow@64267
   622
    using \<open>finite A\<close> by (subst sum.union_disjoint[symmetric]) auto
nipkow@64267
   623
  also have "\<dots> = sum g A" by (simp add: insert_absorb[OF \<open>a \<in> A\<close>])
wenzelm@63654
   624
  finally show ?thesis
wenzelm@63654
   625
    by (auto simp add: add_right_mono add_strict_left_mono)
haftmann@54744
   626
qed
haftmann@54744
   627
nipkow@64267
   628
lemma sum_mono_inv:
Andreas@63561
   629
  fixes f g :: "'i \<Rightarrow> 'a :: ordered_cancel_comm_monoid_add"
nipkow@64267
   630
  assumes eq: "sum f I = sum g I"
Andreas@63561
   631
  assumes le: "\<And>i. i \<in> I \<Longrightarrow> f i \<le> g i"
Andreas@63561
   632
  assumes i: "i \<in> I"
Andreas@63561
   633
  assumes I: "finite I"
Andreas@63561
   634
  shows "f i = g i"
wenzelm@63654
   635
proof (rule ccontr)
wenzelm@63654
   636
  assume "\<not> ?thesis"
Andreas@63561
   637
  with le[OF i] have "f i < g i" by simp
wenzelm@63654
   638
  with i have "\<exists>i\<in>I. f i < g i" ..
nipkow@64267
   639
  from sum_strict_mono_ex1[OF I _ this] le have "sum f I < sum g I"
wenzelm@63654
   640
    by blast
Andreas@63561
   641
  with eq show False by simp
Andreas@63561
   642
qed
Andreas@63561
   643
nipkow@64267
   644
lemma member_le_sum:
lp15@63938
   645
  fixes f :: "_ \<Rightarrow> 'b::{semiring_1, ordered_comm_monoid_add}"
lp15@66112
   646
  assumes "i \<in> A"
lp15@66112
   647
    and le: "\<And>x. x \<in> A - {i} \<Longrightarrow> 0 \<le> f x"
lp15@63938
   648
    and "finite A"
nipkow@64267
   649
  shows "f i \<le> sum f A"
lp15@63938
   650
proof -
nipkow@64267
   651
  have "f i \<le> sum f (A \<inter> {i})"
lp15@63938
   652
    by (simp add: assms)
lp15@63938
   653
  also have "... = (\<Sum>x\<in>A. if x \<in> {i} then f x else 0)"
nipkow@64267
   654
    using assms sum.inter_restrict by blast
nipkow@64267
   655
  also have "... \<le> sum f A"
nipkow@64267
   656
    apply (rule sum_mono)
lp15@63938
   657
    apply (auto simp: le)
lp15@63938
   658
    done
lp15@63938
   659
  finally show ?thesis .
lp15@63938
   660
qed
lp15@63938
   661
nipkow@64267
   662
lemma sum_negf: "(\<Sum>x\<in>A. - f x) = - (\<Sum>x\<in>A. f x)"
wenzelm@63654
   663
  for f :: "'b \<Rightarrow> 'a::ab_group_add"
wenzelm@63915
   664
  by (induct A rule: infinite_finite_induct) auto
haftmann@54744
   665
nipkow@64267
   666
lemma sum_subtractf: "(\<Sum>x\<in>A. f x - g x) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
wenzelm@63654
   667
  for f g :: "'b \<Rightarrow>'a::ab_group_add"
nipkow@64267
   668
  using sum.distrib [of f "- g" A] by (simp add: sum_negf)
haftmann@54744
   669
nipkow@64267
   670
lemma sum_subtractf_nat:
wenzelm@63654
   671
  "(\<And>x. x \<in> A \<Longrightarrow> g x \<le> f x) \<Longrightarrow> (\<Sum>x\<in>A. f x - g x) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
wenzelm@63654
   672
  for f g :: "'a \<Rightarrow> nat"
nipkow@64267
   673
  by (induct A rule: infinite_finite_induct) (auto simp: sum_mono)
hoelzl@59416
   674
wenzelm@63654
   675
context ordered_comm_monoid_add
wenzelm@63654
   676
begin
wenzelm@63654
   677
lp15@65680
   678
lemma sum_nonneg: "(\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x) \<Longrightarrow> 0 \<le> sum f A"
wenzelm@63915
   679
proof (induct A rule: infinite_finite_induct)
wenzelm@63915
   680
  case infinite
wenzelm@63915
   681
  then show ?case by simp
haftmann@54744
   682
next
wenzelm@63915
   683
  case empty
wenzelm@63915
   684
  then show ?case by simp
wenzelm@63915
   685
next
wenzelm@63915
   686
  case (insert x F)
nipkow@64267
   687
  then have "0 + 0 \<le> f x + sum f F" by (blast intro: add_mono)
wenzelm@63915
   688
  with insert show ?case by simp
haftmann@54744
   689
qed
haftmann@54744
   690
lp15@65680
   691
lemma sum_nonpos: "(\<And>x. x \<in> A \<Longrightarrow> f x \<le> 0) \<Longrightarrow> sum f A \<le> 0"
wenzelm@63915
   692
proof (induct A rule: infinite_finite_induct)
wenzelm@63915
   693
  case infinite
wenzelm@63915
   694
  then show ?case by simp
haftmann@54744
   695
next
wenzelm@63915
   696
  case empty
wenzelm@63915
   697
  then show ?case by simp
wenzelm@63915
   698
next
wenzelm@63915
   699
  case (insert x F)
nipkow@64267
   700
  then have "f x + sum f F \<le> 0 + 0" by (blast intro: add_mono)
wenzelm@63915
   701
  with insert show ?case by simp
haftmann@54744
   702
qed
haftmann@54744
   703
nipkow@64267
   704
lemma sum_nonneg_eq_0_iff:
lp15@65680
   705
  "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x) \<Longrightarrow> sum f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
nipkow@64267
   706
  by (induct set: finite) (simp_all add: add_nonneg_eq_0_iff sum_nonneg)
hoelzl@62376
   707
nipkow@64267
   708
lemma sum_nonneg_0:
hoelzl@62376
   709
  "finite s \<Longrightarrow> (\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0) \<Longrightarrow> (\<Sum> i \<in> s. f i) = 0 \<Longrightarrow> i \<in> s \<Longrightarrow> f i = 0"
nipkow@64267
   710
  by (simp add: sum_nonneg_eq_0_iff)
hoelzl@62376
   711
nipkow@64267
   712
lemma sum_nonneg_leq_bound:
haftmann@54744
   713
  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
   714
  shows "f i \<le> B"
haftmann@54744
   715
proof -
wenzelm@63654
   716
  from assms have "f i \<le> f i + (\<Sum>i \<in> s - {i}. f i)"
nipkow@64267
   717
    by (intro add_increasing2 sum_nonneg) auto
hoelzl@62376
   718
  also have "\<dots> = B"
nipkow@64267
   719
    using sum.remove[of s i f] assms by simp
hoelzl@62376
   720
  finally show ?thesis by auto
haftmann@54744
   721
qed
haftmann@54744
   722
nipkow@64267
   723
lemma sum_mono2:
wenzelm@63654
   724
  assumes fin: "finite B"
wenzelm@63654
   725
    and sub: "A \<subseteq> B"
wenzelm@63654
   726
    and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
nipkow@64267
   727
  shows "sum f A \<le> sum f B"
haftmann@54744
   728
proof -
nipkow@64267
   729
  have "sum f A \<le> sum f A + sum f (B-A)"
lp15@65680
   730
    by (auto intro: add_increasing2 [OF sum_nonneg] nn)
nipkow@64267
   731
  also from fin finite_subset[OF sub fin] have "\<dots> = sum f (A \<union> (B-A))"
nipkow@64267
   732
    by (simp add: sum.union_disjoint del: Un_Diff_cancel)
wenzelm@63654
   733
  also from sub have "A \<union> (B-A) = B" by blast
haftmann@54744
   734
  finally show ?thesis .
haftmann@54744
   735
qed
haftmann@54744
   736
nipkow@64267
   737
lemma sum_le_included:
haftmann@57418
   738
  assumes "finite s" "finite t"
haftmann@57418
   739
  and "\<forall>y\<in>t. 0 \<le> g y" "(\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<le> g y)"
nipkow@64267
   740
  shows "sum f s \<le> sum g t"
haftmann@57418
   741
proof -
nipkow@64267
   742
  have "sum f s \<le> sum (\<lambda>y. sum g {x. x\<in>t \<and> i x = y}) s"
nipkow@64267
   743
  proof (rule sum_mono)
wenzelm@63654
   744
    fix y
wenzelm@63654
   745
    assume "y \<in> s"
haftmann@57418
   746
    with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto
nipkow@64267
   747
    with assms show "f y \<le> sum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")
nipkow@64267
   748
      using order_trans[of "?A (i z)" "sum g {z}" "?B (i z)", intro]
nipkow@64267
   749
      by (auto intro!: sum_mono2)
haftmann@57418
   750
  qed
nipkow@64267
   751
  also have "\<dots> \<le> sum (\<lambda>y. sum g {x. x\<in>t \<and> i x = y}) (i ` t)"
nipkow@64267
   752
    using assms(2-4) by (auto intro!: sum_mono2 sum_nonneg)
nipkow@64267
   753
  also have "\<dots> \<le> sum g t"
nipkow@64267
   754
    using assms by (auto simp: sum_image_gen[symmetric])
haftmann@57418
   755
  finally show ?thesis .
haftmann@57418
   756
qed
haftmann@57418
   757
wenzelm@63654
   758
end
wenzelm@63654
   759
nipkow@64267
   760
lemma (in canonically_ordered_monoid_add) sum_eq_0_iff [simp]:
nipkow@64267
   761
  "finite F \<Longrightarrow> (sum f F = 0) = (\<forall>a\<in>F. f a = 0)"
nipkow@64267
   762
  by (intro ballI sum_nonneg_eq_0_iff zero_le)
hoelzl@62376
   763
nipkow@64267
   764
lemma sum_distrib_left: "r * sum f A = sum (\<lambda>n. r * f n) A"
wenzelm@63915
   765
  for f :: "'a \<Rightarrow> 'b::semiring_0"
wenzelm@63915
   766
proof (induct A rule: infinite_finite_induct)
wenzelm@63915
   767
  case infinite
wenzelm@63915
   768
  then show ?case by simp
haftmann@54744
   769
next
wenzelm@63915
   770
  case empty
wenzelm@63915
   771
  then show ?case by simp
wenzelm@63915
   772
next
wenzelm@63915
   773
  case insert
wenzelm@63915
   774
  then show ?case by (simp add: distrib_left)
haftmann@54744
   775
qed
haftmann@54744
   776
nipkow@64267
   777
lemma sum_distrib_right: "sum f A * r = (\<Sum>n\<in>A. f n * r)"
wenzelm@63654
   778
  for r :: "'a::semiring_0"
wenzelm@63915
   779
proof (induct A rule: infinite_finite_induct)
wenzelm@63915
   780
  case infinite
wenzelm@63915
   781
  then show ?case by simp
wenzelm@63654
   782
next
wenzelm@63915
   783
  case empty
wenzelm@63915
   784
  then show ?case by simp
wenzelm@63915
   785
next
wenzelm@63915
   786
  case insert
wenzelm@63915
   787
  then show ?case by (simp add: distrib_right)
wenzelm@63654
   788
qed
wenzelm@63654
   789
nipkow@64267
   790
lemma sum_divide_distrib: "sum f A / r = (\<Sum>n\<in>A. f n / r)"
wenzelm@63654
   791
  for r :: "'a::field"
wenzelm@63915
   792
proof (induct A rule: infinite_finite_induct)
wenzelm@63915
   793
  case infinite
wenzelm@63915
   794
  then show ?case by simp
haftmann@54744
   795
next
wenzelm@63915
   796
  case empty
wenzelm@63915
   797
  then show ?case by simp
wenzelm@63915
   798
next
wenzelm@63915
   799
  case insert
wenzelm@63915
   800
  then show ?case by (simp add: add_divide_distrib)
haftmann@54744
   801
qed
haftmann@54744
   802
nipkow@64267
   803
lemma sum_abs[iff]: "\<bar>sum f A\<bar> \<le> sum (\<lambda>i. \<bar>f i\<bar>) A"
wenzelm@63654
   804
  for f :: "'a \<Rightarrow> 'b::ordered_ab_group_add_abs"
wenzelm@63915
   805
proof (induct A rule: infinite_finite_induct)
wenzelm@63915
   806
  case infinite
wenzelm@63915
   807
  then show ?case by simp
wenzelm@63654
   808
next
wenzelm@63915
   809
  case empty
wenzelm@63915
   810
  then show ?case by simp
wenzelm@63915
   811
next
wenzelm@63915
   812
  case insert
wenzelm@63915
   813
  then show ?case by (auto intro: abs_triangle_ineq order_trans)
wenzelm@63654
   814
qed
wenzelm@63654
   815
nipkow@64267
   816
lemma sum_abs_ge_zero[iff]: "0 \<le> sum (\<lambda>i. \<bar>f i\<bar>) A"
wenzelm@63654
   817
  for f :: "'a \<Rightarrow> 'b::ordered_ab_group_add_abs"
nipkow@64267
   818
  by (simp add: sum_nonneg)
wenzelm@63654
   819
nipkow@64267
   820
lemma abs_sum_abs[simp]: "\<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar> = (\<Sum>a\<in>A. \<bar>f a\<bar>)"
wenzelm@63654
   821
  for f :: "'a \<Rightarrow> 'b::ordered_ab_group_add_abs"
wenzelm@63915
   822
proof (induct A rule: infinite_finite_induct)
wenzelm@63915
   823
  case infinite
wenzelm@63915
   824
  then show ?case by simp
wenzelm@63915
   825
next
wenzelm@63915
   826
  case empty
wenzelm@63915
   827
  then show ?case by simp
haftmann@54744
   828
next
wenzelm@63915
   829
  case (insert a A)
wenzelm@63915
   830
  then have "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
wenzelm@63915
   831
  also from insert have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>" by simp
wenzelm@63915
   832
  also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>" by (simp del: abs_of_nonneg)
wenzelm@63915
   833
  also from insert have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" by simp
wenzelm@63915
   834
  finally show ?case .
haftmann@54744
   835
qed
haftmann@54744
   836
nipkow@64267
   837
lemma sum_diff1_ring:
wenzelm@63654
   838
  fixes f :: "'b \<Rightarrow> 'a::ring"
wenzelm@63654
   839
  assumes "finite A" "a \<in> A"
nipkow@64267
   840
  shows "sum f (A - {a}) = sum f A - (f a)"
nipkow@64267
   841
  unfolding sum.remove [OF assms] by auto
haftmann@54744
   842
nipkow@64267
   843
lemma sum_product:
wenzelm@63654
   844
  fixes f :: "'a \<Rightarrow> 'b::semiring_0"
nipkow@64267
   845
  shows "sum f A * sum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
nipkow@64267
   846
  by (simp add: sum_distrib_left sum_distrib_right) (rule sum.commute)
haftmann@54744
   847
nipkow@64267
   848
lemma sum_mult_sum_if_inj:
wenzelm@63654
   849
  fixes f :: "'a \<Rightarrow> 'b::semiring_0"
wenzelm@63654
   850
  shows "inj_on (\<lambda>(a, b). f a * g b) (A \<times> B) \<Longrightarrow>
nipkow@64267
   851
    sum f A * sum g B = sum id {f a * g b |a b. a \<in> A \<and> b \<in> B}"
nipkow@64267
   852
  by(auto simp: sum_product sum.cartesian_product intro!: sum.reindex_cong[symmetric])
haftmann@54744
   853
nipkow@64267
   854
lemma sum_SucD: "sum f A = Suc n \<Longrightarrow> \<exists>a\<in>A. 0 < f a"
wenzelm@63915
   855
  by (induct A rule: infinite_finite_induct) auto
haftmann@54744
   856
nipkow@64267
   857
lemma sum_eq_Suc0_iff:
nipkow@64267
   858
  "finite A \<Longrightarrow> sum f A = Suc 0 \<longleftrightarrow> (\<exists>a\<in>A. f a = Suc 0 \<and> (\<forall>b\<in>A. a \<noteq> b \<longrightarrow> f b = 0))"
wenzelm@63915
   859
  by (induct A rule: finite_induct) (auto simp add: add_is_1)
haftmann@54744
   860
nipkow@64267
   861
lemmas sum_eq_1_iff = sum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
haftmann@54744
   862
nipkow@64267
   863
lemma sum_Un_nat:
nipkow@64267
   864
  "finite A \<Longrightarrow> finite B \<Longrightarrow> sum f (A \<union> B) = sum f A + sum f B - sum f (A \<inter> B)"
wenzelm@63654
   865
  for f :: "'a \<Rightarrow> nat"
wenzelm@61799
   866
  \<comment> \<open>For the natural numbers, we have subtraction.\<close>
nipkow@64267
   867
  by (subst sum.union_inter [symmetric]) (auto simp: algebra_simps)
haftmann@54744
   868
nipkow@64267
   869
lemma sum_diff1_nat: "sum f (A - {a}) = (if a \<in> A then sum f A - f a else sum f A)"
wenzelm@63654
   870
  for f :: "'a \<Rightarrow> nat"
wenzelm@63915
   871
proof (induct A rule: infinite_finite_induct)
wenzelm@63915
   872
  case infinite
wenzelm@63915
   873
  then show ?case by simp
wenzelm@63915
   874
next
wenzelm@63915
   875
  case empty
wenzelm@63915
   876
  then show ?case by simp
wenzelm@63915
   877
next
wenzelm@63915
   878
  case insert
wenzelm@63915
   879
  then show ?case
wenzelm@63915
   880
    apply (auto simp: insert_Diff_if)
wenzelm@63654
   881
    apply (drule mk_disjoint_insert)
wenzelm@63654
   882
    apply auto
wenzelm@63654
   883
    done
wenzelm@63654
   884
qed
haftmann@54744
   885
nipkow@64267
   886
lemma sum_diff_nat:
wenzelm@63654
   887
  fixes f :: "'a \<Rightarrow> nat"
wenzelm@63654
   888
  assumes "finite B" and "B \<subseteq> A"
nipkow@64267
   889
  shows "sum f (A - B) = sum f A - sum f B"
wenzelm@63654
   890
  using assms
haftmann@54744
   891
proof induct
wenzelm@63654
   892
  case empty
wenzelm@63654
   893
  then show ?case by simp
haftmann@54744
   894
next
wenzelm@63654
   895
  case (insert x F)
nipkow@64267
   896
  note IH = \<open>F \<subseteq> A \<Longrightarrow> sum f (A - F) = sum f A - sum f F\<close>
wenzelm@63654
   897
  from \<open>x \<notin> F\<close> \<open>insert x F \<subseteq> A\<close> have "x \<in> A - F" by simp
nipkow@64267
   898
  then have A: "sum f ((A - F) - {x}) = sum f (A - F) - f x"
nipkow@64267
   899
    by (simp add: sum_diff1_nat)
wenzelm@63654
   900
  from \<open>insert x F \<subseteq> A\<close> have "F \<subseteq> A" by simp
nipkow@64267
   901
  with IH have "sum f (A - F) = sum f A - sum f F" by simp
nipkow@64267
   902
  with A have B: "sum f ((A - F) - {x}) = sum f A - sum f F - f x"
haftmann@54744
   903
    by simp
wenzelm@63654
   904
  from \<open>x \<notin> F\<close> have "A - insert x F = (A - F) - {x}" by auto
nipkow@64267
   905
  with B have C: "sum f (A - insert x F) = sum f A - sum f F - f x"
haftmann@54744
   906
    by simp
nipkow@64267
   907
  from \<open>finite F\<close> \<open>x \<notin> F\<close> have "sum f (insert x F) = sum f F + f x"
wenzelm@63654
   908
    by simp
nipkow@64267
   909
  with C have "sum f (A - insert x F) = sum f A - sum f (insert x F)"
haftmann@54744
   910
    by simp
wenzelm@63654
   911
  then show ?case by simp
haftmann@54744
   912
qed
haftmann@54744
   913
nipkow@64267
   914
lemma sum_comp_morphism:
nipkow@64267
   915
  "h 0 = 0 \<Longrightarrow> (\<And>x y. h (x + y) = h x + h y) \<Longrightarrow> sum (h \<circ> g) A = h (sum g A)"
wenzelm@63915
   916
  by (induct A rule: infinite_finite_induct) simp_all
haftmann@54744
   917
nipkow@64267
   918
lemma (in comm_semiring_1) dvd_sum: "(\<And>a. a \<in> A \<Longrightarrow> d dvd f a) \<Longrightarrow> d dvd sum f A"
haftmann@59010
   919
  by (induct A rule: infinite_finite_induct) simp_all
haftmann@59010
   920
nipkow@64267
   921
lemma (in ordered_comm_monoid_add) sum_pos:
nipkow@64267
   922
  "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> 0 < f i) \<Longrightarrow> 0 < sum f I"
hoelzl@62377
   923
  by (induct I rule: finite_ne_induct) (auto intro: add_pos_pos)
hoelzl@62377
   924
nipkow@64267
   925
lemma (in ordered_comm_monoid_add) sum_pos2:
hoelzl@62377
   926
  assumes I: "finite I" "i \<in> I" "0 < f i" "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
nipkow@64267
   927
  shows "0 < sum f I"
lp15@60974
   928
proof -
nipkow@64267
   929
  have "0 < f i + sum f (I - {i})"
nipkow@64267
   930
    using assms by (intro add_pos_nonneg sum_nonneg) auto
nipkow@64267
   931
  also have "\<dots> = sum f I"
nipkow@64267
   932
    using assms by (simp add: sum.remove)
lp15@60974
   933
  finally show ?thesis .
lp15@60974
   934
qed
lp15@60974
   935
nipkow@64267
   936
lemma sum_cong_Suc:
eberlm@61524
   937
  assumes "0 \<notin> A" "\<And>x. Suc x \<in> A \<Longrightarrow> f (Suc x) = g (Suc x)"
nipkow@64267
   938
  shows "sum f A = sum g A"
nipkow@64267
   939
proof (rule sum.cong)
wenzelm@63654
   940
  fix x
wenzelm@63654
   941
  assume "x \<in> A"
wenzelm@63654
   942
  with assms(1) show "f x = g x"
wenzelm@63654
   943
    by (cases x) (auto intro!: assms(2))
eberlm@61524
   944
qed simp_all
eberlm@61524
   945
haftmann@54744
   946
nipkow@64267
   947
subsubsection \<open>Cardinality as special case of @{const sum}\<close>
haftmann@54744
   948
nipkow@64267
   949
lemma card_eq_sum: "card A = sum (\<lambda>x. 1) A"
haftmann@54744
   950
proof -
haftmann@54744
   951
  have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)"
haftmann@54744
   952
    by (simp add: fun_eq_iff)
haftmann@54744
   953
  then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) = Finite_Set.fold (\<lambda>_. Suc)"
haftmann@54744
   954
    by (rule arg_cong)
haftmann@54744
   955
  then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
haftmann@54744
   956
    by (blast intro: fun_cong)
wenzelm@63654
   957
  then show ?thesis
nipkow@64267
   958
    by (simp add: card.eq_fold sum.eq_fold)
haftmann@54744
   959
qed
haftmann@54744
   960
nipkow@64267
   961
lemma sum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
wenzelm@63915
   962
  by (induct A rule: infinite_finite_induct) (auto simp: algebra_simps)
haftmann@54744
   963
nipkow@64267
   964
lemma sum_Suc: "sum (\<lambda>x. Suc(f x)) A = sum f A + card A"
nipkow@64267
   965
  using sum.distrib[of f "\<lambda>_. 1" A] by simp
nipkow@58349
   966
nipkow@64267
   967
lemma sum_bounded_above:
wenzelm@63654
   968
  fixes K :: "'a::{semiring_1,ordered_comm_monoid_add}"
wenzelm@63654
   969
  assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> K"
nipkow@64267
   970
  shows "sum f A \<le> of_nat (card A) * K"
haftmann@54744
   971
proof (cases "finite A")
haftmann@54744
   972
  case True
wenzelm@63654
   973
  then show ?thesis
nipkow@64267
   974
    using le sum_mono[where K=A and g = "\<lambda>x. K"] by simp
haftmann@54744
   975
next
wenzelm@63654
   976
  case False
wenzelm@63654
   977
  then show ?thesis by simp
haftmann@54744
   978
qed
haftmann@54744
   979
nipkow@64267
   980
lemma sum_bounded_above_strict:
wenzelm@63654
   981
  fixes K :: "'a::{ordered_cancel_comm_monoid_add,semiring_1}"
wenzelm@63654
   982
  assumes "\<And>i. i\<in>A \<Longrightarrow> f i < K" "card A > 0"
nipkow@64267
   983
  shows "sum f A < of_nat (card A) * K"
nipkow@64267
   984
  using assms sum_strict_mono[where A=A and g = "\<lambda>x. K"]
wenzelm@63654
   985
  by (simp add: card_gt_0_iff)
lp15@60974
   986
nipkow@64267
   987
lemma sum_bounded_below:
wenzelm@63654
   988
  fixes K :: "'a::{semiring_1,ordered_comm_monoid_add}"
wenzelm@63654
   989
  assumes le: "\<And>i. i\<in>A \<Longrightarrow> K \<le> f i"
nipkow@64267
   990
  shows "of_nat (card A) * K \<le> sum f A"
lp15@60974
   991
proof (cases "finite A")
lp15@60974
   992
  case True
wenzelm@63654
   993
  then show ?thesis
nipkow@64267
   994
    using le sum_mono[where K=A and f = "\<lambda>x. K"] by simp
lp15@60974
   995
next
wenzelm@63654
   996
  case False
wenzelm@63654
   997
  then show ?thesis by simp
lp15@60974
   998
qed
lp15@60974
   999
haftmann@54744
  1000
lemma card_UN_disjoint:
haftmann@54744
  1001
  assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
haftmann@54744
  1002
    and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
haftmann@54744
  1003
  shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
haftmann@54744
  1004
proof -
wenzelm@63654
  1005
  have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)"
wenzelm@63654
  1006
    by simp
wenzelm@63654
  1007
  with assms show ?thesis
nipkow@64267
  1008
    by (simp add: card_eq_sum sum.UNION_disjoint del: sum_constant)
haftmann@54744
  1009
qed
haftmann@54744
  1010
haftmann@54744
  1011
lemma card_Union_disjoint:
wenzelm@63654
  1012
  "finite C \<Longrightarrow> \<forall>A\<in>C. finite A \<Longrightarrow> \<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {} \<Longrightarrow>
nipkow@64267
  1013
    card (\<Union>C) = sum card C"
wenzelm@63654
  1014
  by (frule card_UN_disjoint [of C id]) simp_all
haftmann@54744
  1015
nipkow@64267
  1016
lemma sum_multicount_gen:
haftmann@57418
  1017
  assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
nipkow@64267
  1018
  shows "sum (\<lambda>i. (card {j\<in>t. R i j})) s = sum k t"
wenzelm@63654
  1019
    (is "?l = ?r")
haftmann@57418
  1020
proof-
nipkow@64267
  1021
  have "?l = sum (\<lambda>i. sum (\<lambda>x.1) {j\<in>t. R i j}) s"
wenzelm@63654
  1022
    by auto
wenzelm@63654
  1023
  also have "\<dots> = ?r"
nipkow@64267
  1024
    unfolding sum.commute_restrict [OF assms(1-2)]
haftmann@57418
  1025
    using assms(3) by auto
haftmann@57418
  1026
  finally show ?thesis .
haftmann@57418
  1027
qed
haftmann@57418
  1028
nipkow@64267
  1029
lemma sum_multicount:
haftmann@57418
  1030
  assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
nipkow@64267
  1031
  shows "sum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
haftmann@57418
  1032
proof-
nipkow@64267
  1033
  have "?l = sum (\<lambda>i. k) T"
nipkow@64267
  1034
    by (rule sum_multicount_gen) (auto simp: assms)
haftmann@57512
  1035
  also have "\<dots> = ?r" by (simp add: mult.commute)
haftmann@57418
  1036
  finally show ?thesis by auto
haftmann@57418
  1037
qed
haftmann@57418
  1038
wenzelm@63654
  1039
wenzelm@60758
  1040
subsubsection \<open>Cardinality of products\<close>
haftmann@54744
  1041
haftmann@54744
  1042
lemma card_SigmaI [simp]:
wenzelm@63654
  1043
  "finite A \<Longrightarrow> \<forall>a\<in>A. finite (B a) \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
nipkow@64267
  1044
  by (simp add: card_eq_sum sum.Sigma del: sum_constant)
haftmann@54744
  1045
haftmann@54744
  1046
(*
haftmann@54744
  1047
lemma SigmaI_insert: "y \<notin> A ==>
wenzelm@61943
  1048
  (SIGMA x:(insert y A). B x) = (({y} \<times> (B y)) \<union> (SIGMA x: A. B x))"
haftmann@54744
  1049
  by auto
haftmann@54744
  1050
*)
haftmann@54744
  1051
wenzelm@63654
  1052
lemma card_cartesian_product: "card (A \<times> B) = card A * card B"
haftmann@54744
  1053
  by (cases "finite A \<and> finite B")
haftmann@54744
  1054
    (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
haftmann@54744
  1055
wenzelm@63654
  1056
lemma card_cartesian_product_singleton:  "card ({x} \<times> A) = card A"
wenzelm@63654
  1057
  by (simp add: card_cartesian_product)
haftmann@54744
  1058
haftmann@54744
  1059
wenzelm@60758
  1060
subsection \<open>Generalized product over a set\<close>
haftmann@54744
  1061
haftmann@54744
  1062
context comm_monoid_mult
haftmann@54744
  1063
begin
haftmann@54744
  1064
nipkow@64272
  1065
sublocale prod: comm_monoid_set times 1
nipkow@64272
  1066
  defines prod = prod.F ..
haftmann@54744
  1067
nipkow@64272
  1068
abbreviation Prod ("\<Prod>_" [1000] 999)
nipkow@64272
  1069
  where "\<Prod>A \<equiv> prod (\<lambda>x. x) A"
haftmann@54744
  1070
haftmann@54744
  1071
end
haftmann@54744
  1072
wenzelm@61955
  1073
syntax (ASCII)
nipkow@64272
  1074
  "_prod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(4PROD _:_./ _)" [0, 51, 10] 10)
wenzelm@61955
  1075
syntax
nipkow@64272
  1076
  "_prod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(2\<Prod>_\<in>_./ _)" [0, 51, 10] 10)
wenzelm@61799
  1077
translations \<comment> \<open>Beware of argument permutation!\<close>
nipkow@64272
  1078
  "\<Prod>i\<in>A. b" == "CONST prod (\<lambda>i. b) A"
haftmann@54744
  1079
wenzelm@61955
  1080
text \<open>Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter \<open>\<Prod>x|P. e\<close>.\<close>
haftmann@54744
  1081
wenzelm@61955
  1082
syntax (ASCII)
nipkow@64272
  1083
  "_qprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(4PROD _ |/ _./ _)" [0, 0, 10] 10)
haftmann@54744
  1084
syntax
nipkow@64272
  1085
  "_qprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(2\<Prod>_ | (_)./ _)" [0, 0, 10] 10)
haftmann@54744
  1086
translations
nipkow@64272
  1087
  "\<Prod>x|P. t" => "CONST prod (\<lambda>x. t) {x. P}"
haftmann@54744
  1088
haftmann@59010
  1089
context comm_monoid_mult
haftmann@59010
  1090
begin
haftmann@59010
  1091
nipkow@64272
  1092
lemma prod_dvd_prod: "(\<And>a. a \<in> A \<Longrightarrow> f a dvd g a) \<Longrightarrow> prod f A dvd prod g A"
haftmann@59010
  1093
proof (induct A rule: infinite_finite_induct)
wenzelm@63654
  1094
  case infinite
wenzelm@63654
  1095
  then show ?case by (auto intro: dvdI)
wenzelm@63654
  1096
next
wenzelm@63654
  1097
  case empty
wenzelm@63654
  1098
  then show ?case by (auto intro: dvdI)
haftmann@59010
  1099
next
wenzelm@63654
  1100
  case (insert a A)
nipkow@64272
  1101
  then have "f a dvd g a" and "prod f A dvd prod g A"
wenzelm@63654
  1102
    by simp_all
nipkow@64272
  1103
  then obtain r s where "g a = f a * r" and "prod g A = prod f A * s"
wenzelm@63654
  1104
    by (auto elim!: dvdE)
nipkow@64272
  1105
  then have "g a * prod g A = f a * prod f A * (r * s)"
wenzelm@63654
  1106
    by (simp add: ac_simps)
wenzelm@63654
  1107
  with insert.hyps show ?case
wenzelm@63654
  1108
    by (auto intro: dvdI)
haftmann@59010
  1109
qed
haftmann@59010
  1110
nipkow@64272
  1111
lemma prod_dvd_prod_subset: "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> prod f A dvd prod f B"
nipkow@64272
  1112
  by (auto simp add: prod.subset_diff ac_simps intro: dvdI)
haftmann@59010
  1113
haftmann@59010
  1114
end
haftmann@59010
  1115
haftmann@54744
  1116
wenzelm@60758
  1117
subsubsection \<open>Properties in more restricted classes of structures\<close>
haftmann@54744
  1118
lp15@65687
  1119
context linordered_nonzero_semiring
lp15@65687
  1120
begin
lp15@65687
  1121
  
lp15@65687
  1122
lemma prod_ge_1: "(\<And>x. x \<in> A \<Longrightarrow> 1 \<le> f x) \<Longrightarrow> 1 \<le> prod f A"
lp15@65687
  1123
proof (induct A rule: infinite_finite_induct)
lp15@65687
  1124
  case infinite
lp15@65687
  1125
  then show ?case by simp
lp15@65687
  1126
next
lp15@65687
  1127
  case empty
lp15@65687
  1128
  then show ?case by simp
lp15@65687
  1129
next
lp15@65687
  1130
  case (insert x F)
lp15@65687
  1131
  have "1 * 1 \<le> f x * prod f F"
lp15@65687
  1132
    by (rule mult_mono') (use insert in auto)
lp15@65687
  1133
  with insert show ?case by simp
lp15@65687
  1134
qed
lp15@65687
  1135
lp15@65687
  1136
lemma prod_le_1:
lp15@65687
  1137
  fixes f :: "'b \<Rightarrow> 'a"
lp15@65687
  1138
  assumes "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x \<and> f x \<le> 1"
lp15@65687
  1139
  shows "prod f A \<le> 1"
lp15@65687
  1140
    using assms
lp15@65687
  1141
proof (induct A rule: infinite_finite_induct)
lp15@65687
  1142
  case infinite
lp15@65687
  1143
  then show ?case by simp
lp15@65687
  1144
next
lp15@65687
  1145
  case empty
lp15@65687
  1146
  then show ?case by simp
lp15@65687
  1147
next
lp15@65687
  1148
  case (insert x F)
lp15@65687
  1149
  then show ?case by (force simp: mult.commute intro: dest: mult_le_one)
lp15@65687
  1150
qed
lp15@65687
  1151
lp15@65687
  1152
end
lp15@65687
  1153
haftmann@59010
  1154
context comm_semiring_1
haftmann@59010
  1155
begin
haftmann@54744
  1156
nipkow@64272
  1157
lemma dvd_prod_eqI [intro]:
haftmann@59010
  1158
  assumes "finite A" and "a \<in> A" and "b = f a"
nipkow@64272
  1159
  shows "b dvd prod f A"
haftmann@59010
  1160
proof -
nipkow@64272
  1161
  from \<open>finite A\<close> have "prod f (insert a (A - {a})) = f a * prod f (A - {a})"
nipkow@64272
  1162
    by (intro prod.insert) auto
wenzelm@63654
  1163
  also from \<open>a \<in> A\<close> have "insert a (A - {a}) = A"
wenzelm@63654
  1164
    by blast
nipkow@64272
  1165
  finally have "prod f A = f a * prod f (A - {a})" .
wenzelm@63654
  1166
  with \<open>b = f a\<close> show ?thesis
wenzelm@63654
  1167
    by simp
haftmann@59010
  1168
qed
haftmann@54744
  1169
nipkow@64272
  1170
lemma dvd_prodI [intro]: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> f a dvd prod f A"
wenzelm@63654
  1171
  by auto
haftmann@54744
  1172
nipkow@64272
  1173
lemma prod_zero:
haftmann@59010
  1174
  assumes "finite A" and "\<exists>a\<in>A. f a = 0"
nipkow@64272
  1175
  shows "prod f A = 0"
wenzelm@63654
  1176
  using assms
wenzelm@63654
  1177
proof (induct A)
wenzelm@63654
  1178
  case empty
wenzelm@63654
  1179
  then show ?case by simp
haftmann@59010
  1180
next
haftmann@59010
  1181
  case (insert a A)
haftmann@59010
  1182
  then have "f a = 0 \<or> (\<exists>a\<in>A. f a = 0)" by simp
nipkow@64272
  1183
  then have "f a * prod f A = 0" by rule (simp_all add: insert)
haftmann@59010
  1184
  with insert show ?case by simp
haftmann@59010
  1185
qed
haftmann@54744
  1186
nipkow@64272
  1187
lemma prod_dvd_prod_subset2:
haftmann@59010
  1188
  assumes "finite B" and "A \<subseteq> B" and "\<And>a. a \<in> A \<Longrightarrow> f a dvd g a"
nipkow@64272
  1189
  shows "prod f A dvd prod g B"
haftmann@59010
  1190
proof -
nipkow@64272
  1191
  from assms have "prod f A dvd prod g A"
nipkow@64272
  1192
    by (auto intro: prod_dvd_prod)
nipkow@64272
  1193
  moreover from assms have "prod g A dvd prod g B"
nipkow@64272
  1194
    by (auto intro: prod_dvd_prod_subset)
haftmann@59010
  1195
  ultimately show ?thesis by (rule dvd_trans)
haftmann@59010
  1196
qed
haftmann@59010
  1197
haftmann@59010
  1198
end
haftmann@59010
  1199
nipkow@64272
  1200
lemma (in semidom) prod_zero_iff [simp]:
haftmann@63924
  1201
  fixes f :: "'b \<Rightarrow> 'a"
haftmann@59010
  1202
  assumes "finite A"
nipkow@64272
  1203
  shows "prod f A = 0 \<longleftrightarrow> (\<exists>a\<in>A. f a = 0)"
haftmann@59010
  1204
  using assms by (induct A) (auto simp: no_zero_divisors)
haftmann@59010
  1205
nipkow@64272
  1206
lemma (in semidom_divide) prod_diff1:
haftmann@60353
  1207
  assumes "finite A" and "f a \<noteq> 0"
nipkow@64272
  1208
  shows "prod f (A - {a}) = (if a \<in> A then prod f A div f a else prod f A)"
haftmann@60353
  1209
proof (cases "a \<notin> A")
wenzelm@63654
  1210
  case True
wenzelm@63654
  1211
  then show ?thesis by simp
haftmann@60353
  1212
next
wenzelm@63654
  1213
  case False
wenzelm@63654
  1214
  with assms show ?thesis
wenzelm@63654
  1215
  proof induct
wenzelm@63654
  1216
    case empty
wenzelm@63654
  1217
    then show ?case by simp
haftmann@60353
  1218
  next
haftmann@60353
  1219
    case (insert b B)
haftmann@60353
  1220
    then show ?case
haftmann@60353
  1221
    proof (cases "a = b")
wenzelm@63654
  1222
      case True
wenzelm@63654
  1223
      with insert show ?thesis by simp
haftmann@60353
  1224
    next
wenzelm@63654
  1225
      case False
wenzelm@63654
  1226
      with insert have "a \<in> B" by simp
wenzelm@63040
  1227
      define C where "C = B - {a}"
wenzelm@63654
  1228
      with \<open>finite B\<close> \<open>a \<in> B\<close> have "B = insert a C" "finite C" "a \<notin> C"
wenzelm@63654
  1229
        by auto
wenzelm@63654
  1230
      with insert show ?thesis
wenzelm@63654
  1231
        by (auto simp add: insert_commute ac_simps)
haftmann@60353
  1232
    qed
haftmann@60353
  1233
  qed
haftmann@60353
  1234
qed
haftmann@54744
  1235
nipkow@64267
  1236
lemma sum_zero_power [simp]: "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
wenzelm@63654
  1237
  for c :: "nat \<Rightarrow> 'a::division_ring"
wenzelm@63654
  1238
  by (induct A rule: infinite_finite_induct) auto
haftmann@62481
  1239
nipkow@64267
  1240
lemma sum_zero_power' [simp]:
wenzelm@63654
  1241
  "(\<Sum>i\<in>A. c i * 0^i / d i) = (if finite A \<and> 0 \<in> A then c 0 / d 0 else 0)"
wenzelm@63654
  1242
  for c :: "nat \<Rightarrow> 'a::field"
nipkow@64267
  1243
  using sum_zero_power [of "\<lambda>i. c i / d i" A] by auto
haftmann@62481
  1244
lp15@65680
  1245
lemma (in field) prod_inversef: "prod (inverse \<circ> f) A = inverse (prod f A)"
lp15@65680
  1246
 proof (cases "finite A")
lp15@65680
  1247
   case True
lp15@65680
  1248
   then show ?thesis
lp15@65680
  1249
     by (induct A rule: finite_induct) simp_all
lp15@65680
  1250
 next
lp15@65680
  1251
   case False
lp15@65680
  1252
   then show ?thesis
lp15@65680
  1253
     by auto
lp15@65680
  1254
 qed
haftmann@59010
  1255
lp15@65680
  1256
lemma (in field) prod_dividef: "(\<Prod>x\<in>A. f x / g x) = prod f A / prod g A"
lp15@65680
  1257
  using prod_inversef [of g A] by (simp add: divide_inverse prod.distrib)
haftmann@54744
  1258
nipkow@64272
  1259
lemma prod_Un:
haftmann@59010
  1260
  fixes f :: "'b \<Rightarrow> 'a :: field"
haftmann@59010
  1261
  assumes "finite A" and "finite B"
wenzelm@63654
  1262
    and "\<forall>x\<in>A \<inter> B. f x \<noteq> 0"
nipkow@64272
  1263
  shows "prod f (A \<union> B) = prod f A * prod f B / prod f (A \<inter> B)"
haftmann@59010
  1264
proof -
nipkow@64272
  1265
  from assms have "prod f A * prod f B = prod f (A \<union> B) * prod f (A \<inter> B)"
nipkow@64272
  1266
    by (simp add: prod.union_inter [symmetric, of A B])
wenzelm@63654
  1267
  with assms show ?thesis
wenzelm@63654
  1268
    by simp
haftmann@59010
  1269
qed
haftmann@54744
  1270
nipkow@64272
  1271
lemma (in linordered_semidom) prod_nonneg: "(\<forall>a\<in>A. 0 \<le> f a) \<Longrightarrow> 0 \<le> prod f A"
haftmann@59010
  1272
  by (induct A rule: infinite_finite_induct) simp_all
haftmann@59010
  1273
nipkow@64272
  1274
lemma (in linordered_semidom) prod_pos: "(\<forall>a\<in>A. 0 < f a) \<Longrightarrow> 0 < prod f A"
haftmann@59010
  1275
  by (induct A rule: infinite_finite_induct) simp_all
haftmann@59010
  1276
nipkow@64272
  1277
lemma (in linordered_semidom) prod_mono:
nipkow@64272
  1278
  "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i \<Longrightarrow> prod f A \<le> prod g A"
nipkow@64272
  1279
  by (induct A rule: infinite_finite_induct) (auto intro!: prod_nonneg mult_mono)
haftmann@54744
  1280
nipkow@64272
  1281
lemma (in linordered_semidom) prod_mono_strict:
wenzelm@63654
  1282
  assumes "finite A" "\<forall>i\<in>A. 0 \<le> f i \<and> f i < g i" "A \<noteq> {}"
nipkow@64272
  1283
  shows "prod f A < prod g A"
wenzelm@63654
  1284
  using assms
wenzelm@63654
  1285
proof (induct A rule: finite_induct)
wenzelm@63654
  1286
  case empty
wenzelm@63654
  1287
  then show ?case by simp
wenzelm@63654
  1288
next
wenzelm@63654
  1289
  case insert
nipkow@64272
  1290
  then show ?case by (force intro: mult_strict_mono' prod_nonneg)
wenzelm@63654
  1291
qed
lp15@60974
  1292
nipkow@64272
  1293
lemma (in linordered_field) abs_prod: "\<bar>prod f A\<bar> = (\<Prod>x\<in>A. \<bar>f x\<bar>)"
haftmann@59010
  1294
  by (induct A rule: infinite_finite_induct) (simp_all add: abs_mult)
haftmann@54744
  1295
nipkow@64272
  1296
lemma prod_eq_1_iff [simp]: "finite A \<Longrightarrow> prod f A = 1 \<longleftrightarrow> (\<forall>a\<in>A. f a = 1)"
wenzelm@63654
  1297
  for f :: "'a \<Rightarrow> nat"
haftmann@59010
  1298
  by (induct A rule: finite_induct) simp_all
haftmann@54744
  1299
nipkow@64272
  1300
lemma prod_pos_nat_iff [simp]: "finite A \<Longrightarrow> prod f A > 0 \<longleftrightarrow> (\<forall>a\<in>A. f a > 0)"
wenzelm@63654
  1301
  for f :: "'a \<Rightarrow> nat"
nipkow@64272
  1302
  using prod_zero_iff by (simp del: neq0_conv add: zero_less_iff_neq_zero)
haftmann@54744
  1303
nipkow@64272
  1304
lemma prod_constant: "(\<Prod>x\<in> A. y) = y ^ card A"
wenzelm@63654
  1305
  for y :: "'a::comm_monoid_mult"
haftmann@62481
  1306
  by (induct A rule: infinite_finite_induct) simp_all
haftmann@62481
  1307
nipkow@64272
  1308
lemma prod_power_distrib: "prod f A ^ n = prod (\<lambda>x. (f x) ^ n) A"
wenzelm@63654
  1309
  for f :: "'a \<Rightarrow> 'b::comm_semiring_1"
wenzelm@63654
  1310
  by (induct A rule: infinite_finite_induct) (auto simp add: power_mult_distrib)
haftmann@62481
  1311
nipkow@64267
  1312
lemma power_sum: "c ^ (\<Sum>a\<in>A. f a) = (\<Prod>a\<in>A. c ^ f a)"
haftmann@62481
  1313
  by (induct A rule: infinite_finite_induct) (simp_all add: power_add)
haftmann@62481
  1314
nipkow@64272
  1315
lemma prod_gen_delta:
wenzelm@63654
  1316
  fixes b :: "'b \<Rightarrow> 'a::comm_monoid_mult"
wenzelm@63654
  1317
  assumes fin: "finite S"
nipkow@64272
  1318
  shows "prod (\<lambda>k. if k = a then b k else c) S =
wenzelm@63654
  1319
    (if a \<in> S then b a * c ^ (card S - 1) else c ^ card S)"
wenzelm@63654
  1320
proof -
haftmann@62481
  1321
  let ?f = "(\<lambda>k. if k=a then b k else c)"
wenzelm@63654
  1322
  show ?thesis
wenzelm@63654
  1323
  proof (cases "a \<in> S")
wenzelm@63654
  1324
    case False
wenzelm@63654
  1325
    then have "\<forall> k\<in> S. ?f k = c" by simp
nipkow@64272
  1326
    with False show ?thesis by (simp add: prod_constant)
wenzelm@63654
  1327
  next
wenzelm@63654
  1328
    case True
haftmann@62481
  1329
    let ?A = "S - {a}"
haftmann@62481
  1330
    let ?B = "{a}"
wenzelm@63654
  1331
    from True have eq: "S = ?A \<union> ?B" by blast
wenzelm@63654
  1332
    have disjoint: "?A \<inter> ?B = {}" by simp
wenzelm@63654
  1333
    from fin have fin': "finite ?A" "finite ?B" by auto
nipkow@64272
  1334
    have f_A0: "prod ?f ?A = prod (\<lambda>i. c) ?A"
nipkow@64272
  1335
      by (rule prod.cong) auto
wenzelm@63654
  1336
    from fin True have card_A: "card ?A = card S - 1" by auto
nipkow@64272
  1337
    have f_A1: "prod ?f ?A = c ^ card ?A"
nipkow@64272
  1338
      unfolding f_A0 by (rule prod_constant)
nipkow@64272
  1339
    have "prod ?f ?A * prod ?f ?B = prod ?f S"
nipkow@64272
  1340
      using prod.union_disjoint[OF fin' disjoint, of ?f, unfolded eq[symmetric]]
haftmann@62481
  1341
      by simp
wenzelm@63654
  1342
    with True card_A show ?thesis
nipkow@64272
  1343
      by (simp add: f_A1 field_simps cong add: prod.cong cong del: if_weak_cong)
wenzelm@63654
  1344
  qed
haftmann@62481
  1345
qed
haftmann@62481
  1346
nipkow@64267
  1347
lemma sum_image_le:
lp15@63952
  1348
  fixes g :: "'a \<Rightarrow> 'b::ordered_ab_group_add"
lp15@63952
  1349
  assumes "finite I" "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> g(f i)"
nipkow@64267
  1350
    shows "sum g (f ` I) \<le> sum (g \<circ> f) I"
lp15@63952
  1351
  using assms
lp15@63952
  1352
proof induction
lp15@63952
  1353
  case empty
lp15@63952
  1354
  then show ?case by auto
lp15@63952
  1355
next
lp15@63952
  1356
  case (insert x F) then
nipkow@64267
  1357
  have "sum g (f ` insert x F) = sum g (insert (f x) (f ` F))" by simp
nipkow@64267
  1358
  also have "\<dots> \<le> g (f x) + sum g (f ` F)"
nipkow@64267
  1359
    by (simp add: insert sum.insert_if)
nipkow@64267
  1360
  also have "\<dots>  \<le> sum (g \<circ> f) (insert x F)"
lp15@63952
  1361
    using insert by auto
lp15@63952
  1362
  finally show ?case .
lp15@63952
  1363
qed
lp15@63952
  1364
 
haftmann@54744
  1365
end