author | wenzelm |
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parent 72302 | d7d90ed4c74e |
child 75607 | 3c544d64c218 |
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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||
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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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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) |
60040
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diff
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|
205 |
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) |
1fa1023b13b9
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59506
diff
changeset
|
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) |
1fa1023b13b9
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parents:
59506
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changeset
|
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) |
1fa1023b13b9
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parents:
59506
diff
changeset
|
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) |
1fa1023b13b9
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parents:
59506
diff
changeset
|
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) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
59506
diff
changeset
|
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) |
1fa1023b13b9
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parents:
59506
diff
changeset
|
211 |
lemma INFM_E: "INFM x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> thesis) \<Longrightarrow> thesis" by (fact frequentlyE) |
1fa1023b13b9
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changeset
|
212 |
lemma MOST_I: "(\<And>x. P x) \<Longrightarrow> MOST x. P x" by (rule eventuallyI) |
1fa1023b13b9
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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 |
|
60040
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|
220 |
text \<open> |
1fa1023b13b9
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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>. |
60040
1fa1023b13b9
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changeset
|
223 |
\<close> |
1fa1023b13b9
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changeset
|
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 |
||
60040
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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)" |
|
71813 | 247 |
proof (induction n arbitrary: S) |
248 |
case 0 |
|
249 |
then have "enumerate S 0 \<le> enumerate S (Suc 0)" |
|
250 |
by (simp add: enumerate_0 Least_le enumerate_in_set) |
|
251 |
moreover have "enumerate (S - {enumerate S 0}) 0 \<in> S - {enumerate S 0}" |
|
252 |
by (meson "0.prems" enumerate_in_set infinite_remove) |
|
253 |
then have "enumerate S 0 \<noteq> enumerate (S - {enumerate S 0}) 0" |
|
254 |
by auto |
|
255 |
ultimately show ?case |
|
256 |
by (simp add: enumerate_Suc') |
|
257 |
next |
|
258 |
case (Suc n) |
|
259 |
then show ?case |
|
260 |
by (simp add: enumerate_Suc') |
|
261 |
qed |
|
20809 | 262 |
|
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
59506
diff
changeset
|
263 |
lemma enumerate_mono: "m < n \<Longrightarrow> infinite S \<Longrightarrow> enumerate S m < enumerate S n" |
64967 | 264 |
by (induct m n rule: less_Suc_induct) (auto intro: enumerate_step) |
20809 | 265 |
|
72095
cfb6c22a5636
lemmas about sets and the enumerate operator
paulson <lp15@cam.ac.uk>
parents:
72090
diff
changeset
|
266 |
lemma enumerate_mono_iff [simp]: |
cfb6c22a5636
lemmas about sets and the enumerate operator
paulson <lp15@cam.ac.uk>
parents:
72090
diff
changeset
|
267 |
"infinite S \<Longrightarrow> enumerate S m < enumerate S n \<longleftrightarrow> m < n" |
cfb6c22a5636
lemmas about sets and the enumerate operator
paulson <lp15@cam.ac.uk>
parents:
72090
diff
changeset
|
268 |
by (metis enumerate_mono less_asym less_linear) |
cfb6c22a5636
lemmas about sets and the enumerate operator
paulson <lp15@cam.ac.uk>
parents:
72090
diff
changeset
|
269 |
|
50134 | 270 |
lemma le_enumerate: |
271 |
assumes S: "infinite S" |
|
272 |
shows "n \<le> enumerate S n" |
|
61810 | 273 |
using S |
50134 | 274 |
proof (induct n) |
53239 | 275 |
case 0 |
276 |
then show ?case by simp |
|
277 |
next |
|
50134 | 278 |
case (Suc n) |
279 |
then have "n \<le> enumerate S n" by simp |
|
60500 | 280 |
also note enumerate_mono[of n "Suc n", OF _ \<open>infinite S\<close>] |
50134 | 281 |
finally show ?case by simp |
53239 | 282 |
qed |
50134 | 283 |
|
69516
09bb8f470959
most of Topology_Euclidean_Space (now Elementary_Topology) requires fewer dependencies
immler
parents:
68406
diff
changeset
|
284 |
lemma infinite_enumerate: |
09bb8f470959
most of Topology_Euclidean_Space (now Elementary_Topology) requires fewer dependencies
immler
parents:
68406
diff
changeset
|
285 |
assumes fS: "infinite S" |
09bb8f470959
most of Topology_Euclidean_Space (now Elementary_Topology) requires fewer dependencies
immler
parents:
68406
diff
changeset
|
286 |
shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (\<forall>n. r n \<in> S)" |
09bb8f470959
most of Topology_Euclidean_Space (now Elementary_Topology) requires fewer dependencies
immler
parents:
68406
diff
changeset
|
287 |
unfolding strict_mono_def |
72095
cfb6c22a5636
lemmas about sets and the enumerate operator
paulson <lp15@cam.ac.uk>
parents:
72090
diff
changeset
|
288 |
using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by blast |
69516
09bb8f470959
most of Topology_Euclidean_Space (now Elementary_Topology) requires fewer dependencies
immler
parents:
68406
diff
changeset
|
289 |
|
50134 | 290 |
lemma enumerate_Suc'': |
291 |
fixes S :: "'a::wellorder set" |
|
53239 | 292 |
assumes "infinite S" |
293 |
shows "enumerate S (Suc n) = (LEAST s. s \<in> S \<and> enumerate S n < s)" |
|
294 |
using assms |
|
50134 | 295 |
proof (induct n arbitrary: S) |
296 |
case 0 |
|
53239 | 297 |
then have "\<forall>s \<in> S. enumerate S 0 \<le> s" |
50134 | 298 |
by (auto simp: enumerate.simps intro: Least_le) |
299 |
then show ?case |
|
300 |
unfolding enumerate_Suc' enumerate_0[of "S - {enumerate S 0}"] |
|
53239 | 301 |
by (intro arg_cong[where f = Least] ext) auto |
50134 | 302 |
next |
303 |
case (Suc n S) |
|
304 |
show ?case |
|
60500 | 305 |
using enumerate_mono[OF zero_less_Suc \<open>infinite S\<close>, of n] \<open>infinite S\<close> |
50134 | 306 |
apply (subst (1 2) enumerate_Suc') |
307 |
apply (subst Suc) |
|
64967 | 308 |
apply (use \<open>infinite S\<close> in simp) |
53239 | 309 |
apply (intro arg_cong[where f = Least] ext) |
68406 | 310 |
apply (auto simp flip: enumerate_Suc') |
53239 | 311 |
done |
50134 | 312 |
qed |
313 |
||
314 |
lemma enumerate_Ex: |
|
64967 | 315 |
fixes S :: "nat set" |
316 |
assumes S: "infinite S" |
|
317 |
and s: "s \<in> S" |
|
318 |
shows "\<exists>n. enumerate S n = s" |
|
319 |
using s |
|
50134 | 320 |
proof (induct s rule: less_induct) |
321 |
case (less s) |
|
322 |
show ?case |
|
64967 | 323 |
proof (cases "\<exists>y\<in>S. y < s") |
324 |
case True |
|
50134 | 325 |
let ?y = "Max {s'\<in>S. s' < s}" |
64967 | 326 |
from True have y: "\<And>x. ?y < x \<longleftrightarrow> (\<forall>s'\<in>S. s' < s \<longrightarrow> s' < x)" |
53239 | 327 |
by (subst Max_less_iff) auto |
328 |
then have y_in: "?y \<in> {s'\<in>S. s' < s}" |
|
329 |
by (intro Max_in) auto |
|
330 |
with less.hyps[of ?y] obtain n where "enumerate S n = ?y" |
|
331 |
by auto |
|
50134 | 332 |
with S have "enumerate S (Suc n) = s" |
333 |
by (auto simp: y less enumerate_Suc'' intro!: Least_equality) |
|
64967 | 334 |
then show ?thesis by auto |
50134 | 335 |
next |
64967 | 336 |
case False |
50134 | 337 |
then have "\<forall>t\<in>S. s \<le> t" by auto |
60500 | 338 |
with \<open>s \<in> S\<close> show ?thesis |
50134 | 339 |
by (auto intro!: exI[of _ 0] Least_equality simp: enumerate_0) |
340 |
qed |
|
341 |
qed |
|
342 |
||
71813 | 343 |
lemma inj_enumerate: |
344 |
fixes S :: "'a::wellorder set" |
|
345 |
assumes S: "infinite S" |
|
346 |
shows "inj (enumerate S)" |
|
347 |
unfolding inj_on_def |
|
348 |
proof clarsimp |
|
349 |
show "\<And>x y. enumerate S x = enumerate S y \<Longrightarrow> x = y" |
|
350 |
by (metis neq_iff enumerate_mono[OF _ \<open>infinite S\<close>]) |
|
351 |
qed |
|
352 |
||
353 |
text \<open>To generalise this, we'd need a condition that all initial segments were finite\<close> |
|
50134 | 354 |
lemma bij_enumerate: |
355 |
fixes S :: "nat set" |
|
356 |
assumes S: "infinite S" |
|
357 |
shows "bij_betw (enumerate S) UNIV S" |
|
358 |
proof - |
|
71813 | 359 |
have "\<forall>s \<in> S. \<exists>i. enumerate S i = s" |
50134 | 360 |
using enumerate_Ex[OF S] by auto |
71813 | 361 |
moreover note \<open>infinite S\<close> inj_enumerate |
50134 | 362 |
ultimately show ?thesis |
363 |
unfolding bij_betw_def by (auto intro: enumerate_in_set) |
|
364 |
qed |
|
365 |
||
72097 | 366 |
lemma |
367 |
fixes S :: "nat set" |
|
368 |
assumes S: "infinite S" |
|
369 |
shows range_enumerate: "range (enumerate S) = S" |
|
370 |
and strict_mono_enumerate: "strict_mono (enumerate S)" |
|
371 |
by (auto simp add: bij_betw_imp_surj_on bij_enumerate assms strict_mono_def) |
|
372 |
||
64967 | 373 |
text \<open>A pair of weird and wonderful lemmas from HOL Light.\<close> |
63492
a662e8139804
More advanced theorems about retracts, homotopies., etc
paulson <lp15@cam.ac.uk>
parents:
61945
diff
changeset
|
374 |
lemma finite_transitivity_chain: |
a662e8139804
More advanced theorems about retracts, homotopies., etc
paulson <lp15@cam.ac.uk>
parents:
61945
diff
changeset
|
375 |
assumes "finite A" |
64967 | 376 |
and R: "\<And>x. \<not> R x x" "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z" |
377 |
and A: "\<And>x. x \<in> A \<Longrightarrow> \<exists>y. y \<in> A \<and> R x y" |
|
63492
a662e8139804
More advanced theorems about retracts, homotopies., etc
paulson <lp15@cam.ac.uk>
parents:
61945
diff
changeset
|
378 |
shows "A = {}" |
64967 | 379 |
using \<open>finite A\<close> A |
380 |
proof (induct A) |
|
381 |
case empty |
|
382 |
then show ?case by simp |
|
383 |
next |
|
63492
a662e8139804
More advanced theorems about retracts, homotopies., etc
paulson <lp15@cam.ac.uk>
parents:
61945
diff
changeset
|
384 |
case (insert a A) |
71840
8ed78bb0b915
Tuned some proofs in HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71827
diff
changeset
|
385 |
have False |
8ed78bb0b915
Tuned some proofs in HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71827
diff
changeset
|
386 |
using R(1)[of a] R(2)[of _ a] insert(3,4) by blast |
8ed78bb0b915
Tuned some proofs in HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71827
diff
changeset
|
387 |
thus ?case .. |
64967 | 388 |
qed |
63492
a662e8139804
More advanced theorems about retracts, homotopies., etc
paulson <lp15@cam.ac.uk>
parents:
61945
diff
changeset
|
389 |
|
a662e8139804
More advanced theorems about retracts, homotopies., etc
paulson <lp15@cam.ac.uk>
parents:
61945
diff
changeset
|
390 |
corollary Union_maximal_sets: |
a662e8139804
More advanced theorems about retracts, homotopies., etc
paulson <lp15@cam.ac.uk>
parents:
61945
diff
changeset
|
391 |
assumes "finite \<F>" |
64967 | 392 |
shows "\<Union>{T \<in> \<F>. \<forall>U\<in>\<F>. \<not> T \<subset> U} = \<Union>\<F>" |
63492
a662e8139804
More advanced theorems about retracts, homotopies., etc
paulson <lp15@cam.ac.uk>
parents:
61945
diff
changeset
|
393 |
(is "?lhs = ?rhs") |
a662e8139804
More advanced theorems about retracts, homotopies., etc
paulson <lp15@cam.ac.uk>
parents:
61945
diff
changeset
|
394 |
proof |
64967 | 395 |
show "?lhs \<subseteq> ?rhs" by force |
63492
a662e8139804
More advanced theorems about retracts, homotopies., etc
paulson <lp15@cam.ac.uk>
parents:
61945
diff
changeset
|
396 |
show "?rhs \<subseteq> ?lhs" |
a662e8139804
More advanced theorems about retracts, homotopies., etc
paulson <lp15@cam.ac.uk>
parents:
61945
diff
changeset
|
397 |
proof (rule Union_subsetI) |
a662e8139804
More advanced theorems about retracts, homotopies., etc
paulson <lp15@cam.ac.uk>
parents:
61945
diff
changeset
|
398 |
fix S |
a662e8139804
More advanced theorems about retracts, homotopies., etc
paulson <lp15@cam.ac.uk>
parents:
61945
diff
changeset
|
399 |
assume "S \<in> \<F>" |
64967 | 400 |
have "{T \<in> \<F>. S \<subseteq> T} = {}" |
401 |
if "\<not> (\<exists>y. y \<in> {T \<in> \<F>. \<forall>U\<in>\<F>. \<not> T \<subset> U} \<and> S \<subseteq> y)" |
|
71813 | 402 |
proof - |
403 |
have \<section>: "\<And>x. x \<in> \<F> \<and> S \<subseteq> x \<Longrightarrow> \<exists>y. y \<in> \<F> \<and> S \<subseteq> y \<and> x \<subset> y" |
|
404 |
using that by (blast intro: dual_order.trans psubset_imp_subset) |
|
405 |
show ?thesis |
|
406 |
proof (rule finite_transitivity_chain [of _ "\<lambda>T U. S \<subseteq> T \<and> T \<subset> U"]) |
|
407 |
qed (use assms in \<open>auto intro: \<section>\<close>) |
|
408 |
qed |
|
64967 | 409 |
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" |
410 |
by blast |
|
63492
a662e8139804
More advanced theorems about retracts, homotopies., etc
paulson <lp15@cam.ac.uk>
parents:
61945
diff
changeset
|
411 |
qed |
64967 | 412 |
qed |
63492
a662e8139804
More advanced theorems about retracts, homotopies., etc
paulson <lp15@cam.ac.uk>
parents:
61945
diff
changeset
|
413 |
|
71827 | 414 |
subsection \<open>Properties of @{term enumerate} on finite sets\<close> |
415 |
||
416 |
lemma finite_enumerate_in_set: "\<lbrakk>finite S; n < card S\<rbrakk> \<Longrightarrow> enumerate S n \<in> S" |
|
417 |
proof (induction n arbitrary: S) |
|
418 |
case 0 |
|
419 |
then show ?case |
|
72302
d7d90ed4c74e
fixed some remarkably ugly proofs
paulson <lp15@cam.ac.uk>
parents:
72097
diff
changeset
|
420 |
by (metis all_not_in_conv card.empty enumerate.simps(1) not_less0 wellorder_Least_lemma(1)) |
71827 | 421 |
next |
422 |
case (Suc n) |
|
423 |
show ?case |
|
424 |
using Suc.prems Suc.IH [of "S - {LEAST n. n \<in> S}"] |
|
425 |
apply (simp add: enumerate.simps) |
|
426 |
by (metis Diff_empty Diff_insert0 Suc_lessD card.remove less_Suc_eq) |
|
427 |
qed |
|
428 |
||
429 |
lemma finite_enumerate_step: "\<lbrakk>finite S; Suc n < card S\<rbrakk> \<Longrightarrow> enumerate S n < enumerate S (Suc n)" |
|
430 |
proof (induction n arbitrary: S) |
|
431 |
case 0 |
|
432 |
then have "enumerate S 0 \<le> enumerate S (Suc 0)" |
|
433 |
by (simp add: Least_le enumerate.simps(1) finite_enumerate_in_set) |
|
434 |
moreover have "enumerate (S - {enumerate S 0}) 0 \<in> S - {enumerate S 0}" |
|
435 |
by (metis 0 Suc_lessD Suc_less_eq card_Suc_Diff1 enumerate_in_set finite_enumerate_in_set) |
|
436 |
then have "enumerate S 0 \<noteq> enumerate (S - {enumerate S 0}) 0" |
|
437 |
by auto |
|
438 |
ultimately show ?case |
|
439 |
by (simp add: enumerate_Suc') |
|
440 |
next |
|
441 |
case (Suc n) |
|
442 |
then show ?case |
|
443 |
by (simp add: enumerate_Suc' finite_enumerate_in_set) |
|
444 |
qed |
|
445 |
||
446 |
lemma finite_enumerate_mono: "\<lbrakk>m < n; finite S; n < card S\<rbrakk> \<Longrightarrow> enumerate S m < enumerate S n" |
|
447 |
by (induct m n rule: less_Suc_induct) (auto intro: finite_enumerate_step) |
|
448 |
||
72095
cfb6c22a5636
lemmas about sets and the enumerate operator
paulson <lp15@cam.ac.uk>
parents:
72090
diff
changeset
|
449 |
lemma finite_enumerate_mono_iff [simp]: |
cfb6c22a5636
lemmas about sets and the enumerate operator
paulson <lp15@cam.ac.uk>
parents:
72090
diff
changeset
|
450 |
"\<lbrakk>finite S; m < card S; n < card S\<rbrakk> \<Longrightarrow> enumerate S m < enumerate S n \<longleftrightarrow> m < n" |
cfb6c22a5636
lemmas about sets and the enumerate operator
paulson <lp15@cam.ac.uk>
parents:
72090
diff
changeset
|
451 |
by (metis finite_enumerate_mono less_asym less_linear) |
71827 | 452 |
|
453 |
lemma finite_le_enumerate: |
|
454 |
assumes "finite S" "n < card S" |
|
455 |
shows "n \<le> enumerate S n" |
|
456 |
using assms |
|
457 |
proof (induction n) |
|
458 |
case 0 |
|
459 |
then show ?case by simp |
|
460 |
next |
|
461 |
case (Suc n) |
|
462 |
then have "n \<le> enumerate S n" by simp |
|
463 |
also note finite_enumerate_mono[of n "Suc n", OF _ \<open>finite S\<close>] |
|
464 |
finally show ?case |
|
465 |
using Suc.prems(2) Suc_leI by blast |
|
466 |
qed |
|
467 |
||
468 |
lemma finite_enumerate: |
|
469 |
assumes fS: "finite S" |
|
470 |
shows "\<exists>r::nat\<Rightarrow>nat. strict_mono_on r {..<card S} \<and> (\<forall>n<card S. r n \<in> S)" |
|
471 |
unfolding strict_mono_def |
|
472 |
using finite_enumerate_in_set[OF fS] finite_enumerate_mono[of _ _ S] fS |
|
473 |
by (metis lessThan_iff strict_mono_on_def) |
|
474 |
||
475 |
lemma finite_enumerate_Suc'': |
|
476 |
fixes S :: "'a::wellorder set" |
|
477 |
assumes "finite S" "Suc n < card S" |
|
478 |
shows "enumerate S (Suc n) = (LEAST s. s \<in> S \<and> enumerate S n < s)" |
|
479 |
using assms |
|
480 |
proof (induction n arbitrary: S) |
|
481 |
case 0 |
|
482 |
then have "\<forall>s \<in> S. enumerate S 0 \<le> s" |
|
483 |
by (auto simp: enumerate.simps intro: Least_le) |
|
484 |
then show ?case |
|
485 |
unfolding enumerate_Suc' enumerate_0[of "S - {enumerate S 0}"] |
|
486 |
by (metis Diff_iff dual_order.strict_iff_order singletonD singletonI) |
|
487 |
next |
|
488 |
case (Suc n S) |
|
489 |
then have "Suc n < card (S - {enumerate S 0})" |
|
490 |
using Suc.prems(2) finite_enumerate_in_set by force |
|
491 |
then show ?case |
|
492 |
apply (subst (1 2) enumerate_Suc') |
|
493 |
apply (simp add: Suc) |
|
494 |
apply (intro arg_cong[where f = Least] HOL.ext) |
|
495 |
using finite_enumerate_mono[OF zero_less_Suc \<open>finite S\<close>, of n] Suc.prems |
|
496 |
by (auto simp flip: enumerate_Suc') |
|
497 |
qed |
|
498 |
||
499 |
lemma finite_enumerate_initial_segment: |
|
500 |
fixes S :: "'a::wellorder set" |
|
72090 | 501 |
assumes "finite S" and n: "n < card (S \<inter> {..<s})" |
71827 | 502 |
shows "enumerate (S \<inter> {..<s}) n = enumerate S n" |
503 |
using n |
|
504 |
proof (induction n) |
|
505 |
case 0 |
|
506 |
have "(LEAST n. n \<in> S \<and> n < s) = (LEAST n. n \<in> S)" |
|
507 |
proof (rule Least_equality) |
|
508 |
have "\<exists>t. t \<in> S \<and> t < s" |
|
509 |
by (metis "0" card_gt_0_iff disjoint_iff_not_equal lessThan_iff) |
|
510 |
then show "(LEAST n. n \<in> S) \<in> S \<and> (LEAST n. n \<in> S) < s" |
|
511 |
by (meson LeastI Least_le le_less_trans) |
|
512 |
qed (simp add: Least_le) |
|
513 |
then show ?case |
|
514 |
by (auto simp: enumerate_0) |
|
515 |
next |
|
516 |
case (Suc n) |
|
517 |
then have less_card: "Suc n < card S" |
|
518 |
by (meson assms(1) card_mono inf_sup_ord(1) leD le_less_linear order.trans) |
|
72090 | 519 |
obtain T where T: "T \<in> {s \<in> S. enumerate S n < s}" |
71827 | 520 |
by (metis Infinite_Set.enumerate_step enumerate_in_set finite_enumerate_in_set finite_enumerate_step less_card mem_Collect_eq) |
72090 | 521 |
have "(LEAST x. x \<in> S \<and> x < s \<and> enumerate S n < x) = (LEAST x. x \<in> S \<and> enumerate S n < x)" |
71827 | 522 |
(is "_ = ?r") |
523 |
proof (intro Least_equality conjI) |
|
524 |
show "?r \<in> S" |
|
72090 | 525 |
by (metis (mono_tags, lifting) LeastI mem_Collect_eq T) |
526 |
have "\<not> s \<le> ?r" |
|
527 |
using not_less_Least [of _ "\<lambda>x. x \<in> S \<and> enumerate S n < x"] Suc assms |
|
528 |
by (metis (mono_tags, lifting) Int_Collect Suc_lessD finite_Int finite_enumerate_in_set finite_enumerate_step lessThan_def less_le_trans) |
|
529 |
then show "?r < s" |
|
530 |
by auto |
|
71827 | 531 |
show "enumerate S n < ?r" |
72090 | 532 |
by (metis (no_types, lifting) LeastI mem_Collect_eq T) |
71827 | 533 |
qed (auto simp: Least_le) |
534 |
then show ?case |
|
535 |
using Suc assms by (simp add: finite_enumerate_Suc'' less_card) |
|
536 |
qed |
|
537 |
||
538 |
lemma finite_enumerate_Ex: |
|
539 |
fixes S :: "'a::wellorder set" |
|
540 |
assumes S: "finite S" |
|
541 |
and s: "s \<in> S" |
|
542 |
shows "\<exists>n<card S. enumerate S n = s" |
|
543 |
using s S |
|
544 |
proof (induction s arbitrary: S rule: less_induct) |
|
545 |
case (less s) |
|
546 |
show ?case |
|
547 |
proof (cases "\<exists>y\<in>S. y < s") |
|
548 |
case True |
|
549 |
let ?T = "S \<inter> {..<s}" |
|
550 |
have "finite ?T" |
|
551 |
using less.prems(2) by blast |
|
552 |
have TS: "card ?T < card S" |
|
553 |
using less.prems by (blast intro: psubset_card_mono [OF \<open>finite S\<close>]) |
|
554 |
from True have y: "\<And>x. Max ?T < x \<longleftrightarrow> (\<forall>s'\<in>S. s' < s \<longrightarrow> s' < x)" |
|
555 |
by (subst Max_less_iff) (auto simp: \<open>finite ?T\<close>) |
|
556 |
then have y_in: "Max ?T \<in> {s'\<in>S. s' < s}" |
|
557 |
using Max_in \<open>finite ?T\<close> by fastforce |
|
558 |
with less.IH[of "Max ?T" ?T] obtain n where n: "enumerate ?T n = Max ?T" "n < card ?T" |
|
559 |
using \<open>finite ?T\<close> by blast |
|
560 |
then have "Suc n < card S" |
|
561 |
using TS less_trans_Suc by blast |
|
562 |
with S n have "enumerate S (Suc n) = s" |
|
563 |
by (subst finite_enumerate_Suc'') (auto simp: y finite_enumerate_initial_segment less finite_enumerate_Suc'' intro!: Least_equality) |
|
564 |
then show ?thesis |
|
565 |
using \<open>Suc n < card S\<close> by blast |
|
566 |
next |
|
567 |
case False |
|
568 |
then have "\<forall>t\<in>S. s \<le> t" by auto |
|
569 |
moreover have "0 < card S" |
|
570 |
using card_0_eq less.prems by blast |
|
571 |
ultimately show ?thesis |
|
572 |
using \<open>s \<in> S\<close> |
|
573 |
by (auto intro!: exI[of _ 0] Least_equality simp: enumerate_0) |
|
574 |
qed |
|
575 |
qed |
|
576 |
||
72095
cfb6c22a5636
lemmas about sets and the enumerate operator
paulson <lp15@cam.ac.uk>
parents:
72090
diff
changeset
|
577 |
lemma finite_enum_subset: |
cfb6c22a5636
lemmas about sets and the enumerate operator
paulson <lp15@cam.ac.uk>
parents:
72090
diff
changeset
|
578 |
assumes "\<And>i. i < card X \<Longrightarrow> enumerate X i = enumerate Y i" and "finite X" "finite Y" "card X \<le> card Y" |
cfb6c22a5636
lemmas about sets and the enumerate operator
paulson <lp15@cam.ac.uk>
parents:
72090
diff
changeset
|
579 |
shows "X \<subseteq> Y" |
cfb6c22a5636
lemmas about sets and the enumerate operator
paulson <lp15@cam.ac.uk>
parents:
72090
diff
changeset
|
580 |
by (metis assms finite_enumerate_Ex finite_enumerate_in_set less_le_trans subsetI) |
cfb6c22a5636
lemmas about sets and the enumerate operator
paulson <lp15@cam.ac.uk>
parents:
72090
diff
changeset
|
581 |
|
cfb6c22a5636
lemmas about sets and the enumerate operator
paulson <lp15@cam.ac.uk>
parents:
72090
diff
changeset
|
582 |
lemma finite_enum_ext: |
cfb6c22a5636
lemmas about sets and the enumerate operator
paulson <lp15@cam.ac.uk>
parents:
72090
diff
changeset
|
583 |
assumes "\<And>i. i < card X \<Longrightarrow> enumerate X i = enumerate Y i" and "finite X" "finite Y" "card X = card Y" |
cfb6c22a5636
lemmas about sets and the enumerate operator
paulson <lp15@cam.ac.uk>
parents:
72090
diff
changeset
|
584 |
shows "X = Y" |
cfb6c22a5636
lemmas about sets and the enumerate operator
paulson <lp15@cam.ac.uk>
parents:
72090
diff
changeset
|
585 |
by (intro antisym finite_enum_subset) (auto simp: assms) |
cfb6c22a5636
lemmas about sets and the enumerate operator
paulson <lp15@cam.ac.uk>
parents:
72090
diff
changeset
|
586 |
|
71827 | 587 |
lemma finite_bij_enumerate: |
588 |
fixes S :: "'a::wellorder set" |
|
589 |
assumes S: "finite S" |
|
590 |
shows "bij_betw (enumerate S) {..<card S} S" |
|
591 |
proof - |
|
592 |
have "\<And>n m. \<lbrakk>n \<noteq> m; n < card S; m < card S\<rbrakk> \<Longrightarrow> enumerate S n \<noteq> enumerate S m" |
|
593 |
using finite_enumerate_mono[OF _ \<open>finite S\<close>] by (auto simp: neq_iff) |
|
594 |
then have "inj_on (enumerate S) {..<card S}" |
|
595 |
by (auto simp: inj_on_def) |
|
596 |
moreover have "\<forall>s \<in> S. \<exists>i<card S. enumerate S i = s" |
|
597 |
using finite_enumerate_Ex[OF S] by auto |
|
598 |
moreover note \<open>finite S\<close> |
|
599 |
ultimately show ?thesis |
|
600 |
unfolding bij_betw_def by (auto intro: finite_enumerate_in_set) |
|
601 |
qed |
|
602 |
||
603 |
lemma ex_bij_betw_strict_mono_card: |
|
604 |
fixes M :: "'a::wellorder set" |
|
605 |
assumes "finite M" |
|
606 |
obtains h where "bij_betw h {..<card M} M" and "strict_mono_on h {..<card M}" |
|
607 |
proof |
|
608 |
show "bij_betw (enumerate M) {..<card M} M" |
|
609 |
by (simp add: assms finite_bij_enumerate) |
|
610 |
show "strict_mono_on (enumerate M) {..<card M}" |
|
611 |
by (simp add: assms finite_enumerate_mono strict_mono_on_def) |
|
612 |
qed |
|
613 |
||
20809 | 614 |
end |