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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 nonempty, 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 (ST)"


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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 (ST)"


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have "S  (insert x T) = (ST)  {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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nonempty 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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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
