author | wenzelm |
Sat, 12 Jan 2013 14:53:56 +0100 | |
changeset 50843 | 1465521b92a1 |
parent 50134 | 13211e07d931 |
child 53239 | 2f21813cf2f0 |
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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header {* Infinite Sets and Related Concepts *} |
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theory Infinite_Set |
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Main is (Complex_Main) base entry point in library theories
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imports Main |
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begin |
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subsection "Infinite Sets" |
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text {* |
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Some elementary facts about infinite sets, mostly by Stefan Merz. |
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Beware! Because "infinite" merely abbreviates a negation, these |
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lemmas may not work well with @{text "blast"}. |
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*} |
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abbreviation |
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infinite :: "'a set \<Rightarrow> bool" where |
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"infinite S == \<not> finite S" |
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text {* |
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Infinite sets are non-empty, and if we remove some elements from an |
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infinite set, the result is still infinite. |
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*} |
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lemma infinite_imp_nonempty: "infinite S ==> S \<noteq> {}" |
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by auto |
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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: "infinite S \<Longrightarrow> infinite (S \<union> T)" |
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by simp |
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lemma infinite_Un: "infinite (S \<union> T) \<longleftrightarrow> infinite S \<or> infinite 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 |
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assume "finite T" |
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with T have "finite S" by (simp add: finite_subset) |
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with S 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 numbers is |
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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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then show "\<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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then show ?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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then show ?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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then show "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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then have "S \<subseteq> {..k}" by auto |
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then show ?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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then show "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 ?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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then have "S \<subseteq> {..m}" |
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by (auto simp add: sym [OF linorder_not_less]) |
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with `?lhs` 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 ?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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then have "\<forall>n. m<n \<longrightarrow> n\<notin>S" by auto |
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with `?rhs` 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 ?lhs |
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show ?rhs |
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proof |
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fix m |
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from `?lhs` 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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then have "m\<le>n \<and> n\<in>S" by simp |
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then show "\<exists>n. m \<le> n \<and> n \<in> S" .. |
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qed |
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next |
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assume ?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 `?rhs` obtain n where "Suc m \<le> n \<and> n\<in>S" |
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by blast |
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then have "m<n \<and> n\<in>S" by simp |
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then show "\<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 to know |
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that for any number larger than some @{text k}, there is some larger |
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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 - |
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{ |
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fix m have "\<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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then show ?thesis .. |
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qed |
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} |
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then show ?thesis |
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by (auto simp add: infinite_nat_iff_unbounded) |
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qed |
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(* duplicates Finite_Set.infinite_UNIV_nat *) |
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lemma nat_infinite: "infinite (UNIV :: nat set)" |
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by (auto simp add: infinite_nat_iff_unbounded) |
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lemma nat_not_finite: "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 we |
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show that a set @{text S} is infinite if and only if there exists an |
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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 "finite (range f)" and "inj f" |
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then 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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lemma int_infinite [simp]: |
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shows "infinite (UNIV::int set)" |
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proof - |
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from inj_int have "infinite (range int)" by (rule range_inj_infinite) |
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moreover |
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have "range int \<subseteq> (UNIV::int set)" by simp |
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ultimately show "infinite (UNIV::int set)" by (simp add: infinite_super) |
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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 an |
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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: "!!x y. x < (y::'a::linorder) ==> 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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then show ?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)" then show "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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then have "Sseq n \<noteq> {}" by auto |
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then show "\<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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fix i j :: nat |
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assume "i < j" |
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show "pick i \<noteq> pick j" |
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proof |
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assume eq: "pick i = pick j" |
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from `i < j` 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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lemma 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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by (auto simp add: infinite_countable_subset range_inj_infinite infinite_super) |
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text {* |
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For any function with infinite domain and finite range there is some |
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element that is the image of infinitely many domain elements. In |
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particular, any infinite sequence of elements from a finite set |
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contains some element that occurs infinitely often. |
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*} |
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lemma 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> ?thesis" |
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with img have "finite (UN y:f`A. f -` {y})" by blast |
20809 | 343 |
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 by (simp add: infinite_super) |
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qed |
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lemma inf_img_fin_domE: |
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assumes "finite (f`A)" and "infinite A" |
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obtains y where "y \<in> f`A" and "infinite (f -` {y})" |
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using assms by (blast dest: inf_img_fin_dom) |
20809 | 352 |
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subsection "Infinitely Many and Almost All" |
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text {* |
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357 |
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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*} |
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definition |
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Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "INFM " 10) where |
20809 | 364 |
"Inf_many P = infinite {x. P x}" |
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more robust syntax for definition/abbreviation/notation;
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definition |
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Alm_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "MOST " 10) where |
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"Alm_all P = (\<not> (INFM x. \<not> P x))" |
20809 | 369 |
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21210 | 370 |
notation (xsymbols) |
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Inf_many (binder "\<exists>\<^sub>\<infinity>" 10) and |
20809 | 372 |
Alm_all (binder "\<forall>\<^sub>\<infinity>" 10) |
373 |
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21210 | 374 |
notation (HTML output) |
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Inf_many (binder "\<exists>\<^sub>\<infinity>" 10) and |
20809 | 376 |
Alm_all (binder "\<forall>\<^sub>\<infinity>" 10) |
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34112 | 378 |
lemma INFM_iff_infinite: "(INFM x. P x) \<longleftrightarrow> infinite {x. P x}" |
379 |
unfolding Inf_many_def .. |
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381 |
lemma MOST_iff_cofinite: "(MOST x. P x) \<longleftrightarrow> finite {x. \<not> P x}" |
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382 |
unfolding Alm_all_def Inf_many_def by simp |
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384 |
(* legacy name *) |
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385 |
lemmas MOST_iff_finiteNeg = MOST_iff_cofinite |
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387 |
lemma not_INFM [simp]: "\<not> (INFM x. P x) \<longleftrightarrow> (MOST x. \<not> P x)" |
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388 |
unfolding Alm_all_def not_not .. |
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20809 | 389 |
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34112 | 390 |
lemma not_MOST [simp]: "\<not> (MOST x. P x) \<longleftrightarrow> (INFM x. \<not> P x)" |
391 |
unfolding Alm_all_def not_not .. |
|
392 |
||
393 |
lemma INFM_const [simp]: "(INFM x::'a. P) \<longleftrightarrow> P \<and> infinite (UNIV::'a set)" |
|
394 |
unfolding Inf_many_def by simp |
|
395 |
||
396 |
lemma MOST_const [simp]: "(MOST x::'a. P) \<longleftrightarrow> P \<or> finite (UNIV::'a set)" |
|
397 |
unfolding Alm_all_def by simp |
|
398 |
||
399 |
lemma INFM_EX: "(\<exists>\<^sub>\<infinity>x. P x) \<Longrightarrow> (\<exists>x. P x)" |
|
400 |
by (erule contrapos_pp, simp) |
|
20809 | 401 |
|
402 |
lemma ALL_MOST: "\<forall>x. P x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. P x" |
|
34112 | 403 |
by simp |
404 |
||
405 |
lemma INFM_E: assumes "INFM x. P x" obtains x where "P x" |
|
406 |
using INFM_EX [OF assms] by (rule exE) |
|
407 |
||
408 |
lemma MOST_I: assumes "\<And>x. P x" shows "MOST x. P x" |
|
409 |
using assms by simp |
|
20809 | 410 |
|
27407
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
411 |
lemma INFM_mono: |
20809 | 412 |
assumes inf: "\<exists>\<^sub>\<infinity>x. P x" and q: "\<And>x. P x \<Longrightarrow> Q x" |
413 |
shows "\<exists>\<^sub>\<infinity>x. Q x" |
|
414 |
proof - |
|
415 |
from inf have "infinite {x. P x}" unfolding Inf_many_def . |
|
416 |
moreover from q have "{x. P x} \<subseteq> {x. Q x}" by auto |
|
417 |
ultimately show ?thesis |
|
418 |
by (simp add: Inf_many_def infinite_super) |
|
419 |
qed |
|
420 |
||
421 |
lemma MOST_mono: "\<forall>\<^sub>\<infinity>x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> \<forall>\<^sub>\<infinity>x. Q x" |
|
27407
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
422 |
unfolding Alm_all_def by (blast intro: INFM_mono) |
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
423 |
|
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
424 |
lemma INFM_disj_distrib: |
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
425 |
"(\<exists>\<^sub>\<infinity>x. P x \<or> Q x) \<longleftrightarrow> (\<exists>\<^sub>\<infinity>x. P x) \<or> (\<exists>\<^sub>\<infinity>x. Q x)" |
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
426 |
unfolding Inf_many_def by (simp add: Collect_disj_eq) |
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
427 |
|
34112 | 428 |
lemma INFM_imp_distrib: |
429 |
"(INFM x. P x \<longrightarrow> Q x) \<longleftrightarrow> ((MOST x. P x) \<longrightarrow> (INFM x. Q x))" |
|
430 |
by (simp only: imp_conv_disj INFM_disj_distrib not_MOST) |
|
431 |
||
27407
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
432 |
lemma MOST_conj_distrib: |
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
433 |
"(\<forall>\<^sub>\<infinity>x. P x \<and> Q x) \<longleftrightarrow> (\<forall>\<^sub>\<infinity>x. P x) \<and> (\<forall>\<^sub>\<infinity>x. Q x)" |
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
434 |
unfolding Alm_all_def by (simp add: INFM_disj_distrib del: disj_not1) |
20809 | 435 |
|
34112 | 436 |
lemma MOST_conjI: |
437 |
"MOST x. P x \<Longrightarrow> MOST x. Q x \<Longrightarrow> MOST x. P x \<and> Q x" |
|
438 |
by (simp add: MOST_conj_distrib) |
|
439 |
||
34113 | 440 |
lemma INFM_conjI: |
441 |
"INFM x. P x \<Longrightarrow> MOST x. Q x \<Longrightarrow> INFM x. P x \<and> Q x" |
|
442 |
unfolding MOST_iff_cofinite INFM_iff_infinite |
|
443 |
apply (drule (1) Diff_infinite_finite) |
|
444 |
apply (simp add: Collect_conj_eq Collect_neg_eq) |
|
445 |
done |
|
446 |
||
27407
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
447 |
lemma MOST_rev_mp: |
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
448 |
assumes "\<forall>\<^sub>\<infinity>x. P x" and "\<forall>\<^sub>\<infinity>x. P x \<longrightarrow> Q x" |
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
449 |
shows "\<forall>\<^sub>\<infinity>x. Q x" |
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
450 |
proof - |
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
451 |
have "\<forall>\<^sub>\<infinity>x. P x \<and> (P x \<longrightarrow> Q x)" |
34112 | 452 |
using assms by (rule MOST_conjI) |
27407
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
453 |
thus ?thesis by (rule MOST_mono) simp |
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
454 |
qed |
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
455 |
|
34112 | 456 |
lemma MOST_imp_iff: |
457 |
assumes "MOST x. P x" |
|
458 |
shows "(MOST x. P x \<longrightarrow> Q x) \<longleftrightarrow> (MOST x. Q x)" |
|
459 |
proof |
|
460 |
assume "MOST x. P x \<longrightarrow> Q x" |
|
461 |
with assms show "MOST x. Q x" by (rule MOST_rev_mp) |
|
462 |
next |
|
463 |
assume "MOST x. Q x" |
|
464 |
then show "MOST x. P x \<longrightarrow> Q x" by (rule MOST_mono) simp |
|
465 |
qed |
|
27407
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
466 |
|
34112 | 467 |
lemma INFM_MOST_simps [simp]: |
468 |
"\<And>P Q. (INFM x. P x \<and> Q) \<longleftrightarrow> (INFM x. P x) \<and> Q" |
|
469 |
"\<And>P Q. (INFM x. P \<and> Q x) \<longleftrightarrow> P \<and> (INFM x. Q x)" |
|
470 |
"\<And>P Q. (MOST x. P x \<or> Q) \<longleftrightarrow> (MOST x. P x) \<or> Q" |
|
471 |
"\<And>P Q. (MOST x. P \<or> Q x) \<longleftrightarrow> P \<or> (MOST x. Q x)" |
|
472 |
"\<And>P Q. (MOST x. P x \<longrightarrow> Q) \<longleftrightarrow> ((INFM x. P x) \<longrightarrow> Q)" |
|
473 |
"\<And>P Q. (MOST x. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (MOST x. Q x))" |
|
474 |
unfolding Alm_all_def Inf_many_def |
|
475 |
by (simp_all add: Collect_conj_eq) |
|
476 |
||
477 |
text {* Properties of quantifiers with injective functions. *} |
|
478 |
||
479 |
lemma INFM_inj: |
|
480 |
"INFM x. P (f x) \<Longrightarrow> inj f \<Longrightarrow> INFM x. P x" |
|
481 |
unfolding INFM_iff_infinite |
|
482 |
by (clarify, drule (1) finite_vimageI, simp) |
|
27407
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
483 |
|
34112 | 484 |
lemma MOST_inj: |
485 |
"MOST x. P x \<Longrightarrow> inj f \<Longrightarrow> MOST x. P (f x)" |
|
486 |
unfolding MOST_iff_cofinite |
|
487 |
by (drule (1) finite_vimageI, simp) |
|
488 |
||
489 |
text {* Properties of quantifiers with singletons. *} |
|
490 |
||
491 |
lemma not_INFM_eq [simp]: |
|
492 |
"\<not> (INFM x. x = a)" |
|
493 |
"\<not> (INFM x. a = x)" |
|
494 |
unfolding INFM_iff_infinite by simp_all |
|
495 |
||
496 |
lemma MOST_neq [simp]: |
|
497 |
"MOST x. x \<noteq> a" |
|
498 |
"MOST x. a \<noteq> x" |
|
499 |
unfolding MOST_iff_cofinite by simp_all |
|
27407
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
500 |
|
34112 | 501 |
lemma INFM_neq [simp]: |
502 |
"(INFM x::'a. x \<noteq> a) \<longleftrightarrow> infinite (UNIV::'a set)" |
|
503 |
"(INFM x::'a. a \<noteq> x) \<longleftrightarrow> infinite (UNIV::'a set)" |
|
504 |
unfolding INFM_iff_infinite by simp_all |
|
505 |
||
506 |
lemma MOST_eq [simp]: |
|
507 |
"(MOST x::'a. x = a) \<longleftrightarrow> finite (UNIV::'a set)" |
|
508 |
"(MOST x::'a. a = x) \<longleftrightarrow> finite (UNIV::'a set)" |
|
509 |
unfolding MOST_iff_cofinite by simp_all |
|
510 |
||
511 |
lemma MOST_eq_imp: |
|
512 |
"MOST x. x = a \<longrightarrow> P x" |
|
513 |
"MOST x. a = x \<longrightarrow> P x" |
|
514 |
unfolding MOST_iff_cofinite by simp_all |
|
515 |
||
516 |
text {* Properties of quantifiers over the naturals. *} |
|
27407
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
517 |
|
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
518 |
lemma INFM_nat: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) = (\<forall>m. \<exists>n. m<n \<and> P n)" |
20809 | 519 |
by (simp add: Inf_many_def infinite_nat_iff_unbounded) |
520 |
||
27407
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
521 |
lemma INFM_nat_le: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) = (\<forall>m. \<exists>n. m\<le>n \<and> P n)" |
20809 | 522 |
by (simp add: Inf_many_def infinite_nat_iff_unbounded_le) |
523 |
||
524 |
lemma MOST_nat: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) = (\<exists>m. \<forall>n. m<n \<longrightarrow> P n)" |
|
27407
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
525 |
by (simp add: Alm_all_def INFM_nat) |
20809 | 526 |
|
527 |
lemma MOST_nat_le: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) = (\<exists>m. \<forall>n. m\<le>n \<longrightarrow> P n)" |
|
27407
68e111812b83
rename lemmas INF_foo to INFM_foo; add new lemmas about MOST and INFM
huffman
parents:
27368
diff
changeset
|
528 |
by (simp add: Alm_all_def INFM_nat_le) |
20809 | 529 |
|
530 |
||
531 |
subsection "Enumeration of an Infinite Set" |
|
532 |
||
533 |
text {* |
|
534 |
The set's element type must be wellordered (e.g. the natural numbers). |
|
535 |
*} |
|
536 |
||
34941 | 537 |
primrec (in wellorder) enumerate :: "'a set \<Rightarrow> nat \<Rightarrow> 'a" where |
538 |
enumerate_0: "enumerate S 0 = (LEAST n. n \<in> S)" |
|
539 |
| enumerate_Suc: "enumerate S (Suc n) = enumerate (S - {LEAST n. n \<in> S}) n" |
|
20809 | 540 |
|
541 |
lemma enumerate_Suc': |
|
542 |
"enumerate S (Suc n) = enumerate (S - {enumerate S 0}) n" |
|
543 |
by simp |
|
544 |
||
545 |
lemma enumerate_in_set: "infinite S \<Longrightarrow> enumerate S n : S" |
|
29901 | 546 |
apply (induct n arbitrary: S) |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44454
diff
changeset
|
547 |
apply (fastforce intro: LeastI dest!: infinite_imp_nonempty) |
29901 | 548 |
apply simp |
44454 | 549 |
apply (metis DiffE infinite_remove) |
29901 | 550 |
done |
20809 | 551 |
|
552 |
declare enumerate_0 [simp del] enumerate_Suc [simp del] |
|
553 |
||
554 |
lemma enumerate_step: "infinite S \<Longrightarrow> enumerate S n < enumerate S (Suc n)" |
|
555 |
apply (induct n arbitrary: S) |
|
556 |
apply (rule order_le_neq_trans) |
|
557 |
apply (simp add: enumerate_0 Least_le enumerate_in_set) |
|
558 |
apply (simp only: enumerate_Suc') |
|
559 |
apply (subgoal_tac "enumerate (S - {enumerate S 0}) 0 : S - {enumerate S 0}") |
|
560 |
apply (blast intro: sym) |
|
561 |
apply (simp add: enumerate_in_set del: Diff_iff) |
|
562 |
apply (simp add: enumerate_Suc') |
|
563 |
done |
|
564 |
||
565 |
lemma enumerate_mono: "m<n \<Longrightarrow> infinite S \<Longrightarrow> enumerate S m < enumerate S n" |
|
566 |
apply (erule less_Suc_induct) |
|
567 |
apply (auto intro: enumerate_step) |
|
568 |
done |
|
569 |
||
570 |
||
50134 | 571 |
lemma le_enumerate: |
572 |
assumes S: "infinite S" |
|
573 |
shows "n \<le> enumerate S n" |
|
574 |
using S |
|
575 |
proof (induct n) |
|
576 |
case (Suc n) |
|
577 |
then have "n \<le> enumerate S n" by simp |
|
578 |
also note enumerate_mono[of n "Suc n", OF _ `infinite S`] |
|
579 |
finally show ?case by simp |
|
580 |
qed simp |
|
581 |
||
582 |
lemma enumerate_Suc'': |
|
583 |
fixes S :: "'a::wellorder set" |
|
584 |
shows "infinite S \<Longrightarrow> enumerate S (Suc n) = (LEAST s. s \<in> S \<and> enumerate S n < s)" |
|
585 |
proof (induct n arbitrary: S) |
|
586 |
case 0 |
|
587 |
then have "\<forall>s\<in>S. enumerate S 0 \<le> s" |
|
588 |
by (auto simp: enumerate.simps intro: Least_le) |
|
589 |
then show ?case |
|
590 |
unfolding enumerate_Suc' enumerate_0[of "S - {enumerate S 0}"] |
|
591 |
by (intro arg_cong[where f=Least] ext) auto |
|
592 |
next |
|
593 |
case (Suc n S) |
|
594 |
show ?case |
|
595 |
using enumerate_mono[OF zero_less_Suc `infinite S`, of n] `infinite S` |
|
596 |
apply (subst (1 2) enumerate_Suc') |
|
597 |
apply (subst Suc) |
|
598 |
apply (insert `infinite S`, simp) |
|
599 |
by (intro arg_cong[where f=Least] ext) |
|
600 |
(auto simp: enumerate_Suc'[symmetric]) |
|
601 |
qed |
|
602 |
||
603 |
lemma enumerate_Ex: |
|
604 |
assumes S: "infinite (S::nat set)" |
|
605 |
shows "s \<in> S \<Longrightarrow> \<exists>n. enumerate S n = s" |
|
606 |
proof (induct s rule: less_induct) |
|
607 |
case (less s) |
|
608 |
show ?case |
|
609 |
proof cases |
|
610 |
let ?y = "Max {s'\<in>S. s' < s}" |
|
611 |
assume "\<exists>y\<in>S. y < s" |
|
612 |
then have y: "\<And>x. ?y < x \<longleftrightarrow> (\<forall>s'\<in>S. s' < s \<longrightarrow> s' < x)" by (subst Max_less_iff) auto |
|
613 |
then have y_in: "?y \<in> {s'\<in>S. s' < s}" by (intro Max_in) auto |
|
614 |
with less.hyps[of ?y] obtain n where "enumerate S n = ?y" by auto |
|
615 |
with S have "enumerate S (Suc n) = s" |
|
616 |
by (auto simp: y less enumerate_Suc'' intro!: Least_equality) |
|
617 |
then show ?case by auto |
|
618 |
next |
|
619 |
assume *: "\<not> (\<exists>y\<in>S. y < s)" |
|
620 |
then have "\<forall>t\<in>S. s \<le> t" by auto |
|
621 |
with `s \<in> S` show ?thesis |
|
622 |
by (auto intro!: exI[of _ 0] Least_equality simp: enumerate_0) |
|
623 |
qed |
|
624 |
qed |
|
625 |
||
626 |
lemma bij_enumerate: |
|
627 |
fixes S :: "nat set" |
|
628 |
assumes S: "infinite S" |
|
629 |
shows "bij_betw (enumerate S) UNIV S" |
|
630 |
proof - |
|
631 |
have "\<And>n m. n \<noteq> m \<Longrightarrow> enumerate S n \<noteq> enumerate S m" |
|
632 |
using enumerate_mono[OF _ `infinite S`] by (auto simp: neq_iff) |
|
633 |
then have "inj (enumerate S)" |
|
634 |
by (auto simp: inj_on_def) |
|
635 |
moreover have "\<forall>s\<in>S. \<exists>i. enumerate S i = s" |
|
636 |
using enumerate_Ex[OF S] by auto |
|
637 |
moreover note `infinite S` |
|
638 |
ultimately show ?thesis |
|
639 |
unfolding bij_betw_def by (auto intro: enumerate_in_set) |
|
640 |
qed |
|
641 |
||
20809 | 642 |
subsection "Miscellaneous" |
643 |
||
644 |
text {* |
|
645 |
A few trivial lemmas about sets that contain at most one element. |
|
646 |
These simplify the reasoning about deterministic automata. |
|
647 |
*} |
|
648 |
||
649 |
definition |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21256
diff
changeset
|
650 |
atmost_one :: "'a set \<Rightarrow> bool" where |
20809 | 651 |
"atmost_one S = (\<forall>x y. x\<in>S \<and> y\<in>S \<longrightarrow> x=y)" |
652 |
||
653 |
lemma atmost_one_empty: "S = {} \<Longrightarrow> atmost_one S" |
|
654 |
by (simp add: atmost_one_def) |
|
655 |
||
656 |
lemma atmost_one_singleton: "S = {x} \<Longrightarrow> atmost_one S" |
|
657 |
by (simp add: atmost_one_def) |
|
658 |
||
659 |
lemma atmost_one_unique [elim]: "atmost_one S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> y = x" |
|
660 |
by (simp add: atmost_one_def) |
|
661 |
||
662 |
end |
|
46783 | 663 |