author | paulson <lp15@cam.ac.uk> |
Wed, 17 Apr 2019 21:53:45 +0100 | |
changeset 70179 | 269dcea7426c |
parent 70178 | 4900351361b0 |
child 71813 | b11d7ffb48e0 |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Infinite_Set.thy |
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Author: Stephan Merz |
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*) |
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section \<open>Infinite Sets and Related Concepts\<close> |
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theory Infinite_Set |
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imports Main |
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begin |
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subsection \<open>The set of natural numbers is infinite\<close> |
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lemma infinite_nat_iff_unbounded_le: "infinite S \<longleftrightarrow> (\<forall>m. \<exists>n\<ge>m. n \<in> S)" |
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for S :: "nat set" |
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using frequently_cofinite[of "\<lambda>x. x \<in> S"] |
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by (simp add: cofinite_eq_sequentially frequently_def eventually_sequentially) |
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lemma infinite_nat_iff_unbounded: "infinite S \<longleftrightarrow> (\<forall>m. \<exists>n>m. n \<in> S)" |
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for S :: "nat set" |
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using frequently_cofinite[of "\<lambda>x. x \<in> S"] |
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by (simp add: cofinite_eq_sequentially frequently_def eventually_at_top_dense) |
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lemma finite_nat_iff_bounded: "finite S \<longleftrightarrow> (\<exists>k. S \<subseteq> {..<k})" |
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for S :: "nat set" |
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using infinite_nat_iff_unbounded_le[of S] by (simp add: subset_eq) (metis not_le) |
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lemma finite_nat_iff_bounded_le: "finite S \<longleftrightarrow> (\<exists>k. S \<subseteq> {.. k})" |
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for S :: "nat set" |
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using infinite_nat_iff_unbounded[of S] by (simp add: subset_eq) (metis not_le) |
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lemma finite_nat_bounded: "finite S \<Longrightarrow> \<exists>k. S \<subseteq> {..<k}" |
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for S :: "nat set" |
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by (simp add: finite_nat_iff_bounded) |
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text \<open> |
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For a set of natural numbers to be infinite, it is enough to know |
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that for any number larger than some \<open>k\<close>, there is some larger |
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number that is an element of the set. |
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\<close> |
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lemma unbounded_k_infinite: "\<forall>m>k. \<exists>n>m. n \<in> S \<Longrightarrow> infinite (S::nat set)" |
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apply (clarsimp simp add: finite_nat_set_iff_bounded) |
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apply (drule_tac x="Suc (max m k)" in spec) |
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using less_Suc_eq apply fastforce |
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done |
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lemma nat_not_finite: "finite (UNIV::nat set) \<Longrightarrow> R" |
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by simp |
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lemma range_inj_infinite: |
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fixes f :: "nat \<Rightarrow> 'a" |
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assumes "inj f" |
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shows "infinite (range f)" |
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proof |
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assume "finite (range f)" |
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from this assms have "finite (UNIV::nat set)" |
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by (rule finite_imageD) |
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then show False by simp |
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qed |
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subsection \<open>The set of integers is also infinite\<close> |
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lemma infinite_int_iff_infinite_nat_abs: "infinite S \<longleftrightarrow> infinite ((nat \<circ> abs) ` S)" |
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for S :: "int set" |
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proof (unfold Not_eq_iff, rule iffI) |
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assume "finite ((nat \<circ> abs) ` S)" |
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then have "finite (nat ` (abs ` S))" |
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by (simp add: image_image cong: image_cong) |
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moreover have "inj_on nat (abs ` S)" |
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by (rule inj_onI) auto |
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ultimately have "finite (abs ` S)" |
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by (rule finite_imageD) |
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then show "finite S" |
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by (rule finite_image_absD) |
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qed simp |
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proposition infinite_int_iff_unbounded_le: "infinite S \<longleftrightarrow> (\<forall>m. \<exists>n. \<bar>n\<bar> \<ge> m \<and> n \<in> S)" |
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for S :: "int set" |
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by (simp add: infinite_int_iff_infinite_nat_abs infinite_nat_iff_unbounded_le o_def image_def) |
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(metis abs_ge_zero nat_le_eq_zle le_nat_iff) |
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proposition infinite_int_iff_unbounded: "infinite S \<longleftrightarrow> (\<forall>m. \<exists>n. \<bar>n\<bar> > m \<and> n \<in> S)" |
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for S :: "int set" |
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by (simp add: infinite_int_iff_infinite_nat_abs infinite_nat_iff_unbounded o_def image_def) |
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(metis (full_types) nat_le_iff nat_mono not_le) |
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proposition finite_int_iff_bounded: "finite S \<longleftrightarrow> (\<exists>k. abs ` S \<subseteq> {..<k})" |
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for S :: "int set" |
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using infinite_int_iff_unbounded_le[of S] by (simp add: subset_eq) (metis not_le) |
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proposition finite_int_iff_bounded_le: "finite S \<longleftrightarrow> (\<exists>k. abs ` S \<subseteq> {.. k})" |
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for S :: "int set" |
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using infinite_int_iff_unbounded[of S] by (simp add: subset_eq) (metis not_le) |
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subsection \<open>Infinitely Many and Almost All\<close> |
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text \<open> |
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We often need to reason about the existence of infinitely many |
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(resp., all but finitely many) objects satisfying some predicate, so |
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we introduce corresponding binders and their proof rules. |
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\<close> |
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lemma not_INFM [simp]: "\<not> (INFM x. P x) \<longleftrightarrow> (MOST x. \<not> P x)" |
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by (rule not_frequently) |
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lemma not_MOST [simp]: "\<not> (MOST x. P x) \<longleftrightarrow> (INFM x. \<not> P x)" |
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by (rule not_eventually) |
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lemma INFM_const [simp]: "(INFM x::'a. P) \<longleftrightarrow> P \<and> infinite (UNIV::'a set)" |
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by (simp add: frequently_const_iff) |
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lemma MOST_const [simp]: "(MOST x::'a. P) \<longleftrightarrow> P \<or> finite (UNIV::'a set)" |
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by (simp add: eventually_const_iff) |
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lemma INFM_imp_distrib: "(INFM x. P x \<longrightarrow> Q x) \<longleftrightarrow> ((MOST x. P x) \<longrightarrow> (INFM x. Q x))" |
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by (rule frequently_imp_iff) |
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lemma MOST_imp_iff: "MOST x. P x \<Longrightarrow> (MOST x. P x \<longrightarrow> Q x) \<longleftrightarrow> (MOST x. Q x)" |
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by (auto intro: eventually_rev_mp eventually_mono) |
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lemma INFM_conjI: "INFM x. P x \<Longrightarrow> MOST x. Q x \<Longrightarrow> INFM x. P x \<and> Q x" |
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by (rule frequently_rev_mp[of P]) (auto elim: eventually_mono) |
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text \<open>Properties of quantifiers with injective functions.\<close> |
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lemma INFM_inj: "INFM x. P (f x) \<Longrightarrow> inj f \<Longrightarrow> INFM x. P x" |
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using finite_vimageI[of "{x. P x}" f] by (auto simp: frequently_cofinite) |
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lemma MOST_inj: "MOST x. P x \<Longrightarrow> inj f \<Longrightarrow> MOST x. P (f x)" |
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using finite_vimageI[of "{x. \<not> P x}" f] by (auto simp: eventually_cofinite) |
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text \<open>Properties of quantifiers with singletons.\<close> |
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lemma not_INFM_eq [simp]: |
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"\<not> (INFM x. x = a)" |
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"\<not> (INFM x. a = x)" |
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unfolding frequently_cofinite by simp_all |
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lemma MOST_neq [simp]: |
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"MOST x. x \<noteq> a" |
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"MOST x. a \<noteq> x" |
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unfolding eventually_cofinite by simp_all |
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lemma INFM_neq [simp]: |
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"(INFM x::'a. x \<noteq> a) \<longleftrightarrow> infinite (UNIV::'a set)" |
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"(INFM x::'a. a \<noteq> x) \<longleftrightarrow> infinite (UNIV::'a set)" |
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unfolding frequently_cofinite by simp_all |
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lemma MOST_eq [simp]: |
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"(MOST x::'a. x = a) \<longleftrightarrow> finite (UNIV::'a set)" |
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"(MOST x::'a. a = x) \<longleftrightarrow> finite (UNIV::'a set)" |
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unfolding eventually_cofinite by simp_all |
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lemma MOST_eq_imp: |
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"MOST x. x = a \<longrightarrow> P x" |
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"MOST x. a = x \<longrightarrow> P x" |
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unfolding eventually_cofinite by simp_all |
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text \<open>Properties of quantifiers over the naturals.\<close> |
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lemma MOST_nat: "(\<forall>\<^sub>\<infinity>n. P n) \<longleftrightarrow> (\<exists>m. \<forall>n>m. P n)" |
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for P :: "nat \<Rightarrow> bool" |
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by (auto simp add: eventually_cofinite finite_nat_iff_bounded_le subset_eq simp flip: not_le) |
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lemma MOST_nat_le: "(\<forall>\<^sub>\<infinity>n. P n) \<longleftrightarrow> (\<exists>m. \<forall>n\<ge>m. P n)" |
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for P :: "nat \<Rightarrow> bool" |
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by (auto simp add: eventually_cofinite finite_nat_iff_bounded subset_eq simp flip: not_le) |
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lemma INFM_nat: "(\<exists>\<^sub>\<infinity>n. P n) \<longleftrightarrow> (\<forall>m. \<exists>n>m. P n)" |
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for P :: "nat \<Rightarrow> bool" |
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by (simp add: frequently_cofinite infinite_nat_iff_unbounded) |
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lemma INFM_nat_le: "(\<exists>\<^sub>\<infinity>n. P n) \<longleftrightarrow> (\<forall>m. \<exists>n\<ge>m. P n)" |
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for P :: "nat \<Rightarrow> bool" |
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by (simp add: frequently_cofinite infinite_nat_iff_unbounded_le) |
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lemma MOST_INFM: "infinite (UNIV::'a set) \<Longrightarrow> MOST x::'a. P x \<Longrightarrow> INFM x::'a. P x" |
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by (simp add: eventually_frequently) |
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lemma MOST_Suc_iff: "(MOST n. P (Suc n)) \<longleftrightarrow> (MOST n. P n)" |
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by (simp add: cofinite_eq_sequentially) |
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lemma MOST_SucI: "MOST n. P n \<Longrightarrow> MOST n. P (Suc n)" |
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and MOST_SucD: "MOST n. P (Suc n) \<Longrightarrow> MOST n. P n" |
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by (simp_all add: MOST_Suc_iff) |
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lemma MOST_ge_nat: "MOST n::nat. m \<le> n" |
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by (simp add: cofinite_eq_sequentially) |
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\<comment> \<open>legacy names\<close> |
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lemma Inf_many_def: "Inf_many P \<longleftrightarrow> infinite {x. P x}" by (fact frequently_cofinite) |
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lemma Alm_all_def: "Alm_all P \<longleftrightarrow> \<not> (INFM x. \<not> P x)" by simp |
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lemma INFM_iff_infinite: "(INFM x. P x) \<longleftrightarrow> infinite {x. P x}" by (fact frequently_cofinite) |
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lemma MOST_iff_cofinite: "(MOST x. P x) \<longleftrightarrow> finite {x. \<not> P x}" by (fact eventually_cofinite) |
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lemma INFM_EX: "(\<exists>\<^sub>\<infinity>x. P x) \<Longrightarrow> (\<exists>x. P x)" by (fact frequently_ex) |
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202 |
lemma ALL_MOST: "\<forall>x. P x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. P x" by (fact always_eventually) |
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203 |
lemma INFM_mono: "\<exists>\<^sub>\<infinity>x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> \<exists>\<^sub>\<infinity>x. Q x" by (fact frequently_elim1) |
61810 | 204 |
lemma MOST_mono: "\<forall>\<^sub>\<infinity>x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> \<forall>\<^sub>\<infinity>x. Q x" by (fact eventually_mono) |
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lemma INFM_disj_distrib: "(\<exists>\<^sub>\<infinity>x. P x \<or> Q x) \<longleftrightarrow> (\<exists>\<^sub>\<infinity>x. P x) \<or> (\<exists>\<^sub>\<infinity>x. Q x)" by (fact frequently_disj_iff) |
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206 |
lemma MOST_rev_mp: "\<forall>\<^sub>\<infinity>x. P x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. P x \<longrightarrow> Q x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. Q x" by (fact eventually_rev_mp) |
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207 |
lemma MOST_conj_distrib: "(\<forall>\<^sub>\<infinity>x. P x \<and> Q x) \<longleftrightarrow> (\<forall>\<^sub>\<infinity>x. P x) \<and> (\<forall>\<^sub>\<infinity>x. Q x)" by (fact eventually_conj_iff) |
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208 |
lemma MOST_conjI: "MOST x. P x \<Longrightarrow> MOST x. Q x \<Longrightarrow> MOST x. P x \<and> Q x" by (fact eventually_conj) |
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209 |
lemma INFM_finite_Bex_distrib: "finite A \<Longrightarrow> (INFM y. \<exists>x\<in>A. P x y) \<longleftrightarrow> (\<exists>x\<in>A. INFM y. P x y)" by (fact frequently_bex_finite_distrib) |
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210 |
lemma MOST_finite_Ball_distrib: "finite A \<Longrightarrow> (MOST y. \<forall>x\<in>A. P x y) \<longleftrightarrow> (\<forall>x\<in>A. MOST y. P x y)" by (fact eventually_ball_finite_distrib) |
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211 |
lemma INFM_E: "INFM x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> thesis) \<Longrightarrow> thesis" by (fact frequentlyE) |
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212 |
lemma MOST_I: "(\<And>x. P x) \<Longrightarrow> MOST x. P x" by (rule eventuallyI) |
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213 |
lemmas MOST_iff_finiteNeg = MOST_iff_cofinite |
20809 | 214 |
|
215 |
||
64967 | 216 |
subsection \<open>Enumeration of an Infinite Set\<close> |
217 |
||
218 |
text \<open>The set's element type must be wellordered (e.g. the natural numbers).\<close> |
|
20809 | 219 |
|
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220 |
text \<open> |
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221 |
Could be generalized to |
69593 | 222 |
\<^prop>\<open>enumerate' S n = (SOME t. t \<in> s \<and> finite {s\<in>S. s < t} \<and> card {s\<in>S. s < t} = n)\<close>. |
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223 |
\<close> |
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224 |
|
53239 | 225 |
primrec (in wellorder) enumerate :: "'a set \<Rightarrow> nat \<Rightarrow> 'a" |
64967 | 226 |
where |
227 |
enumerate_0: "enumerate S 0 = (LEAST n. n \<in> S)" |
|
228 |
| enumerate_Suc: "enumerate S (Suc n) = enumerate (S - {LEAST n. n \<in> S}) n" |
|
20809 | 229 |
|
53239 | 230 |
lemma enumerate_Suc': "enumerate S (Suc n) = enumerate (S - {enumerate S 0}) n" |
20809 | 231 |
by simp |
232 |
||
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233 |
lemma enumerate_in_set: "infinite S \<Longrightarrow> enumerate S n \<in> S" |
64967 | 234 |
proof (induct n arbitrary: S) |
235 |
case 0 |
|
236 |
then show ?case |
|
237 |
by (fastforce intro: LeastI dest!: infinite_imp_nonempty) |
|
238 |
next |
|
239 |
case (Suc n) |
|
240 |
then show ?case |
|
241 |
by simp (metis DiffE infinite_remove) |
|
242 |
qed |
|
20809 | 243 |
|
244 |
declare enumerate_0 [simp del] enumerate_Suc [simp del] |
|
245 |
||
246 |
lemma enumerate_step: "infinite S \<Longrightarrow> enumerate S n < enumerate S (Suc n)" |
|
247 |
apply (induct n arbitrary: S) |
|
248 |
apply (rule order_le_neq_trans) |
|
249 |
apply (simp add: enumerate_0 Least_le enumerate_in_set) |
|
250 |
apply (simp only: enumerate_Suc') |
|
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251 |
apply (subgoal_tac "enumerate (S - {enumerate S 0}) 0 \<in> S - {enumerate S 0}") |
20809 | 252 |
apply (blast intro: sym) |
253 |
apply (simp add: enumerate_in_set del: Diff_iff) |
|
254 |
apply (simp add: enumerate_Suc') |
|
255 |
done |
|
256 |
||
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257 |
lemma enumerate_mono: "m < n \<Longrightarrow> infinite S \<Longrightarrow> enumerate S m < enumerate S n" |
64967 | 258 |
by (induct m n rule: less_Suc_induct) (auto intro: enumerate_step) |
20809 | 259 |
|
50134 | 260 |
lemma le_enumerate: |
261 |
assumes S: "infinite S" |
|
262 |
shows "n \<le> enumerate S n" |
|
61810 | 263 |
using S |
50134 | 264 |
proof (induct n) |
53239 | 265 |
case 0 |
266 |
then show ?case by simp |
|
267 |
next |
|
50134 | 268 |
case (Suc n) |
269 |
then have "n \<le> enumerate S n" by simp |
|
60500 | 270 |
also note enumerate_mono[of n "Suc n", OF _ \<open>infinite S\<close>] |
50134 | 271 |
finally show ?case by simp |
53239 | 272 |
qed |
50134 | 273 |
|
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274 |
lemma infinite_enumerate: |
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275 |
assumes fS: "infinite S" |
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276 |
shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (\<forall>n. r n \<in> S)" |
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277 |
unfolding strict_mono_def |
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using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto |
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279 |
|
50134 | 280 |
lemma enumerate_Suc'': |
281 |
fixes S :: "'a::wellorder set" |
|
53239 | 282 |
assumes "infinite S" |
283 |
shows "enumerate S (Suc n) = (LEAST s. s \<in> S \<and> enumerate S n < s)" |
|
284 |
using assms |
|
50134 | 285 |
proof (induct n arbitrary: S) |
286 |
case 0 |
|
53239 | 287 |
then have "\<forall>s \<in> S. enumerate S 0 \<le> s" |
50134 | 288 |
by (auto simp: enumerate.simps intro: Least_le) |
289 |
then show ?case |
|
290 |
unfolding enumerate_Suc' enumerate_0[of "S - {enumerate S 0}"] |
|
53239 | 291 |
by (intro arg_cong[where f = Least] ext) auto |
50134 | 292 |
next |
293 |
case (Suc n S) |
|
294 |
show ?case |
|
60500 | 295 |
using enumerate_mono[OF zero_less_Suc \<open>infinite S\<close>, of n] \<open>infinite S\<close> |
50134 | 296 |
apply (subst (1 2) enumerate_Suc') |
297 |
apply (subst Suc) |
|
64967 | 298 |
apply (use \<open>infinite S\<close> in simp) |
53239 | 299 |
apply (intro arg_cong[where f = Least] ext) |
68406 | 300 |
apply (auto simp flip: enumerate_Suc') |
53239 | 301 |
done |
50134 | 302 |
qed |
303 |
||
304 |
lemma enumerate_Ex: |
|
64967 | 305 |
fixes S :: "nat set" |
306 |
assumes S: "infinite S" |
|
307 |
and s: "s \<in> S" |
|
308 |
shows "\<exists>n. enumerate S n = s" |
|
309 |
using s |
|
50134 | 310 |
proof (induct s rule: less_induct) |
311 |
case (less s) |
|
312 |
show ?case |
|
64967 | 313 |
proof (cases "\<exists>y\<in>S. y < s") |
314 |
case True |
|
50134 | 315 |
let ?y = "Max {s'\<in>S. s' < s}" |
64967 | 316 |
from True have y: "\<And>x. ?y < x \<longleftrightarrow> (\<forall>s'\<in>S. s' < s \<longrightarrow> s' < x)" |
53239 | 317 |
by (subst Max_less_iff) auto |
318 |
then have y_in: "?y \<in> {s'\<in>S. s' < s}" |
|
319 |
by (intro Max_in) auto |
|
320 |
with less.hyps[of ?y] obtain n where "enumerate S n = ?y" |
|
321 |
by auto |
|
50134 | 322 |
with S have "enumerate S (Suc n) = s" |
323 |
by (auto simp: y less enumerate_Suc'' intro!: Least_equality) |
|
64967 | 324 |
then show ?thesis by auto |
50134 | 325 |
next |
64967 | 326 |
case False |
50134 | 327 |
then have "\<forall>t\<in>S. s \<le> t" by auto |
60500 | 328 |
with \<open>s \<in> S\<close> show ?thesis |
50134 | 329 |
by (auto intro!: exI[of _ 0] Least_equality simp: enumerate_0) |
330 |
qed |
|
331 |
qed |
|
332 |
||
333 |
lemma bij_enumerate: |
|
334 |
fixes S :: "nat set" |
|
335 |
assumes S: "infinite S" |
|
336 |
shows "bij_betw (enumerate S) UNIV S" |
|
337 |
proof - |
|
338 |
have "\<And>n m. n \<noteq> m \<Longrightarrow> enumerate S n \<noteq> enumerate S m" |
|
60500 | 339 |
using enumerate_mono[OF _ \<open>infinite S\<close>] by (auto simp: neq_iff) |
50134 | 340 |
then have "inj (enumerate S)" |
341 |
by (auto simp: inj_on_def) |
|
53239 | 342 |
moreover have "\<forall>s \<in> S. \<exists>i. enumerate S i = s" |
50134 | 343 |
using enumerate_Ex[OF S] by auto |
60500 | 344 |
moreover note \<open>infinite S\<close> |
50134 | 345 |
ultimately show ?thesis |
346 |
unfolding bij_betw_def by (auto intro: enumerate_in_set) |
|
347 |
qed |
|
348 |
||
64967 | 349 |
text \<open>A pair of weird and wonderful lemmas from HOL Light.\<close> |
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|
350 |
lemma finite_transitivity_chain: |
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|
351 |
assumes "finite A" |
64967 | 352 |
and R: "\<And>x. \<not> R x x" "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z" |
353 |
and A: "\<And>x. x \<in> A \<Longrightarrow> \<exists>y. y \<in> A \<and> R x y" |
|
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|
354 |
shows "A = {}" |
64967 | 355 |
using \<open>finite A\<close> A |
356 |
proof (induct A) |
|
357 |
case empty |
|
358 |
then show ?case by simp |
|
359 |
next |
|
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diff
changeset
|
360 |
case (insert a A) |
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More advanced theorems about retracts, homotopies., etc
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diff
changeset
|
361 |
with R show ?case |
64967 | 362 |
by (metis empty_iff insert_iff) (* somewhat slow *) |
363 |
qed |
|
63492
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diff
changeset
|
364 |
|
a662e8139804
More advanced theorems about retracts, homotopies., etc
paulson <lp15@cam.ac.uk>
parents:
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diff
changeset
|
365 |
corollary Union_maximal_sets: |
a662e8139804
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changeset
|
366 |
assumes "finite \<F>" |
64967 | 367 |
shows "\<Union>{T \<in> \<F>. \<forall>U\<in>\<F>. \<not> T \<subset> U} = \<Union>\<F>" |
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changeset
|
368 |
(is "?lhs = ?rhs") |
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|
369 |
proof |
64967 | 370 |
show "?lhs \<subseteq> ?rhs" by force |
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changeset
|
371 |
show "?rhs \<subseteq> ?lhs" |
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changeset
|
372 |
proof (rule Union_subsetI) |
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changeset
|
373 |
fix S |
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diff
changeset
|
374 |
assume "S \<in> \<F>" |
64967 | 375 |
have "{T \<in> \<F>. S \<subseteq> T} = {}" |
376 |
if "\<not> (\<exists>y. y \<in> {T \<in> \<F>. \<forall>U\<in>\<F>. \<not> T \<subset> U} \<and> S \<subseteq> y)" |
|
63492
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diff
changeset
|
377 |
apply (rule finite_transitivity_chain [of _ "\<lambda>T U. S \<subseteq> T \<and> T \<subset> U"]) |
64967 | 378 |
apply (use assms that in auto) |
379 |
apply (blast intro: dual_order.trans psubset_imp_subset) |
|
380 |
done |
|
381 |
with \<open>S \<in> \<F>\<close> show "\<exists>y. y \<in> {T \<in> \<F>. \<forall>U\<in>\<F>. \<not> T \<subset> U} \<and> S \<subseteq> y" |
|
382 |
by blast |
|
63492
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diff
changeset
|
383 |
qed |
64967 | 384 |
qed |
63492
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diff
changeset
|
385 |
|
20809 | 386 |
end |