author | paulson |
Wed, 13 Jul 2005 15:06:20 +0200 | |
changeset 16796 | 140f1e0ea846 |
parent 16733 | 236dfafbeb63 |
child 19313 | 45c9fc22904b |
permissions | -rw-r--r-- |
14442 | 1 |
(* Title: HOL/Infnite_Set.thy |
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ID: $Id$ |
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Author: Stephan Merz |
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*) |
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header {* Infnite Sets and Related Concepts*} |
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theory Infinite_Set |
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linear arithmetic now takes "&" in assumptions apart.
nipkow
parents:
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diff
changeset
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imports Hilbert_Choice Binomial |
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begin |
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subsection "Infinite Sets" |
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text {* Some elementary facts about infinite sets, by Stefan Merz. *} |
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syntax |
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infinite :: "'a set \<Rightarrow> bool" |
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translations |
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"infinite S" == "S \<notin> Finites" |
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text {* |
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Infinite sets are non-empty, and if we remove some elements |
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from an infinite set, the result is still infinite. |
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*} |
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lemma infinite_nonempty: |
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"\<not> (infinite {})" |
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by simp |
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lemma infinite_remove: |
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"infinite S \<Longrightarrow> infinite (S - {a})" |
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by simp |
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lemma Diff_infinite_finite: |
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assumes T: "finite T" and S: "infinite S" |
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shows "infinite (S-T)" |
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using T |
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proof (induct) |
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from S |
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show "infinite (S - {})" by auto |
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next |
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fix T x |
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assume ih: "infinite (S-T)" |
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have "S - (insert x T) = (S-T) - {x}" |
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by (rule Diff_insert) |
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with ih |
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show "infinite (S - (insert x T))" |
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by (simp add: infinite_remove) |
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qed |
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lemma Un_infinite: |
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"infinite S \<Longrightarrow> infinite (S \<union> T)" |
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by simp |
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lemma infinite_super: |
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assumes T: "S \<subseteq> T" and S: "infinite S" |
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shows "infinite T" |
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proof (rule ccontr) |
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assume "\<not>(infinite T)" |
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with T |
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have "finite S" by (simp add: finite_subset) |
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with S |
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show False by simp |
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qed |
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text {* |
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As a concrete example, we prove that the set of natural |
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numbers is infinite. |
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*} |
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lemma finite_nat_bounded: |
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assumes S: "finite (S::nat set)" |
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shows "\<exists>k. S \<subseteq> {..<k}" (is "\<exists>k. ?bounded S k") |
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using S |
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proof (induct) |
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have "?bounded {} 0" by simp |
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thus "\<exists>k. ?bounded {} k" .. |
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next |
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fix S x |
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assume "\<exists>k. ?bounded S k" |
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then obtain k where k: "?bounded S k" .. |
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show "\<exists>k. ?bounded (insert x S) k" |
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proof (cases "x<k") |
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case True |
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with k show ?thesis by auto |
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next |
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case False |
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with k have "?bounded S (Suc x)" by auto |
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thus ?thesis by auto |
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qed |
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qed |
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lemma finite_nat_iff_bounded: |
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"finite (S::nat set) = (\<exists>k. S \<subseteq> {..<k})" (is "?lhs = ?rhs") |
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proof |
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assume ?lhs |
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thus ?rhs by (rule finite_nat_bounded) |
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next |
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assume ?rhs |
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then obtain k where "S \<subseteq> {..<k}" .. |
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thus "finite S" |
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by (rule finite_subset, simp) |
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qed |
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lemma finite_nat_iff_bounded_le: |
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"finite (S::nat set) = (\<exists>k. S \<subseteq> {..k})" (is "?lhs = ?rhs") |
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proof |
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assume ?lhs |
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then obtain k where "S \<subseteq> {..<k}" |
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by (blast dest: finite_nat_bounded) |
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hence "S \<subseteq> {..k}" by auto |
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thus ?rhs .. |
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next |
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assume ?rhs |
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then obtain k where "S \<subseteq> {..k}" .. |
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thus "finite S" |
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by (rule finite_subset, simp) |
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qed |
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lemma infinite_nat_iff_unbounded: |
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"infinite (S::nat set) = (\<forall>m. \<exists>n. m<n \<and> n\<in>S)" |
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(is "?lhs = ?rhs") |
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proof |
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assume inf: ?lhs |
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show ?rhs |
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proof (rule ccontr) |
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assume "\<not> ?rhs" |
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then obtain m where m: "\<forall>n. m<n \<longrightarrow> n\<notin>S" by blast |
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hence "S \<subseteq> {..m}" |
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by (auto simp add: sym[OF linorder_not_less]) |
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with inf show "False" |
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by (simp add: finite_nat_iff_bounded_le) |
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qed |
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next |
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assume unbounded: ?rhs |
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show ?lhs |
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proof |
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assume "finite S" |
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then obtain m where "S \<subseteq> {..m}" |
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by (auto simp add: finite_nat_iff_bounded_le) |
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hence "\<forall>n. m<n \<longrightarrow> n\<notin>S" by auto |
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with unbounded show "False" by blast |
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qed |
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qed |
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lemma infinite_nat_iff_unbounded_le: |
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"infinite (S::nat set) = (\<forall>m. \<exists>n. m\<le>n \<and> n\<in>S)" |
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(is "?lhs = ?rhs") |
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proof |
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assume inf: ?lhs |
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show ?rhs |
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proof |
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fix m |
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from inf obtain n where "m<n \<and> n\<in>S" |
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by (auto simp add: infinite_nat_iff_unbounded) |
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hence "m\<le>n \<and> n\<in>S" by auto |
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thus "\<exists>n. m \<le> n \<and> n \<in> S" .. |
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qed |
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next |
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assume unbounded: ?rhs |
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show ?lhs |
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proof (auto simp add: infinite_nat_iff_unbounded) |
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fix m |
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from unbounded obtain n where "(Suc m)\<le>n \<and> n\<in>S" |
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by blast |
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hence "m<n \<and> n\<in>S" by auto |
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thus "\<exists>n. m < n \<and> n \<in> S" .. |
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qed |
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qed |
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text {* |
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For a set of natural numbers to be infinite, it is enough |
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to know that for any number larger than some @{text k}, there |
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is some larger number that is an element of the set. |
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*} |
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lemma unbounded_k_infinite: |
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assumes k: "\<forall>m. k<m \<longrightarrow> (\<exists>n. m<n \<and> n\<in>S)" |
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shows "infinite (S::nat set)" |
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proof (auto simp add: infinite_nat_iff_unbounded) |
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fix m show "\<exists>n. m<n \<and> n\<in>S" |
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proof (cases "k<m") |
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case True |
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with k show ?thesis by blast |
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next |
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case False |
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from k obtain n where "Suc k < n \<and> n\<in>S" by auto |
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with False have "m<n \<and> n\<in>S" by auto |
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thus ?thesis .. |
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qed |
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qed |
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theorem nat_infinite [simp]: |
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"infinite (UNIV :: nat set)" |
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by (auto simp add: infinite_nat_iff_unbounded) |
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theorem nat_not_finite [elim]: |
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"finite (UNIV::nat set) \<Longrightarrow> R" |
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by simp |
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text {* |
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Every infinite set contains a countable subset. More precisely |
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we show that a set @{text S} is infinite if and only if there exists |
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an injective function from the naturals into @{text S}. |
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*} |
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lemma range_inj_infinite: |
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"inj (f::nat \<Rightarrow> 'a) \<Longrightarrow> infinite (range f)" |
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proof |
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assume "inj f" |
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and "finite (range f)" |
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hence "finite (UNIV::nat set)" |
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by (auto intro: finite_imageD simp del: nat_infinite) |
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thus "False" by simp |
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qed |
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text {* |
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The ``only if'' direction is harder because it requires the |
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construction of a sequence of pairwise different elements of |
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an infinite set @{text S}. The idea is to construct a sequence of |
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non-empty and infinite subsets of @{text S} obtained by successively |
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removing elements of @{text S}. |
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*} |
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lemma linorder_injI: |
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assumes hyp: "\<forall>x y. x < (y::'a::linorder) \<longrightarrow> f x \<noteq> f y" |
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shows "inj f" |
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proof (rule inj_onI) |
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fix x y |
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assume f_eq: "f x = f y" |
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show "x = y" |
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proof (rule linorder_cases) |
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assume "x < y" |
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with hyp have "f x \<noteq> f y" by blast |
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with f_eq show ?thesis by simp |
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next |
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assume "x = y" |
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thus ?thesis . |
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next |
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assume "y < x" |
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with hyp have "f y \<noteq> f x" by blast |
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with f_eq show ?thesis by simp |
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qed |
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qed |
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lemma infinite_countable_subset: |
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assumes inf: "infinite (S::'a set)" |
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shows "\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S" |
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proof - |
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def Sseq \<equiv> "nat_rec S (\<lambda>n T. T - {SOME e. e \<in> T})" |
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def pick \<equiv> "\<lambda>n. (SOME e. e \<in> Sseq n)" |
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have Sseq_inf: "\<And>n. infinite (Sseq n)" |
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proof - |
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fix n |
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show "infinite (Sseq n)" |
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proof (induct n) |
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from inf show "infinite (Sseq 0)" |
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by (simp add: Sseq_def) |
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next |
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fix n |
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assume "infinite (Sseq n)" thus "infinite (Sseq (Suc n))" |
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by (simp add: Sseq_def infinite_remove) |
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qed |
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qed |
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have Sseq_S: "\<And>n. Sseq n \<subseteq> S" |
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proof - |
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fix n |
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show "Sseq n \<subseteq> S" |
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by (induct n, auto simp add: Sseq_def) |
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qed |
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have Sseq_pick: "\<And>n. pick n \<in> Sseq n" |
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proof - |
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fix n |
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show "pick n \<in> Sseq n" |
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proof (unfold pick_def, rule someI_ex) |
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from Sseq_inf have "infinite (Sseq n)" . |
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hence "Sseq n \<noteq> {}" by auto |
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thus "\<exists>x. x \<in> Sseq n" by auto |
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qed |
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qed |
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with Sseq_S have rng: "range pick \<subseteq> S" |
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by auto |
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have pick_Sseq_gt: "\<And>n m. pick n \<notin> Sseq (n + Suc m)" |
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proof - |
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fix n m |
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show "pick n \<notin> Sseq (n + Suc m)" |
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by (induct m, auto simp add: Sseq_def pick_def) |
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qed |
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have pick_pick: "\<And>n m. pick n \<noteq> pick (n + Suc m)" |
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proof - |
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fix n m |
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from Sseq_pick have "pick (n + Suc m) \<in> Sseq (n + Suc m)" . |
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moreover from pick_Sseq_gt |
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have "pick n \<notin> Sseq (n + Suc m)" . |
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ultimately show "pick n \<noteq> pick (n + Suc m)" |
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by auto |
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qed |
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have inj: "inj pick" |
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proof (rule linorder_injI) |
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show "\<forall>i j. i<(j::nat) \<longrightarrow> pick i \<noteq> pick j" |
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proof (clarify) |
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fix i j |
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assume ij: "i<(j::nat)" |
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and eq: "pick i = pick j" |
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from ij obtain k where "j = i + (Suc k)" |
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by (auto simp add: less_iff_Suc_add) |
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with pick_pick have "pick i \<noteq> pick j" by simp |
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with eq show "False" by simp |
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qed |
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qed |
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from rng inj show ?thesis by auto |
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qed |
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theorem infinite_iff_countable_subset: |
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"infinite S = (\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S)" |
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(is "?lhs = ?rhs") |
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by (auto simp add: infinite_countable_subset |
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range_inj_infinite infinite_super) |
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text {* |
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For any function with infinite domain and finite range |
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there is some element that is the image of infinitely |
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many domain elements. In particular, any infinite sequence |
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of elements from a finite set contains some element that |
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occurs infinitely often. |
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*} |
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theorem inf_img_fin_dom: |
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assumes img: "finite (f`A)" and dom: "infinite A" |
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shows "\<exists>y \<in> f`A. infinite (f -` {y})" |
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proof (rule ccontr) |
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assume "\<not> (\<exists>y\<in>f ` A. infinite (f -` {y}))" |
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with img have "finite (UN y:f`A. f -` {y})" |
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by (blast intro: finite_UN_I) |
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moreover have "A \<subseteq> (UN y:f`A. f -` {y})" by auto |
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moreover note dom |
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ultimately show "False" |
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by (simp add: infinite_super) |
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qed |
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theorems inf_img_fin_domE = inf_img_fin_dom[THEN bexE] |
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subsection "Infinitely Many and Almost All" |
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text {* |
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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, |
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so we introduce corresponding binders and their proof rules. |
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*} |
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consts |
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Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "INF " 10) |
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Alm_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "MOST " 10) |
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defs |
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INF_def: "Inf_many P \<equiv> infinite {x. P x}" |
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MOST_def: "Alm_all P \<equiv> \<not>(INF x. \<not> P x)" |
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syntax (xsymbols) |
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"MOST " :: "[idts, bool] \<Rightarrow> bool" ("(3\<forall>\<^sub>\<infinity>_./ _)" [0,10] 10) |
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"INF " :: "[idts, bool] \<Rightarrow> bool" ("(3\<exists>\<^sub>\<infinity>_./ _)" [0,10] 10) |
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syntax (HTML output) |
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"MOST " :: "[idts, bool] \<Rightarrow> bool" ("(3\<forall>\<^sub>\<infinity>_./ _)" [0,10] 10) |
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"INF " :: "[idts, bool] \<Rightarrow> bool" ("(3\<exists>\<^sub>\<infinity>_./ _)" [0,10] 10) |
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lemma INF_EX: |
369 |
"(\<exists>\<^sub>\<infinity>x. P x) \<Longrightarrow> (\<exists>x. P x)" |
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proof (unfold INF_def, rule ccontr) |
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assume inf: "infinite {x. P x}" |
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and notP: "\<not>(\<exists>x. P x)" |
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from notP have "{x. P x} = {}" by simp |
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hence "finite {x. P x}" by simp |
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with inf show "False" by simp |
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qed |
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377 |
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lemma MOST_iff_finiteNeg: |
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"(\<forall>\<^sub>\<infinity>x. P x) = finite {x. \<not> P x}" |
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by (simp add: MOST_def INF_def) |
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lemma ALL_MOST: |
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383 |
"\<forall>x. P x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. P x" |
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by (simp add: MOST_iff_finiteNeg) |
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385 |
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386 |
lemma INF_mono: |
|
387 |
assumes inf: "\<exists>\<^sub>\<infinity>x. P x" and q: "\<And>x. P x \<Longrightarrow> Q x" |
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shows "\<exists>\<^sub>\<infinity>x. Q x" |
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proof - |
|
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from inf have "infinite {x. P x}" by (unfold INF_def) |
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moreover from q have "{x. P x} \<subseteq> {x. Q x}" by auto |
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392 |
ultimately show ?thesis |
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by (simp add: INF_def infinite_super) |
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qed |
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lemma MOST_mono: |
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397 |
"\<lbrakk> \<forall>\<^sub>\<infinity>x. P x; \<And>x. P x \<Longrightarrow> Q x \<rbrakk> \<Longrightarrow> \<forall>\<^sub>\<infinity>x. Q x" |
|
398 |
by (unfold MOST_def, blast intro: INF_mono) |
|
399 |
||
400 |
lemma INF_nat: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) = (\<forall>m. \<exists>n. m<n \<and> P n)" |
|
401 |
by (simp add: INF_def infinite_nat_iff_unbounded) |
|
402 |
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403 |
lemma INF_nat_le: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) = (\<forall>m. \<exists>n. m\<le>n \<and> P n)" |
|
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by (simp add: INF_def infinite_nat_iff_unbounded_le) |
|
405 |
||
406 |
lemma MOST_nat: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) = (\<exists>m. \<forall>n. m<n \<longrightarrow> P n)" |
|
407 |
by (simp add: MOST_def INF_nat) |
|
408 |
||
409 |
lemma MOST_nat_le: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) = (\<exists>m. \<forall>n. m\<le>n \<longrightarrow> P n)" |
|
410 |
by (simp add: MOST_def INF_nat_le) |
|
411 |
||
412 |
||
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subsection "Miscellaneous" |
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414 |
||
415 |
text {* |
|
416 |
A few trivial lemmas about sets that contain at most one element. |
|
417 |
These simplify the reasoning about deterministic automata. |
|
418 |
*} |
|
419 |
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420 |
constdefs |
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421 |
atmost_one :: "'a set \<Rightarrow> bool" |
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"atmost_one S \<equiv> \<forall>x y. x\<in>S \<and> y\<in>S \<longrightarrow> x=y" |
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423 |
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lemma atmost_one_empty: "S={} \<Longrightarrow> atmost_one S" |
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by (simp add: atmost_one_def) |
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426 |
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lemma atmost_one_singleton: "S = {x} \<Longrightarrow> atmost_one S" |
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by (simp add: atmost_one_def) |
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429 |
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lemma atmost_one_unique [elim]: "\<lbrakk> atmost_one S; x \<in> S; y \<in> S \<rbrakk> \<Longrightarrow> y=x" |
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by (simp add: atmost_one_def) |
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432 |
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end |