src/HOL/Lattices_Big.thy
author nipkow
Tue Aug 15 09:29:35 2017 +0200 (22 months ago)
changeset 66425 8756322dc5de
parent 66364 fa3247e6ee4b
child 67036 783c901a62cb
permissions -rw-r--r--
added Min_mset and Max_mset
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(*  Title:      HOL/Lattices_Big.thy
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    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
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                with contributions by Jeremy Avigad
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*)
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section \<open>Big infimum (minimum) and supremum (maximum) over finite (non-empty) sets\<close>
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theory Lattices_Big
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  imports Option
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begin
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subsection \<open>Generic lattice operations over a set\<close>
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subsubsection \<open>Without neutral element\<close>
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locale semilattice_set = semilattice
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begin
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interpretation comp_fun_idem f
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  by standard (simp_all add: fun_eq_iff left_commute)
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definition F :: "'a set \<Rightarrow> 'a"
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where
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  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)"
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lemma eq_fold:
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  assumes "finite A"
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  shows "F (insert x A) = Finite_Set.fold f x A"
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proof (rule sym)
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  let ?f = "\<lambda>x y. Some (case y of None \<Rightarrow> x | Some z \<Rightarrow> f x z)"
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  interpret comp_fun_idem "?f"
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    by standard (simp_all add: fun_eq_iff commute left_commute split: option.split)
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  from assms show "Finite_Set.fold f x A = F (insert x A)"
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  proof induct
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    case empty then show ?case by (simp add: eq_fold')
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  next
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    case (insert y B) then show ?case by (simp add: insert_commute [of x] eq_fold')
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  qed
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qed
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lemma singleton [simp]:
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  "F {x} = x"
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  by (simp add: eq_fold)
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lemma insert_not_elem:
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  assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
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  shows "F (insert x A) = x \<^bold>* F A"
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proof -
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  from \<open>A \<noteq> {}\<close> obtain b where "b \<in> A" by blast
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  then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
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  with \<open>finite A\<close> and \<open>x \<notin> A\<close>
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    have "finite (insert x B)" and "b \<notin> insert x B" by auto
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  then have "F (insert b (insert x B)) = x \<^bold>* F (insert b B)"
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    by (simp add: eq_fold)
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  then show ?thesis by (simp add: * insert_commute)
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qed
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lemma in_idem:
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  assumes "finite A" and "x \<in> A"
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  shows "x \<^bold>* F A = F A"
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proof -
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  from assms have "A \<noteq> {}" by auto
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  with \<open>finite A\<close> show ?thesis using \<open>x \<in> A\<close>
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    by (induct A rule: finite_ne_induct) (auto simp add: ac_simps insert_not_elem)
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qed
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lemma insert [simp]:
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  assumes "finite A" and "A \<noteq> {}"
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  shows "F (insert x A) = x \<^bold>* F A"
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  using assms by (cases "x \<in> A") (simp_all add: insert_absorb in_idem insert_not_elem)
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lemma union:
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  assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
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  shows "F (A \<union> B) = F A \<^bold>* F B"
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  using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
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lemma remove:
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  assumes "finite A" and "x \<in> A"
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  shows "F A = (if A - {x} = {} then x else x \<^bold>* F (A - {x}))"
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proof -
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  from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
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  with assms 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 (insert x A) = (if A - {x} = {} then x else x \<^bold>* F (A - {x}))"
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  using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
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lemma subset:
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  assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
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  shows "F B \<^bold>* F A = F A"
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proof -
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  from assms have "A \<noteq> {}" and "finite B" by (auto dest: finite_subset)
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  with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
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qed
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lemma closed:
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  assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x \<^bold>* y \<in> {x, y}"
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  shows "F A \<in> A"
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using \<open>finite A\<close> \<open>A \<noteq> {}\<close> proof (induct rule: finite_ne_induct)
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  case singleton then show ?case by simp
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next
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  case insert with elem show ?case by force
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qed
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lemma hom_commute:
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  assumes hom: "\<And>x y. h (x \<^bold>* y) = h x \<^bold>* h y"
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  and N: "finite N" "N \<noteq> {}"
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  shows "h (F N) = F (h ` N)"
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using N proof (induct rule: finite_ne_induct)
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  case singleton thus ?case by simp
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next
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  case (insert n N)
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  then have "h (F (insert n N)) = h (n \<^bold>* F N)" by simp
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  also have "\<dots> = h n \<^bold>* h (F N)" by (rule hom)
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  also have "h (F N) = F (h ` N)" by (rule insert)
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  also have "h n \<^bold>* \<dots> = F (insert (h n) (h ` N))"
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    using insert by simp
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  also have "insert (h n) (h ` N) = h ` insert n N" by simp
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  finally show ?case .
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qed
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lemma infinite: "\<not> finite A \<Longrightarrow> F A = the None"
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  unfolding eq_fold' by (cases "finite (UNIV::'a set)") (auto intro: finite_subset fold_infinite)
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end
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locale semilattice_order_set = binary?: semilattice_order + semilattice_set
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begin
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lemma bounded_iff:
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  assumes "finite A" and "A \<noteq> {}"
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  shows "x \<^bold>\<le> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<^bold>\<le> a)"
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  using assms by (induct rule: finite_ne_induct) (simp_all add: bounded_iff)
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lemma boundedI:
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  assumes "finite A"
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  assumes "A \<noteq> {}"
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  assumes "\<And>a. a \<in> A \<Longrightarrow> x \<^bold>\<le> a"
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  shows "x \<^bold>\<le> F A"
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  using assms by (simp add: bounded_iff)
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lemma boundedE:
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  assumes "finite A" and "A \<noteq> {}" and "x \<^bold>\<le> F A"
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  obtains "\<And>a. a \<in> A \<Longrightarrow> x \<^bold>\<le> a"
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  using assms by (simp add: bounded_iff)
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lemma coboundedI:
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  assumes "finite A"
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    and "a \<in> A"
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  shows "F A \<^bold>\<le> a"
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proof -
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  from assms have "A \<noteq> {}" by auto
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  from \<open>finite A\<close> \<open>A \<noteq> {}\<close> \<open>a \<in> A\<close> show ?thesis
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  proof (induct rule: finite_ne_induct)
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    case singleton thus ?case by (simp add: refl)
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  next
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    case (insert x B)
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    from insert have "a = x \<or> a \<in> B" by simp
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    then show ?case using insert by (auto intro: coboundedI2)
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  qed
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qed
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lemma antimono:
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  assumes "A \<subseteq> B" and "A \<noteq> {}" and "finite B"
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  shows "F B \<^bold>\<le> F A"
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proof (cases "A = B")
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  case True then show ?thesis by (simp add: refl)
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next
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  case False
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  have B: "B = A \<union> (B - A)" using \<open>A \<subseteq> B\<close> by blast
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  then have "F B = F (A \<union> (B - A))" by simp
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  also have "\<dots> = F A \<^bold>* F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
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  also have "\<dots> \<^bold>\<le> F A" by simp
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  finally show ?thesis .
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qed
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end
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subsubsection \<open>With neutral element\<close>
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locale semilattice_neutr_set = semilattice_neutr
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begin
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interpretation comp_fun_idem f
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  by standard (simp_all add: fun_eq_iff left_commute)
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definition F :: "'a set \<Rightarrow> 'a"
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where
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  eq_fold: "F A = Finite_Set.fold f \<^bold>1 A"
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lemma infinite [simp]:
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  "\<not> finite A \<Longrightarrow> F A = \<^bold>1"
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  by (simp add: eq_fold)
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lemma empty [simp]:
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  "F {} = \<^bold>1"
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  by (simp add: eq_fold)
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lemma insert [simp]:
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  assumes "finite A"
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  shows "F (insert x A) = x \<^bold>* F A"
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  using assms by (simp add: eq_fold)
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lemma in_idem:
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  assumes "finite A" and "x \<in> A"
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  shows "x \<^bold>* F A = F A"
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proof -
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  from assms have "A \<noteq> {}" by auto
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  with \<open>finite A\<close> show ?thesis using \<open>x \<in> A\<close>
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    by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
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qed
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lemma union:
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  assumes "finite A" and "finite B"
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  shows "F (A \<union> B) = F A \<^bold>* F B"
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  using assms by (induct A) (simp_all add: ac_simps)
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lemma remove:
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  assumes "finite A" and "x \<in> A"
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  shows "F A = x \<^bold>* F (A - {x})"
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proof -
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  from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
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  with assms 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 (insert x A) = x \<^bold>* F (A - {x})"
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  using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
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lemma subset:
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  assumes "finite A" and "B \<subseteq> A"
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  shows "F B \<^bold>* F A = F A"
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proof -
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  from assms have "finite B" by (auto dest: finite_subset)
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  with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
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qed
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lemma closed:
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  assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x \<^bold>* y \<in> {x, y}"
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  shows "F A \<in> A"
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using \<open>finite A\<close> \<open>A \<noteq> {}\<close> proof (induct rule: finite_ne_induct)
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  case singleton then show ?case by simp
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next
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  case insert with elem show ?case by force
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qed
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end
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locale semilattice_order_neutr_set = binary?: semilattice_neutr_order + semilattice_neutr_set
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begin
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lemma bounded_iff:
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  assumes "finite A"
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  shows "x \<^bold>\<le> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<^bold>\<le> a)"
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  using assms by (induct A) (simp_all add: bounded_iff)
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lemma boundedI:
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  assumes "finite A"
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  assumes "\<And>a. a \<in> A \<Longrightarrow> x \<^bold>\<le> a"
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  shows "x \<^bold>\<le> F A"
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  using assms by (simp add: bounded_iff)
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lemma boundedE:
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  assumes "finite A" and "x \<^bold>\<le> F A"
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  obtains "\<And>a. a \<in> A \<Longrightarrow> x \<^bold>\<le> a"
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  using assms by (simp add: bounded_iff)
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lemma coboundedI:
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  assumes "finite A"
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    and "a \<in> A"
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  shows "F A \<^bold>\<le> a"
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proof -
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  from assms have "A \<noteq> {}" by auto
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  from \<open>finite A\<close> \<open>A \<noteq> {}\<close> \<open>a \<in> A\<close> show ?thesis
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  proof (induct rule: finite_ne_induct)
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    case singleton thus ?case by (simp add: refl)
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  next
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    case (insert x B)
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    from insert have "a = x \<or> a \<in> B" by simp
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    then show ?case using insert by (auto intro: coboundedI2)
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  qed
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qed
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lemma antimono:
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  assumes "A \<subseteq> B" and "finite B"
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  shows "F B \<^bold>\<le> F A"
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proof (cases "A = B")
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  case True then show ?thesis by (simp add: refl)
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next
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  case False
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  have B: "B = A \<union> (B - A)" using \<open>A \<subseteq> B\<close> by blast
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  then have "F B = F (A \<union> (B - A))" by simp
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  also have "\<dots> = F A \<^bold>* F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
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  also have "\<dots> \<^bold>\<le> F A" by simp
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  finally show ?thesis .
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qed
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end
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subsection \<open>Lattice operations on finite sets\<close>
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context semilattice_inf
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begin
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sublocale Inf_fin: semilattice_order_set inf less_eq less
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defines
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  Inf_fin ("\<Sqinter>\<^sub>f\<^sub>i\<^sub>n_" [900] 900) = Inf_fin.F ..
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end
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context semilattice_sup
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begin
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sublocale Sup_fin: semilattice_order_set sup greater_eq greater
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defines
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  Sup_fin ("\<Squnion>\<^sub>f\<^sub>i\<^sub>n_" [900] 900) = Sup_fin.F ..
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   323
end
haftmann@54868
   324
haftmann@54744
   325
wenzelm@60758
   326
subsection \<open>Infimum and Supremum over non-empty sets\<close>
haftmann@54744
   327
haftmann@54744
   328
context lattice
haftmann@54744
   329
begin
haftmann@54744
   330
haftmann@54745
   331
lemma Inf_fin_le_Sup_fin [simp]: 
haftmann@54745
   332
  assumes "finite A" and "A \<noteq> {}"
haftmann@54745
   333
  shows "\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA \<le> \<Squnion>\<^sub>f\<^sub>i\<^sub>nA"
haftmann@54745
   334
proof -
wenzelm@60758
   335
  from \<open>A \<noteq> {}\<close> obtain a where "a \<in> A" by blast
wenzelm@60758
   336
  with \<open>finite A\<close> have "\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA \<le> a" by (rule Inf_fin.coboundedI)
wenzelm@60758
   337
  moreover from \<open>finite A\<close> \<open>a \<in> A\<close> have "a \<le> \<Squnion>\<^sub>f\<^sub>i\<^sub>nA" by (rule Sup_fin.coboundedI)
haftmann@54745
   338
  ultimately show ?thesis by (rule order_trans)
haftmann@54745
   339
qed
haftmann@54744
   340
haftmann@54744
   341
lemma sup_Inf_absorb [simp]:
haftmann@54745
   342
  "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> \<Sqinter>\<^sub>f\<^sub>i\<^sub>nA \<squnion> a = a"
haftmann@54745
   343
  by (rule sup_absorb2) (rule Inf_fin.coboundedI)
haftmann@54744
   344
haftmann@54744
   345
lemma inf_Sup_absorb [simp]:
haftmann@54745
   346
  "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> a \<sqinter> \<Squnion>\<^sub>f\<^sub>i\<^sub>nA = a"
haftmann@54745
   347
  by (rule inf_absorb1) (rule Sup_fin.coboundedI)
haftmann@54744
   348
haftmann@54744
   349
end
haftmann@54744
   350
haftmann@54744
   351
context distrib_lattice
haftmann@54744
   352
begin
haftmann@54744
   353
haftmann@54744
   354
lemma sup_Inf1_distrib:
haftmann@54744
   355
  assumes "finite A"
haftmann@54744
   356
    and "A \<noteq> {}"
haftmann@54744
   357
  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@54744
   358
using assms by (simp add: image_def Inf_fin.hom_commute [where h="sup x", OF sup_inf_distrib1])
haftmann@54744
   359
  (rule arg_cong [where f="Inf_fin"], blast)
haftmann@54744
   360
haftmann@54744
   361
lemma sup_Inf2_distrib:
haftmann@54744
   362
  assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
haftmann@54744
   363
  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@54744
   364
using A proof (induct rule: finite_ne_induct)
haftmann@54744
   365
  case singleton then show ?case
haftmann@54744
   366
    by (simp add: sup_Inf1_distrib [OF B])
haftmann@54744
   367
next
haftmann@54744
   368
  case (insert x A)
haftmann@54744
   369
  have finB: "finite {sup x b |b. b \<in> B}"
haftmann@54744
   370
    by (rule finite_surj [where f = "sup x", OF B(1)], auto)
haftmann@54744
   371
  have finAB: "finite {sup a b |a b. a \<in> A \<and> b \<in> B}"
haftmann@54744
   372
  proof -
haftmann@54744
   373
    have "{sup a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {sup a b})"
haftmann@54744
   374
      by blast
haftmann@54744
   375
    thus ?thesis by(simp add: insert(1) B(1))
haftmann@54744
   376
  qed
haftmann@54744
   377
  have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
haftmann@54744
   378
  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)"
haftmann@54744
   379
    using insert by simp
haftmann@54744
   380
  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)
haftmann@54744
   381
  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})"
haftmann@54744
   382
    using insert by(simp add:sup_Inf1_distrib[OF B])
haftmann@54744
   383
  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})"
haftmann@54744
   384
    (is "_ = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n?M")
haftmann@54744
   385
    using B insert
haftmann@54744
   386
    by (simp add: Inf_fin.union [OF finB _ finAB ne])
haftmann@54744
   387
  also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"
haftmann@54744
   388
    by blast
haftmann@54744
   389
  finally show ?case .
haftmann@54744
   390
qed
haftmann@54744
   391
haftmann@54744
   392
lemma inf_Sup1_distrib:
haftmann@54744
   393
  assumes "finite A" and "A \<noteq> {}"
haftmann@54744
   394
  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@54744
   395
using assms by (simp add: image_def Sup_fin.hom_commute [where h="inf x", OF inf_sup_distrib1])
haftmann@54744
   396
  (rule arg_cong [where f="Sup_fin"], blast)
haftmann@54744
   397
haftmann@54744
   398
lemma inf_Sup2_distrib:
haftmann@54744
   399
  assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
haftmann@54744
   400
  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@54744
   401
using A proof (induct rule: finite_ne_induct)
haftmann@54744
   402
  case singleton thus ?case
haftmann@54744
   403
    by(simp add: inf_Sup1_distrib [OF B])
haftmann@54744
   404
next
haftmann@54744
   405
  case (insert x A)
haftmann@54744
   406
  have finB: "finite {inf x b |b. b \<in> B}"
haftmann@54744
   407
    by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto)
haftmann@54744
   408
  have finAB: "finite {inf a b |a b. a \<in> A \<and> b \<in> B}"
haftmann@54744
   409
  proof -
haftmann@54744
   410
    have "{inf a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {inf a b})"
haftmann@54744
   411
      by blast
haftmann@54744
   412
    thus ?thesis by(simp add: insert(1) B(1))
haftmann@54744
   413
  qed
haftmann@54744
   414
  have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
haftmann@54744
   415
  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)"
haftmann@54744
   416
    using insert by simp
haftmann@54744
   417
  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)
haftmann@54744
   418
  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})"
haftmann@54744
   419
    using insert by(simp add:inf_Sup1_distrib[OF B])
haftmann@54744
   420
  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})"
haftmann@54744
   421
    (is "_ = \<Squnion>\<^sub>f\<^sub>i\<^sub>n?M")
haftmann@54744
   422
    using B insert
haftmann@54744
   423
    by (simp add: Sup_fin.union [OF finB _ finAB ne])
haftmann@54744
   424
  also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"
haftmann@54744
   425
    by blast
haftmann@54744
   426
  finally show ?case .
haftmann@54744
   427
qed
haftmann@54744
   428
haftmann@54744
   429
end
haftmann@54744
   430
haftmann@54744
   431
context complete_lattice
haftmann@54744
   432
begin
haftmann@54744
   433
haftmann@54744
   434
lemma Inf_fin_Inf:
haftmann@54744
   435
  assumes "finite A" and "A \<noteq> {}"
haftmann@54868
   436
  shows "\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA = \<Sqinter>A"
haftmann@54744
   437
proof -
haftmann@54744
   438
  from assms obtain b B where "A = insert b B" and "finite B" by auto
haftmann@54744
   439
  then show ?thesis
haftmann@54744
   440
    by (simp add: Inf_fin.eq_fold inf_Inf_fold_inf inf.commute [of b])
haftmann@54744
   441
qed
haftmann@54744
   442
haftmann@54744
   443
lemma Sup_fin_Sup:
haftmann@54744
   444
  assumes "finite A" and "A \<noteq> {}"
haftmann@54868
   445
  shows "\<Squnion>\<^sub>f\<^sub>i\<^sub>nA = \<Squnion>A"
haftmann@54744
   446
proof -
haftmann@54744
   447
  from assms obtain b B where "A = insert b B" and "finite B" by auto
haftmann@54744
   448
  then show ?thesis
haftmann@54744
   449
    by (simp add: Sup_fin.eq_fold sup_Sup_fold_sup sup.commute [of b])
haftmann@54744
   450
qed
haftmann@54744
   451
haftmann@54744
   452
end
haftmann@54744
   453
haftmann@54744
   454
wenzelm@60758
   455
subsection \<open>Minimum and Maximum over non-empty sets\<close>
haftmann@54744
   456
haftmann@54744
   457
context linorder
haftmann@54744
   458
begin
haftmann@54744
   459
wenzelm@61605
   460
sublocale Min: semilattice_order_set min less_eq less
wenzelm@61605
   461
  + Max: semilattice_order_set max greater_eq greater
haftmann@61776
   462
defines
haftmann@61776
   463
  Min = Min.F and Max = Max.F ..
haftmann@54864
   464
haftmann@54864
   465
end
haftmann@54864
   466
wenzelm@60758
   467
text \<open>An aside: @{const Min}/@{const Max} on linear orders as special case of @{const Inf_fin}/@{const Sup_fin}\<close>
haftmann@54864
   468
haftmann@54864
   469
lemma Inf_fin_Min:
haftmann@54864
   470
  "Inf_fin = (Min :: 'a::{semilattice_inf, linorder} set \<Rightarrow> 'a)"
haftmann@54864
   471
  by (simp add: Inf_fin_def Min_def inf_min)
haftmann@54864
   472
haftmann@54864
   473
lemma Sup_fin_Max:
haftmann@54864
   474
  "Sup_fin = (Max :: 'a::{semilattice_sup, linorder} set \<Rightarrow> 'a)"
haftmann@54864
   475
  by (simp add: Sup_fin_def Max_def sup_max)
haftmann@54864
   476
haftmann@54864
   477
context linorder
haftmann@54864
   478
begin
haftmann@54864
   479
haftmann@54744
   480
lemma dual_min:
haftmann@54744
   481
  "ord.min greater_eq = max"
haftmann@54744
   482
  by (auto simp add: ord.min_def max_def fun_eq_iff)
haftmann@54744
   483
haftmann@54744
   484
lemma dual_max:
haftmann@54744
   485
  "ord.max greater_eq = min"
haftmann@54744
   486
  by (auto simp add: ord.max_def min_def fun_eq_iff)
haftmann@54744
   487
haftmann@54744
   488
lemma dual_Min:
haftmann@54744
   489
  "linorder.Min greater_eq = Max"
haftmann@54744
   490
proof -
wenzelm@61605
   491
  interpret dual: linorder greater_eq greater by (fact dual_linorder)
haftmann@54744
   492
  show ?thesis by (simp add: dual.Min_def dual_min Max_def)
haftmann@54744
   493
qed
haftmann@54744
   494
haftmann@54744
   495
lemma dual_Max:
haftmann@54744
   496
  "linorder.Max greater_eq = Min"
haftmann@54744
   497
proof -
wenzelm@61605
   498
  interpret dual: linorder greater_eq greater by (fact dual_linorder)
haftmann@54744
   499
  show ?thesis by (simp add: dual.Max_def dual_max Min_def)
haftmann@54744
   500
qed
haftmann@54744
   501
haftmann@54744
   502
lemmas Min_singleton = Min.singleton
haftmann@54744
   503
lemmas Max_singleton = Max.singleton
haftmann@54744
   504
lemmas Min_insert = Min.insert
haftmann@54744
   505
lemmas Max_insert = Max.insert
haftmann@54744
   506
lemmas Min_Un = Min.union
haftmann@54744
   507
lemmas Max_Un = Max.union
haftmann@54744
   508
lemmas hom_Min_commute = Min.hom_commute
haftmann@54744
   509
lemmas hom_Max_commute = Max.hom_commute
haftmann@54744
   510
haftmann@54744
   511
lemma Min_in [simp]:
haftmann@54744
   512
  assumes "finite A" and "A \<noteq> {}"
haftmann@54744
   513
  shows "Min A \<in> A"
haftmann@54744
   514
  using assms by (auto simp add: min_def Min.closed)
haftmann@54744
   515
haftmann@54744
   516
lemma Max_in [simp]:
haftmann@54744
   517
  assumes "finite A" and "A \<noteq> {}"
haftmann@54744
   518
  shows "Max A \<in> A"
haftmann@54744
   519
  using assms by (auto simp add: max_def Max.closed)
haftmann@54744
   520
haftmann@58467
   521
lemma Min_insert2:
haftmann@58467
   522
  assumes "finite A" and min: "\<And>b. b \<in> A \<Longrightarrow> a \<le> b"
haftmann@58467
   523
  shows "Min (insert a A) = a"
haftmann@58467
   524
proof (cases "A = {}")
wenzelm@63915
   525
  case True
wenzelm@63915
   526
  then show ?thesis by simp
haftmann@58467
   527
next
wenzelm@63915
   528
  case False
wenzelm@63915
   529
  with \<open>finite A\<close> have "Min (insert a A) = min a (Min A)"
haftmann@58467
   530
    by simp
wenzelm@60758
   531
  moreover from \<open>finite A\<close> \<open>A \<noteq> {}\<close> min have "a \<le> Min A" by simp
haftmann@58467
   532
  ultimately show ?thesis by (simp add: min.absorb1)
haftmann@58467
   533
qed
haftmann@58467
   534
haftmann@58467
   535
lemma Max_insert2:
haftmann@58467
   536
  assumes "finite A" and max: "\<And>b. b \<in> A \<Longrightarrow> b \<le> a"
haftmann@58467
   537
  shows "Max (insert a A) = a"
haftmann@58467
   538
proof (cases "A = {}")
wenzelm@63915
   539
  case True
wenzelm@63915
   540
  then show ?thesis by simp
haftmann@58467
   541
next
wenzelm@63915
   542
  case False
wenzelm@63915
   543
  with \<open>finite A\<close> have "Max (insert a A) = max a (Max A)"
haftmann@58467
   544
    by simp
wenzelm@60758
   545
  moreover from \<open>finite A\<close> \<open>A \<noteq> {}\<close> max have "Max A \<le> a" by simp
haftmann@58467
   546
  ultimately show ?thesis by (simp add: max.absorb1)
haftmann@58467
   547
qed
haftmann@58467
   548
haftmann@54744
   549
lemma Min_le [simp]:
haftmann@54744
   550
  assumes "finite A" and "x \<in> A"
haftmann@54744
   551
  shows "Min A \<le> x"
haftmann@54744
   552
  using assms by (fact Min.coboundedI)
haftmann@54744
   553
haftmann@54744
   554
lemma Max_ge [simp]:
haftmann@54744
   555
  assumes "finite A" and "x \<in> A"
haftmann@54744
   556
  shows "x \<le> Max A"
haftmann@54744
   557
  using assms by (fact Max.coboundedI)
haftmann@54744
   558
haftmann@54744
   559
lemma Min_eqI:
haftmann@54744
   560
  assumes "finite A"
haftmann@54744
   561
  assumes "\<And>y. y \<in> A \<Longrightarrow> y \<ge> x"
haftmann@54744
   562
    and "x \<in> A"
haftmann@54744
   563
  shows "Min A = x"
haftmann@54744
   564
proof (rule antisym)
wenzelm@60758
   565
  from \<open>x \<in> A\<close> have "A \<noteq> {}" by auto
haftmann@54744
   566
  with assms show "Min A \<ge> x" by simp
haftmann@54744
   567
next
haftmann@54744
   568
  from assms show "x \<ge> Min A" by simp
haftmann@54744
   569
qed
haftmann@54744
   570
haftmann@54744
   571
lemma Max_eqI:
haftmann@54744
   572
  assumes "finite A"
haftmann@54744
   573
  assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
haftmann@54744
   574
    and "x \<in> A"
haftmann@54744
   575
  shows "Max A = x"
haftmann@54744
   576
proof (rule antisym)
wenzelm@60758
   577
  from \<open>x \<in> A\<close> have "A \<noteq> {}" by auto
haftmann@54744
   578
  with assms show "Max A \<le> x" by simp
haftmann@54744
   579
next
haftmann@54744
   580
  from assms show "x \<le> Max A" by simp
haftmann@54744
   581
qed
haftmann@54744
   582
nipkow@66425
   583
lemma eq_Min_iff:
nipkow@66425
   584
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> m = Min A  \<longleftrightarrow>  m \<in> A \<and> (\<forall>a \<in> A. m \<le> a)"
nipkow@66425
   585
by (meson Min_in Min_le antisym)
nipkow@66425
   586
nipkow@66425
   587
lemma Min_eq_iff:
nipkow@66425
   588
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> Min A = m  \<longleftrightarrow>  m \<in> A \<and> (\<forall>a \<in> A. m \<le> a)"
nipkow@66425
   589
by (meson Min_in Min_le antisym)
nipkow@66425
   590
nipkow@66425
   591
lemma eq_Max_iff:
nipkow@66425
   592
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> m = Max A  \<longleftrightarrow>  m \<in> A \<and> (\<forall>a \<in> A. a \<le> m)"
nipkow@66425
   593
by (meson Max_in Max_ge antisym)
nipkow@66425
   594
nipkow@66425
   595
lemma Max_eq_iff:
nipkow@66425
   596
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> Max A = m  \<longleftrightarrow>  m \<in> A \<and> (\<forall>a \<in> A. a \<le> m)"
nipkow@66425
   597
by (meson Max_in Max_ge antisym)
nipkow@66425
   598
haftmann@54744
   599
context
haftmann@54744
   600
  fixes A :: "'a set"
haftmann@54744
   601
  assumes fin_nonempty: "finite A" "A \<noteq> {}"
haftmann@54744
   602
begin
haftmann@54744
   603
haftmann@54744
   604
lemma Min_ge_iff [simp]:
haftmann@54744
   605
  "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
haftmann@54744
   606
  using fin_nonempty by (fact Min.bounded_iff)
haftmann@54744
   607
haftmann@54744
   608
lemma Max_le_iff [simp]:
haftmann@54744
   609
  "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
haftmann@54744
   610
  using fin_nonempty by (fact Max.bounded_iff)
haftmann@54744
   611
haftmann@54744
   612
lemma Min_gr_iff [simp]:
haftmann@54744
   613
  "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
haftmann@54744
   614
  using fin_nonempty  by (induct rule: finite_ne_induct) simp_all
haftmann@54744
   615
haftmann@54744
   616
lemma Max_less_iff [simp]:
haftmann@54744
   617
  "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
haftmann@54744
   618
  using fin_nonempty by (induct rule: finite_ne_induct) simp_all
haftmann@54744
   619
haftmann@54744
   620
lemma Min_le_iff:
haftmann@54744
   621
  "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
haftmann@54744
   622
  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_le_iff_disj)
haftmann@54744
   623
haftmann@54744
   624
lemma Max_ge_iff:
haftmann@54744
   625
  "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
haftmann@54744
   626
  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: le_max_iff_disj)
haftmann@54744
   627
haftmann@54744
   628
lemma Min_less_iff:
haftmann@54744
   629
  "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
haftmann@54744
   630
  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_less_iff_disj)
haftmann@54744
   631
haftmann@54744
   632
lemma Max_gr_iff:
haftmann@54744
   633
  "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
haftmann@54744
   634
  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: less_max_iff_disj)
haftmann@54744
   635
haftmann@54744
   636
end
haftmann@54744
   637
nipkow@57800
   638
lemma Max_eq_if:
nipkow@57800
   639
  assumes "finite A"  "finite B"  "\<forall>a\<in>A. \<exists>b\<in>B. a \<le> b"  "\<forall>b\<in>B. \<exists>a\<in>A. b \<le> a"
nipkow@57800
   640
  shows "Max A = Max B"
nipkow@57800
   641
proof cases
nipkow@57800
   642
  assume "A = {}" thus ?thesis using assms by simp
nipkow@57800
   643
next
nipkow@57800
   644
  assume "A \<noteq> {}" thus ?thesis using assms
nipkow@57800
   645
    by(blast intro: antisym Max_in Max_ge_iff[THEN iffD2])
nipkow@57800
   646
qed
nipkow@57800
   647
haftmann@54744
   648
lemma Min_antimono:
haftmann@54744
   649
  assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
haftmann@54744
   650
  shows "Min N \<le> Min M"
haftmann@54744
   651
  using assms by (fact Min.antimono)
haftmann@54744
   652
haftmann@54744
   653
lemma Max_mono:
haftmann@54744
   654
  assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
haftmann@54744
   655
  shows "Max M \<le> Max N"
haftmann@54744
   656
  using assms by (fact Max.antimono)
haftmann@54744
   657
wenzelm@56140
   658
end
wenzelm@56140
   659
wenzelm@56140
   660
context linorder  (* FIXME *)
wenzelm@56140
   661
begin
wenzelm@56140
   662
haftmann@54744
   663
lemma mono_Min_commute:
haftmann@54744
   664
  assumes "mono f"
haftmann@54744
   665
  assumes "finite A" and "A \<noteq> {}"
haftmann@54744
   666
  shows "f (Min A) = Min (f ` A)"
haftmann@54744
   667
proof (rule linorder_class.Min_eqI [symmetric])
wenzelm@60758
   668
  from \<open>finite A\<close> show "finite (f ` A)" by simp
haftmann@54744
   669
  from assms show "f (Min A) \<in> f ` A" by simp
haftmann@54744
   670
  fix x
haftmann@54744
   671
  assume "x \<in> f ` A"
haftmann@54744
   672
  then obtain y where "y \<in> A" and "x = f y" ..
haftmann@54744
   673
  with assms have "Min A \<le> y" by auto
wenzelm@60758
   674
  with \<open>mono f\<close> have "f (Min A) \<le> f y" by (rule monoE)
wenzelm@60758
   675
  with \<open>x = f y\<close> show "f (Min A) \<le> x" by simp
haftmann@54744
   676
qed
haftmann@54744
   677
haftmann@54744
   678
lemma mono_Max_commute:
haftmann@54744
   679
  assumes "mono f"
haftmann@54744
   680
  assumes "finite A" and "A \<noteq> {}"
haftmann@54744
   681
  shows "f (Max A) = Max (f ` A)"
haftmann@54744
   682
proof (rule linorder_class.Max_eqI [symmetric])
wenzelm@60758
   683
  from \<open>finite A\<close> show "finite (f ` A)" by simp
haftmann@54744
   684
  from assms show "f (Max A) \<in> f ` A" by simp
haftmann@54744
   685
  fix x
haftmann@54744
   686
  assume "x \<in> f ` A"
haftmann@54744
   687
  then obtain y where "y \<in> A" and "x = f y" ..
haftmann@54744
   688
  with assms have "y \<le> Max A" by auto
wenzelm@60758
   689
  with \<open>mono f\<close> have "f y \<le> f (Max A)" by (rule monoE)
wenzelm@60758
   690
  with \<open>x = f y\<close> show "x \<le> f (Max A)" by simp
haftmann@54744
   691
qed
haftmann@54744
   692
haftmann@54744
   693
lemma finite_linorder_max_induct [consumes 1, case_names empty insert]:
haftmann@54744
   694
  assumes fin: "finite A"
haftmann@54744
   695
  and empty: "P {}" 
haftmann@54744
   696
  and insert: "\<And>b A. finite A \<Longrightarrow> \<forall>a\<in>A. a < b \<Longrightarrow> P A \<Longrightarrow> P (insert b A)"
haftmann@54744
   697
  shows "P A"
haftmann@54744
   698
using fin empty insert
haftmann@54744
   699
proof (induct rule: finite_psubset_induct)
haftmann@54744
   700
  case (psubset A)
haftmann@54744
   701
  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 
haftmann@54744
   702
  have fin: "finite A" by fact 
haftmann@54744
   703
  have empty: "P {}" by fact
haftmann@54744
   704
  have step: "\<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. a < b; P A\<rbrakk> \<Longrightarrow> P (insert b A)" by fact
haftmann@54744
   705
  show "P A"
haftmann@54744
   706
  proof (cases "A = {}")
haftmann@54744
   707
    assume "A = {}" 
wenzelm@60758
   708
    then show "P A" using \<open>P {}\<close> by simp
haftmann@54744
   709
  next
haftmann@54744
   710
    let ?B = "A - {Max A}" 
haftmann@54744
   711
    let ?A = "insert (Max A) ?B"
wenzelm@60758
   712
    have "finite ?B" using \<open>finite A\<close> by simp
haftmann@54744
   713
    assume "A \<noteq> {}"
wenzelm@60758
   714
    with \<open>finite A\<close> have "Max A : A" by auto
haftmann@54744
   715
    then have A: "?A = A" using insert_Diff_single insert_absorb by auto
wenzelm@60758
   716
    then have "P ?B" using \<open>P {}\<close> step IH [of ?B] by blast
haftmann@54744
   717
    moreover 
wenzelm@60758
   718
    have "\<forall>a\<in>?B. a < Max A" using Max_ge [OF \<open>finite A\<close>] by fastforce
wenzelm@60758
   719
    ultimately show "P A" using A insert_Diff_single step [OF \<open>finite ?B\<close>] by fastforce
haftmann@54744
   720
  qed
haftmann@54744
   721
qed
haftmann@54744
   722
haftmann@54744
   723
lemma finite_linorder_min_induct [consumes 1, case_names empty insert]:
haftmann@54744
   724
  "\<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@54744
   725
  by (rule linorder.finite_linorder_max_induct [OF dual_linorder])
haftmann@54744
   726
haftmann@54744
   727
lemma Least_Min:
haftmann@54744
   728
  assumes "finite {a. P a}" and "\<exists>a. P a"
haftmann@54744
   729
  shows "(LEAST a. P a) = Min {a. P a}"
haftmann@54744
   730
proof -
haftmann@54744
   731
  { fix A :: "'a set"
haftmann@54744
   732
    assume A: "finite A" "A \<noteq> {}"
haftmann@54744
   733
    have "(LEAST a. a \<in> A) = Min A"
haftmann@54744
   734
    using A proof (induct A rule: finite_ne_induct)
haftmann@54744
   735
      case singleton show ?case by (rule Least_equality) simp_all
haftmann@54744
   736
    next
haftmann@54744
   737
      case (insert a A)
haftmann@54744
   738
      have "(LEAST b. b = a \<or> b \<in> A) = min a (LEAST a. a \<in> A)"
haftmann@54744
   739
        by (auto intro!: Least_equality simp add: min_def not_le Min_le_iff insert.hyps dest!: less_imp_le)
haftmann@54744
   740
      with insert show ?case by simp
haftmann@54744
   741
    qed
haftmann@54744
   742
  } from this [of "{a. P a}"] assms show ?thesis by simp
haftmann@54744
   743
qed
haftmann@54744
   744
hoelzl@59000
   745
lemma infinite_growing:
hoelzl@59000
   746
  assumes "X \<noteq> {}"
hoelzl@59000
   747
  assumes *: "\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>X. y > x"
hoelzl@59000
   748
  shows "\<not> finite X"
hoelzl@59000
   749
proof
hoelzl@59000
   750
  assume "finite X"
wenzelm@60758
   751
  with \<open>X \<noteq> {}\<close> have "Max X \<in> X" "\<forall>x\<in>X. x \<le> Max X"
hoelzl@59000
   752
    by auto
hoelzl@59000
   753
  with *[of "Max X"] show False
hoelzl@59000
   754
    by auto
hoelzl@59000
   755
qed
hoelzl@59000
   756
haftmann@54744
   757
end
haftmann@54744
   758
haftmann@54744
   759
context linordered_ab_semigroup_add
haftmann@54744
   760
begin
haftmann@54744
   761
haftmann@54744
   762
lemma add_Min_commute:
haftmann@54744
   763
  fixes k
haftmann@54744
   764
  assumes "finite N" and "N \<noteq> {}"
haftmann@54744
   765
  shows "k + Min N = Min {k + m | m. m \<in> N}"
haftmann@54744
   766
proof -
haftmann@54744
   767
  have "\<And>x y. k + min x y = min (k + x) (k + y)"
haftmann@54744
   768
    by (simp add: min_def not_le)
haftmann@54744
   769
      (blast intro: antisym less_imp_le add_left_mono)
haftmann@54744
   770
  with assms show ?thesis
haftmann@54744
   771
    using hom_Min_commute [of "plus k" N]
haftmann@54744
   772
    by simp (blast intro: arg_cong [where f = Min])
haftmann@54744
   773
qed
haftmann@54744
   774
haftmann@54744
   775
lemma add_Max_commute:
haftmann@54744
   776
  fixes k
haftmann@54744
   777
  assumes "finite N" and "N \<noteq> {}"
haftmann@54744
   778
  shows "k + Max N = Max {k + m | m. m \<in> N}"
haftmann@54744
   779
proof -
haftmann@54744
   780
  have "\<And>x y. k + max x y = max (k + x) (k + y)"
haftmann@54744
   781
    by (simp add: max_def not_le)
haftmann@54744
   782
      (blast intro: antisym less_imp_le add_left_mono)
haftmann@54744
   783
  with assms show ?thesis
haftmann@54744
   784
    using hom_Max_commute [of "plus k" N]
haftmann@54744
   785
    by simp (blast intro: arg_cong [where f = Max])
haftmann@54744
   786
qed
haftmann@54744
   787
haftmann@54744
   788
end
haftmann@54744
   789
haftmann@54744
   790
context linordered_ab_group_add
haftmann@54744
   791
begin
haftmann@54744
   792
haftmann@54744
   793
lemma minus_Max_eq_Min [simp]:
haftmann@54744
   794
  "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Max S = Min (uminus ` S)"
haftmann@54744
   795
  by (induct S rule: finite_ne_induct) (simp_all add: minus_max_eq_min)
haftmann@54744
   796
haftmann@54744
   797
lemma minus_Min_eq_Max [simp]:
haftmann@54744
   798
  "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Min S = Max (uminus ` S)"
haftmann@54744
   799
  by (induct S rule: finite_ne_induct) (simp_all add: minus_min_eq_max)
haftmann@54744
   800
haftmann@54744
   801
end
haftmann@54744
   802
haftmann@54744
   803
context complete_linorder
haftmann@54744
   804
begin
haftmann@54744
   805
haftmann@54744
   806
lemma Min_Inf:
haftmann@54744
   807
  assumes "finite A" and "A \<noteq> {}"
haftmann@54744
   808
  shows "Min A = Inf A"
haftmann@54744
   809
proof -
haftmann@54744
   810
  from assms obtain b B where "A = insert b B" and "finite B" by auto
haftmann@54744
   811
  then show ?thesis
haftmann@54744
   812
    by (simp add: Min.eq_fold complete_linorder_inf_min [symmetric] inf_Inf_fold_inf inf.commute [of b])
haftmann@54744
   813
qed
haftmann@54744
   814
haftmann@54744
   815
lemma Max_Sup:
haftmann@54744
   816
  assumes "finite A" and "A \<noteq> {}"
haftmann@54744
   817
  shows "Max A = Sup A"
haftmann@54744
   818
proof -
haftmann@54744
   819
  from assms obtain b B where "A = insert b B" and "finite B" by auto
haftmann@54744
   820
  then show ?thesis
haftmann@54744
   821
    by (simp add: Max.eq_fold complete_linorder_sup_max [symmetric] sup_Sup_fold_sup sup.commute [of b])
haftmann@54744
   822
qed
haftmann@54744
   823
haftmann@54744
   824
end
haftmann@54744
   825
nipkow@65817
   826
nipkow@65817
   827
subsection \<open>Arg Min\<close>
nipkow@65817
   828
nipkow@65842
   829
definition is_arg_min :: "('a \<Rightarrow> 'b::ord) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool" where
nipkow@65842
   830
"is_arg_min f P x = (P x \<and> \<not>(\<exists>y. P y \<and> f y < f x))"
nipkow@65842
   831
nipkow@65842
   832
definition arg_min :: "('a \<Rightarrow> 'b::ord) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a" where
nipkow@65842
   833
"arg_min f P = (SOME x. is_arg_min f P x)"
nipkow@65842
   834
nipkow@65842
   835
abbreviation arg_min_on :: "('a \<Rightarrow> 'b::ord) \<Rightarrow> 'a set \<Rightarrow> 'a" where
nipkow@65842
   836
"arg_min_on f S \<equiv> arg_min f (\<lambda>x. x \<in> S)"
nipkow@65842
   837
nipkow@65951
   838
syntax
nipkow@65951
   839
  "_arg_min" :: "('a \<Rightarrow> 'b) \<Rightarrow> pttrn \<Rightarrow> bool \<Rightarrow> 'a"
nipkow@65951
   840
    ("(3ARG'_MIN _ _./ _)" [0, 0, 10] 10)
nipkow@65951
   841
translations
nipkow@65951
   842
  "ARG_MIN f x. P" \<rightleftharpoons> "CONST arg_min f (\<lambda>x. P)"
nipkow@65951
   843
nipkow@65842
   844
lemma is_arg_min_linorder: fixes f :: "'a \<Rightarrow> 'b :: linorder"
nipkow@65842
   845
shows "is_arg_min f P x = (P x \<and> (\<forall>y. P y \<longrightarrow> f x \<le> f y))"
nipkow@65842
   846
by(auto simp add: is_arg_min_def)
nipkow@65817
   847
nipkow@65951
   848
lemma arg_minI:
nipkow@65951
   849
  "\<lbrakk> P x;
nipkow@65951
   850
    \<And>y. P y \<Longrightarrow> \<not> f y < f x;
nipkow@65951
   851
    \<And>x. \<lbrakk> P x; \<forall>y. P y \<longrightarrow> \<not> f y < f x \<rbrakk> \<Longrightarrow> Q x \<rbrakk>
nipkow@65951
   852
  \<Longrightarrow> Q (arg_min f P)"
nipkow@65951
   853
apply (simp add: arg_min_def is_arg_min_def)
nipkow@65951
   854
apply (rule someI2_ex)
nipkow@65951
   855
 apply blast
nipkow@65951
   856
apply blast
nipkow@65951
   857
done
nipkow@65951
   858
nipkow@65951
   859
lemma arg_min_equality:
nipkow@65952
   860
  "\<lbrakk> P k; \<And>x. P x \<Longrightarrow> f k \<le> f x \<rbrakk> \<Longrightarrow> f (arg_min f P) = f k"
nipkow@65951
   861
  for f :: "_ \<Rightarrow> 'a::order"
nipkow@65951
   862
apply (rule arg_minI)
nipkow@65951
   863
  apply assumption
nipkow@65951
   864
 apply (simp add: less_le_not_le)
nipkow@65951
   865
by (metis le_less)
nipkow@65951
   866
nipkow@65952
   867
lemma wf_linord_ex_has_least:
nipkow@65952
   868
  "\<lbrakk> wf r; \<forall>x y. (x, y) \<in> r\<^sup>+ \<longleftrightarrow> (y, x) \<notin> r\<^sup>*; P k \<rbrakk>
nipkow@65952
   869
   \<Longrightarrow> \<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> (m x, m y) \<in> r\<^sup>*)"
nipkow@65952
   870
apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
nipkow@65952
   871
apply (drule_tac x = "m ` Collect P" in spec)
nipkow@65952
   872
by force
nipkow@65952
   873
nipkow@65952
   874
lemma ex_has_least_nat: "P k \<Longrightarrow> \<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> m x \<le> m y)"
nipkow@65952
   875
  for m :: "'a \<Rightarrow> nat"
nipkow@65952
   876
apply (simp only: pred_nat_trancl_eq_le [symmetric])
nipkow@65952
   877
apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
nipkow@65952
   878
 apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le)
nipkow@65952
   879
by assumption
nipkow@65952
   880
nipkow@65951
   881
lemma arg_min_nat_lemma:
nipkow@65951
   882
  "P k \<Longrightarrow> P(arg_min m P) \<and> (\<forall>y. P y \<longrightarrow> m (arg_min m P) \<le> m y)"
nipkow@65842
   883
  for m :: "'a \<Rightarrow> nat"
nipkow@65842
   884
apply (simp add: arg_min_def is_arg_min_linorder)
nipkow@65842
   885
apply (rule someI_ex)
nipkow@65842
   886
apply (erule ex_has_least_nat)
nipkow@65842
   887
done
nipkow@65842
   888
nipkow@65842
   889
lemmas arg_min_natI = arg_min_nat_lemma [THEN conjunct1]
nipkow@65817
   890
nipkow@65951
   891
lemma is_arg_min_arg_min_nat: fixes m :: "'a \<Rightarrow> nat"
nipkow@65951
   892
shows "P x \<Longrightarrow> is_arg_min m P (arg_min m P)"
nipkow@65951
   893
by (metis arg_min_nat_lemma is_arg_min_linorder)
nipkow@65951
   894
nipkow@65842
   895
lemma arg_min_nat_le: "P x \<Longrightarrow> m (arg_min m P) \<le> m x"
nipkow@65842
   896
  for m :: "'a \<Rightarrow> nat"
nipkow@65842
   897
by (rule arg_min_nat_lemma [THEN conjunct2, THEN spec, THEN mp])
nipkow@65842
   898
nipkow@65842
   899
lemma ex_min_if_finite:
nipkow@65842
   900
  "\<lbrakk> finite S; S \<noteq> {} \<rbrakk> \<Longrightarrow> \<exists>m\<in>S. \<not>(\<exists>x\<in>S. x < (m::'a::order))"
nipkow@65842
   901
by(induction rule: finite.induct) (auto intro: order.strict_trans)
nipkow@65842
   902
nipkow@65842
   903
lemma ex_is_arg_min_if_finite: fixes f :: "'a \<Rightarrow> 'b :: order"
nipkow@65842
   904
shows "\<lbrakk> finite S; S \<noteq> {} \<rbrakk> \<Longrightarrow> \<exists>x. is_arg_min f (\<lambda>x. x : S) x"
nipkow@65842
   905
unfolding is_arg_min_def
nipkow@65842
   906
using ex_min_if_finite[of "f ` S"]
nipkow@65842
   907
by auto
nipkow@65817
   908
nipkow@65817
   909
lemma arg_min_SOME_Min:
nipkow@65842
   910
  "finite S \<Longrightarrow> arg_min_on f S = (SOME y. y \<in> S \<and> f y = Min(f ` S))"
nipkow@65842
   911
unfolding arg_min_def is_arg_min_linorder
nipkow@65817
   912
apply(rule arg_cong[where f = Eps])
nipkow@65817
   913
apply (auto simp: fun_eq_iff intro: Min_eqI[symmetric])
nipkow@65817
   914
done
nipkow@65817
   915
nipkow@65842
   916
lemma arg_min_if_finite: fixes f :: "'a \<Rightarrow> 'b :: order"
nipkow@65842
   917
assumes "finite S" "S \<noteq> {}"
nipkow@65842
   918
shows  "arg_min_on f S \<in> S" and "\<not>(\<exists>x\<in>S. f x < f (arg_min_on f S))"
nipkow@65842
   919
using ex_is_arg_min_if_finite[OF assms, of f]
nipkow@65842
   920
unfolding arg_min_def is_arg_min_def
nipkow@65842
   921
by(auto dest!: someI_ex)
nipkow@65817
   922
nipkow@65817
   923
lemma arg_min_least: fixes f :: "'a \<Rightarrow> 'b :: linorder"
nipkow@65842
   924
shows "\<lbrakk> finite S;  S \<noteq> {};  y \<in> S \<rbrakk> \<Longrightarrow> f(arg_min_on f S) \<le> f y"
nipkow@65817
   925
by(simp add: arg_min_SOME_Min inv_into_def2[symmetric] f_inv_into_f)
nipkow@65817
   926
nipkow@65817
   927
lemma arg_min_inj_eq: fixes f :: "'a \<Rightarrow> 'b :: order"
nipkow@65842
   928
shows "\<lbrakk> inj_on f {x. P x}; P a; \<forall>y. P y \<longrightarrow> f a \<le> f y \<rbrakk> \<Longrightarrow> arg_min f P = a"
nipkow@65842
   929
apply(simp add: arg_min_def is_arg_min_def)
nipkow@65817
   930
apply(rule someI2[of _ a])
nipkow@65817
   931
 apply (simp add: less_le_not_le)
nipkow@65842
   932
by (metis inj_on_eq_iff less_le mem_Collect_eq)
nipkow@65817
   933
nipkow@65954
   934
nipkow@65954
   935
subsection \<open>Arg Max\<close>
nipkow@65954
   936
nipkow@65954
   937
definition is_arg_max :: "('a \<Rightarrow> 'b::ord) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool" where
nipkow@65954
   938
"is_arg_max f P x = (P x \<and> \<not>(\<exists>y. P y \<and> f y > f x))"
nipkow@65954
   939
nipkow@65954
   940
definition arg_max :: "('a \<Rightarrow> 'b::ord) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a" where
nipkow@65954
   941
"arg_max f P = (SOME x. is_arg_max f P x)"
nipkow@65954
   942
nipkow@65954
   943
abbreviation arg_max_on :: "('a \<Rightarrow> 'b::ord) \<Rightarrow> 'a set \<Rightarrow> 'a" where
nipkow@65954
   944
"arg_max_on f S \<equiv> arg_max f (\<lambda>x. x \<in> S)"
nipkow@65954
   945
nipkow@65954
   946
syntax
nipkow@65954
   947
  "_arg_max" :: "('a \<Rightarrow> 'b) \<Rightarrow> pttrn \<Rightarrow> bool \<Rightarrow> 'a"
nipkow@65954
   948
    ("(3ARG'_MAX _ _./ _)" [0, 0, 10] 10)
nipkow@65954
   949
translations
nipkow@65954
   950
  "ARG_MAX f x. P" \<rightleftharpoons> "CONST arg_max f (\<lambda>x. P)"
nipkow@65954
   951
nipkow@65954
   952
lemma is_arg_max_linorder: fixes f :: "'a \<Rightarrow> 'b :: linorder"
nipkow@65954
   953
shows "is_arg_max f P x = (P x \<and> (\<forall>y. P y \<longrightarrow> f x \<ge> f y))"
nipkow@65954
   954
by(auto simp add: is_arg_max_def)
nipkow@65954
   955
nipkow@65954
   956
lemma arg_maxI:
nipkow@65954
   957
  "P x \<Longrightarrow>
nipkow@65954
   958
    (\<And>y. P y \<Longrightarrow> \<not> f y > f x) \<Longrightarrow>
nipkow@65954
   959
    (\<And>x. P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> \<not> f y > f x \<Longrightarrow> Q x) \<Longrightarrow>
nipkow@65954
   960
    Q (arg_max f P)"
nipkow@65954
   961
apply (simp add: arg_max_def is_arg_max_def)
nipkow@65954
   962
apply (rule someI2_ex)
nipkow@65954
   963
 apply blast
nipkow@65954
   964
apply blast
nipkow@65954
   965
done
nipkow@65954
   966
nipkow@65954
   967
lemma arg_max_equality:
nipkow@65954
   968
  "\<lbrakk> P k; \<And>x. P x \<Longrightarrow> f x \<le> f k \<rbrakk> \<Longrightarrow> f (arg_max f P) = f k"
nipkow@65954
   969
  for f :: "_ \<Rightarrow> 'a::order"
nipkow@65954
   970
apply (rule arg_maxI [where f = f])
nipkow@65954
   971
  apply assumption
nipkow@65954
   972
 apply (simp add: less_le_not_le)
nipkow@65954
   973
by (metis le_less)
nipkow@65954
   974
nipkow@65954
   975
lemma ex_has_greatest_nat_lemma:
nipkow@65954
   976
  "P k \<Longrightarrow> \<forall>x. P x \<longrightarrow> (\<exists>y. P y \<and> \<not> f y \<le> f x) \<Longrightarrow> \<exists>y. P y \<and> \<not> f y < f k + n"
nipkow@65954
   977
  for f :: "'a \<Rightarrow> nat"
nipkow@65954
   978
by (induct n) (force simp: le_Suc_eq)+
nipkow@65954
   979
nipkow@65954
   980
lemma ex_has_greatest_nat:
nipkow@65954
   981
  "P k \<Longrightarrow> \<forall>y. P y \<longrightarrow> f y < b \<Longrightarrow> \<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> f y \<le> f x)"
nipkow@65954
   982
  for f :: "'a \<Rightarrow> nat"
nipkow@65954
   983
apply (rule ccontr)
nipkow@65954
   984
apply (cut_tac P = P and n = "b - f k" in ex_has_greatest_nat_lemma)
nipkow@65954
   985
  apply (subgoal_tac [3] "f k \<le> b")
nipkow@65954
   986
   apply auto
nipkow@65954
   987
done
nipkow@65954
   988
nipkow@65954
   989
lemma arg_max_nat_lemma:
nipkow@65954
   990
  "\<lbrakk> P k;  \<forall>y. P y \<longrightarrow> f y < b \<rbrakk>
nipkow@65954
   991
  \<Longrightarrow> P (arg_max f P) \<and> (\<forall>y. P y \<longrightarrow> f y \<le> f (arg_max f P))"
nipkow@65954
   992
  for f :: "'a \<Rightarrow> nat"
nipkow@65954
   993
apply (simp add: arg_max_def is_arg_max_linorder)
nipkow@65954
   994
apply (rule someI_ex)
nipkow@65954
   995
apply (erule (1) ex_has_greatest_nat)
nipkow@65954
   996
done
nipkow@65954
   997
nipkow@65954
   998
lemmas arg_max_natI = arg_max_nat_lemma [THEN conjunct1]
nipkow@65954
   999
nipkow@65954
  1000
lemma arg_max_nat_le: "P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> f y < b \<Longrightarrow> f x \<le> f (arg_max f P)"
nipkow@65954
  1001
  for f :: "'a \<Rightarrow> nat"
nipkow@65954
  1002
by (blast dest: arg_max_nat_lemma [THEN conjunct2, THEN spec, of P])
nipkow@65954
  1003
haftmann@54744
  1004
end