src/HOL/Big_Operators.thy
author wenzelm
Tue Sep 03 01:12:40 2013 +0200 (2013-09-03)
changeset 53374 a14d2a854c02
parent 53174 71a2702da5e0
child 54147 97a8ff4e4ac9
permissions -rw-r--r--
tuned proofs -- clarified flow of facts wrt. calculation;
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(*  Title:      HOL/Big_Operators.thy
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    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
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                with contributions by Jeremy Avigad
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*)
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header {* Big operators and finite (non-empty) sets *}
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theory Big_Operators
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imports Finite_Set Option Metis
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begin
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subsection {* Generic monoid operation over a set *}
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no_notation times (infixl "*" 70)
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no_notation Groups.one ("1")
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locale comm_monoid_set = comm_monoid
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begin
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interpretation comp_fun_commute f
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  by default (simp add: fun_eq_iff left_commute)
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interpretation comp_fun_commute "f \<circ> g"
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  by (rule comp_comp_fun_commute)
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definition F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
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where
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  eq_fold: "F g A = Finite_Set.fold (f \<circ> g) 1 A"
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lemma infinite [simp]:
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  "\<not> finite A \<Longrightarrow> F g A = 1"
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  by (simp add: eq_fold)
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lemma empty [simp]:
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  "F g {} = 1"
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  by (simp add: eq_fold)
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lemma insert [simp]:
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  assumes "finite A" and "x \<notin> A"
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  shows "F g (insert x A) = g x * F g A"
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  using assms by (simp add: eq_fold)
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lemma remove:
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  assumes "finite A" and "x \<in> A"
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  shows "F g A = g x * F g (A - {x})"
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proof -
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  from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
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    by (auto dest: mk_disjoint_insert)
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  moreover from `finite A` A have "finite B" by simp
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  ultimately show ?thesis by simp
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qed
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lemma insert_remove:
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  assumes "finite A"
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  shows "F g (insert x A) = g x * F g (A - {x})"
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  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
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lemma neutral:
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  assumes "\<forall>x\<in>A. g x = 1"
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  shows "F g A = 1"
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  using assms by (induct A rule: infinite_finite_induct) simp_all
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lemma neutral_const [simp]:
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  "F (\<lambda>_. 1) A = 1"
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  by (simp add: neutral)
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lemma union_inter:
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  assumes "finite A" and "finite B"
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  shows "F g (A \<union> B) * F g (A \<inter> B) = F g A * F g B"
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  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
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using assms proof (induct A)
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  case empty then show ?case by simp
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next
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  case (insert x A) then show ?case
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    by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
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qed
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corollary union_inter_neutral:
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  assumes "finite A" and "finite B"
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  and I0: "\<forall>x \<in> A \<inter> B. g x = 1"
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  shows "F g (A \<union> B) = F g A * F g B"
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  using assms by (simp add: union_inter [symmetric] neutral)
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corollary union_disjoint:
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  assumes "finite A" and "finite B"
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  assumes "A \<inter> B = {}"
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  shows "F g (A \<union> B) = F g A * F g B"
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  using assms by (simp add: union_inter_neutral)
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lemma subset_diff:
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  "B \<subseteq> A \<Longrightarrow> finite A \<Longrightarrow> F g A = F g (A - B) * F g B"
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  by (metis Diff_partition union_disjoint Diff_disjoint finite_Un inf_commute sup_commute)
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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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proof (cases "finite A")
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  case True
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  then have "\<And>C. C \<subseteq> A \<longrightarrow> (\<forall>x\<in>C. g x = h x) \<longrightarrow> F g C = F h C"
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  proof induct
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    case empty then show ?case by simp
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  next
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    case (insert x F) then show ?case apply -
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    apply (simp add: subset_insert_iff, clarify)
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    apply (subgoal_tac "finite C")
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      prefer 2 apply (blast dest: finite_subset [rotated])
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    apply (subgoal_tac "C = insert x (C - {x})")
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      prefer 2 apply blast
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    apply (erule ssubst)
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    apply (simp add: Ball_def del: insert_Diff_single)
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    done
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  qed
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  with `A = B` g_h show ?thesis by simp
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next
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  case False
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  with `A = B` show ?thesis by simp
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qed
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lemma strong_cong [cong]:
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  assumes "A = B" "\<And>x. x \<in> B =simp=> g x = h x"
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  shows "F (\<lambda>x. g x) A = F (\<lambda>x. h x) B"
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  by (rule cong) (insert assms, simp_all add: simp_implies_def)
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lemma UNION_disjoint:
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  assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
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  and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
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  shows "F g (UNION I A) = F (\<lambda>x. F g (A x)) I"
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apply (insert assms)
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apply (induct rule: finite_induct)
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apply simp
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apply atomize
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apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
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 prefer 2 apply blast
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apply (subgoal_tac "A x Int UNION Fa A = {}")
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 prefer 2 apply blast
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apply (simp add: union_disjoint)
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done
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lemma Union_disjoint:
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  assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
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  shows "F g (Union C) = F (F g) C"
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proof cases
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  assume "finite C"
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  from UNION_disjoint [OF this assms]
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  show ?thesis
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    by (simp add: SUP_def)
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qed (auto dest: finite_UnionD intro: infinite)
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lemma distrib:
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  "F (\<lambda>x. g x * h x) A = F g A * F h A"
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  using assms by (induct A rule: infinite_finite_induct) (simp_all add: assoc commute left_commute)
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lemma Sigma:
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  "finite A \<Longrightarrow> \<forall>x\<in>A. finite (B x) \<Longrightarrow> F (\<lambda>x. F (g x) (B x)) A = F (split g) (SIGMA x:A. B x)"
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apply (subst Sigma_def)
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apply (subst UNION_disjoint, assumption, simp)
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 apply blast
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apply (rule cong)
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apply rule
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apply (simp add: fun_eq_iff)
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apply (subst UNION_disjoint, simp, simp)
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 apply blast
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apply (simp add: comp_def)
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done
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lemma related: 
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  assumes Re: "R 1 1" 
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  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
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  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
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  shows "R (F h S) (F g S)"
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  using fS by (rule finite_subset_induct) (insert assms, auto)
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lemma eq_general:
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  assumes h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y" 
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  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
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  shows "F f1 S = F f2 S'"
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proof-
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  from h f12 have hS: "h ` S = S'" by blast
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  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
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    from f12 h H  have "x = y" by auto }
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  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
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  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
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  from hS have "F f2 S' = F f2 (h ` S)" by simp
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  also have "\<dots> = F (f2 o h) S" using reindex [OF hinj, of f2] .
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  also have "\<dots> = F f1 S " using th cong [of _ _ "f2 o h" f1]
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    by blast
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  finally show ?thesis ..
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qed
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lemma eq_general_reverses:
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  assumes kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
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  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"
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  shows "F j S = F g T"
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  (* metis solves it, but not yet available here *)
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  apply (rule eq_general [of T S h g j])
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  apply (rule ballI)
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  apply (frule kh)
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  apply (rule ex1I[])
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  apply blast
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  apply clarsimp
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  apply (drule hk) apply simp
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  apply (rule sym)
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  apply (erule conjunct1[OF conjunct2[OF hk]])
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  apply (rule ballI)
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  apply (drule hk)
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  apply blast
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  done
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lemma mono_neutral_cong_left:
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  assumes "finite T" and "S \<subseteq> T" and "\<forall>i \<in> T - S. h i = 1"
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  and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g S = F h T"
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proof-
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  have eq: "T = S \<union> (T - S)" using `S \<subseteq> T` by blast
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  have d: "S \<inter> (T - S) = {}" using `S \<subseteq> T` by blast
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  from `finite T` `S \<subseteq> T` have f: "finite S" "finite (T - S)"
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    by (auto intro: finite_subset)
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  show ?thesis using assms(4)
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    by (simp add: union_disjoint [OF f d, unfolded eq [symmetric]] neutral [OF assms(3)])
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qed
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lemma mono_neutral_cong_right:
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  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1; \<And>x. x \<in> S \<Longrightarrow> g x = h x \<rbrakk>
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   \<Longrightarrow> F g T = F h S"
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  by (auto intro!: mono_neutral_cong_left [symmetric])
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lemma mono_neutral_left:
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  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g S = F g T"
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  by (blast intro: mono_neutral_cong_left)
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lemma mono_neutral_right:
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  "\<lbrakk> finite T;  S \<subseteq> T;  \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g T = F g S"
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  by (blast intro!: mono_neutral_left [symmetric])
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lemma delta: 
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  assumes fS: "finite S"
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  shows "F (\<lambda>k. if k = a then b k else 1) S = (if a \<in> S then b a else 1)"
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proof-
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  let ?f = "(\<lambda>k. if k=a then b k else 1)"
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  { assume a: "a \<notin> S"
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    hence "\<forall>k\<in>S. ?f k = 1" by simp
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    hence ?thesis  using a by simp }
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  moreover
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  { assume a: "a \<in> S"
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    let ?A = "S - {a}"
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    let ?B = "{a}"
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    have eq: "S = ?A \<union> ?B" using a by blast 
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    have dj: "?A \<inter> ?B = {}" by simp
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    from fS have fAB: "finite ?A" "finite ?B" by auto  
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    have "F ?f S = F ?f ?A * F ?f ?B"
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      using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]]
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      by simp
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    then have ?thesis using a by simp }
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  ultimately show ?thesis by blast
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qed
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lemma delta': 
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  assumes fS: "finite S"
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  shows "F (\<lambda>k. if a = k then b k else 1) S = (if a \<in> S then b a else 1)"
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  using delta [OF fS, of a b, symmetric] by (auto intro: cong)
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lemma If_cases:
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  fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
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  assumes fA: "finite A"
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  shows "F (\<lambda>x. if P x then h x else g x) A =
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    F h (A \<inter> {x. P x}) * F g (A \<inter> - {x. P x})"
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proof -
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  have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}" 
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          "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}" 
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    by blast+
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  from fA 
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  have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
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  let ?g = "\<lambda>x. if P x then h x else g x"
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  from union_disjoint [OF f a(2), of ?g] a(1)
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  show ?thesis
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    by (subst (1 2) cong) simp_all
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qed
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lemma cartesian_product:
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   "F (\<lambda>x. F (g x) B) A = F (split g) (A <*> B)"
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apply (rule sym)
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apply (cases "finite A") 
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 apply (cases "finite B") 
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  apply (simp add: Sigma)
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 apply (cases "A={}", simp)
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 apply simp
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apply (auto intro: infinite dest: finite_cartesian_productD2)
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apply (cases "B = {}") apply (auto intro: infinite dest: finite_cartesian_productD1)
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done
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end
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notation times (infixl "*" 70)
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notation Groups.one ("1")
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subsection {* Generalized summation over a set *}
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context comm_monoid_add
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begin
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haftmann@51738
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definition setsum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
haftmann@51489
   314
where
haftmann@51489
   315
  "setsum = comm_monoid_set.F plus 0"
haftmann@26041
   316
haftmann@51738
   317
sublocale setsum!: comm_monoid_set plus 0
haftmann@51489
   318
where
haftmann@51546
   319
  "comm_monoid_set.F plus 0 = setsum"
haftmann@51489
   320
proof -
haftmann@51489
   321
  show "comm_monoid_set plus 0" ..
haftmann@51489
   322
  then interpret setsum!: comm_monoid_set plus 0 .
haftmann@51546
   323
  from setsum_def show "comm_monoid_set.F plus 0 = setsum" by rule
haftmann@51489
   324
qed
nipkow@15402
   325
wenzelm@19535
   326
abbreviation
haftmann@51489
   327
  Setsum ("\<Sum>_" [1000] 999) where
haftmann@51489
   328
  "\<Sum>A \<equiv> setsum (%x. x) A"
wenzelm@19535
   329
haftmann@51738
   330
end
haftmann@51738
   331
nipkow@15402
   332
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
nipkow@15402
   333
written @{text"\<Sum>x\<in>A. e"}. *}
nipkow@15402
   334
nipkow@15402
   335
syntax
paulson@17189
   336
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
nipkow@15402
   337
syntax (xsymbols)
paulson@17189
   338
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402
   339
syntax (HTML output)
paulson@17189
   340
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402
   341
nipkow@15402
   342
translations -- {* Beware of argument permutation! *}
nipkow@28853
   343
  "SUM i:A. b" == "CONST setsum (%i. b) A"
nipkow@28853
   344
  "\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A"
nipkow@15402
   345
nipkow@15402
   346
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
nipkow@15402
   347
 @{text"\<Sum>x|P. e"}. *}
nipkow@15402
   348
nipkow@15402
   349
syntax
paulson@17189
   350
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
nipkow@15402
   351
syntax (xsymbols)
paulson@17189
   352
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
nipkow@15402
   353
syntax (HTML output)
paulson@17189
   354
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
nipkow@15402
   355
nipkow@15402
   356
translations
nipkow@28853
   357
  "SUM x|P. t" => "CONST setsum (%x. t) {x. P}"
nipkow@28853
   358
  "\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}"
nipkow@15402
   359
nipkow@15402
   360
print_translation {*
nipkow@15402
   361
let
wenzelm@35115
   362
  fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) $ Abs (y, Ty, P)] =
wenzelm@35115
   363
        if x <> y then raise Match
wenzelm@35115
   364
        else
wenzelm@35115
   365
          let
wenzelm@49660
   366
            val x' = Syntax_Trans.mark_bound_body (x, Tx);
wenzelm@35115
   367
            val t' = subst_bound (x', t);
wenzelm@35115
   368
            val P' = subst_bound (x', P);
wenzelm@49660
   369
          in
wenzelm@49660
   370
            Syntax.const @{syntax_const "_qsetsum"} $ Syntax_Trans.mark_bound_abs (x, Tx) $ P' $ t'
wenzelm@49660
   371
          end
wenzelm@35115
   372
    | setsum_tr' _ = raise Match;
wenzelm@52143
   373
in [(@{const_syntax setsum}, K setsum_tr')] end
nipkow@15402
   374
*}
nipkow@15402
   375
haftmann@51489
   376
text {* TODO These are candidates for generalization *}
nipkow@15402
   377
haftmann@51489
   378
context comm_monoid_add
haftmann@51489
   379
begin
nipkow@15402
   380
haftmann@51489
   381
lemma setsum_reindex_id: 
haftmann@35816
   382
  "inj_on f B ==> setsum f B = setsum id (f ` B)"
haftmann@51489
   383
  by (simp add: setsum.reindex)
nipkow@15402
   384
haftmann@51489
   385
lemma setsum_reindex_nonzero:
chaieb@29674
   386
  assumes fS: "finite S"
haftmann@51489
   387
  and nz: "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x = f y \<Longrightarrow> h (f x) = 0"
haftmann@51489
   388
  shows "setsum h (f ` S) = setsum (h \<circ> f) S"
haftmann@51489
   389
using nz proof (induct rule: finite_induct [OF fS])
chaieb@29674
   390
  case 1 thus ?case by simp
chaieb@29674
   391
next
chaieb@29674
   392
  case (2 x F) 
nipkow@48849
   393
  { assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
chaieb@29674
   394
    then obtain y where y: "y \<in> F" "f x = f y" by auto 
chaieb@29674
   395
    from "2.hyps" y have xy: "x \<noteq> y" by auto
haftmann@51489
   396
    from "2.prems" [of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
chaieb@29674
   397
    have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
chaieb@29674
   398
    also have "\<dots> = setsum (h o f) (insert x F)" 
haftmann@35816
   399
      unfolding setsum.insert[OF `finite F` `x\<notin>F`]
haftmann@35816
   400
      using h0
haftmann@51489
   401
      apply (simp cong del: setsum.strong_cong)
chaieb@29674
   402
      apply (rule "2.hyps"(3))
chaieb@29674
   403
      apply (rule_tac y="y" in  "2.prems")
chaieb@29674
   404
      apply simp_all
chaieb@29674
   405
      done
nipkow@48849
   406
    finally have ?case . }
chaieb@29674
   407
  moreover
nipkow@48849
   408
  { assume fxF: "f x \<notin> f ` F"
chaieb@29674
   409
    have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)" 
chaieb@29674
   410
      using fxF "2.hyps" by simp 
chaieb@29674
   411
    also have "\<dots> = setsum (h o f) (insert x F)"
haftmann@35816
   412
      unfolding setsum.insert[OF `finite F` `x\<notin>F`]
haftmann@51489
   413
      apply (simp cong del: setsum.strong_cong)
haftmann@35816
   414
      apply (rule cong [OF refl [of "op + (h (f x))"]])
chaieb@29674
   415
      apply (rule "2.hyps"(3))
chaieb@29674
   416
      apply (rule_tac y="y" in  "2.prems")
chaieb@29674
   417
      apply simp_all
chaieb@29674
   418
      done
nipkow@48849
   419
    finally have ?case . }
chaieb@29674
   420
  ultimately show ?case by blast
chaieb@29674
   421
qed
chaieb@29674
   422
haftmann@51489
   423
lemma setsum_cong2:
haftmann@51489
   424
  "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> setsum f A = setsum g A"
haftmann@51489
   425
  by (auto intro: setsum.cong)
nipkow@15554
   426
nipkow@48849
   427
lemma setsum_reindex_cong:
nipkow@28853
   428
   "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|] 
nipkow@28853
   429
    ==> setsum h B = setsum g A"
haftmann@51489
   430
  by (simp add: setsum.reindex)
chaieb@29674
   431
chaieb@30260
   432
lemma setsum_restrict_set:
chaieb@30260
   433
  assumes fA: "finite A"
chaieb@30260
   434
  shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"
chaieb@30260
   435
proof-
chaieb@30260
   436
  from fA have fab: "finite (A \<inter> B)" by auto
chaieb@30260
   437
  have aba: "A \<inter> B \<subseteq> A" by blast
chaieb@30260
   438
  let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"
haftmann@51489
   439
  from setsum.mono_neutral_left [OF fA aba, of ?g]
chaieb@30260
   440
  show ?thesis by simp
chaieb@30260
   441
qed
chaieb@30260
   442
nipkow@15402
   443
lemma setsum_Union_disjoint:
hoelzl@44937
   444
  assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"
hoelzl@44937
   445
  shows "setsum f (Union C) = setsum (setsum f) C"
haftmann@51489
   446
  using assms by (fact setsum.Union_disjoint)
nipkow@15402
   447
haftmann@51489
   448
lemma setsum_cartesian_product:
haftmann@51489
   449
  "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
haftmann@51489
   450
  by (fact setsum.cartesian_product)
nipkow@15402
   451
haftmann@51489
   452
lemma setsum_UNION_zero:
nipkow@48893
   453
  assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"
nipkow@48893
   454
  and f0: "\<And>T1 T2 x. T1\<in>S \<Longrightarrow> T2\<in>S \<Longrightarrow> T1 \<noteq> T2 \<Longrightarrow> x \<in> T1 \<Longrightarrow> x \<in> T2 \<Longrightarrow> f x = 0"
nipkow@48893
   455
  shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"
nipkow@48893
   456
  using fSS f0
nipkow@48893
   457
proof(induct rule: finite_induct[OF fS])
nipkow@48893
   458
  case 1 thus ?case by simp
nipkow@48893
   459
next
nipkow@48893
   460
  case (2 T F)
nipkow@48893
   461
  then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F" 
nipkow@48893
   462
    and H: "setsum f (\<Union> F) = setsum (setsum f) F" by auto
nipkow@48893
   463
  from fTF have fUF: "finite (\<Union>F)" by auto
nipkow@48893
   464
  from "2.prems" TF fTF
nipkow@48893
   465
  show ?case 
haftmann@51489
   466
    by (auto simp add: H [symmetric] intro: setsum.union_inter_neutral [OF fTF(1) fUF, of f])
haftmann@51489
   467
qed
haftmann@51489
   468
haftmann@51489
   469
text {* Commuting outer and inner summation *}
haftmann@51489
   470
haftmann@51489
   471
lemma setsum_commute:
haftmann@51489
   472
  "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
haftmann@51489
   473
proof (simp add: setsum_cartesian_product)
haftmann@51489
   474
  have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
haftmann@51489
   475
    (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
haftmann@51489
   476
    (is "?s = _")
haftmann@51489
   477
    apply (simp add: setsum.reindex [where h = "%(i, j). (j, i)"] swap_inj_on)
haftmann@51489
   478
    apply (simp add: split_def)
haftmann@51489
   479
    done
haftmann@51489
   480
  also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
haftmann@51489
   481
    (is "_ = ?t")
haftmann@51489
   482
    apply (simp add: swap_product)
haftmann@51489
   483
    done
haftmann@51489
   484
  finally show "?s = ?t" .
haftmann@51489
   485
qed
haftmann@51489
   486
haftmann@51489
   487
lemma setsum_Plus:
haftmann@51489
   488
  fixes A :: "'a set" and B :: "'b set"
haftmann@51489
   489
  assumes fin: "finite A" "finite B"
haftmann@51489
   490
  shows "setsum f (A <+> B) = setsum (f \<circ> Inl) A + setsum (f \<circ> Inr) B"
haftmann@51489
   491
proof -
haftmann@51489
   492
  have "A <+> B = Inl ` A \<union> Inr ` B" by auto
haftmann@51489
   493
  moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"
haftmann@51489
   494
    by auto
haftmann@51489
   495
  moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto
haftmann@51489
   496
  moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)
haftmann@51489
   497
  ultimately show ?thesis using fin by(simp add: setsum.union_disjoint setsum.reindex)
nipkow@48893
   498
qed
nipkow@48893
   499
haftmann@51489
   500
end
haftmann@51489
   501
haftmann@51489
   502
text {* TODO These are legacy *}
haftmann@51489
   503
haftmann@51489
   504
lemma setsum_empty:
haftmann@51489
   505
  "setsum f {} = 0"
haftmann@51489
   506
  by (fact setsum.empty)
haftmann@51489
   507
haftmann@51489
   508
lemma setsum_insert:
haftmann@51489
   509
  "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
haftmann@51489
   510
  by (fact setsum.insert)
haftmann@51489
   511
haftmann@51489
   512
lemma setsum_infinite:
haftmann@51489
   513
  "~ finite A ==> setsum f A = 0"
haftmann@51489
   514
  by (fact setsum.infinite)
haftmann@51489
   515
haftmann@51489
   516
lemma setsum_reindex:
haftmann@51489
   517
  "inj_on f B \<Longrightarrow> setsum h (f ` B) = setsum (h \<circ> f) B"
haftmann@51489
   518
  by (fact setsum.reindex)
haftmann@51489
   519
haftmann@51489
   520
lemma setsum_cong:
haftmann@51489
   521
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
haftmann@51489
   522
  by (fact setsum.cong)
haftmann@51489
   523
haftmann@51489
   524
lemma strong_setsum_cong:
haftmann@51489
   525
  "A = B ==> (!!x. x:B =simp=> f x = g x)
haftmann@51489
   526
   ==> setsum (%x. f x) A = setsum (%x. g x) B"
haftmann@51489
   527
  by (fact setsum.strong_cong)
haftmann@51489
   528
haftmann@51489
   529
lemmas setsum_0 = setsum.neutral_const
haftmann@51489
   530
lemmas setsum_0' = setsum.neutral
haftmann@51489
   531
haftmann@51489
   532
lemma setsum_Un_Int: "finite A ==> finite B ==>
haftmann@51489
   533
  setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
haftmann@51489
   534
  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
haftmann@51489
   535
  by (fact setsum.union_inter)
haftmann@51489
   536
haftmann@51489
   537
lemma setsum_Un_disjoint: "finite A ==> finite B
haftmann@51489
   538
  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
haftmann@51489
   539
  by (fact setsum.union_disjoint)
haftmann@51489
   540
haftmann@51489
   541
lemma setsum_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
haftmann@51489
   542
    setsum f A = setsum f (A - B) + setsum f B"
haftmann@51489
   543
  by (fact setsum.subset_diff)
haftmann@51489
   544
haftmann@51489
   545
lemma setsum_mono_zero_left: 
haftmann@51489
   546
  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 0 \<rbrakk> \<Longrightarrow> setsum f S = setsum f T"
haftmann@51489
   547
  by (fact setsum.mono_neutral_left)
haftmann@51489
   548
haftmann@51489
   549
lemmas setsum_mono_zero_right = setsum.mono_neutral_right
haftmann@51489
   550
haftmann@51489
   551
lemma setsum_mono_zero_cong_left: 
haftmann@51489
   552
  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 0; \<And>x. x \<in> S \<Longrightarrow> f x = g x \<rbrakk>
haftmann@51489
   553
  \<Longrightarrow> setsum f S = setsum g T"
haftmann@51489
   554
  by (fact setsum.mono_neutral_cong_left)
haftmann@51489
   555
haftmann@51489
   556
lemmas setsum_mono_zero_cong_right = setsum.mono_neutral_cong_right
haftmann@51489
   557
haftmann@51489
   558
lemma setsum_delta: "finite S \<Longrightarrow>
haftmann@51489
   559
  setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
haftmann@51489
   560
  by (fact setsum.delta)
haftmann@51489
   561
haftmann@51489
   562
lemma setsum_delta': "finite S \<Longrightarrow>
haftmann@51489
   563
  setsum (\<lambda>k. if a = k then b k else 0) S = (if a\<in> S then b a else 0)"
haftmann@51489
   564
  by (fact setsum.delta')
haftmann@51489
   565
haftmann@51489
   566
lemma setsum_cases:
haftmann@51489
   567
  assumes "finite A"
haftmann@51489
   568
  shows "setsum (\<lambda>x. if P x then f x else g x) A =
haftmann@51489
   569
         setsum f (A \<inter> {x. P x}) + setsum g (A \<inter> - {x. P x})"
haftmann@51489
   570
  using assms by (fact setsum.If_cases)
haftmann@51489
   571
haftmann@51489
   572
(*But we can't get rid of finite I. If infinite, although the rhs is 0, 
haftmann@51489
   573
  the lhs need not be, since UNION I A could still be finite.*)
haftmann@51489
   574
lemma setsum_UN_disjoint:
haftmann@51489
   575
  assumes "finite I" and "ALL i:I. finite (A i)"
haftmann@51489
   576
    and "ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}"
haftmann@51489
   577
  shows "setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
haftmann@51489
   578
  using assms by (fact setsum.UNION_disjoint)
haftmann@51489
   579
haftmann@51489
   580
(*But we can't get rid of finite A. If infinite, although the lhs is 0, 
haftmann@51489
   581
  the rhs need not be, since SIGMA A B could still be finite.*)
haftmann@51489
   582
lemma setsum_Sigma:
haftmann@51489
   583
  assumes "finite A" and  "ALL x:A. finite (B x)"
haftmann@51489
   584
  shows "(\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"
haftmann@51489
   585
  using assms by (fact setsum.Sigma)
haftmann@51489
   586
haftmann@51489
   587
lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
haftmann@51489
   588
  by (fact setsum.distrib)
haftmann@51489
   589
haftmann@51489
   590
lemma setsum_Un_zero:  
haftmann@51489
   591
  "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 0 \<rbrakk> \<Longrightarrow>
haftmann@51489
   592
  setsum f (S \<union> T) = setsum f S + setsum f T"
haftmann@51489
   593
  by (fact setsum.union_inter_neutral)
haftmann@51489
   594
haftmann@51489
   595
lemma setsum_eq_general_reverses:
haftmann@51489
   596
  assumes fS: "finite S" and fT: "finite T"
haftmann@51489
   597
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
haftmann@51489
   598
  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"
haftmann@51489
   599
  shows "setsum f S = setsum g T"
haftmann@51489
   600
  using kh hk by (fact setsum.eq_general_reverses)
haftmann@51489
   601
nipkow@15402
   602
nipkow@15402
   603
subsubsection {* Properties in more restricted classes of structures *}
nipkow@15402
   604
nipkow@15402
   605
lemma setsum_Un: "finite A ==> finite B ==>
nipkow@28853
   606
  (setsum f (A Un B) :: 'a :: ab_group_add) =
nipkow@28853
   607
   setsum f A + setsum f B - setsum f (A Int B)"
nipkow@29667
   608
by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
nipkow@15402
   609
haftmann@49715
   610
lemma setsum_Un2:
haftmann@49715
   611
  assumes "finite (A \<union> B)"
haftmann@49715
   612
  shows "setsum f (A \<union> B) = setsum f (A - B) + setsum f (B - A) + setsum f (A \<inter> B)"
haftmann@49715
   613
proof -
haftmann@49715
   614
  have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
haftmann@49715
   615
    by auto
haftmann@49715
   616
  with assms show ?thesis by simp (subst setsum_Un_disjoint, auto)+
haftmann@49715
   617
qed
haftmann@49715
   618
nipkow@15402
   619
lemma setsum_diff1: "finite A \<Longrightarrow>
nipkow@15402
   620
  (setsum f (A - {a}) :: ('a::ab_group_add)) =
nipkow@15402
   621
  (if a:A then setsum f A - f a else setsum f A)"
nipkow@28853
   622
by (erule finite_induct) (auto simp add: insert_Diff_if)
nipkow@28853
   623
nipkow@15402
   624
lemma setsum_diff:
nipkow@15402
   625
  assumes le: "finite A" "B \<subseteq> A"
nipkow@15402
   626
  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
nipkow@15402
   627
proof -
nipkow@15402
   628
  from le have finiteB: "finite B" using finite_subset by auto
nipkow@15402
   629
  show ?thesis using finiteB le
wenzelm@21575
   630
  proof induct
wenzelm@19535
   631
    case empty
wenzelm@19535
   632
    thus ?case by auto
wenzelm@19535
   633
  next
wenzelm@19535
   634
    case (insert x F)
wenzelm@19535
   635
    thus ?case using le finiteB 
wenzelm@19535
   636
      by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
nipkow@15402
   637
  qed
wenzelm@19535
   638
qed
nipkow@15402
   639
nipkow@15402
   640
lemma setsum_mono:
haftmann@35028
   641
  assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))"
nipkow@15402
   642
  shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
nipkow@15402
   643
proof (cases "finite K")
nipkow@15402
   644
  case True
nipkow@15402
   645
  thus ?thesis using le
wenzelm@19535
   646
  proof induct
nipkow@15402
   647
    case empty
nipkow@15402
   648
    thus ?case by simp
nipkow@15402
   649
  next
nipkow@15402
   650
    case insert
nipkow@44890
   651
    thus ?case using add_mono by fastforce
nipkow@15402
   652
  qed
nipkow@15402
   653
next
haftmann@51489
   654
  case False then show ?thesis by simp
nipkow@15402
   655
qed
nipkow@15402
   656
nipkow@15554
   657
lemma setsum_strict_mono:
haftmann@35028
   658
  fixes f :: "'a \<Rightarrow> 'b::{ordered_cancel_ab_semigroup_add,comm_monoid_add}"
wenzelm@19535
   659
  assumes "finite A"  "A \<noteq> {}"
wenzelm@19535
   660
    and "!!x. x:A \<Longrightarrow> f x < g x"
wenzelm@19535
   661
  shows "setsum f A < setsum g A"
wenzelm@41550
   662
  using assms
nipkow@15554
   663
proof (induct rule: finite_ne_induct)
nipkow@15554
   664
  case singleton thus ?case by simp
nipkow@15554
   665
next
nipkow@15554
   666
  case insert thus ?case by (auto simp: add_strict_mono)
nipkow@15554
   667
qed
nipkow@15554
   668
nipkow@46699
   669
lemma setsum_strict_mono_ex1:
nipkow@46699
   670
fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_cancel_ab_semigroup_add}"
nipkow@46699
   671
assumes "finite A" and "ALL x:A. f x \<le> g x" and "EX a:A. f a < g a"
nipkow@46699
   672
shows "setsum f A < setsum g A"
nipkow@46699
   673
proof-
nipkow@46699
   674
  from assms(3) obtain a where a: "a:A" "f a < g a" by blast
nipkow@46699
   675
  have "setsum f A = setsum f ((A-{a}) \<union> {a})"
nipkow@46699
   676
    by(simp add:insert_absorb[OF `a:A`])
nipkow@46699
   677
  also have "\<dots> = setsum f (A-{a}) + setsum f {a}"
nipkow@46699
   678
    using `finite A` by(subst setsum_Un_disjoint) auto
nipkow@46699
   679
  also have "setsum f (A-{a}) \<le> setsum g (A-{a})"
nipkow@46699
   680
    by(rule setsum_mono)(simp add: assms(2))
nipkow@46699
   681
  also have "setsum f {a} < setsum g {a}" using a by simp
nipkow@46699
   682
  also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"
nipkow@46699
   683
    using `finite A` by(subst setsum_Un_disjoint[symmetric]) auto
nipkow@46699
   684
  also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF `a:A`])
nipkow@46699
   685
  finally show ?thesis by (metis add_right_mono add_strict_left_mono)
nipkow@46699
   686
qed
nipkow@46699
   687
nipkow@15535
   688
lemma setsum_negf:
wenzelm@19535
   689
  "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
nipkow@15535
   690
proof (cases "finite A")
berghofe@22262
   691
  case True thus ?thesis by (induct set: finite) auto
nipkow@15535
   692
next
haftmann@51489
   693
  case False thus ?thesis by simp
nipkow@15535
   694
qed
nipkow@15402
   695
nipkow@15535
   696
lemma setsum_subtractf:
wenzelm@19535
   697
  "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
wenzelm@19535
   698
    setsum f A - setsum g A"
nipkow@15535
   699
proof (cases "finite A")
nipkow@15535
   700
  case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
nipkow@15535
   701
next
haftmann@51489
   702
  case False thus ?thesis by simp
nipkow@15535
   703
qed
nipkow@15402
   704
nipkow@15535
   705
lemma setsum_nonneg:
haftmann@35028
   706
  assumes nn: "\<forall>x\<in>A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
wenzelm@19535
   707
  shows "0 \<le> setsum f A"
nipkow@15535
   708
proof (cases "finite A")
nipkow@15535
   709
  case True thus ?thesis using nn
wenzelm@21575
   710
  proof induct
wenzelm@19535
   711
    case empty then show ?case by simp
wenzelm@19535
   712
  next
wenzelm@19535
   713
    case (insert x F)
wenzelm@19535
   714
    then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
wenzelm@19535
   715
    with insert show ?case by simp
wenzelm@19535
   716
  qed
nipkow@15535
   717
next
haftmann@51489
   718
  case False thus ?thesis by simp
nipkow@15535
   719
qed
nipkow@15402
   720
nipkow@15535
   721
lemma setsum_nonpos:
haftmann@35028
   722
  assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})"
wenzelm@19535
   723
  shows "setsum f A \<le> 0"
nipkow@15535
   724
proof (cases "finite A")
nipkow@15535
   725
  case True thus ?thesis using np
wenzelm@21575
   726
  proof induct
wenzelm@19535
   727
    case empty then show ?case by simp
wenzelm@19535
   728
  next
wenzelm@19535
   729
    case (insert x F)
wenzelm@19535
   730
    then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
wenzelm@19535
   731
    with insert show ?case by simp
wenzelm@19535
   732
  qed
nipkow@15535
   733
next
haftmann@51489
   734
  case False thus ?thesis by simp
nipkow@15535
   735
qed
nipkow@15402
   736
hoelzl@36622
   737
lemma setsum_nonneg_leq_bound:
hoelzl@36622
   738
  fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
hoelzl@36622
   739
  assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"
hoelzl@36622
   740
  shows "f i \<le> B"
hoelzl@36622
   741
proof -
hoelzl@36622
   742
  have "0 \<le> (\<Sum> i \<in> s - {i}. f i)" and "0 \<le> f i"
hoelzl@36622
   743
    using assms by (auto intro!: setsum_nonneg)
hoelzl@36622
   744
  moreover
hoelzl@36622
   745
  have "(\<Sum> i \<in> s - {i}. f i) + f i = B"
hoelzl@36622
   746
    using assms by (simp add: setsum_diff1)
hoelzl@36622
   747
  ultimately show ?thesis by auto
hoelzl@36622
   748
qed
hoelzl@36622
   749
hoelzl@36622
   750
lemma setsum_nonneg_0:
hoelzl@36622
   751
  fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
hoelzl@36622
   752
  assumes "finite s" and pos: "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"
hoelzl@36622
   753
  and "(\<Sum> i \<in> s. f i) = 0" and i: "i \<in> s"
hoelzl@36622
   754
  shows "f i = 0"
hoelzl@36622
   755
  using setsum_nonneg_leq_bound[OF assms] pos[OF i] by auto
hoelzl@36622
   756
nipkow@15539
   757
lemma setsum_mono2:
haftmann@36303
   758
fixes f :: "'a \<Rightarrow> 'b :: ordered_comm_monoid_add"
nipkow@15539
   759
assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
nipkow@15539
   760
shows "setsum f A \<le> setsum f B"
nipkow@15539
   761
proof -
nipkow@15539
   762
  have "setsum f A \<le> setsum f A + setsum f (B-A)"
nipkow@15539
   763
    by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
nipkow@15539
   764
  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
nipkow@15539
   765
    by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
nipkow@15539
   766
  also have "A \<union> (B-A) = B" using sub by blast
nipkow@15539
   767
  finally show ?thesis .
nipkow@15539
   768
qed
nipkow@15542
   769
avigad@16775
   770
lemma setsum_mono3: "finite B ==> A <= B ==> 
avigad@16775
   771
    ALL x: B - A. 
haftmann@35028
   772
      0 <= ((f x)::'a::{comm_monoid_add,ordered_ab_semigroup_add}) ==>
avigad@16775
   773
        setsum f A <= setsum f B"
avigad@16775
   774
  apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
avigad@16775
   775
  apply (erule ssubst)
avigad@16775
   776
  apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
avigad@16775
   777
  apply simp
avigad@16775
   778
  apply (rule add_left_mono)
avigad@16775
   779
  apply (erule setsum_nonneg)
avigad@16775
   780
  apply (subst setsum_Un_disjoint [THEN sym])
avigad@16775
   781
  apply (erule finite_subset, assumption)
avigad@16775
   782
  apply (rule finite_subset)
avigad@16775
   783
  prefer 2
avigad@16775
   784
  apply assumption
haftmann@32698
   785
  apply (auto simp add: sup_absorb2)
avigad@16775
   786
done
avigad@16775
   787
ballarin@19279
   788
lemma setsum_right_distrib: 
huffman@22934
   789
  fixes f :: "'a => ('b::semiring_0)"
nipkow@15402
   790
  shows "r * setsum f A = setsum (%n. r * f n) A"
nipkow@15402
   791
proof (cases "finite A")
nipkow@15402
   792
  case True
nipkow@15402
   793
  thus ?thesis
wenzelm@21575
   794
  proof induct
nipkow@15402
   795
    case empty thus ?case by simp
nipkow@15402
   796
  next
webertj@49962
   797
    case (insert x A) thus ?case by (simp add: distrib_left)
nipkow@15402
   798
  qed
nipkow@15402
   799
next
haftmann@51489
   800
  case False thus ?thesis by simp
nipkow@15402
   801
qed
nipkow@15402
   802
ballarin@17149
   803
lemma setsum_left_distrib:
huffman@22934
   804
  "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
ballarin@17149
   805
proof (cases "finite A")
ballarin@17149
   806
  case True
ballarin@17149
   807
  then show ?thesis
ballarin@17149
   808
  proof induct
ballarin@17149
   809
    case empty thus ?case by simp
ballarin@17149
   810
  next
webertj@49962
   811
    case (insert x A) thus ?case by (simp add: distrib_right)
ballarin@17149
   812
  qed
ballarin@17149
   813
next
haftmann@51489
   814
  case False thus ?thesis by simp
ballarin@17149
   815
qed
ballarin@17149
   816
ballarin@17149
   817
lemma setsum_divide_distrib:
ballarin@17149
   818
  "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
ballarin@17149
   819
proof (cases "finite A")
ballarin@17149
   820
  case True
ballarin@17149
   821
  then show ?thesis
ballarin@17149
   822
  proof induct
ballarin@17149
   823
    case empty thus ?case by simp
ballarin@17149
   824
  next
ballarin@17149
   825
    case (insert x A) thus ?case by (simp add: add_divide_distrib)
ballarin@17149
   826
  qed
ballarin@17149
   827
next
haftmann@51489
   828
  case False thus ?thesis by simp
ballarin@17149
   829
qed
ballarin@17149
   830
nipkow@15535
   831
lemma setsum_abs[iff]: 
haftmann@35028
   832
  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
nipkow@15402
   833
  shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
nipkow@15535
   834
proof (cases "finite A")
nipkow@15535
   835
  case True
nipkow@15535
   836
  thus ?thesis
wenzelm@21575
   837
  proof induct
nipkow@15535
   838
    case empty thus ?case by simp
nipkow@15535
   839
  next
nipkow@15535
   840
    case (insert x A)
nipkow@15535
   841
    thus ?case by (auto intro: abs_triangle_ineq order_trans)
nipkow@15535
   842
  qed
nipkow@15402
   843
next
haftmann@51489
   844
  case False thus ?thesis by simp
nipkow@15402
   845
qed
nipkow@15402
   846
nipkow@15535
   847
lemma setsum_abs_ge_zero[iff]: 
haftmann@35028
   848
  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
nipkow@15402
   849
  shows "0 \<le> setsum (%i. abs(f i)) A"
nipkow@15535
   850
proof (cases "finite A")
nipkow@15535
   851
  case True
nipkow@15535
   852
  thus ?thesis
wenzelm@21575
   853
  proof induct
nipkow@15535
   854
    case empty thus ?case by simp
nipkow@15535
   855
  next
huffman@36977
   856
    case (insert x A) thus ?case by auto
nipkow@15535
   857
  qed
nipkow@15402
   858
next
haftmann@51489
   859
  case False thus ?thesis by simp
nipkow@15402
   860
qed
nipkow@15402
   861
nipkow@15539
   862
lemma abs_setsum_abs[simp]: 
haftmann@35028
   863
  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
nipkow@15539
   864
  shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
nipkow@15539
   865
proof (cases "finite A")
nipkow@15539
   866
  case True
nipkow@15539
   867
  thus ?thesis
wenzelm@21575
   868
  proof induct
nipkow@15539
   869
    case empty thus ?case by simp
nipkow@15539
   870
  next
nipkow@15539
   871
    case (insert a A)
nipkow@15539
   872
    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
nipkow@15539
   873
    also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
avigad@16775
   874
    also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
avigad@16775
   875
      by (simp del: abs_of_nonneg)
nipkow@15539
   876
    also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
nipkow@15539
   877
    finally show ?case .
nipkow@15539
   878
  qed
nipkow@15539
   879
next
haftmann@51489
   880
  case False thus ?thesis by simp
nipkow@31080
   881
qed
nipkow@31080
   882
haftmann@51489
   883
lemma setsum_diff1'[rule_format]:
haftmann@51489
   884
  "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
haftmann@51489
   885
apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"])
haftmann@51489
   886
apply (auto simp add: insert_Diff_if add_ac)
haftmann@51489
   887
done
ballarin@17149
   888
haftmann@51489
   889
lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
haftmann@51489
   890
  shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
haftmann@51489
   891
unfolding setsum_diff1'[OF assms] by auto
ballarin@17149
   892
ballarin@19279
   893
lemma setsum_product:
huffman@22934
   894
  fixes f :: "'a => ('b::semiring_0)"
ballarin@19279
   895
  shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
ballarin@19279
   896
  by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)
ballarin@19279
   897
nipkow@34223
   898
lemma setsum_mult_setsum_if_inj:
nipkow@34223
   899
fixes f :: "'a => ('b::semiring_0)"
nipkow@34223
   900
shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
nipkow@34223
   901
  setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
nipkow@34223
   902
by(auto simp: setsum_product setsum_cartesian_product
nipkow@34223
   903
        intro!:  setsum_reindex_cong[symmetric])
nipkow@34223
   904
haftmann@51489
   905
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
haftmann@51489
   906
apply (case_tac "finite A")
haftmann@51489
   907
 prefer 2 apply simp
haftmann@51489
   908
apply (erule rev_mp)
haftmann@51489
   909
apply (erule finite_induct, auto)
haftmann@51489
   910
done
haftmann@51489
   911
haftmann@51489
   912
lemma setsum_eq_0_iff [simp]:
haftmann@51489
   913
  "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
haftmann@51489
   914
  by (induct set: finite) auto
haftmann@51489
   915
haftmann@51489
   916
lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
haftmann@51489
   917
  setsum f A = Suc 0 \<longleftrightarrow> (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
haftmann@51489
   918
apply(erule finite_induct)
haftmann@51489
   919
apply (auto simp add:add_is_1)
haftmann@51489
   920
done
haftmann@51489
   921
haftmann@51489
   922
lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
haftmann@51489
   923
haftmann@51489
   924
lemma setsum_Un_nat: "finite A ==> finite B ==>
haftmann@51489
   925
  (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
haftmann@51489
   926
  -- {* For the natural numbers, we have subtraction. *}
haftmann@51489
   927
by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
haftmann@51489
   928
haftmann@51489
   929
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
haftmann@51489
   930
  (if a:A then setsum f A - f a else setsum f A)"
haftmann@51489
   931
apply (case_tac "finite A")
haftmann@51489
   932
 prefer 2 apply simp
haftmann@51489
   933
apply (erule finite_induct)
haftmann@51489
   934
 apply (auto simp add: insert_Diff_if)
haftmann@51489
   935
apply (drule_tac a = a in mk_disjoint_insert, auto)
haftmann@51489
   936
done
haftmann@51489
   937
haftmann@51489
   938
lemma setsum_diff_nat: 
haftmann@51489
   939
assumes "finite B" and "B \<subseteq> A"
haftmann@51489
   940
shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
haftmann@51489
   941
using assms
haftmann@51489
   942
proof induct
haftmann@51489
   943
  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
haftmann@51489
   944
next
haftmann@51489
   945
  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
haftmann@51489
   946
    and xFinA: "insert x F \<subseteq> A"
haftmann@51489
   947
    and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
haftmann@51489
   948
  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
haftmann@51489
   949
  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
haftmann@51489
   950
    by (simp add: setsum_diff1_nat)
haftmann@51489
   951
  from xFinA have "F \<subseteq> A" by simp
haftmann@51489
   952
  with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
haftmann@51489
   953
  with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
haftmann@51489
   954
    by simp
haftmann@51489
   955
  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
haftmann@51489
   956
  with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
haftmann@51489
   957
    by simp
haftmann@51489
   958
  from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
haftmann@51489
   959
  with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
haftmann@51489
   960
    by simp
haftmann@51489
   961
  thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
haftmann@51489
   962
qed
haftmann@51489
   963
haftmann@51600
   964
lemma setsum_comp_morphism:
haftmann@51600
   965
  assumes "h 0 = 0" and "\<And>x y. h (x + y) = h x + h y"
haftmann@51600
   966
  shows "setsum (h \<circ> g) A = h (setsum g A)"
haftmann@51600
   967
proof (cases "finite A")
haftmann@51600
   968
  case False then show ?thesis by (simp add: assms)
haftmann@51600
   969
next
haftmann@51600
   970
  case True then show ?thesis by (induct A) (simp_all add: assms)
haftmann@51600
   971
qed
haftmann@51600
   972
haftmann@51489
   973
haftmann@51489
   974
subsubsection {* Cardinality as special case of @{const setsum} *}
haftmann@51489
   975
haftmann@51489
   976
lemma card_eq_setsum:
haftmann@51489
   977
  "card A = setsum (\<lambda>x. 1) A"
haftmann@51489
   978
proof -
haftmann@51489
   979
  have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)"
haftmann@51489
   980
    by (simp add: fun_eq_iff)
haftmann@51489
   981
  then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) = Finite_Set.fold (\<lambda>_. Suc)"
haftmann@51489
   982
    by (rule arg_cong)
haftmann@51489
   983
  then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
haftmann@51489
   984
    by (blast intro: fun_cong)
haftmann@51489
   985
  then show ?thesis by (simp add: card.eq_fold setsum.eq_fold)
haftmann@51489
   986
qed
haftmann@51489
   987
haftmann@51489
   988
lemma setsum_constant [simp]:
haftmann@51489
   989
  "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
haftmann@35722
   990
apply (cases "finite A")
haftmann@35722
   991
apply (erule finite_induct)
haftmann@35722
   992
apply (auto simp add: algebra_simps)
haftmann@35722
   993
done
haftmann@35722
   994
haftmann@35722
   995
lemma setsum_bounded:
haftmann@35722
   996
  assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"
haftmann@51489
   997
  shows "setsum f A \<le> of_nat (card A) * K"
haftmann@35722
   998
proof (cases "finite A")
haftmann@35722
   999
  case True
haftmann@35722
  1000
  thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
haftmann@35722
  1001
next
haftmann@51489
  1002
  case False thus ?thesis by simp
haftmann@35722
  1003
qed
haftmann@35722
  1004
haftmann@35722
  1005
lemma card_UN_disjoint:
haftmann@46629
  1006
  assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
haftmann@46629
  1007
    and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
haftmann@46629
  1008
  shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
haftmann@46629
  1009
proof -
haftmann@46629
  1010
  have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)" by simp
haftmann@46629
  1011
  with assms show ?thesis by (simp add: card_eq_setsum setsum_UN_disjoint del: setsum_constant)
haftmann@46629
  1012
qed
haftmann@35722
  1013
haftmann@35722
  1014
lemma card_Union_disjoint:
haftmann@35722
  1015
  "finite C ==> (ALL A:C. finite A) ==>
haftmann@35722
  1016
   (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {})
haftmann@35722
  1017
   ==> card (Union C) = setsum card C"
haftmann@35722
  1018
apply (frule card_UN_disjoint [of C id])
hoelzl@44937
  1019
apply (simp_all add: SUP_def id_def)
haftmann@35722
  1020
done
haftmann@35722
  1021
haftmann@35722
  1022
haftmann@35722
  1023
subsubsection {* Cardinality of products *}
haftmann@35722
  1024
haftmann@35722
  1025
lemma card_SigmaI [simp]:
haftmann@35722
  1026
  "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
haftmann@35722
  1027
  \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
haftmann@35722
  1028
by(simp add: card_eq_setsum setsum_Sigma del:setsum_constant)
haftmann@35722
  1029
haftmann@35722
  1030
(*
haftmann@35722
  1031
lemma SigmaI_insert: "y \<notin> A ==>
haftmann@35722
  1032
  (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
haftmann@35722
  1033
  by auto
haftmann@35722
  1034
*)
haftmann@35722
  1035
haftmann@35722
  1036
lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
haftmann@35722
  1037
  by (cases "finite A \<and> finite B")
haftmann@35722
  1038
    (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
haftmann@35722
  1039
haftmann@35722
  1040
lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
haftmann@35722
  1041
by (simp add: card_cartesian_product)
haftmann@35722
  1042
ballarin@17149
  1043
nipkow@15402
  1044
subsection {* Generalized product over a set *}
nipkow@15402
  1045
haftmann@51738
  1046
context comm_monoid_mult
haftmann@51738
  1047
begin
haftmann@51738
  1048
haftmann@51738
  1049
definition setprod :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
haftmann@51489
  1050
where
haftmann@51489
  1051
  "setprod = comm_monoid_set.F times 1"
haftmann@35816
  1052
haftmann@51738
  1053
sublocale setprod!: comm_monoid_set times 1
haftmann@51489
  1054
where
haftmann@51546
  1055
  "comm_monoid_set.F times 1 = setprod"
haftmann@51489
  1056
proof -
haftmann@51489
  1057
  show "comm_monoid_set times 1" ..
haftmann@51489
  1058
  then interpret setprod!: comm_monoid_set times 1 .
haftmann@51546
  1059
  from setprod_def show "comm_monoid_set.F times 1 = setprod" by rule
haftmann@51489
  1060
qed
nipkow@15402
  1061
wenzelm@19535
  1062
abbreviation
haftmann@51489
  1063
  Setprod ("\<Prod>_" [1000] 999) where
haftmann@51489
  1064
  "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"
wenzelm@19535
  1065
haftmann@51738
  1066
end
haftmann@51738
  1067
nipkow@15402
  1068
syntax
paulson@17189
  1069
  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
nipkow@15402
  1070
syntax (xsymbols)
paulson@17189
  1071
  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402
  1072
syntax (HTML output)
paulson@17189
  1073
  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
nipkow@16550
  1074
nipkow@16550
  1075
translations -- {* Beware of argument permutation! *}
nipkow@28853
  1076
  "PROD i:A. b" == "CONST setprod (%i. b) A" 
nipkow@28853
  1077
  "\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A" 
nipkow@16550
  1078
nipkow@16550
  1079
text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
nipkow@16550
  1080
 @{text"\<Prod>x|P. e"}. *}
nipkow@16550
  1081
nipkow@16550
  1082
syntax
paulson@17189
  1083
  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
nipkow@16550
  1084
syntax (xsymbols)
paulson@17189
  1085
  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
nipkow@16550
  1086
syntax (HTML output)
paulson@17189
  1087
  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
nipkow@16550
  1088
nipkow@15402
  1089
translations
nipkow@28853
  1090
  "PROD x|P. t" => "CONST setprod (%x. t) {x. P}"
nipkow@28853
  1091
  "\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"
nipkow@16550
  1092
haftmann@51489
  1093
text {* TODO These are candidates for generalization *}
haftmann@51489
  1094
haftmann@51489
  1095
context comm_monoid_mult
haftmann@51489
  1096
begin
haftmann@51489
  1097
haftmann@51489
  1098
lemma setprod_reindex_id:
haftmann@51489
  1099
  "inj_on f B ==> setprod f B = setprod id (f ` B)"
haftmann@51489
  1100
  by (auto simp add: setprod.reindex)
haftmann@51489
  1101
haftmann@51489
  1102
lemma setprod_reindex_cong:
haftmann@51489
  1103
  "inj_on f A ==> B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
haftmann@51489
  1104
  by (frule setprod.reindex, simp)
haftmann@51489
  1105
haftmann@51489
  1106
lemma strong_setprod_reindex_cong:
haftmann@51489
  1107
  assumes i: "inj_on f A"
haftmann@51489
  1108
  and B: "B = f ` A" and eq: "\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x"
haftmann@51489
  1109
  shows "setprod h B = setprod g A"
haftmann@51489
  1110
proof-
haftmann@51489
  1111
  have "setprod h B = setprod (h o f) A"
haftmann@51489
  1112
    by (simp add: B setprod.reindex [OF i, of h])
haftmann@51489
  1113
  then show ?thesis apply simp
haftmann@51489
  1114
    apply (rule setprod.cong)
haftmann@51489
  1115
    apply simp
haftmann@51489
  1116
    by (simp add: eq)
haftmann@51489
  1117
qed
haftmann@51489
  1118
haftmann@51489
  1119
lemma setprod_Union_disjoint:
haftmann@51489
  1120
  assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}" 
haftmann@51489
  1121
  shows "setprod f (Union C) = setprod (setprod f) C"
haftmann@51489
  1122
  using assms by (fact setprod.Union_disjoint)
haftmann@51489
  1123
haftmann@51489
  1124
text{*Here we can eliminate the finiteness assumptions, by cases.*}
haftmann@51489
  1125
lemma setprod_cartesian_product:
haftmann@51489
  1126
  "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
haftmann@51489
  1127
  by (fact setprod.cartesian_product)
haftmann@51489
  1128
haftmann@51489
  1129
lemma setprod_Un2:
haftmann@51489
  1130
  assumes "finite (A \<union> B)"
haftmann@51489
  1131
  shows "setprod f (A \<union> B) = setprod f (A - B) * setprod f (B - A) * setprod f (A \<inter> B)"
haftmann@51489
  1132
proof -
haftmann@51489
  1133
  have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
haftmann@51489
  1134
    by auto
haftmann@51489
  1135
  with assms show ?thesis by simp (subst setprod.union_disjoint, auto)+
haftmann@51489
  1136
qed
haftmann@51489
  1137
haftmann@51489
  1138
end
haftmann@51489
  1139
haftmann@51489
  1140
text {* TODO These are legacy *}
haftmann@51489
  1141
haftmann@35816
  1142
lemma setprod_empty: "setprod f {} = 1"
haftmann@35816
  1143
  by (fact setprod.empty)
nipkow@15402
  1144
haftmann@35816
  1145
lemma setprod_insert: "[| finite A; a \<notin> A |] ==>
nipkow@15402
  1146
    setprod f (insert a A) = f a * setprod f A"
haftmann@35816
  1147
  by (fact setprod.insert)
nipkow@15402
  1148
haftmann@35816
  1149
lemma setprod_infinite: "~ finite A ==> setprod f A = 1"
haftmann@35816
  1150
  by (fact setprod.infinite)
paulson@15409
  1151
nipkow@15402
  1152
lemma setprod_reindex:
haftmann@51489
  1153
  "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
haftmann@51489
  1154
  by (fact setprod.reindex)
nipkow@15402
  1155
nipkow@15402
  1156
lemma setprod_cong:
nipkow@15402
  1157
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
haftmann@51489
  1158
  by (fact setprod.cong)
nipkow@15402
  1159
nipkow@48849
  1160
lemma strong_setprod_cong:
berghofe@16632
  1161
  "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
haftmann@51489
  1162
  by (fact setprod.strong_cong)
nipkow@15402
  1163
haftmann@51489
  1164
lemma setprod_Un_one:
haftmann@51489
  1165
  "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 1 \<rbrakk>
haftmann@51489
  1166
  \<Longrightarrow> setprod f (S \<union> T) = setprod f S  * setprod f T"
haftmann@51489
  1167
  by (fact setprod.union_inter_neutral)
chaieb@29674
  1168
haftmann@51489
  1169
lemmas setprod_1 = setprod.neutral_const
haftmann@51489
  1170
lemmas setprod_1' = setprod.neutral
nipkow@15402
  1171
nipkow@15402
  1172
lemma setprod_Un_Int: "finite A ==> finite B
nipkow@15402
  1173
    ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
haftmann@51489
  1174
  by (fact setprod.union_inter)
nipkow@15402
  1175
nipkow@15402
  1176
lemma setprod_Un_disjoint: "finite A ==> finite B
nipkow@15402
  1177
  ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
haftmann@51489
  1178
  by (fact setprod.union_disjoint)
nipkow@48849
  1179
nipkow@48849
  1180
lemma setprod_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
nipkow@48849
  1181
    setprod f A = setprod f (A - B) * setprod f B"
haftmann@51489
  1182
  by (fact setprod.subset_diff)
nipkow@15402
  1183
nipkow@48849
  1184
lemma setprod_mono_one_left:
nipkow@48849
  1185
  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 1 \<rbrakk> \<Longrightarrow> setprod f S = setprod f T"
haftmann@51489
  1186
  by (fact setprod.mono_neutral_left)
nipkow@30837
  1187
haftmann@51489
  1188
lemmas setprod_mono_one_right = setprod.mono_neutral_right
nipkow@30837
  1189
nipkow@48849
  1190
lemma setprod_mono_one_cong_left: 
nipkow@48849
  1191
  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1; \<And>x. x \<in> S \<Longrightarrow> f x = g x \<rbrakk>
nipkow@48849
  1192
  \<Longrightarrow> setprod f S = setprod g T"
haftmann@51489
  1193
  by (fact setprod.mono_neutral_cong_left)
nipkow@48849
  1194
haftmann@51489
  1195
lemmas setprod_mono_one_cong_right = setprod.mono_neutral_cong_right
chaieb@29674
  1196
nipkow@48849
  1197
lemma setprod_delta: "finite S \<Longrightarrow>
nipkow@48849
  1198
  setprod (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"
haftmann@51489
  1199
  by (fact setprod.delta)
chaieb@29674
  1200
nipkow@48849
  1201
lemma setprod_delta': "finite S \<Longrightarrow>
nipkow@48849
  1202
  setprod (\<lambda>k. if a = k then b k else 1) S = (if a\<in> S then b a else 1)"
haftmann@51489
  1203
  by (fact setprod.delta')
chaieb@29674
  1204
nipkow@15402
  1205
lemma setprod_UN_disjoint:
nipkow@15402
  1206
    "finite I ==> (ALL i:I. finite (A i)) ==>
nipkow@15402
  1207
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
nipkow@15402
  1208
      setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
haftmann@51489
  1209
  by (fact setprod.UNION_disjoint)
nipkow@15402
  1210
nipkow@15402
  1211
lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@16550
  1212
    (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
paulson@17189
  1213
    (\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
haftmann@51489
  1214
  by (fact setprod.Sigma)
nipkow@15402
  1215
haftmann@51489
  1216
lemma setprod_timesf: "setprod (\<lambda>x. f x * g x) A = setprod f A * setprod g A"
haftmann@51489
  1217
  by (fact setprod.distrib)
nipkow@15402
  1218
nipkow@15402
  1219
nipkow@15402
  1220
subsubsection {* Properties in more restricted classes of structures *}
nipkow@15402
  1221
nipkow@15402
  1222
lemma setprod_zero:
huffman@23277
  1223
     "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0"
nipkow@28853
  1224
apply (induct set: finite, force, clarsimp)
nipkow@28853
  1225
apply (erule disjE, auto)
nipkow@28853
  1226
done
nipkow@15402
  1227
haftmann@51489
  1228
lemma setprod_zero_iff[simp]: "finite A ==> 
haftmann@51489
  1229
  (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) =
haftmann@51489
  1230
  (EX x: A. f x = 0)"
haftmann@51489
  1231
by (erule finite_induct, auto simp:no_zero_divisors)
haftmann@51489
  1232
haftmann@51489
  1233
lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
haftmann@51489
  1234
  (setprod f (A Un B) :: 'a ::{field})
haftmann@51489
  1235
   = setprod f A * setprod f B / setprod f (A Int B)"
haftmann@51489
  1236
by (subst setprod_Un_Int [symmetric], auto)
haftmann@51489
  1237
nipkow@15402
  1238
lemma setprod_nonneg [rule_format]:
haftmann@35028
  1239
   "(ALL x: A. (0::'a::linordered_semidom) \<le> f x) --> 0 \<le> setprod f A"
huffman@30841
  1240
by (cases "finite A", induct set: finite, simp_all add: mult_nonneg_nonneg)
huffman@30841
  1241
haftmann@35028
  1242
lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::linordered_semidom) < f x)
nipkow@28853
  1243
  --> 0 < setprod f A"
huffman@30841
  1244
by (cases "finite A", induct set: finite, simp_all add: mult_pos_pos)
nipkow@15402
  1245
nipkow@15402
  1246
lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
nipkow@28853
  1247
  (setprod f (A - {a}) :: 'a :: {field}) =
nipkow@28853
  1248
  (if a:A then setprod f A / f a else setprod f A)"
haftmann@36303
  1249
  by (erule finite_induct) (auto simp add: insert_Diff_if)
nipkow@15402
  1250
paulson@31906
  1251
lemma setprod_inversef: 
haftmann@36409
  1252
  fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
paulson@31906
  1253
  shows "finite A ==> setprod (inverse \<circ> f) A = inverse (setprod f A)"
nipkow@28853
  1254
by (erule finite_induct) auto
nipkow@15402
  1255
nipkow@15402
  1256
lemma setprod_dividef:
haftmann@36409
  1257
  fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
wenzelm@31916
  1258
  shows "finite A
nipkow@28853
  1259
    ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
nipkow@28853
  1260
apply (subgoal_tac
nipkow@15402
  1261
         "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
nipkow@28853
  1262
apply (erule ssubst)
nipkow@28853
  1263
apply (subst divide_inverse)
nipkow@28853
  1264
apply (subst setprod_timesf)
nipkow@28853
  1265
apply (subst setprod_inversef, assumption+, rule refl)
nipkow@28853
  1266
apply (rule setprod_cong, rule refl)
nipkow@28853
  1267
apply (subst divide_inverse, auto)
nipkow@28853
  1268
done
nipkow@28853
  1269
nipkow@29925
  1270
lemma setprod_dvd_setprod [rule_format]: 
nipkow@29925
  1271
    "(ALL x : A. f x dvd g x) \<longrightarrow> setprod f A dvd setprod g A"
nipkow@29925
  1272
  apply (cases "finite A")
nipkow@29925
  1273
  apply (induct set: finite)
nipkow@29925
  1274
  apply (auto simp add: dvd_def)
nipkow@29925
  1275
  apply (rule_tac x = "k * ka" in exI)
nipkow@29925
  1276
  apply (simp add: algebra_simps)
nipkow@29925
  1277
done
nipkow@29925
  1278
nipkow@29925
  1279
lemma setprod_dvd_setprod_subset:
nipkow@29925
  1280
  "finite B \<Longrightarrow> A <= B \<Longrightarrow> setprod f A dvd setprod f B"
nipkow@29925
  1281
  apply (subgoal_tac "setprod f B = setprod f A * setprod f (B - A)")
nipkow@29925
  1282
  apply (unfold dvd_def, blast)
nipkow@29925
  1283
  apply (subst setprod_Un_disjoint [symmetric])
nipkow@29925
  1284
  apply (auto elim: finite_subset intro: setprod_cong)
nipkow@29925
  1285
done
nipkow@29925
  1286
nipkow@29925
  1287
lemma setprod_dvd_setprod_subset2:
nipkow@29925
  1288
  "finite B \<Longrightarrow> A <= B \<Longrightarrow> ALL x : A. (f x::'a::comm_semiring_1) dvd g x \<Longrightarrow> 
nipkow@29925
  1289
      setprod f A dvd setprod g B"
nipkow@29925
  1290
  apply (rule dvd_trans)
nipkow@29925
  1291
  apply (rule setprod_dvd_setprod, erule (1) bspec)
nipkow@29925
  1292
  apply (erule (1) setprod_dvd_setprod_subset)
nipkow@29925
  1293
done
nipkow@29925
  1294
nipkow@29925
  1295
lemma dvd_setprod: "finite A \<Longrightarrow> i:A \<Longrightarrow> 
nipkow@29925
  1296
    (f i ::'a::comm_semiring_1) dvd setprod f A"
nipkow@29925
  1297
by (induct set: finite) (auto intro: dvd_mult)
nipkow@29925
  1298
nipkow@29925
  1299
lemma dvd_setsum [rule_format]: "(ALL i : A. d dvd f i) \<longrightarrow> 
nipkow@29925
  1300
    (d::'a::comm_semiring_1) dvd (SUM x : A. f x)"
nipkow@29925
  1301
  apply (cases "finite A")
nipkow@29925
  1302
  apply (induct set: finite)
nipkow@29925
  1303
  apply auto
nipkow@29925
  1304
done
nipkow@29925
  1305
hoelzl@35171
  1306
lemma setprod_mono:
hoelzl@35171
  1307
  fixes f :: "'a \<Rightarrow> 'b\<Colon>linordered_semidom"
hoelzl@35171
  1308
  assumes "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i"
hoelzl@35171
  1309
  shows "setprod f A \<le> setprod g A"
hoelzl@35171
  1310
proof (cases "finite A")
hoelzl@35171
  1311
  case True
hoelzl@35171
  1312
  hence ?thesis "setprod f A \<ge> 0" using subset_refl[of A]
hoelzl@35171
  1313
  proof (induct A rule: finite_subset_induct)
hoelzl@35171
  1314
    case (insert a F)
hoelzl@35171
  1315
    thus "setprod f (insert a F) \<le> setprod g (insert a F)" "0 \<le> setprod f (insert a F)"
hoelzl@35171
  1316
      unfolding setprod_insert[OF insert(1,3)]
hoelzl@35171
  1317
      using assms[rule_format,OF insert(2)] insert
hoelzl@35171
  1318
      by (auto intro: mult_mono mult_nonneg_nonneg)
hoelzl@35171
  1319
  qed auto
hoelzl@35171
  1320
  thus ?thesis by simp
hoelzl@35171
  1321
qed auto
hoelzl@35171
  1322
hoelzl@35171
  1323
lemma abs_setprod:
hoelzl@35171
  1324
  fixes f :: "'a \<Rightarrow> 'b\<Colon>{linordered_field,abs}"
hoelzl@35171
  1325
  shows "abs (setprod f A) = setprod (\<lambda>x. abs (f x)) A"
hoelzl@35171
  1326
proof (cases "finite A")
hoelzl@35171
  1327
  case True thus ?thesis
huffman@35216
  1328
    by induct (auto simp add: field_simps abs_mult)
hoelzl@35171
  1329
qed auto
hoelzl@35171
  1330
haftmann@31017
  1331
lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
nipkow@28853
  1332
apply (erule finite_induct)
huffman@35216
  1333
apply auto
nipkow@28853
  1334
done
nipkow@15402
  1335
chaieb@29674
  1336
lemma setprod_gen_delta:
chaieb@29674
  1337
  assumes fS: "finite S"
haftmann@51489
  1338
  shows "setprod (\<lambda>k. if k=a then b k else c) S = (if a \<in> S then (b a ::'a::comm_monoid_mult) * c^ (card S - 1) else c^ card S)"
chaieb@29674
  1339
proof-
chaieb@29674
  1340
  let ?f = "(\<lambda>k. if k=a then b k else c)"
chaieb@29674
  1341
  {assume a: "a \<notin> S"
chaieb@29674
  1342
    hence "\<forall> k\<in> S. ?f k = c" by simp
nipkow@48849
  1343
    hence ?thesis  using a setprod_constant[OF fS, of c] by simp }
chaieb@29674
  1344
  moreover 
chaieb@29674
  1345
  {assume a: "a \<in> S"
chaieb@29674
  1346
    let ?A = "S - {a}"
chaieb@29674
  1347
    let ?B = "{a}"
chaieb@29674
  1348
    have eq: "S = ?A \<union> ?B" using a by blast 
chaieb@29674
  1349
    have dj: "?A \<inter> ?B = {}" by simp
chaieb@29674
  1350
    from fS have fAB: "finite ?A" "finite ?B" by auto  
chaieb@29674
  1351
    have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
chaieb@29674
  1352
      apply (rule setprod_cong) by auto
chaieb@29674
  1353
    have cA: "card ?A = card S - 1" using fS a by auto
chaieb@29674
  1354
    have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
chaieb@29674
  1355
    have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
chaieb@29674
  1356
      using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
chaieb@29674
  1357
      by simp
chaieb@29674
  1358
    then have ?thesis using a cA
haftmann@36349
  1359
      by (simp add: fA1 field_simps cong add: setprod_cong cong del: if_weak_cong)}
chaieb@29674
  1360
  ultimately show ?thesis by blast
chaieb@29674
  1361
qed
chaieb@29674
  1362
haftmann@51489
  1363
lemma setprod_eq_1_iff [simp]:
haftmann@51489
  1364
  "finite F ==> setprod f F = 1 \<longleftrightarrow> (ALL a:F. f a = (1::nat))"
haftmann@51489
  1365
  by (induct set: finite) auto
chaieb@29674
  1366
haftmann@51489
  1367
lemma setprod_pos_nat:
haftmann@51489
  1368
  "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0"
haftmann@51489
  1369
using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
haftmann@51489
  1370
haftmann@51489
  1371
lemma setprod_pos_nat_iff[simp]:
haftmann@51489
  1372
  "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))"
haftmann@51489
  1373
using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
haftmann@51489
  1374
haftmann@51489
  1375
haftmann@51489
  1376
subsection {* Generic lattice operations over a set *}
haftmann@35816
  1377
haftmann@35816
  1378
no_notation times (infixl "*" 70)
haftmann@35816
  1379
no_notation Groups.one ("1")
haftmann@35816
  1380
haftmann@51489
  1381
haftmann@51489
  1382
subsubsection {* Without neutral element *}
haftmann@51489
  1383
haftmann@51489
  1384
locale semilattice_set = semilattice
haftmann@51489
  1385
begin
haftmann@51489
  1386
haftmann@51738
  1387
interpretation comp_fun_idem f
haftmann@51738
  1388
  by default (simp_all add: fun_eq_iff left_commute)
haftmann@51738
  1389
haftmann@51489
  1390
definition F :: "'a set \<Rightarrow> 'a"
haftmann@51489
  1391
where
haftmann@51489
  1392
  eq_fold': "F A = the (Finite_Set.fold (\<lambda>x y. Some (case y of None \<Rightarrow> x | Some z \<Rightarrow> f x z)) None A)"
haftmann@51489
  1393
haftmann@51489
  1394
lemma eq_fold:
haftmann@51489
  1395
  assumes "finite A"
haftmann@51489
  1396
  shows "F (insert x A) = Finite_Set.fold f x A"
haftmann@51489
  1397
proof (rule sym)
haftmann@51489
  1398
  let ?f = "\<lambda>x y. Some (case y of None \<Rightarrow> x | Some z \<Rightarrow> f x z)"
haftmann@51489
  1399
  interpret comp_fun_idem "?f"
haftmann@51489
  1400
    by default (simp_all add: fun_eq_iff commute left_commute split: option.split)
haftmann@51489
  1401
  from assms show "Finite_Set.fold f x A = F (insert x A)"
haftmann@51489
  1402
  proof induct
haftmann@51489
  1403
    case empty then show ?case by (simp add: eq_fold')
haftmann@51489
  1404
  next
haftmann@51489
  1405
    case (insert y B) then show ?case by (simp add: insert_commute [of x] eq_fold')
haftmann@51489
  1406
  qed
haftmann@51489
  1407
qed
haftmann@51489
  1408
haftmann@51489
  1409
lemma singleton [simp]:
haftmann@51489
  1410
  "F {x} = x"
haftmann@51489
  1411
  by (simp add: eq_fold)
haftmann@51489
  1412
haftmann@51489
  1413
lemma insert_not_elem:
haftmann@51489
  1414
  assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
haftmann@51489
  1415
  shows "F (insert x A) = x * F A"
haftmann@51489
  1416
proof -
haftmann@51489
  1417
  from `A \<noteq> {}` obtain b where "b \<in> A" by blast
haftmann@51489
  1418
  then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@51489
  1419
  with `finite A` and `x \<notin> A`
haftmann@51489
  1420
    have "finite (insert x B)" and "b \<notin> insert x B" by auto
haftmann@51489
  1421
  then have "F (insert b (insert x B)) = x * F (insert b B)"
haftmann@51489
  1422
    by (simp add: eq_fold)
haftmann@51489
  1423
  then show ?thesis by (simp add: * insert_commute)
haftmann@51489
  1424
qed
haftmann@51489
  1425
haftmann@51586
  1426
lemma in_idem:
haftmann@51489
  1427
  assumes "finite A" and "x \<in> A"
haftmann@51489
  1428
  shows "x * F A = F A"
haftmann@51489
  1429
proof -
haftmann@51489
  1430
  from assms have "A \<noteq> {}" by auto
haftmann@51489
  1431
  with `finite A` show ?thesis using `x \<in> A`
haftmann@51489
  1432
    by (induct A rule: finite_ne_induct) (auto simp add: ac_simps insert_not_elem)
haftmann@51489
  1433
qed
haftmann@51489
  1434
haftmann@51489
  1435
lemma insert [simp]:
haftmann@51489
  1436
  assumes "finite A" and "A \<noteq> {}"
haftmann@51489
  1437
  shows "F (insert x A) = x * F A"
haftmann@51586
  1438
  using assms by (cases "x \<in> A") (simp_all add: insert_absorb in_idem insert_not_elem)
haftmann@51489
  1439
haftmann@51489
  1440
lemma union:
haftmann@51489
  1441
  assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
haftmann@51489
  1442
  shows "F (A \<union> B) = F A * F B"
haftmann@51489
  1443
  using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
haftmann@51489
  1444
haftmann@51489
  1445
lemma remove:
haftmann@51489
  1446
  assumes "finite A" and "x \<in> A"
haftmann@51489
  1447
  shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
haftmann@51489
  1448
proof -
haftmann@51489
  1449
  from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@51489
  1450
  with assms show ?thesis by simp
haftmann@51489
  1451
qed
haftmann@51489
  1452
haftmann@51489
  1453
lemma insert_remove:
haftmann@51489
  1454
  assumes "finite A"
haftmann@51489
  1455
  shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
haftmann@51489
  1456
  using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
haftmann@51489
  1457
haftmann@51489
  1458
lemma subset:
haftmann@51489
  1459
  assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
haftmann@51489
  1460
  shows "F B * F A = F A"
haftmann@51489
  1461
proof -
haftmann@51489
  1462
  from assms have "A \<noteq> {}" and "finite B" by (auto dest: finite_subset)
haftmann@51489
  1463
  with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
haftmann@51489
  1464
qed
haftmann@51489
  1465
haftmann@51489
  1466
lemma closed:
haftmann@51489
  1467
  assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
haftmann@51489
  1468
  shows "F A \<in> A"
haftmann@51489
  1469
using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
haftmann@51489
  1470
  case singleton then show ?case by simp
haftmann@51489
  1471
next
haftmann@51489
  1472
  case insert with elem show ?case by force
haftmann@51489
  1473
qed
haftmann@51489
  1474
haftmann@51489
  1475
lemma hom_commute:
haftmann@51489
  1476
  assumes hom: "\<And>x y. h (x * y) = h x * h y"
haftmann@51489
  1477
  and N: "finite N" "N \<noteq> {}"
haftmann@51489
  1478
  shows "h (F N) = F (h ` N)"
haftmann@51489
  1479
using N proof (induct rule: finite_ne_induct)
haftmann@51489
  1480
  case singleton thus ?case by simp
haftmann@51489
  1481
next
haftmann@51489
  1482
  case (insert n N)
haftmann@51489
  1483
  then have "h (F (insert n N)) = h (n * F N)" by simp
haftmann@51489
  1484
  also have "\<dots> = h n * h (F N)" by (rule hom)
haftmann@51489
  1485
  also have "h (F N) = F (h ` N)" by (rule insert)
haftmann@51489
  1486
  also have "h n * \<dots> = F (insert (h n) (h ` N))"
haftmann@51489
  1487
    using insert by simp
haftmann@51489
  1488
  also have "insert (h n) (h ` N) = h ` insert n N" by simp
haftmann@51489
  1489
  finally show ?case .
haftmann@51489
  1490
qed
haftmann@51489
  1491
haftmann@51489
  1492
end
haftmann@51489
  1493
haftmann@51489
  1494
locale semilattice_order_set = semilattice_order + semilattice_set
haftmann@51489
  1495
begin
haftmann@51489
  1496
haftmann@51489
  1497
lemma bounded_iff:
haftmann@51489
  1498
  assumes "finite A" and "A \<noteq> {}"
haftmann@51489
  1499
  shows "x \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> a)"
haftmann@51489
  1500
  using assms by (induct rule: finite_ne_induct) (simp_all add: bounded_iff)
haftmann@51489
  1501
haftmann@51489
  1502
lemma boundedI:
haftmann@51489
  1503
  assumes "finite A"
haftmann@51489
  1504
  assumes "A \<noteq> {}"
haftmann@51489
  1505
  assumes "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
haftmann@51489
  1506
  shows "x \<preceq> F A"
haftmann@51489
  1507
  using assms by (simp add: bounded_iff)
haftmann@51489
  1508
haftmann@51489
  1509
lemma boundedE:
haftmann@51489
  1510
  assumes "finite A" and "A \<noteq> {}" and "x \<preceq> F A"
haftmann@51489
  1511
  obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
haftmann@51489
  1512
  using assms by (simp add: bounded_iff)
haftmann@35816
  1513
haftmann@51489
  1514
lemma coboundedI:
haftmann@51489
  1515
  assumes "finite A"
haftmann@51489
  1516
    and "a \<in> A"
haftmann@51489
  1517
  shows "F A \<preceq> a"
haftmann@51489
  1518
proof -
haftmann@51489
  1519
  from assms have "A \<noteq> {}" by auto
haftmann@51489
  1520
  from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
haftmann@51489
  1521
  proof (induct rule: finite_ne_induct)
haftmann@51489
  1522
    case singleton thus ?case by (simp add: refl)
haftmann@51489
  1523
  next
haftmann@51489
  1524
    case (insert x B)
haftmann@51489
  1525
    from insert have "a = x \<or> a \<in> B" by simp
haftmann@51489
  1526
    then show ?case using insert by (auto intro: coboundedI2)
haftmann@51489
  1527
  qed
haftmann@51489
  1528
qed
haftmann@51489
  1529
haftmann@51489
  1530
lemma antimono:
haftmann@51489
  1531
  assumes "A \<subseteq> B" and "A \<noteq> {}" and "finite B"
haftmann@51489
  1532
  shows "F B \<preceq> F A"
haftmann@51489
  1533
proof (cases "A = B")
haftmann@51489
  1534
  case True then show ?thesis by (simp add: refl)
haftmann@51489
  1535
next
haftmann@51489
  1536
  case False
haftmann@51489
  1537
  have B: "B = A \<union> (B - A)" using `A \<subseteq> B` by blast
haftmann@51489
  1538
  then have "F B = F (A \<union> (B - A))" by simp
haftmann@51489
  1539
  also have "\<dots> = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
haftmann@51489
  1540
  also have "\<dots> \<preceq> F A" by simp
haftmann@51489
  1541
  finally show ?thesis .
haftmann@51489
  1542
qed
haftmann@51489
  1543
haftmann@51489
  1544
end
haftmann@51489
  1545
haftmann@51489
  1546
haftmann@51489
  1547
subsubsection {* With neutral element *}
haftmann@51489
  1548
haftmann@51489
  1549
locale semilattice_neutr_set = semilattice_neutr
haftmann@51489
  1550
begin
haftmann@51489
  1551
haftmann@51738
  1552
interpretation comp_fun_idem f
haftmann@51738
  1553
  by default (simp_all add: fun_eq_iff left_commute)
haftmann@51738
  1554
haftmann@51489
  1555
definition F :: "'a set \<Rightarrow> 'a"
haftmann@51489
  1556
where
haftmann@51489
  1557
  eq_fold: "F A = Finite_Set.fold f 1 A"
haftmann@51489
  1558
haftmann@51489
  1559
lemma infinite [simp]:
haftmann@51489
  1560
  "\<not> finite A \<Longrightarrow> F A = 1"
haftmann@51489
  1561
  by (simp add: eq_fold)
haftmann@51489
  1562
haftmann@51489
  1563
lemma empty [simp]:
haftmann@51489
  1564
  "F {} = 1"
haftmann@51489
  1565
  by (simp add: eq_fold)
haftmann@51489
  1566
haftmann@51489
  1567
lemma insert [simp]:
haftmann@51489
  1568
  assumes "finite A"
haftmann@51489
  1569
  shows "F (insert x A) = x * F A"
haftmann@51738
  1570
  using assms by (simp add: eq_fold)
haftmann@51489
  1571
haftmann@51586
  1572
lemma in_idem:
haftmann@51489
  1573
  assumes "finite A" and "x \<in> A"
haftmann@51489
  1574
  shows "x * F A = F A"
haftmann@51489
  1575
proof -
haftmann@51489
  1576
  from assms have "A \<noteq> {}" by auto
haftmann@51489
  1577
  with `finite A` show ?thesis using `x \<in> A`
haftmann@51489
  1578
    by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
haftmann@51489
  1579
qed
haftmann@51489
  1580
haftmann@51489
  1581
lemma union:
haftmann@51489
  1582
  assumes "finite A" and "finite B"
haftmann@51489
  1583
  shows "F (A \<union> B) = F A * F B"
haftmann@51489
  1584
  using assms by (induct A) (simp_all add: ac_simps)
haftmann@51489
  1585
haftmann@51489
  1586
lemma remove:
haftmann@51489
  1587
  assumes "finite A" and "x \<in> A"
haftmann@51489
  1588
  shows "F A = x * F (A - {x})"
haftmann@51489
  1589
proof -
haftmann@51489
  1590
  from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@51489
  1591
  with assms show ?thesis by simp
haftmann@51489
  1592
qed
haftmann@51489
  1593
haftmann@51489
  1594
lemma insert_remove:
haftmann@51489
  1595
  assumes "finite A"
haftmann@51489
  1596
  shows "F (insert x A) = x * F (A - {x})"
haftmann@51489
  1597
  using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
haftmann@51489
  1598
haftmann@51489
  1599
lemma subset:
haftmann@51489
  1600
  assumes "finite A" and "B \<subseteq> A"
haftmann@51489
  1601
  shows "F B * F A = F A"
haftmann@51489
  1602
proof -
haftmann@51489
  1603
  from assms have "finite B" by (auto dest: finite_subset)
haftmann@51489
  1604
  with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
haftmann@51489
  1605
qed
haftmann@51489
  1606
haftmann@51489
  1607
lemma closed:
haftmann@51489
  1608
  assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
haftmann@51489
  1609
  shows "F A \<in> A"
haftmann@51489
  1610
using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
haftmann@51489
  1611
  case singleton then show ?case by simp
haftmann@51489
  1612
next
haftmann@51489
  1613
  case insert with elem show ?case by force
haftmann@51489
  1614
qed
haftmann@51489
  1615
haftmann@51489
  1616
end
haftmann@51489
  1617
haftmann@51489
  1618
locale semilattice_order_neutr_set = semilattice_neutr_order + semilattice_neutr_set
haftmann@51489
  1619
begin
haftmann@51489
  1620
haftmann@51489
  1621
lemma bounded_iff:
haftmann@51489
  1622
  assumes "finite A"
haftmann@51489
  1623
  shows "x \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> a)"
haftmann@51489
  1624
  using assms by (induct A) (simp_all add: bounded_iff)
haftmann@51489
  1625
haftmann@51489
  1626
lemma boundedI:
haftmann@51489
  1627
  assumes "finite A"
haftmann@51489
  1628
  assumes "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
haftmann@51489
  1629
  shows "x \<preceq> F A"
haftmann@51489
  1630
  using assms by (simp add: bounded_iff)
haftmann@51489
  1631
haftmann@51489
  1632
lemma boundedE:
haftmann@51489
  1633
  assumes "finite A" and "x \<preceq> F A"
haftmann@51489
  1634
  obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
haftmann@51489
  1635
  using assms by (simp add: bounded_iff)
haftmann@51489
  1636
haftmann@51489
  1637
lemma coboundedI:
haftmann@51489
  1638
  assumes "finite A"
haftmann@51489
  1639
    and "a \<in> A"
haftmann@51489
  1640
  shows "F A \<preceq> a"
haftmann@51489
  1641
proof -
haftmann@51489
  1642
  from assms have "A \<noteq> {}" by auto
haftmann@51489
  1643
  from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
haftmann@51489
  1644
  proof (induct rule: finite_ne_induct)
haftmann@51489
  1645
    case singleton thus ?case by (simp add: refl)
haftmann@51489
  1646
  next
haftmann@51489
  1647
    case (insert x B)
haftmann@51489
  1648
    from insert have "a = x \<or> a \<in> B" by simp
haftmann@51489
  1649
    then show ?case using insert by (auto intro: coboundedI2)
haftmann@51489
  1650
  qed
haftmann@51489
  1651
qed
haftmann@51489
  1652
haftmann@51489
  1653
lemma antimono:
haftmann@51489
  1654
  assumes "A \<subseteq> B" and "finite B"
haftmann@51489
  1655
  shows "F B \<preceq> F A"
haftmann@51489
  1656
proof (cases "A = B")
haftmann@51489
  1657
  case True then show ?thesis by (simp add: refl)
haftmann@51489
  1658
next
haftmann@51489
  1659
  case False
haftmann@51489
  1660
  have B: "B = A \<union> (B - A)" using `A \<subseteq> B` by blast
haftmann@51489
  1661
  then have "F B = F (A \<union> (B - A))" by simp
haftmann@51489
  1662
  also have "\<dots> = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
haftmann@51489
  1663
  also have "\<dots> \<preceq> F A" by simp
haftmann@51489
  1664
  finally show ?thesis .
haftmann@51489
  1665
qed
haftmann@51489
  1666
haftmann@51489
  1667
end
haftmann@35816
  1668
haftmann@35816
  1669
notation times (infixl "*" 70)
haftmann@35816
  1670
notation Groups.one ("1")
haftmann@22917
  1671
haftmann@35816
  1672
haftmann@51489
  1673
subsection {* Lattice operations on finite sets *}
haftmann@35816
  1674
haftmann@51489
  1675
text {*
haftmann@51489
  1676
  For historic reasons, there is the sublocale dependency from @{class distrib_lattice}
haftmann@51489
  1677
  to @{class linorder}.  This is badly designed: both should depend on a common abstract
haftmann@51489
  1678
  distributive lattice rather than having this non-subclass dependecy between two
haftmann@51489
  1679
  classes.  But for the moment we have to live with it.  This forces us to setup
haftmann@51489
  1680
  this sublocale dependency simultaneously with the lattice operations on finite
haftmann@51489
  1681
  sets, to avoid garbage.
haftmann@51489
  1682
*}
haftmann@22917
  1683
wenzelm@53174
  1684
definition (in semilattice_inf) Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>\<^sub>f\<^sub>i\<^sub>n_" [900] 900)
haftmann@51489
  1685
where
haftmann@51489
  1686
  "Inf_fin = semilattice_set.F inf"
haftmann@26041
  1687
wenzelm@53174
  1688
definition (in semilattice_sup) Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>\<^sub>f\<^sub>i\<^sub>n_" [900] 900)
haftmann@51489
  1689
where
haftmann@51489
  1690
  "Sup_fin = semilattice_set.F sup"
haftmann@35816
  1691
haftmann@51738
  1692
context linorder
haftmann@51738
  1693
begin
haftmann@51738
  1694
haftmann@51738
  1695
definition Min :: "'a set \<Rightarrow> 'a"
haftmann@51489
  1696
where
haftmann@51489
  1697
  "Min = semilattice_set.F min"
haftmann@35816
  1698
haftmann@51738
  1699
definition Max :: "'a set \<Rightarrow> 'a"
haftmann@51489
  1700
where
haftmann@51489
  1701
  "Max = semilattice_set.F max"
haftmann@51489
  1702
haftmann@51738
  1703
sublocale Min!: semilattice_order_set min less_eq less
haftmann@51540
  1704
  + Max!: semilattice_order_set max greater_eq greater
haftmann@51540
  1705
where
haftmann@51540
  1706
  "semilattice_set.F min = Min"
haftmann@51540
  1707
  and "semilattice_set.F max = Max"
haftmann@51540
  1708
proof -
haftmann@51540
  1709
  show "semilattice_order_set min less_eq less" by default (auto simp add: min_def)
haftmann@52364
  1710
  then interpret Min!: semilattice_order_set min less_eq less .
haftmann@51540
  1711
  show "semilattice_order_set max greater_eq greater" by default (auto simp add: max_def)
haftmann@51540
  1712
  then interpret Max!: semilattice_order_set max greater_eq greater .
haftmann@51540
  1713
  from Min_def show "semilattice_set.F min = Min" by rule
haftmann@51540
  1714
  from Max_def show "semilattice_set.F max = Max" by rule
haftmann@51540
  1715
qed
haftmann@51540
  1716
haftmann@51540
  1717
haftmann@51489
  1718
text {* An aside: @{const min}/@{const max} on linear orders as special case of @{const inf}/@{const sup} *}
haftmann@35816
  1719
haftmann@51738
  1720
sublocale min_max!: distrib_lattice min less_eq less max
haftmann@51489
  1721
where
haftmann@51489
  1722
  "semilattice_inf.Inf_fin min = Min"
haftmann@51489
  1723
  and "semilattice_sup.Sup_fin max = Max"
haftmann@26041
  1724
proof -
haftmann@51489
  1725
  show "class.distrib_lattice min less_eq less max"
haftmann@51489
  1726
  proof
haftmann@51489
  1727
    fix x y z
haftmann@51489
  1728
    show "max x (min y z) = min (max x y) (max x z)"
haftmann@51489
  1729
      by (auto simp add: min_def max_def)
haftmann@51489
  1730
  qed (auto simp add: min_def max_def not_le less_imp_le)
haftmann@51489
  1731
  then interpret min_max!: distrib_lattice min less_eq less max .
haftmann@51489
  1732
  show "semilattice_inf.Inf_fin min = Min"
haftmann@51489
  1733
    by (simp only: min_max.Inf_fin_def Min_def)
haftmann@51489
  1734
  show "semilattice_sup.Sup_fin max = Max"
haftmann@51489
  1735
    by (simp only: min_max.Sup_fin_def Max_def)
haftmann@26041
  1736
qed
haftmann@26041
  1737
haftmann@51489
  1738
lemmas le_maxI1 = min_max.sup_ge1
haftmann@51489
  1739
lemmas le_maxI2 = min_max.sup_ge2
haftmann@51489
  1740
 
haftmann@51489
  1741
lemmas min_ac = min_max.inf_assoc min_max.inf_commute
haftmann@51540
  1742
  min.left_commute
haftmann@51489
  1743
haftmann@51489
  1744
lemmas max_ac = min_max.sup_assoc min_max.sup_commute
haftmann@51540
  1745
  max.left_commute
haftmann@51489
  1746
haftmann@51738
  1747
end
haftmann@51738
  1748
haftmann@51489
  1749
haftmann@51489
  1750
text {* Lattice operations proper *}
haftmann@51489
  1751
haftmann@51489
  1752
sublocale semilattice_inf < Inf_fin!: semilattice_order_set inf less_eq less
haftmann@51489
  1753
where
haftmann@51546
  1754
  "semilattice_set.F inf = Inf_fin"
haftmann@26757
  1755
proof -
haftmann@51489
  1756
  show "semilattice_order_set inf less_eq less" ..
haftmann@52364
  1757
  then interpret Inf_fin!: semilattice_order_set inf less_eq less .
haftmann@51546
  1758
  from Inf_fin_def show "semilattice_set.F inf = Inf_fin" by rule
haftmann@26041
  1759
qed
haftmann@26041
  1760
haftmann@51489
  1761
sublocale semilattice_sup < Sup_fin!: semilattice_order_set sup greater_eq greater
haftmann@51489
  1762
where
haftmann@51546
  1763
  "semilattice_set.F sup = Sup_fin"
haftmann@51489
  1764
proof -
haftmann@51489
  1765
  show "semilattice_order_set sup greater_eq greater" ..
haftmann@51489
  1766
  then interpret Sup_fin!: semilattice_order_set sup greater_eq greater .
haftmann@51546
  1767
  from Sup_fin_def show "semilattice_set.F sup = Sup_fin" by rule
haftmann@51489
  1768
qed
haftmann@35816
  1769
haftmann@51489
  1770
haftmann@51540
  1771
text {* An aside again: @{const Min}/@{const Max} on linear orders as special case of @{const Inf_fin}/@{const Sup_fin} *}
haftmann@51540
  1772
haftmann@51540
  1773
lemma Inf_fin_Min:
haftmann@51540
  1774
  "Inf_fin = (Min :: 'a::{semilattice_inf, linorder} set \<Rightarrow> 'a)"
haftmann@51540
  1775
  by (simp add: Inf_fin_def Min_def inf_min)
haftmann@51540
  1776
haftmann@51540
  1777
lemma Sup_fin_Max:
haftmann@51540
  1778
  "Sup_fin = (Max :: 'a::{semilattice_sup, linorder} set \<Rightarrow> 'a)"
haftmann@51540
  1779
  by (simp add: Sup_fin_def Max_def sup_max)
haftmann@51540
  1780
haftmann@51540
  1781
haftmann@51540
  1782
haftmann@51489
  1783
subsection {* Infimum and Supremum over non-empty sets *}
haftmann@22917
  1784
haftmann@51489
  1785
text {*
haftmann@51489
  1786
  After this non-regular bootstrap, things continue canonically.
haftmann@51489
  1787
*}
haftmann@35816
  1788
haftmann@35816
  1789
context lattice
haftmann@35816
  1790
begin
haftmann@25062
  1791
wenzelm@53174
  1792
lemma Inf_le_Sup [simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>\<^sub>f\<^sub>i\<^sub>nA \<le> \<Squnion>\<^sub>f\<^sub>i\<^sub>nA"
nipkow@15500
  1793
apply(subgoal_tac "EX a. a:A")
nipkow@15500
  1794
prefer 2 apply blast
nipkow@15500
  1795
apply(erule exE)
haftmann@22388
  1796
apply(rule order_trans)
haftmann@51489
  1797
apply(erule (1) Inf_fin.coboundedI)
haftmann@51489
  1798
apply(erule (1) Sup_fin.coboundedI)
nipkow@15500
  1799
done
nipkow@15500
  1800
haftmann@24342
  1801
lemma sup_Inf_absorb [simp]:
wenzelm@53174
  1802
  "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA) = a"
nipkow@15512
  1803
apply(subst sup_commute)
haftmann@51489
  1804
apply(simp add: sup_absorb2 Inf_fin.coboundedI)
nipkow@15504
  1805
done
nipkow@15504
  1806
haftmann@24342
  1807
lemma inf_Sup_absorb [simp]:
wenzelm@53174
  1808
  "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA) = a"
haftmann@51489
  1809
by (simp add: inf_absorb1 Sup_fin.coboundedI)
haftmann@24342
  1810
haftmann@24342
  1811
end
haftmann@24342
  1812
haftmann@24342
  1813
context distrib_lattice
haftmann@24342
  1814
begin
haftmann@24342
  1815
haftmann@24342
  1816
lemma sup_Inf1_distrib:
haftmann@26041
  1817
  assumes "finite A"
haftmann@26041
  1818
    and "A \<noteq> {}"
wenzelm@53174
  1819
  shows "sup x (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA) = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n{sup x a|a. a \<in> A}"
haftmann@51489
  1820
using assms by (simp add: image_def Inf_fin.hom_commute [where h="sup x", OF sup_inf_distrib1])
haftmann@51489
  1821
  (rule arg_cong [where f="Inf_fin"], blast)
nipkow@18423
  1822
haftmann@24342
  1823
lemma sup_Inf2_distrib:
haftmann@24342
  1824
  assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
wenzelm@53174
  1825
  shows "sup (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA) (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nB) = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n{sup a b|a b. a \<in> A \<and> b \<in> B}"
haftmann@24342
  1826
using A proof (induct rule: finite_ne_induct)
haftmann@51489
  1827
  case singleton then show ?case
wenzelm@41550
  1828
    by (simp add: sup_Inf1_distrib [OF B])
nipkow@15500
  1829
next
nipkow@15500
  1830
  case (insert x A)
haftmann@25062
  1831
  have finB: "finite {sup x b |b. b \<in> B}"
haftmann@51489
  1832
    by (rule finite_surj [where f = "sup x", OF B(1)], auto)
haftmann@25062
  1833
  have finAB: "finite {sup a b |a b. a \<in> A \<and> b \<in> B}"
nipkow@15500
  1834
  proof -
haftmann@25062
  1835
    have "{sup a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {sup a b})"
nipkow@15500
  1836
      by blast
berghofe@15517
  1837
    thus ?thesis by(simp add: insert(1) B(1))
nipkow@15500
  1838
  qed
haftmann@25062
  1839
  have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
wenzelm@53174
  1840
  have "sup (\<Sqinter>\<^sub>f\<^sub>i\<^sub>n(insert x A)) (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nB) = sup (inf x (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA)) (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nB)"
wenzelm@41550
  1841
    using insert by simp
wenzelm@53174
  1842
  also have "\<dots> = inf (sup x (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nB)) (sup (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA) (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nB))" by(rule sup_inf_distrib2)
wenzelm@53174
  1843
  also have "\<dots> = inf (\<Sqinter>\<^sub>f\<^sub>i\<^sub>n{sup x b|b. b \<in> B}) (\<Sqinter>\<^sub>f\<^sub>i\<^sub>n{sup a b|a b. a \<in> A \<and> b \<in> B})"
nipkow@15500
  1844
    using insert by(simp add:sup_Inf1_distrib[OF B])
wenzelm@53174
  1845
  also have "\<dots> = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n({sup x b |b. b \<in> B} \<union> {sup a b |a b. a \<in> A \<and> b \<in> B})"
wenzelm@53174
  1846
    (is "_ = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n?M")
nipkow@15500
  1847
    using B insert
haftmann@51489
  1848
    by (simp add: Inf_fin.union [OF finB _ finAB ne])
haftmann@25062
  1849
  also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"
nipkow@15500
  1850
    by blast
nipkow@15500
  1851
  finally show ?case .
nipkow@15500
  1852
qed
nipkow@15500
  1853
haftmann@24342
  1854
lemma inf_Sup1_distrib:
haftmann@26041
  1855
  assumes "finite A" and "A \<noteq> {}"
wenzelm@53174
  1856
  shows "inf x (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA) = \<Squnion>\<^sub>f\<^sub>i\<^sub>n{inf x a|a. a \<in> A}"
haftmann@51489
  1857
using assms by (simp add: image_def Sup_fin.hom_commute [where h="inf x", OF inf_sup_distrib1])
haftmann@51489
  1858
  (rule arg_cong [where f="Sup_fin"], blast)
nipkow@18423
  1859
haftmann@24342
  1860
lemma inf_Sup2_distrib:
haftmann@24342
  1861
  assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
wenzelm@53174
  1862
  shows "inf (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA) (\<Squnion>\<^sub>f\<^sub>i\<^sub>nB) = \<Squnion>\<^sub>f\<^sub>i\<^sub>n{inf a b|a b. a \<in> A \<and> b \<in> B}"
haftmann@24342
  1863
using A proof (induct rule: finite_ne_induct)
nipkow@18423
  1864
  case singleton thus ?case
huffman@44921
  1865
    by(simp add: inf_Sup1_distrib [OF B])
nipkow@18423
  1866
next
nipkow@18423
  1867
  case (insert x A)
haftmann@25062
  1868
  have finB: "finite {inf x b |b. b \<in> B}"
haftmann@25062
  1869
    by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto)
haftmann@25062
  1870
  have finAB: "finite {inf a b |a b. a \<in> A \<and> b \<in> B}"
nipkow@18423
  1871
  proof -
haftmann@25062
  1872
    have "{inf a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {inf a b})"
nipkow@18423
  1873
      by blast
nipkow@18423
  1874
    thus ?thesis by(simp add: insert(1) B(1))
nipkow@18423
  1875
  qed
haftmann@25062
  1876
  have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
wenzelm@53174
  1877
  have "inf (\<Squnion>\<^sub>f\<^sub>i\<^sub>n(insert x A)) (\<Squnion>\<^sub>f\<^sub>i\<^sub>nB) = inf (sup x (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA)) (\<Squnion>\<^sub>f\<^sub>i\<^sub>nB)"
wenzelm@41550
  1878
    using insert by simp
wenzelm@53174
  1879
  also have "\<dots> = sup (inf x (\<Squnion>\<^sub>f\<^sub>i\<^sub>nB)) (inf (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA) (\<Squnion>\<^sub>f\<^sub>i\<^sub>nB))" by(rule inf_sup_distrib2)
wenzelm@53174
  1880
  also have "\<dots> = sup (\<Squnion>\<^sub>f\<^sub>i\<^sub>n{inf x b|b. b \<in> B}) (\<Squnion>\<^sub>f\<^sub>i\<^sub>n{inf a b|a b. a \<in> A \<and> b \<in> B})"
nipkow@18423
  1881
    using insert by(simp add:inf_Sup1_distrib[OF B])
wenzelm@53174
  1882
  also have "\<dots> = \<Squnion>\<^sub>f\<^sub>i\<^sub>n({inf x b |b. b \<in> B} \<union> {inf a b |a b. a \<in> A \<and> b \<in> B})"
wenzelm@53174
  1883
    (is "_ = \<Squnion>\<^sub>f\<^sub>i\<^sub>n?M")
nipkow@18423
  1884
    using B insert
haftmann@51489
  1885
    by (simp add: Sup_fin.union [OF finB _ finAB ne])
haftmann@25062
  1886
  also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"
nipkow@18423
  1887
    by blast
nipkow@18423
  1888
  finally show ?case .
nipkow@18423
  1889
qed
nipkow@18423
  1890
haftmann@24342
  1891
end
haftmann@24342
  1892
haftmann@35719
  1893
context complete_lattice
haftmann@35719
  1894
begin
haftmann@35719
  1895
haftmann@35719
  1896
lemma Inf_fin_Inf:
haftmann@35719
  1897
  assumes "finite A" and "A \<noteq> {}"
wenzelm@53174
  1898
  shows "\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA = Inf A"
haftmann@35719
  1899
proof -
haftmann@51489
  1900
  from assms obtain b B where "A = insert b B" and "finite B" by auto
haftmann@51489
  1901
  then show ?thesis
haftmann@51489
  1902
    by (simp add: Inf_fin.eq_fold inf_Inf_fold_inf inf.commute [of b])
haftmann@35719
  1903
qed
haftmann@35719
  1904
haftmann@35719
  1905
lemma Sup_fin_Sup:
haftmann@35719
  1906
  assumes "finite A" and "A \<noteq> {}"
wenzelm@53174
  1907
  shows "\<Squnion>\<^sub>f\<^sub>i\<^sub>nA = Sup A"
haftmann@35719
  1908
proof -
haftmann@51489
  1909
  from assms obtain b B where "A = insert b B" and "finite B" by auto
haftmann@51489
  1910
  then show ?thesis
haftmann@51489
  1911
    by (simp add: Sup_fin.eq_fold sup_Sup_fold_sup sup.commute [of b])
haftmann@35719
  1912
qed
haftmann@35719
  1913
haftmann@35719
  1914
end
haftmann@35719
  1915
haftmann@22917
  1916
haftmann@51489
  1917
subsection {* Minimum and Maximum over non-empty sets *}
haftmann@22917
  1918
haftmann@24342
  1919
context linorder
haftmann@22917
  1920
begin
haftmann@22917
  1921
haftmann@26041
  1922
lemma dual_min:
haftmann@51489
  1923
  "ord.min greater_eq = max"
wenzelm@46904
  1924
  by (auto simp add: ord.min_def max_def fun_eq_iff)
haftmann@26041
  1925
haftmann@51489
  1926
lemma dual_max:
haftmann@51489
  1927
  "ord.max greater_eq = min"
haftmann@51489
  1928
  by (auto simp add: ord.max_def min_def fun_eq_iff)
haftmann@51489
  1929
haftmann@51489
  1930
lemma dual_Min:
haftmann@51489
  1931
  "linorder.Min greater_eq = Max"
haftmann@26041
  1932
proof -
haftmann@51489
  1933
  interpret dual!: linorder greater_eq greater by (fact dual_linorder)
haftmann@51489
  1934
  show ?thesis by (simp add: dual.Min_def dual_min Max_def)
haftmann@26041
  1935
qed
haftmann@26041
  1936
haftmann@51489
  1937
lemma dual_Max:
haftmann@51489
  1938
  "linorder.Max greater_eq = Min"
haftmann@26041
  1939
proof -
haftmann@51489
  1940
  interpret dual!: linorder greater_eq greater by (fact dual_linorder)
haftmann@51489
  1941
  show ?thesis by (simp add: dual.Max_def dual_max Min_def)
haftmann@26041
  1942
qed
haftmann@26041
  1943
haftmann@51540
  1944
lemmas Min_singleton = Min.singleton
haftmann@51540
  1945
lemmas Max_singleton = Max.singleton
haftmann@51540
  1946
lemmas Min_insert = Min.insert
haftmann@51540
  1947
lemmas Max_insert = Max.insert
haftmann@51540
  1948
lemmas Min_Un = Min.union
haftmann@51540
  1949
lemmas Max_Un = Max.union
haftmann@51540
  1950
lemmas hom_Min_commute = Min.hom_commute
haftmann@51540
  1951
lemmas hom_Max_commute = Max.hom_commute
haftmann@26041
  1952
paulson@24427
  1953
lemma Min_in [simp]:
haftmann@26041
  1954
  assumes "finite A" and "A \<noteq> {}"
haftmann@26041
  1955
  shows "Min A \<in> A"
haftmann@51540
  1956
  using assms by (auto simp add: min_def Min.closed)
nipkow@15392
  1957
paulson@24427
  1958
lemma Max_in [simp]:
haftmann@26041
  1959
  assumes "finite A" and "A \<noteq> {}"
haftmann@26041
  1960
  shows "Max A \<in> A"
haftmann@51540
  1961
  using assms by (auto simp add: max_def Max.closed)
haftmann@26041
  1962
haftmann@26041
  1963
lemma Min_le [simp]:
haftmann@26757
  1964
  assumes "finite A" and "x \<in> A"
haftmann@26041
  1965
  shows "Min A \<le> x"
haftmann@51540
  1966
  using assms by (fact Min.coboundedI)
haftmann@26041
  1967
haftmann@26041
  1968
lemma Max_ge [simp]:
haftmann@26757
  1969
  assumes "finite A" and "x \<in> A"
haftmann@26041
  1970
  shows "x \<le> Max A"
haftmann@51540
  1971
  using assms by (fact Max.coboundedI)
haftmann@26041
  1972
haftmann@30325
  1973
lemma Min_eqI:
haftmann@30325
  1974
  assumes "finite A"
haftmann@30325
  1975
  assumes "\<And>y. y \<in> A \<Longrightarrow> y \<ge> x"
haftmann@30325
  1976
    and "x \<in> A"
haftmann@30325
  1977
  shows "Min A = x"
haftmann@30325
  1978
proof (rule antisym)
haftmann@30325
  1979
  from `x \<in> A` have "A \<noteq> {}" by auto
haftmann@30325
  1980
  with assms show "Min A \<ge> x" by simp
haftmann@30325
  1981
next
haftmann@30325
  1982
  from assms show "x \<ge> Min A" by simp
haftmann@30325
  1983
qed
haftmann@30325
  1984
haftmann@30325
  1985
lemma Max_eqI:
haftmann@30325
  1986
  assumes "finite A"
haftmann@30325
  1987
  assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
haftmann@30325
  1988
    and "x \<in> A"
haftmann@30325
  1989
  shows "Max A = x"
haftmann@30325
  1990
proof (rule antisym)
haftmann@30325
  1991
  from `x \<in> A` have "A \<noteq> {}" by auto
haftmann@30325
  1992
  with assms show "Max A \<le> x" by simp
haftmann@30325
  1993
next
haftmann@30325
  1994
  from assms show "x \<le> Max A" by simp
haftmann@30325
  1995
qed
haftmann@30325
  1996
haftmann@51738
  1997
context
haftmann@51738
  1998
  fixes A :: "'a set"
haftmann@51738
  1999
  assumes fin_nonempty: "finite A" "A \<noteq> {}"
haftmann@51738
  2000
begin
haftmann@51738
  2001
haftmann@51489
  2002
lemma Min_ge_iff [simp, no_atp]:
haftmann@51738
  2003
  "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
haftmann@51738
  2004
  using fin_nonempty by (fact Min.bounded_iff)
haftmann@51489
  2005
haftmann@51489
  2006
lemma Max_le_iff [simp, no_atp]:
haftmann@51738
  2007
  "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
haftmann@51738
  2008
  using fin_nonempty by (fact Max.bounded_iff)
haftmann@51489
  2009
haftmann@51489
  2010
lemma Min_gr_iff [simp, no_atp]:
haftmann@51738
  2011
  "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
haftmann@51738
  2012
  using fin_nonempty  by (induct rule: finite_ne_induct) simp_all
haftmann@51489
  2013
haftmann@51489
  2014
lemma Max_less_iff [simp, no_atp]:
haftmann@51738
  2015
  "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
haftmann@51738
  2016
  using fin_nonempty by (induct rule: finite_ne_induct) simp_all
haftmann@51489
  2017
haftmann@51489
  2018
lemma Min_le_iff [no_atp]:
haftmann@51738
  2019
  "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
haftmann@51738
  2020
  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_le_iff_disj)
haftmann@51489
  2021
haftmann@51489
  2022
lemma Max_ge_iff [no_atp]:
haftmann@51738
  2023
  "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
haftmann@51738
  2024
  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: le_max_iff_disj)
haftmann@51489
  2025
haftmann@51489
  2026
lemma Min_less_iff [no_atp]:
haftmann@51738
  2027
  "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
haftmann@51738
  2028
  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_less_iff_disj)
haftmann@51489
  2029
haftmann@51489
  2030
lemma Max_gr_iff [no_atp]:
haftmann@51738
  2031
  "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
haftmann@51738
  2032
  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: less_max_iff_disj)
haftmann@51738
  2033
haftmann@51738
  2034
end
haftmann@51489
  2035
haftmann@26041
  2036
lemma Min_antimono:
haftmann@26041
  2037
  assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
haftmann@26041
  2038
  shows "Min N \<le> Min M"
haftmann@51540
  2039
  using assms by (fact Min.antimono)
haftmann@26041
  2040
haftmann@26041
  2041
lemma Max_mono:
haftmann@26041
  2042
  assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
haftmann@26041
  2043
  shows "Max M \<le> Max N"
haftmann@51540
  2044
  using assms by (fact Max.antimono)
haftmann@51489
  2045
haftmann@51489
  2046
lemma mono_Min_commute:
haftmann@51489
  2047
  assumes "mono f"
haftmann@51489
  2048
  assumes "finite A" and "A \<noteq> {}"
haftmann@51489
  2049
  shows "f (Min A) = Min (f ` A)"
haftmann@51489
  2050
proof (rule linorder_class.Min_eqI [symmetric])
haftmann@51489
  2051
  from `finite A` show "finite (f ` A)" by simp
haftmann@51489
  2052
  from assms show "f (Min A) \<in> f ` A" by simp
haftmann@51489
  2053
  fix x
haftmann@51489
  2054
  assume "x \<in> f ` A"
haftmann@51489
  2055
  then obtain y where "y \<in> A" and "x = f y" ..
haftmann@51489
  2056
  with assms have "Min A \<le> y" by auto
haftmann@51489
  2057
  with `mono f` have "f (Min A) \<le> f y" by (rule monoE)
haftmann@51489
  2058
  with `x = f y` show "f (Min A) \<le> x" by simp
haftmann@51489
  2059
qed
haftmann@22917
  2060
haftmann@51489
  2061
lemma mono_Max_commute:
haftmann@51489
  2062
  assumes "mono f"
haftmann@51489
  2063
  assumes "finite A" and "A \<noteq> {}"
haftmann@51489
  2064
  shows "f (Max A) = Max (f ` A)"
haftmann@51489
  2065
proof (rule linorder_class.Max_eqI [symmetric])
haftmann@51489
  2066
  from `finite A` show "finite (f ` A)" by simp
haftmann@51489
  2067
  from assms show "f (Max A) \<in> f ` A" by simp
haftmann@51489
  2068
  fix x
haftmann@51489
  2069
  assume "x \<in> f ` A"
haftmann@51489
  2070
  then obtain y where "y \<in> A" and "x = f y" ..
haftmann@51489
  2071
  with assms have "y \<le> Max A" by auto
haftmann@51489
  2072
  with `mono f` have "f y \<le> f (Max A)" by (rule monoE)
haftmann@51489
  2073
  with `x = f y` show "x \<le> f (Max A)" by simp
haftmann@51489
  2074
qed
haftmann@51489
  2075
haftmann@51489
  2076
lemma finite_linorder_max_induct [consumes 1, case_names empty insert]:
haftmann@51489
  2077
  assumes fin: "finite A"
haftmann@51489
  2078
  and empty: "P {}" 
haftmann@51489
  2079
  and insert: "\<And>b A. finite A \<Longrightarrow> \<forall>a\<in>A. a < b \<Longrightarrow> P A \<Longrightarrow> P (insert b A)"
haftmann@51489
  2080
  shows "P A"
urbanc@36079
  2081
using fin empty insert
nipkow@32006
  2082
proof (induct rule: finite_psubset_induct)
urbanc@36079
  2083
  case (psubset A)
urbanc@36079
  2084
  have IH: "\<And>B. \<lbrakk>B < A; P {}; (\<And>A b. \<lbrakk>finite A; \<forall>a\<in>A. a<b; P A\<rbrakk> \<Longrightarrow> P (insert b A))\<rbrakk> \<Longrightarrow> P B" by fact 
urbanc@36079
  2085
  have fin: "finite A" by fact 
urbanc@36079
  2086
  have empty: "P {}" by fact
urbanc@36079
  2087
  have step: "\<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. a < b; P A\<rbrakk> \<Longrightarrow> P (insert b A)" by fact
krauss@26748
  2088
  show "P A"
haftmann@26757
  2089
  proof (cases "A = {}")
urbanc@36079
  2090
    assume "A = {}" 
urbanc@36079
  2091
    then show "P A" using `P {}` by simp
krauss@26748
  2092
  next
urbanc@36079
  2093
    let ?B = "A - {Max A}" 
urbanc@36079
  2094
    let ?A = "insert (Max A) ?B"
urbanc@36079
  2095
    have "finite ?B" using `finite A` by simp
krauss@26748
  2096
    assume "A \<noteq> {}"
krauss@26748
  2097
    with `finite A` have "Max A : A" by auto
urbanc@36079
  2098
    then have A: "?A = A" using insert_Diff_single insert_absorb by auto
haftmann@51489
  2099
    then have "P ?B" using `P {}` step IH [of ?B] by blast
urbanc@36079
  2100
    moreover 
nipkow@44890
  2101
    have "\<forall>a\<in>?B. a < Max A" using Max_ge [OF `finite A`] by fastforce
haftmann@51489
  2102
    ultimately show "P A" using A insert_Diff_single step [OF `finite ?B`] by fastforce
krauss@26748
  2103
  qed
krauss@26748
  2104
qed
krauss@26748
  2105
haftmann@51489
  2106
lemma finite_linorder_min_induct [consumes 1, case_names empty insert]:
haftmann@51489
  2107
  "\<lbrakk>finite A; P {}; \<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. b < a; P A\<rbrakk> \<Longrightarrow> P (insert b A)\<rbrakk> \<Longrightarrow> P A"
haftmann@51489
  2108
  by (rule linorder.finite_linorder_max_induct [OF dual_linorder])
nipkow@32006
  2109
haftmann@52379
  2110
lemma Least_Min:
haftmann@52379
  2111
  assumes "finite {a. P a}" and "\<exists>a. P a"
haftmann@52379
  2112
  shows "(LEAST a. P a) = Min {a. P a}"
haftmann@52379
  2113
proof -
haftmann@52379
  2114
  { fix A :: "'a set"
haftmann@52379
  2115
    assume A: "finite A" "A \<noteq> {}"
haftmann@52379
  2116
    have "(LEAST a. a \<in> A) = Min A"
haftmann@52379
  2117
    using A proof (induct A rule: finite_ne_induct)
haftmann@52379
  2118
      case singleton show ?case by (rule Least_equality) simp_all
haftmann@52379
  2119
    next
haftmann@52379
  2120
      case (insert a A)
haftmann@52379
  2121
      have "(LEAST b. b = a \<or> b \<in> A) = min a (LEAST a. a \<in> A)"
haftmann@52379
  2122
        by (auto intro!: Least_equality simp add: min_def not_le Min_le_iff insert.hyps dest!: less_imp_le)
haftmann@52379
  2123
      with insert show ?case by simp
haftmann@52379
  2124
    qed
haftmann@52379
  2125
  } from this [of "{a. P a}"] assms show ?thesis by simp
haftmann@52379
  2126
qed
haftmann@52379
  2127
haftmann@22917
  2128
end
haftmann@22917
  2129
haftmann@35028
  2130
context linordered_ab_semigroup_add
haftmann@22917
  2131
begin
haftmann@22917
  2132
haftmann@22917
  2133
lemma add_Min_commute:
haftmann@22917
  2134
  fixes k
haftmann@25062
  2135
  assumes "finite N" and "N \<noteq> {}"
haftmann@25062
  2136
  shows "k + Min N = Min {k + m | m. m \<in> N}"
haftmann@25062
  2137
proof -
haftmann@25062
  2138
  have "\<And>x y. k + min x y = min (k + x) (k + y)"
haftmann@25062
  2139
    by (simp add: min_def not_le)
haftmann@25062
  2140
      (blast intro: antisym less_imp_le add_left_mono)
haftmann@25062
  2141
  with assms show ?thesis
haftmann@25062
  2142
    using hom_Min_commute [of "plus k" N]
haftmann@25062
  2143
    by simp (blast intro: arg_cong [where f = Min])
haftmann@25062
  2144
qed
haftmann@22917
  2145
haftmann@22917
  2146
lemma add_Max_commute:
haftmann@22917
  2147
  fixes k
haftmann@25062
  2148
  assumes "finite N" and "N \<noteq> {}"
haftmann@25062
  2149
  shows "k + Max N = Max {k + m | m. m \<in> N}"
haftmann@25062
  2150
proof -
haftmann@25062
  2151
  have "\<And>x y. k + max x y = max (k + x) (k + y)"
haftmann@25062
  2152
    by (simp add: max_def not_le)
haftmann@25062
  2153
      (blast intro: antisym less_imp_le add_left_mono)
haftmann@25062
  2154
  with assms show ?thesis
haftmann@25062
  2155
    using hom_Max_commute [of "plus k" N]
haftmann@25062
  2156
    by simp (blast intro: arg_cong [where f = Max])
haftmann@25062
  2157
qed
haftmann@22917
  2158
haftmann@22917
  2159
end
haftmann@22917
  2160
haftmann@35034
  2161
context linordered_ab_group_add
haftmann@35034
  2162
begin
haftmann@35034
  2163
haftmann@35034
  2164
lemma minus_Max_eq_Min [simp]:
haftmann@51489
  2165
  "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Max S = Min (uminus ` S)"
haftmann@35034
  2166
  by (induct S rule: finite_ne_induct) (simp_all add: minus_max_eq_min)
haftmann@35034
  2167
haftmann@35034
  2168
lemma minus_Min_eq_Max [simp]:
haftmann@51489
  2169
  "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Min S = Max (uminus ` S)"
haftmann@35034
  2170
  by (induct S rule: finite_ne_induct) (simp_all add: minus_min_eq_max)
haftmann@35034
  2171
haftmann@35034
  2172
end
haftmann@35034
  2173
haftmann@51540
  2174
context complete_linorder
haftmann@51540
  2175
begin
haftmann@51540
  2176
haftmann@51540
  2177
lemma Min_Inf:
haftmann@51540
  2178
  assumes "finite A" and "A \<noteq> {}"
haftmann@51540
  2179
  shows "Min A = Inf A"
haftmann@51540
  2180
proof -
haftmann@51540
  2181
  from assms obtain b B where "A = insert b B" and "finite B" by auto
haftmann@51540
  2182
  then show ?thesis
haftmann@51540
  2183
    by (simp add: Min.eq_fold complete_linorder_inf_min [symmetric] inf_Inf_fold_inf inf.commute [of b])
haftmann@51540
  2184
qed
haftmann@51540
  2185
haftmann@51540
  2186
lemma Max_Sup:
haftmann@51540
  2187
  assumes "finite A" and "A \<noteq> {}"
haftmann@51540
  2188
  shows "Max A = Sup A"
haftmann@51540
  2189
proof -
haftmann@51540
  2190
  from assms obtain b B where "A = insert b B" and "finite B" by auto
haftmann@51540
  2191
  then show ?thesis
haftmann@51540
  2192
    by (simp add: Max.eq_fold complete_linorder_sup_max [symmetric] sup_Sup_fold_sup sup.commute [of b])
haftmann@51540
  2193
qed
haftmann@51540
  2194
haftmann@25571
  2195
end
haftmann@51263
  2196
haftmann@51540
  2197
end