src/HOL/Lattices_Big.thy
author hoelzl
Fri Feb 19 13:40:50 2016 +0100 (2016-02-19)
changeset 62378 85ed00c1fe7c
parent 61776 57bb7da5c867
child 63290 9ac558ab0906
permissions -rw-r--r--
generalize more theorems to support enat and ennreal
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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 Finite_Set Option
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begin
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subsection \<open>Generic lattice operations over a set\<close>
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no_notation times (infixl "*" 70)
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no_notation Groups.one ("1")
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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 * 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 * 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 * 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 * 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 * 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 * 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 * 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 * 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 * 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 * y) = h x * 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 * F N)" by simp
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  also have "\<dots> = h n * 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 * \<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 \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> 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 \<preceq> a"
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  shows "x \<preceq> 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 \<preceq> F A"
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  obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> 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 \<preceq> 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 \<preceq> 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 * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
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  also have "\<dots> \<preceq> 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 1 A"
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lemma infinite [simp]:
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  "\<not> finite A \<Longrightarrow> F A = 1"
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  by (simp add: eq_fold)
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lemma empty [simp]:
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  "F {} = 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 * 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 * 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 * 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 * 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 * 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 * 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 * 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 \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> 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 \<preceq> a"
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  shows "x \<preceq> 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 \<preceq> F A"
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  obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> 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 \<preceq> 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 \<preceq> 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 * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
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  also have "\<dots> \<preceq> 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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notation times (infixl "*" 70)
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notation Groups.one ("1")
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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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end
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   332
wenzelm@60758
   333
subsection \<open>Infimum and Supremum over non-empty sets\<close>
haftmann@54744
   334
haftmann@54744
   335
context lattice
haftmann@54744
   336
begin
haftmann@54744
   337
haftmann@54745
   338
lemma Inf_fin_le_Sup_fin [simp]: 
haftmann@54745
   339
  assumes "finite A" and "A \<noteq> {}"
haftmann@54745
   340
  shows "\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA \<le> \<Squnion>\<^sub>f\<^sub>i\<^sub>nA"
haftmann@54745
   341
proof -
wenzelm@60758
   342
  from \<open>A \<noteq> {}\<close> obtain a where "a \<in> A" by blast
wenzelm@60758
   343
  with \<open>finite A\<close> have "\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA \<le> a" by (rule Inf_fin.coboundedI)
wenzelm@60758
   344
  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
   345
  ultimately show ?thesis by (rule order_trans)
haftmann@54745
   346
qed
haftmann@54744
   347
haftmann@54744
   348
lemma sup_Inf_absorb [simp]:
haftmann@54745
   349
  "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> \<Sqinter>\<^sub>f\<^sub>i\<^sub>nA \<squnion> a = a"
haftmann@54745
   350
  by (rule sup_absorb2) (rule Inf_fin.coboundedI)
haftmann@54744
   351
haftmann@54744
   352
lemma inf_Sup_absorb [simp]:
haftmann@54745
   353
  "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> a \<sqinter> \<Squnion>\<^sub>f\<^sub>i\<^sub>nA = a"
haftmann@54745
   354
  by (rule inf_absorb1) (rule Sup_fin.coboundedI)
haftmann@54744
   355
haftmann@54744
   356
end
haftmann@54744
   357
haftmann@54744
   358
context distrib_lattice
haftmann@54744
   359
begin
haftmann@54744
   360
haftmann@54744
   361
lemma sup_Inf1_distrib:
haftmann@54744
   362
  assumes "finite A"
haftmann@54744
   363
    and "A \<noteq> {}"
haftmann@54744
   364
  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
   365
using assms by (simp add: image_def Inf_fin.hom_commute [where h="sup x", OF sup_inf_distrib1])
haftmann@54744
   366
  (rule arg_cong [where f="Inf_fin"], blast)
haftmann@54744
   367
haftmann@54744
   368
lemma sup_Inf2_distrib:
haftmann@54744
   369
  assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
haftmann@54744
   370
  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
   371
using A proof (induct rule: finite_ne_induct)
haftmann@54744
   372
  case singleton then show ?case
haftmann@54744
   373
    by (simp add: sup_Inf1_distrib [OF B])
haftmann@54744
   374
next
haftmann@54744
   375
  case (insert x A)
haftmann@54744
   376
  have finB: "finite {sup x b |b. b \<in> B}"
haftmann@54744
   377
    by (rule finite_surj [where f = "sup x", OF B(1)], auto)
haftmann@54744
   378
  have finAB: "finite {sup a b |a b. a \<in> A \<and> b \<in> B}"
haftmann@54744
   379
  proof -
haftmann@54744
   380
    have "{sup a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {sup a b})"
haftmann@54744
   381
      by blast
haftmann@54744
   382
    thus ?thesis by(simp add: insert(1) B(1))
haftmann@54744
   383
  qed
haftmann@54744
   384
  have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
haftmann@54744
   385
  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
   386
    using insert by simp
haftmann@54744
   387
  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
   388
  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
   389
    using insert by(simp add:sup_Inf1_distrib[OF B])
haftmann@54744
   390
  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
   391
    (is "_ = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n?M")
haftmann@54744
   392
    using B insert
haftmann@54744
   393
    by (simp add: Inf_fin.union [OF finB _ finAB ne])
haftmann@54744
   394
  also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"
haftmann@54744
   395
    by blast
haftmann@54744
   396
  finally show ?case .
haftmann@54744
   397
qed
haftmann@54744
   398
haftmann@54744
   399
lemma inf_Sup1_distrib:
haftmann@54744
   400
  assumes "finite A" and "A \<noteq> {}"
haftmann@54744
   401
  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
   402
using assms by (simp add: image_def Sup_fin.hom_commute [where h="inf x", OF inf_sup_distrib1])
haftmann@54744
   403
  (rule arg_cong [where f="Sup_fin"], blast)
haftmann@54744
   404
haftmann@54744
   405
lemma inf_Sup2_distrib:
haftmann@54744
   406
  assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
haftmann@54744
   407
  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
   408
using A proof (induct rule: finite_ne_induct)
haftmann@54744
   409
  case singleton thus ?case
haftmann@54744
   410
    by(simp add: inf_Sup1_distrib [OF B])
haftmann@54744
   411
next
haftmann@54744
   412
  case (insert x A)
haftmann@54744
   413
  have finB: "finite {inf x b |b. b \<in> B}"
haftmann@54744
   414
    by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto)
haftmann@54744
   415
  have finAB: "finite {inf a b |a b. a \<in> A \<and> b \<in> B}"
haftmann@54744
   416
  proof -
haftmann@54744
   417
    have "{inf a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {inf a b})"
haftmann@54744
   418
      by blast
haftmann@54744
   419
    thus ?thesis by(simp add: insert(1) B(1))
haftmann@54744
   420
  qed
haftmann@54744
   421
  have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
haftmann@54744
   422
  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
   423
    using insert by simp
haftmann@54744
   424
  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
   425
  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
   426
    using insert by(simp add:inf_Sup1_distrib[OF B])
haftmann@54744
   427
  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
   428
    (is "_ = \<Squnion>\<^sub>f\<^sub>i\<^sub>n?M")
haftmann@54744
   429
    using B insert
haftmann@54744
   430
    by (simp add: Sup_fin.union [OF finB _ finAB ne])
haftmann@54744
   431
  also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"
haftmann@54744
   432
    by blast
haftmann@54744
   433
  finally show ?case .
haftmann@54744
   434
qed
haftmann@54744
   435
haftmann@54744
   436
end
haftmann@54744
   437
haftmann@54744
   438
context complete_lattice
haftmann@54744
   439
begin
haftmann@54744
   440
haftmann@54744
   441
lemma Inf_fin_Inf:
haftmann@54744
   442
  assumes "finite A" and "A \<noteq> {}"
haftmann@54868
   443
  shows "\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA = \<Sqinter>A"
haftmann@54744
   444
proof -
haftmann@54744
   445
  from assms obtain b B where "A = insert b B" and "finite B" by auto
haftmann@54744
   446
  then show ?thesis
haftmann@54744
   447
    by (simp add: Inf_fin.eq_fold inf_Inf_fold_inf inf.commute [of b])
haftmann@54744
   448
qed
haftmann@54744
   449
haftmann@54744
   450
lemma Sup_fin_Sup:
haftmann@54744
   451
  assumes "finite A" and "A \<noteq> {}"
haftmann@54868
   452
  shows "\<Squnion>\<^sub>f\<^sub>i\<^sub>nA = \<Squnion>A"
haftmann@54744
   453
proof -
haftmann@54744
   454
  from assms obtain b B where "A = insert b B" and "finite B" by auto
haftmann@54744
   455
  then show ?thesis
haftmann@54744
   456
    by (simp add: Sup_fin.eq_fold sup_Sup_fold_sup sup.commute [of b])
haftmann@54744
   457
qed
haftmann@54744
   458
haftmann@54744
   459
end
haftmann@54744
   460
haftmann@54744
   461
wenzelm@60758
   462
subsection \<open>Minimum and Maximum over non-empty sets\<close>
haftmann@54744
   463
haftmann@54744
   464
context linorder
haftmann@54744
   465
begin
haftmann@54744
   466
wenzelm@61605
   467
sublocale Min: semilattice_order_set min less_eq less
wenzelm@61605
   468
  + Max: semilattice_order_set max greater_eq greater
haftmann@61776
   469
defines
haftmann@61776
   470
  Min = Min.F and Max = Max.F ..
haftmann@54864
   471
haftmann@54864
   472
end
haftmann@54864
   473
wenzelm@60758
   474
text \<open>An aside: @{const Min}/@{const Max} on linear orders as special case of @{const Inf_fin}/@{const Sup_fin}\<close>
haftmann@54864
   475
haftmann@54864
   476
lemma Inf_fin_Min:
haftmann@54864
   477
  "Inf_fin = (Min :: 'a::{semilattice_inf, linorder} set \<Rightarrow> 'a)"
haftmann@54864
   478
  by (simp add: Inf_fin_def Min_def inf_min)
haftmann@54864
   479
haftmann@54864
   480
lemma Sup_fin_Max:
haftmann@54864
   481
  "Sup_fin = (Max :: 'a::{semilattice_sup, linorder} set \<Rightarrow> 'a)"
haftmann@54864
   482
  by (simp add: Sup_fin_def Max_def sup_max)
haftmann@54864
   483
haftmann@54864
   484
context linorder
haftmann@54864
   485
begin
haftmann@54864
   486
haftmann@54744
   487
lemma dual_min:
haftmann@54744
   488
  "ord.min greater_eq = max"
haftmann@54744
   489
  by (auto simp add: ord.min_def max_def fun_eq_iff)
haftmann@54744
   490
haftmann@54744
   491
lemma dual_max:
haftmann@54744
   492
  "ord.max greater_eq = min"
haftmann@54744
   493
  by (auto simp add: ord.max_def min_def fun_eq_iff)
haftmann@54744
   494
haftmann@54744
   495
lemma dual_Min:
haftmann@54744
   496
  "linorder.Min greater_eq = Max"
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.Min_def dual_min Max_def)
haftmann@54744
   500
qed
haftmann@54744
   501
haftmann@54744
   502
lemma dual_Max:
haftmann@54744
   503
  "linorder.Max greater_eq = Min"
haftmann@54744
   504
proof -
wenzelm@61605
   505
  interpret dual: linorder greater_eq greater by (fact dual_linorder)
haftmann@54744
   506
  show ?thesis by (simp add: dual.Max_def dual_max Min_def)
haftmann@54744
   507
qed
haftmann@54744
   508
haftmann@54744
   509
lemmas Min_singleton = Min.singleton
haftmann@54744
   510
lemmas Max_singleton = Max.singleton
haftmann@54744
   511
lemmas Min_insert = Min.insert
haftmann@54744
   512
lemmas Max_insert = Max.insert
haftmann@54744
   513
lemmas Min_Un = Min.union
haftmann@54744
   514
lemmas Max_Un = Max.union
haftmann@54744
   515
lemmas hom_Min_commute = Min.hom_commute
haftmann@54744
   516
lemmas hom_Max_commute = Max.hom_commute
haftmann@54744
   517
haftmann@54744
   518
lemma Min_in [simp]:
haftmann@54744
   519
  assumes "finite A" and "A \<noteq> {}"
haftmann@54744
   520
  shows "Min A \<in> A"
haftmann@54744
   521
  using assms by (auto simp add: min_def Min.closed)
haftmann@54744
   522
haftmann@54744
   523
lemma Max_in [simp]:
haftmann@54744
   524
  assumes "finite A" and "A \<noteq> {}"
haftmann@54744
   525
  shows "Max A \<in> A"
haftmann@54744
   526
  using assms by (auto simp add: max_def Max.closed)
haftmann@54744
   527
haftmann@58467
   528
lemma Min_insert2:
haftmann@58467
   529
  assumes "finite A" and min: "\<And>b. b \<in> A \<Longrightarrow> a \<le> b"
haftmann@58467
   530
  shows "Min (insert a A) = a"
haftmann@58467
   531
proof (cases "A = {}")
haftmann@58467
   532
  case True then show ?thesis by simp
haftmann@58467
   533
next
wenzelm@60758
   534
  case False with \<open>finite A\<close> have "Min (insert a A) = min a (Min A)"
haftmann@58467
   535
    by simp
wenzelm@60758
   536
  moreover from \<open>finite A\<close> \<open>A \<noteq> {}\<close> min have "a \<le> Min A" by simp
haftmann@58467
   537
  ultimately show ?thesis by (simp add: min.absorb1)
haftmann@58467
   538
qed
haftmann@58467
   539
haftmann@58467
   540
lemma Max_insert2:
haftmann@58467
   541
  assumes "finite A" and max: "\<And>b. b \<in> A \<Longrightarrow> b \<le> a"
haftmann@58467
   542
  shows "Max (insert a A) = a"
haftmann@58467
   543
proof (cases "A = {}")
haftmann@58467
   544
  case True then show ?thesis by simp
haftmann@58467
   545
next
wenzelm@60758
   546
  case False with \<open>finite A\<close> have "Max (insert a A) = max a (Max A)"
haftmann@58467
   547
    by simp
wenzelm@60758
   548
  moreover from \<open>finite A\<close> \<open>A \<noteq> {}\<close> max have "Max A \<le> a" by simp
haftmann@58467
   549
  ultimately show ?thesis by (simp add: max.absorb1)
haftmann@58467
   550
qed
haftmann@58467
   551
haftmann@54744
   552
lemma Min_le [simp]:
haftmann@54744
   553
  assumes "finite A" and "x \<in> A"
haftmann@54744
   554
  shows "Min A \<le> x"
haftmann@54744
   555
  using assms by (fact Min.coboundedI)
haftmann@54744
   556
haftmann@54744
   557
lemma Max_ge [simp]:
haftmann@54744
   558
  assumes "finite A" and "x \<in> A"
haftmann@54744
   559
  shows "x \<le> Max A"
haftmann@54744
   560
  using assms by (fact Max.coboundedI)
haftmann@54744
   561
haftmann@54744
   562
lemma Min_eqI:
haftmann@54744
   563
  assumes "finite A"
haftmann@54744
   564
  assumes "\<And>y. y \<in> A \<Longrightarrow> y \<ge> x"
haftmann@54744
   565
    and "x \<in> A"
haftmann@54744
   566
  shows "Min A = x"
haftmann@54744
   567
proof (rule antisym)
wenzelm@60758
   568
  from \<open>x \<in> A\<close> have "A \<noteq> {}" by auto
haftmann@54744
   569
  with assms show "Min A \<ge> x" by simp
haftmann@54744
   570
next
haftmann@54744
   571
  from assms show "x \<ge> Min A" by simp
haftmann@54744
   572
qed
haftmann@54744
   573
haftmann@54744
   574
lemma Max_eqI:
haftmann@54744
   575
  assumes "finite A"
haftmann@54744
   576
  assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
haftmann@54744
   577
    and "x \<in> A"
haftmann@54744
   578
  shows "Max A = x"
haftmann@54744
   579
proof (rule antisym)
wenzelm@60758
   580
  from \<open>x \<in> A\<close> have "A \<noteq> {}" by auto
haftmann@54744
   581
  with assms show "Max A \<le> x" by simp
haftmann@54744
   582
next
haftmann@54744
   583
  from assms show "x \<le> Max A" by simp
haftmann@54744
   584
qed
haftmann@54744
   585
haftmann@54744
   586
context
haftmann@54744
   587
  fixes A :: "'a set"
haftmann@54744
   588
  assumes fin_nonempty: "finite A" "A \<noteq> {}"
haftmann@54744
   589
begin
haftmann@54744
   590
haftmann@54744
   591
lemma Min_ge_iff [simp]:
haftmann@54744
   592
  "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
haftmann@54744
   593
  using fin_nonempty by (fact Min.bounded_iff)
haftmann@54744
   594
haftmann@54744
   595
lemma Max_le_iff [simp]:
haftmann@54744
   596
  "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
haftmann@54744
   597
  using fin_nonempty by (fact Max.bounded_iff)
haftmann@54744
   598
haftmann@54744
   599
lemma Min_gr_iff [simp]:
haftmann@54744
   600
  "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
haftmann@54744
   601
  using fin_nonempty  by (induct rule: finite_ne_induct) simp_all
haftmann@54744
   602
haftmann@54744
   603
lemma Max_less_iff [simp]:
haftmann@54744
   604
  "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
haftmann@54744
   605
  using fin_nonempty by (induct rule: finite_ne_induct) simp_all
haftmann@54744
   606
haftmann@54744
   607
lemma Min_le_iff:
haftmann@54744
   608
  "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
haftmann@54744
   609
  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_le_iff_disj)
haftmann@54744
   610
haftmann@54744
   611
lemma Max_ge_iff:
haftmann@54744
   612
  "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
haftmann@54744
   613
  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: le_max_iff_disj)
haftmann@54744
   614
haftmann@54744
   615
lemma Min_less_iff:
haftmann@54744
   616
  "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
haftmann@54744
   617
  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_less_iff_disj)
haftmann@54744
   618
haftmann@54744
   619
lemma Max_gr_iff:
haftmann@54744
   620
  "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
haftmann@54744
   621
  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: less_max_iff_disj)
haftmann@54744
   622
haftmann@54744
   623
end
haftmann@54744
   624
nipkow@57800
   625
lemma Max_eq_if:
nipkow@57800
   626
  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
   627
  shows "Max A = Max B"
nipkow@57800
   628
proof cases
nipkow@57800
   629
  assume "A = {}" thus ?thesis using assms by simp
nipkow@57800
   630
next
nipkow@57800
   631
  assume "A \<noteq> {}" thus ?thesis using assms
nipkow@57800
   632
    by(blast intro: antisym Max_in Max_ge_iff[THEN iffD2])
nipkow@57800
   633
qed
nipkow@57800
   634
haftmann@54744
   635
lemma Min_antimono:
haftmann@54744
   636
  assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
haftmann@54744
   637
  shows "Min N \<le> Min M"
haftmann@54744
   638
  using assms by (fact Min.antimono)
haftmann@54744
   639
haftmann@54744
   640
lemma Max_mono:
haftmann@54744
   641
  assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
haftmann@54744
   642
  shows "Max M \<le> Max N"
haftmann@54744
   643
  using assms by (fact Max.antimono)
haftmann@54744
   644
wenzelm@56140
   645
end
wenzelm@56140
   646
wenzelm@56140
   647
context linorder  (* FIXME *)
wenzelm@56140
   648
begin
wenzelm@56140
   649
haftmann@54744
   650
lemma mono_Min_commute:
haftmann@54744
   651
  assumes "mono f"
haftmann@54744
   652
  assumes "finite A" and "A \<noteq> {}"
haftmann@54744
   653
  shows "f (Min A) = Min (f ` A)"
haftmann@54744
   654
proof (rule linorder_class.Min_eqI [symmetric])
wenzelm@60758
   655
  from \<open>finite A\<close> show "finite (f ` A)" by simp
haftmann@54744
   656
  from assms show "f (Min A) \<in> f ` A" by simp
haftmann@54744
   657
  fix x
haftmann@54744
   658
  assume "x \<in> f ` A"
haftmann@54744
   659
  then obtain y where "y \<in> A" and "x = f y" ..
haftmann@54744
   660
  with assms have "Min A \<le> y" by auto
wenzelm@60758
   661
  with \<open>mono f\<close> have "f (Min A) \<le> f y" by (rule monoE)
wenzelm@60758
   662
  with \<open>x = f y\<close> show "f (Min A) \<le> x" by simp
haftmann@54744
   663
qed
haftmann@54744
   664
haftmann@54744
   665
lemma mono_Max_commute:
haftmann@54744
   666
  assumes "mono f"
haftmann@54744
   667
  assumes "finite A" and "A \<noteq> {}"
haftmann@54744
   668
  shows "f (Max A) = Max (f ` A)"
haftmann@54744
   669
proof (rule linorder_class.Max_eqI [symmetric])
wenzelm@60758
   670
  from \<open>finite A\<close> show "finite (f ` A)" by simp
haftmann@54744
   671
  from assms show "f (Max A) \<in> f ` A" by simp
haftmann@54744
   672
  fix x
haftmann@54744
   673
  assume "x \<in> f ` A"
haftmann@54744
   674
  then obtain y where "y \<in> A" and "x = f y" ..
haftmann@54744
   675
  with assms have "y \<le> Max A" by auto
wenzelm@60758
   676
  with \<open>mono f\<close> have "f y \<le> f (Max A)" by (rule monoE)
wenzelm@60758
   677
  with \<open>x = f y\<close> show "x \<le> f (Max A)" by simp
haftmann@54744
   678
qed
haftmann@54744
   679
haftmann@54744
   680
lemma finite_linorder_max_induct [consumes 1, case_names empty insert]:
haftmann@54744
   681
  assumes fin: "finite A"
haftmann@54744
   682
  and empty: "P {}" 
haftmann@54744
   683
  and insert: "\<And>b A. finite A \<Longrightarrow> \<forall>a\<in>A. a < b \<Longrightarrow> P A \<Longrightarrow> P (insert b A)"
haftmann@54744
   684
  shows "P A"
haftmann@54744
   685
using fin empty insert
haftmann@54744
   686
proof (induct rule: finite_psubset_induct)
haftmann@54744
   687
  case (psubset A)
haftmann@54744
   688
  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
   689
  have fin: "finite A" by fact 
haftmann@54744
   690
  have empty: "P {}" by fact
haftmann@54744
   691
  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
   692
  show "P A"
haftmann@54744
   693
  proof (cases "A = {}")
haftmann@54744
   694
    assume "A = {}" 
wenzelm@60758
   695
    then show "P A" using \<open>P {}\<close> by simp
haftmann@54744
   696
  next
haftmann@54744
   697
    let ?B = "A - {Max A}" 
haftmann@54744
   698
    let ?A = "insert (Max A) ?B"
wenzelm@60758
   699
    have "finite ?B" using \<open>finite A\<close> by simp
haftmann@54744
   700
    assume "A \<noteq> {}"
wenzelm@60758
   701
    with \<open>finite A\<close> have "Max A : A" by auto
haftmann@54744
   702
    then have A: "?A = A" using insert_Diff_single insert_absorb by auto
wenzelm@60758
   703
    then have "P ?B" using \<open>P {}\<close> step IH [of ?B] by blast
haftmann@54744
   704
    moreover 
wenzelm@60758
   705
    have "\<forall>a\<in>?B. a < Max A" using Max_ge [OF \<open>finite A\<close>] by fastforce
wenzelm@60758
   706
    ultimately show "P A" using A insert_Diff_single step [OF \<open>finite ?B\<close>] by fastforce
haftmann@54744
   707
  qed
haftmann@54744
   708
qed
haftmann@54744
   709
haftmann@54744
   710
lemma finite_linorder_min_induct [consumes 1, case_names empty insert]:
haftmann@54744
   711
  "\<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
   712
  by (rule linorder.finite_linorder_max_induct [OF dual_linorder])
haftmann@54744
   713
haftmann@54744
   714
lemma Least_Min:
haftmann@54744
   715
  assumes "finite {a. P a}" and "\<exists>a. P a"
haftmann@54744
   716
  shows "(LEAST a. P a) = Min {a. P a}"
haftmann@54744
   717
proof -
haftmann@54744
   718
  { fix A :: "'a set"
haftmann@54744
   719
    assume A: "finite A" "A \<noteq> {}"
haftmann@54744
   720
    have "(LEAST a. a \<in> A) = Min A"
haftmann@54744
   721
    using A proof (induct A rule: finite_ne_induct)
haftmann@54744
   722
      case singleton show ?case by (rule Least_equality) simp_all
haftmann@54744
   723
    next
haftmann@54744
   724
      case (insert a A)
haftmann@54744
   725
      have "(LEAST b. b = a \<or> b \<in> A) = min a (LEAST a. a \<in> A)"
haftmann@54744
   726
        by (auto intro!: Least_equality simp add: min_def not_le Min_le_iff insert.hyps dest!: less_imp_le)
haftmann@54744
   727
      with insert show ?case by simp
haftmann@54744
   728
    qed
haftmann@54744
   729
  } from this [of "{a. P a}"] assms show ?thesis by simp
haftmann@54744
   730
qed
haftmann@54744
   731
hoelzl@59000
   732
lemma infinite_growing:
hoelzl@59000
   733
  assumes "X \<noteq> {}"
hoelzl@59000
   734
  assumes *: "\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>X. y > x"
hoelzl@59000
   735
  shows "\<not> finite X"
hoelzl@59000
   736
proof
hoelzl@59000
   737
  assume "finite X"
wenzelm@60758
   738
  with \<open>X \<noteq> {}\<close> have "Max X \<in> X" "\<forall>x\<in>X. x \<le> Max X"
hoelzl@59000
   739
    by auto
hoelzl@59000
   740
  with *[of "Max X"] show False
hoelzl@59000
   741
    by auto
hoelzl@59000
   742
qed
hoelzl@59000
   743
haftmann@54744
   744
end
haftmann@54744
   745
haftmann@54744
   746
context linordered_ab_semigroup_add
haftmann@54744
   747
begin
haftmann@54744
   748
haftmann@54744
   749
lemma add_Min_commute:
haftmann@54744
   750
  fixes k
haftmann@54744
   751
  assumes "finite N" and "N \<noteq> {}"
haftmann@54744
   752
  shows "k + Min N = Min {k + m | m. m \<in> N}"
haftmann@54744
   753
proof -
haftmann@54744
   754
  have "\<And>x y. k + min x y = min (k + x) (k + y)"
haftmann@54744
   755
    by (simp add: min_def not_le)
haftmann@54744
   756
      (blast intro: antisym less_imp_le add_left_mono)
haftmann@54744
   757
  with assms show ?thesis
haftmann@54744
   758
    using hom_Min_commute [of "plus k" N]
haftmann@54744
   759
    by simp (blast intro: arg_cong [where f = Min])
haftmann@54744
   760
qed
haftmann@54744
   761
haftmann@54744
   762
lemma add_Max_commute:
haftmann@54744
   763
  fixes k
haftmann@54744
   764
  assumes "finite N" and "N \<noteq> {}"
haftmann@54744
   765
  shows "k + Max N = Max {k + m | m. m \<in> N}"
haftmann@54744
   766
proof -
haftmann@54744
   767
  have "\<And>x y. k + max x y = max (k + x) (k + y)"
haftmann@54744
   768
    by (simp add: max_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_Max_commute [of "plus k" N]
haftmann@54744
   772
    by simp (blast intro: arg_cong [where f = Max])
haftmann@54744
   773
qed
haftmann@54744
   774
haftmann@54744
   775
end
haftmann@54744
   776
haftmann@54744
   777
context linordered_ab_group_add
haftmann@54744
   778
begin
haftmann@54744
   779
haftmann@54744
   780
lemma minus_Max_eq_Min [simp]:
haftmann@54744
   781
  "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Max S = Min (uminus ` S)"
haftmann@54744
   782
  by (induct S rule: finite_ne_induct) (simp_all add: minus_max_eq_min)
haftmann@54744
   783
haftmann@54744
   784
lemma minus_Min_eq_Max [simp]:
haftmann@54744
   785
  "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Min S = Max (uminus ` S)"
haftmann@54744
   786
  by (induct S rule: finite_ne_induct) (simp_all add: minus_min_eq_max)
haftmann@54744
   787
haftmann@54744
   788
end
haftmann@54744
   789
haftmann@54744
   790
context complete_linorder
haftmann@54744
   791
begin
haftmann@54744
   792
haftmann@54744
   793
lemma Min_Inf:
haftmann@54744
   794
  assumes "finite A" and "A \<noteq> {}"
haftmann@54744
   795
  shows "Min A = Inf A"
haftmann@54744
   796
proof -
haftmann@54744
   797
  from assms obtain b B where "A = insert b B" and "finite B" by auto
haftmann@54744
   798
  then show ?thesis
haftmann@54744
   799
    by (simp add: Min.eq_fold complete_linorder_inf_min [symmetric] inf_Inf_fold_inf inf.commute [of b])
haftmann@54744
   800
qed
haftmann@54744
   801
haftmann@54744
   802
lemma Max_Sup:
haftmann@54744
   803
  assumes "finite A" and "A \<noteq> {}"
haftmann@54744
   804
  shows "Max A = Sup A"
haftmann@54744
   805
proof -
haftmann@54744
   806
  from assms obtain b B where "A = insert b B" and "finite B" by auto
haftmann@54744
   807
  then show ?thesis
haftmann@54744
   808
    by (simp add: Max.eq_fold complete_linorder_sup_max [symmetric] sup_Sup_fold_sup sup.commute [of b])
haftmann@54744
   809
qed
haftmann@54744
   810
haftmann@54744
   811
end
haftmann@54744
   812
haftmann@54744
   813
end