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(* Title: HOL/Infnite_Set.thy
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ID: $Id$
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Author: Stefan Merz
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*)
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header {* Infnite Sets and Related Concepts*}
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theory Infinite_Set = Hilbert_Choice + Finite_Set + SetInterval:
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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 not_less_iff_le])
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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 $k$, 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 $S$ is infinite if and only if there exists
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an injective function from the naturals into $S$.
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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 $S$. The idea is to construct a sequence of
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non-empty and infinite subsets of $S$ obtained by successively
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removing elements of $S$.
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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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390 |
ultimately show ?thesis
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391 |
by (simp add: INF_def infinite_super)
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392 |
qed
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393 |
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394 |
lemma MOST_mono:
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395 |
"\<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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396 |
by (unfold MOST_def, blast intro: INF_mono)
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397 |
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398 |
lemma INF_nat: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) = (\<forall>m. \<exists>n. m<n \<and> P n)"
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399 |
by (simp add: INF_def infinite_nat_iff_unbounded)
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400 |
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401 |
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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402 |
by (simp add: INF_def infinite_nat_iff_unbounded_le)
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403 |
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404 |
lemma MOST_nat: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) = (\<exists>m. \<forall>n. m<n \<longrightarrow> P n)"
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405 |
by (simp add: MOST_def INF_nat)
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406 |
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407 |
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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408 |
by (simp add: MOST_def INF_nat_le)
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409 |
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410 |
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411 |
subsection "Miscellaneous"
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412 |
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413 |
text {*
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414 |
A few trivial lemmas about sets that contain at most one element.
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415 |
These simplify the reasoning about deterministic automata.
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|
416 |
*}
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417 |
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418 |
constdefs
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419 |
atmost_one :: "'a set \<Rightarrow> bool"
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420 |
"atmost_one S \<equiv> \<forall>x y. x\<in>S \<and> y\<in>S \<longrightarrow> x=y"
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421 |
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422 |
lemma atmost_one_empty: "S={} \<Longrightarrow> atmost_one S"
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423 |
by (simp add: atmost_one_def)
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|
424 |
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425 |
lemma atmost_one_singleton: "S = {x} \<Longrightarrow> atmost_one S"
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426 |
by (simp add: atmost_one_def)
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|
427 |
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|
428 |
lemma atmost_one_unique [elim]: "\<lbrakk> atmost_one S; x \<in> S; y \<in> S \<rbrakk> \<Longrightarrow> y=x"
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429 |
by (simp add: atmost_one_def)
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|
430 |
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|
431 |
end
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