src/HOL/Infinite_Set.thy

+(*  Title:      HOL/Infnite_Set.thy
+    ID:         $Id$
+    Author:     Stefan Merz 
+*)
+
+header {* Infnite Sets and Related Concepts*}
+
+theory Infinite_Set = Hilbert_Choice + Finite_Set:
+
+subsection "Infinite Sets"
+
+text {* Some elementary facts about infinite sets, by Stefan Merz. *}
+
+syntax
+  infinite :: "'a set \<Rightarrow> bool"
+translations
+  "infinite S" == "S \<notin> Finites"
+
+text {*
+  Infinite sets are non-empty, and if we remove some elements
+  from an infinite set, the result is still infinite.
+*}
+
+lemma infinite_nonempty:
+  "\<not> (infinite {})"
+by simp
+
+lemma infinite_remove:
+  "infinite S \<Longrightarrow> infinite (S - {a})"
+by simp
+
+lemma Diff_infinite_finite:
+  assumes T: "finite T" and S: "infinite S"
+  shows "infinite (S-T)"
+using T
+proof (induct)
+  from S
+  show "infinite (S - {})" by auto
+next
+  fix T x
+  assume ih: "infinite (S-T)"
+  have "S - (insert x T) = (S-T) - {x}"
+    by (rule Diff_insert)
+  with ih
+  show "infinite (S - (insert x T))"
+    by (simp add: infinite_remove)
+qed
+
+lemma Un_infinite:
+  "infinite S \<Longrightarrow> infinite (S \<union> T)"
+by simp
+
+lemma infinite_super:
+  assumes T: "S \<subseteq> T" and S: "infinite S"
+  shows "infinite T"
+proof (rule ccontr)
+  assume "\<not>(infinite T)"
+  with T
+  have "finite S" by (simp add: finite_subset)
+  with S
+  show False by simp
+qed
+
+text {*
+  As a concrete example, we prove that the set of natural
+  numbers is infinite.
+*}
+
+lemma finite_nat_bounded:
+  assumes S: "finite (S::nat set)"
+  shows "\<exists>k. S \<subseteq> {..k(}" (is "\<exists>k. ?bounded S k")
+using S
+proof (induct)
+  have "?bounded {} 0" by simp
+  thus "\<exists>k. ?bounded {} k" ..
+next
+  fix S x
+  assume "\<exists>k. ?bounded S k"
+  then obtain k where k: "?bounded S k" ..
+  show "\<exists>k. ?bounded (insert x S) k"
+  proof (cases "x<k")
+    case True
+    with k show ?thesis by auto
+  next
+    case False
+    with k have "?bounded S (Suc x)" by auto
+    thus ?thesis by auto
+  qed
+qed
+
+lemma finite_nat_iff_bounded:
+  "finite (S::nat set) = (\<exists>k. S \<subseteq> {..k(})" (is "?lhs = ?rhs")
+proof
+  assume ?lhs
+  thus ?rhs by (rule finite_nat_bounded)
+next
+  assume ?rhs
+  then obtain k where "S \<subseteq> {..k(}" ..
+  thus "finite S"
+    by (rule finite_subset, simp)
+qed
+
+lemma finite_nat_iff_bounded_le:
+  "finite (S::nat set) = (\<exists>k. S \<subseteq> {..k})" (is "?lhs = ?rhs")
+proof
+  assume ?lhs
+  then obtain k where "S \<subseteq> {..k(}" 
+    by (blast dest: finite_nat_bounded)
+  hence "S \<subseteq> {..k}" by auto
+  thus ?rhs ..
+next
+  assume ?rhs
+  then obtain k where "S \<subseteq> {..k}" ..
+  thus "finite S"
+    by (rule finite_subset, simp)
+qed
+
+lemma infinite_nat_iff_unbounded:
+  "infinite (S::nat set) = (\<forall>m. \<exists>n. m<n \<and> n\<in>S)"
+  (is "?lhs = ?rhs")
+proof
+  assume inf: ?lhs
+  show ?rhs
+  proof (rule ccontr)
+    assume "\<not> ?rhs"
+    then obtain m where m: "\<forall>n. m<n \<longrightarrow> n\<notin>S" by blast
+    hence "S \<subseteq> {..m}"
+      by (auto simp add: sym[OF not_less_iff_le])
+    with inf show "False" 
+      by (simp add: finite_nat_iff_bounded_le)
+  qed
+next
+  assume unbounded: ?rhs
+  show ?lhs
+  proof
+    assume "finite S"
+    then obtain m where "S \<subseteq> {..m}"
+      by (auto simp add: finite_nat_iff_bounded_le)
+    hence "\<forall>n. m<n \<longrightarrow> n\<notin>S" by auto
+    with unbounded show "False" by blast
+  qed
+qed
+
+lemma infinite_nat_iff_unbounded_le:
+  "infinite (S::nat set) = (\<forall>m. \<exists>n. m\<le>n \<and> n\<in>S)"
+  (is "?lhs = ?rhs")
+proof
+  assume inf: ?lhs
+  show ?rhs
+  proof
+    fix m
+    from inf obtain n where "m<n \<and> n\<in>S"
+      by (auto simp add: infinite_nat_iff_unbounded)
+    hence "m\<le>n \<and> n\<in>S" by auto
+    thus "\<exists>n. m \<le> n \<and> n \<in> S" ..
+  qed
+next
+  assume unbounded: ?rhs
+  show ?lhs
+  proof (auto simp add: infinite_nat_iff_unbounded)
+    fix m
+    from unbounded obtain n where "(Suc m)\<le>n \<and> n\<in>S"
+      by blast
+    hence "m<n \<and> n\<in>S" by auto
+    thus "\<exists>n. m < n \<and> n \<in> S" ..
+  qed
+qed
+
+text {*
+  For a set of natural numbers to be infinite, it is enough
+  to know that for any number larger than some $k$, there
+  is some larger number that is an element of the set.
+*}
+
+lemma unbounded_k_infinite:
+  assumes k: "\<forall>m. k<m \<longrightarrow> (\<exists>n. m<n \<and> n\<in>S)"
+  shows "infinite (S::nat set)"
+proof (auto simp add: infinite_nat_iff_unbounded)
+  fix m show "\<exists>n. m<n \<and> n\<in>S"
+  proof (cases "k<m")
+    case True
+    with k show ?thesis by blast
+  next
+    case False
+    from k obtain n where "Suc k < n \<and> n\<in>S" by auto
+    with False have "m<n \<and> n\<in>S" by auto
+    thus ?thesis ..
+  qed
+qed
+
+theorem nat_infinite [simp]:
+  "infinite (UNIV :: nat set)"
+by (auto simp add: infinite_nat_iff_unbounded)
+
+theorem nat_not_finite [elim]:
+  "finite (UNIV::nat set) \<Longrightarrow> R"
+by simp
+
+text {*
+  Every infinite set contains a countable subset. More precisely
+  we show that a set $S$ is infinite if and only if there exists 
+  an injective function from the naturals into $S$.
+*}
+
+lemma range_inj_infinite:
+  "inj (f::nat \<Rightarrow> 'a) \<Longrightarrow> infinite (range f)"
+proof
+  assume "inj f"
+    and  "finite (range f)"
+  hence "finite (UNIV::nat set)"
+    by (auto intro: finite_imageD simp del: nat_infinite)
+  thus "False" by simp
+qed
+
+text {*
+  The ``only if'' direction is harder because it requires the
+  construction of a sequence of pairwise different elements of
+  an infinite set $S$. The idea is to construct a sequence of
+  non-empty and infinite subsets of $S$ obtained by successively
+  removing elements of $S$.
+*}
+
+lemma linorder_injI:
+  assumes hyp: "\<forall>x y. x < (y::'a::linorder) \<longrightarrow> f x \<noteq> f y"
+  shows "inj f"
+proof (rule inj_onI)
+  fix x y
+  assume f_eq: "f x = f y"
+  show "x = y"
+  proof (rule linorder_cases)
+    assume "x < y"
+    with hyp have "f x \<noteq> f y" by blast
+    with f_eq show ?thesis by simp
+  next
+    assume "x = y"
+    thus ?thesis .
+  next
+    assume "y < x"
+    with hyp have "f y \<noteq> f x" by blast
+    with f_eq show ?thesis by simp
+  qed
+qed
+
+lemma infinite_countable_subset:
+  assumes inf: "infinite (S::'a set)"
+  shows "\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S"
+proof -
+  def Sseq \<equiv> "nat_rec S (\<lambda>n T. T - {\<epsilon> e. e \<in> T})"
+  def pick \<equiv> "\<lambda>n. (\<epsilon> e. e \<in> Sseq n)"
+  have Sseq_inf: "\<And>n. infinite (Sseq n)"
+  proof -
+    fix n
+    show "infinite (Sseq n)"
+    proof (induct n)
+      from inf show "infinite (Sseq 0)"
+	by (simp add: Sseq_def)
+    next
+      fix n
+      assume "infinite (Sseq n)" thus "infinite (Sseq (Suc n))"
+	by (simp add: Sseq_def infinite_remove)
+    qed
+  qed
+  have Sseq_S: "\<And>n. Sseq n \<subseteq> S"
+  proof -
+    fix n
+    show "Sseq n \<subseteq> S"
+      by (induct n, auto simp add: Sseq_def)
+  qed
+  have Sseq_pick: "\<And>n. pick n \<in> Sseq n"
+  proof -
+    fix n
+    show "pick n \<in> Sseq n"
+    proof (unfold pick_def, rule someI_ex)
+      from Sseq_inf have "infinite (Sseq n)" .
+      hence "Sseq n \<noteq> {}" by auto
+      thus "\<exists>x. x \<in> Sseq n" by auto
+    qed
+  qed
+  with Sseq_S have rng: "range pick \<subseteq> S"
+    by auto
+  have pick_Sseq_gt: "\<And>n m. pick n \<notin> Sseq (n + Suc m)"
+  proof -
+    fix n m
+    show "pick n \<notin> Sseq (n + Suc m)"
+      by (induct m, auto simp add: Sseq_def pick_def)
+  qed
+  have pick_pick: "\<And>n m. pick n \<noteq> pick (n + Suc m)"
+  proof -
+    fix n m
+    from Sseq_pick have "pick (n + Suc m) \<in> Sseq (n + Suc m)" .
+    moreover from pick_Sseq_gt
+    have "pick n \<notin> Sseq (n + Suc m)" .
+    ultimately show "pick n \<noteq> pick (n + Suc m)"
+      by auto
+  qed
+  have inj: "inj pick"
+  proof (rule linorder_injI)
+    show "\<forall>i j. i<(j::nat) \<longrightarrow> pick i \<noteq> pick j"
+    proof (clarify)
+      fix i j
+      assume ij: "i<(j::nat)"
+	and eq: "pick i = pick j"
+      from ij obtain k where "j = i + (Suc k)"
+	by (auto simp add: less_iff_Suc_add)
+      with pick_pick have "pick i \<noteq> pick j" by simp
+      with eq show "False" by simp
+    qed
+  qed
+  from rng inj show ?thesis by auto
+qed
+
+theorem infinite_iff_countable_subset:
+  "infinite S = (\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S)"
+  (is "?lhs = ?rhs")
+by (auto simp add: infinite_countable_subset
+                   range_inj_infinite infinite_super)
+
+text {*
+  For any function with infinite domain and finite range
+  there is some element that is the image of infinitely
+  many domain elements. In particular, any infinite sequence
+  of elements from a finite set contains some element that
+  occurs infinitely often.
+*}
+
+theorem inf_img_fin_dom:
+  assumes img: "finite (f`A)" and dom: "infinite A"
+  shows "\<exists>y \<in> f`A. infinite (f -` {y})"
+proof (rule ccontr)
+  assume "\<not> (\<exists>y\<in>f ` A. infinite (f -` {y}))"
+  with img have "finite (UN y:f`A. f -` {y})"
+    by (blast intro: finite_UN_I)
+  moreover have "A \<subseteq> (UN y:f`A. f -` {y})" by auto
+  moreover note dom
+  ultimately show "False"
+    by (simp add: infinite_super)
+qed
+
+theorems inf_img_fin_domE = inf_img_fin_dom[THEN bexE]
+
+
+subsection "Infinitely Many and Almost All"
+
+text {*
+  We often need to reason about the existence of infinitely many
+  (resp., all but finitely many) objects satisfying some predicate,
+  so we introduce corresponding binders and their proof rules.
+*}
+
+consts
+  Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool"      (binder "INF " 10)
+  Alm_all  :: "('a \<Rightarrow> bool) \<Rightarrow> bool"      (binder "MOST " 10)
+
+defs
+  INF_def:  "Inf_many P \<equiv> infinite {x. P x}"
+  MOST_def: "Alm_all P \<equiv> \<not>(INF x. \<not> P x)"
+
+syntax (xsymbols)
+  "MOST " :: "[idts, bool] \<Rightarrow> bool"       ("(3\<forall>\<^sub>\<infinity>_./ _)" [0,10] 10)
+  "INF "    :: "[idts, bool] \<Rightarrow> bool"     ("(3\<exists>\<^sub>\<infinity>_./ _)" [0,10] 10)
+
+lemma INF_EX:
+  "(\<exists>\<^sub>\<infinity>x. P x) \<Longrightarrow> (\<exists>x. P x)"
+proof (unfold INF_def, rule ccontr)
+  assume inf: "infinite {x. P x}"
+    and notP: "\<not>(\<exists>x. P x)"
+  from notP have "{x. P x} = {}" by simp
+  hence "finite {x. P x}" by simp
+  with inf show "False" by simp
+qed
+
+lemma MOST_iff_finiteNeg:
+  "(\<forall>\<^sub>\<infinity>x. P x) = finite {x. \<not> P x}"
+by (simp add: MOST_def INF_def)
+
+lemma ALL_MOST:
+  "\<forall>x. P x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. P x"
+by (simp add: MOST_iff_finiteNeg)
+
+lemma INF_mono:
+  assumes inf: "\<exists>\<^sub>\<infinity>x. P x" and q: "\<And>x. P x \<Longrightarrow> Q x"
+  shows "\<exists>\<^sub>\<infinity>x. Q x"
+proof -
+  from inf have "infinite {x. P x}" by (unfold INF_def)
+  moreover from q have "{x. P x} \<subseteq> {x. Q x}" by auto
+  ultimately show ?thesis
+    by (simp add: INF_def infinite_super)
+qed
+
+lemma MOST_mono:
+  "\<lbrakk> \<forall>\<^sub>\<infinity>x. P x; \<And>x. P x \<Longrightarrow> Q x \<rbrakk> \<Longrightarrow> \<forall>\<^sub>\<infinity>x. Q x"
+by (unfold MOST_def, blast intro: INF_mono)
+
+lemma INF_nat: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) = (\<forall>m. \<exists>n. m<n \<and> P n)"
+by (simp add: INF_def infinite_nat_iff_unbounded)
+
+lemma INF_nat_le: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) = (\<forall>m. \<exists>n. m\<le>n \<and> P n)"
+by (simp add: INF_def infinite_nat_iff_unbounded_le)
+
+lemma MOST_nat: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) = (\<exists>m. \<forall>n. m<n \<longrightarrow> P n)"
+by (simp add: MOST_def INF_nat)
+
+lemma MOST_nat_le: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) = (\<exists>m. \<forall>n. m\<le>n \<longrightarrow> P n)"
+by (simp add: MOST_def INF_nat_le)
+
+
+subsection "Miscellaneous"
+
+text {*
+  A few trivial lemmas about sets that contain at most one element.
+  These simplify the reasoning about deterministic automata.
+*}
+
+constdefs
+  atmost_one :: "'a set \<Rightarrow> bool"
+  "atmost_one S \<equiv> \<forall>x y. x\<in>S \<and> y\<in>S \<longrightarrow> x=y"
+
+lemma atmost_one_empty: "S={} \<Longrightarrow> atmost_one S"
+by (simp add: atmost_one_def)
+
+lemma atmost_one_singleton: "S = {x} \<Longrightarrow> atmost_one S"
+by (simp add: atmost_one_def)
+
+lemma atmost_one_unique [elim]: "\<lbrakk> atmost_one S; x \<in> S; y \<in> S \<rbrakk> \<Longrightarrow> y=x"
+by (simp add: atmost_one_def)
+
+end