src/HOL/Infinite_Set.thy
author wenzelm
Sat Apr 08 22:51:06 2006 +0200 (2006-04-08)
changeset 19363 667b5ea637dd
parent 19328 e090c939a29b
child 19457 b6eb4b4546fa
permissions -rw-r--r--
refined 'abbreviation';
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(*  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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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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abbreviation
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  infinite :: "'a set \<Rightarrow> bool"
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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
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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:
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  "(\<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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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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  "\<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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lemma INF_mono:
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  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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  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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  "\<lbrakk> \<forall>\<^sub>\<infinity>x. P x; \<And>x. P x \<Longrightarrow> Q x \<rbrakk> \<Longrightarrow> \<forall>\<^sub>\<infinity>x. Q x"
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by (unfold MOST_def, blast intro: INF_mono)
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lemma INF_nat: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) = (\<forall>m. \<exists>n. m<n \<and> P n)"
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by (simp add: INF_def infinite_nat_iff_unbounded)
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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)
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lemma MOST_nat: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) = (\<exists>m. \<forall>n. m<n \<longrightarrow> P n)"
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by (simp add: MOST_def INF_nat)
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lemma MOST_nat_le: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) = (\<exists>m. \<forall>n. m\<le>n \<longrightarrow> P n)"
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by (simp add: MOST_def INF_nat_le)
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subsection "Miscellaneous"
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text {*
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  A few trivial lemmas about sets that contain at most one element.
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  These simplify the reasoning about deterministic automata.
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*}
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constdefs
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  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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lemma atmost_one_empty: "S={} \<Longrightarrow> atmost_one S"
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by (simp add: atmost_one_def)
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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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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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end