diff -r 96d413f78870 -r 6c4fd0b4b63a src/HOL/Library/Infinite_Set.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Infinite_Set.thy	Sun Oct 01 18:29:26 2006 +0200
@@ -0,0 +1,468 @@
+(*  Title:      HOL/Infnite_Set.thy
+    ID:         $Id$
+    Author:     Stephan Merz
+*)
+
+header {* Infinite Sets and Related Concepts *}
+
+theory Infinite_Set
+imports Hilbert_Choice Binomial
+begin
+
+subsection "Infinite Sets"
+
+text {*
+  Some elementary facts about infinite sets, mostly by Stefan Merz.
+  Beware! Because "infinite" merely abbreviates a negation, these
+  lemmas may not work well with @{text "blast"}.
+*}
+
+abbreviation
+  infinite :: "'a set \<Rightarrow> bool"
+  "infinite S == \<not> finite S"
+
+text {*
+  Infinite sets are non-empty, and if we remove some elements from an
+  infinite set, the result is still infinite.
+*}
+
+lemma infinite_imp_nonempty: "infinite S ==> S \<noteq> {}"
+  by auto
+
+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
+  assume "finite 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
+  then show "\<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
+    then show ?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
+  then show ?rhs by (rule finite_nat_bounded)
+next
+  assume ?rhs
+  then obtain k where "S \<subseteq> {..<k}" ..
+  then show "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)
+  then have "S \<subseteq> {..k}" by auto
+  then show ?rhs ..
+next
+  assume ?rhs
+  then obtain k where "S \<subseteq> {..k}" ..
+  then show "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 ?lhs
+  show ?rhs
+  proof (rule ccontr)
+    assume "\<not> ?rhs"
+    then obtain m where m: "\<forall>n. m<n \<longrightarrow> n\<notin>S" by blast
+    then have "S \<subseteq> {..m}"
+      by (auto simp add: sym [OF linorder_not_less])
+    with `?lhs` show False
+      by (simp add: finite_nat_iff_bounded_le)
+  qed
+next
+  assume ?rhs
+  show ?lhs
+  proof
+    assume "finite S"
+    then obtain m where "S \<subseteq> {..m}"
+      by (auto simp add: finite_nat_iff_bounded_le)
+    then have "\<forall>n. m<n \<longrightarrow> n\<notin>S" by auto
+    with `?rhs` 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 ?lhs
+  show ?rhs
+  proof
+    fix m
+    from `?lhs` obtain n where "m<n \<and> n\<in>S"
+      by (auto simp add: infinite_nat_iff_unbounded)
+    then have "m\<le>n \<and> n\<in>S" by simp
+    then show "\<exists>n. m \<le> n \<and> n \<in> S" ..
+  qed
+next
+  assume ?rhs
+  show ?lhs
+  proof (auto simp add: infinite_nat_iff_unbounded)
+    fix m
+    from `?rhs` obtain n where "Suc m \<le> n \<and> n\<in>S"
+      by blast
+    then have "m<n \<and> n\<in>S" by simp
+    then show "\<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 @{text 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 -
+  {
+    fix m have "\<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
+      then show ?thesis ..
+    qed
+  }
+  then show ?thesis
+    by (auto simp add: infinite_nat_iff_unbounded)
+qed
+
+lemma nat_infinite [simp]: "infinite (UNIV :: nat set)"
+  by (auto simp add: infinite_nat_iff_unbounded)
+
+lemma 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 @{text S} is infinite if and only if there exists an
+  injective function from the naturals into @{text S}.
+*}
+
+lemma range_inj_infinite:
+  "inj (f::nat \<Rightarrow> 'a) \<Longrightarrow> infinite (range f)"
+proof
+  assume "inj f"
+    and  "finite (range f)"
+  then have "finite (UNIV::nat set)"
+    by (auto intro: finite_imageD simp del: nat_infinite)
+  then show 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 @{text S}. The idea is to construct a sequence of
+  non-empty and infinite subsets of @{text S} obtained by successively
+  removing elements of @{text S}.
+*}
+
+lemma linorder_injI:
+  assumes hyp: "!!x y. x < (y::'a::linorder) ==> 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"
+    then show ?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 - {SOME e. e \<in> T})"
+  def pick \<equiv> "\<lambda>n. (SOME 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)" then show "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)" .
+      then have "Sseq n \<noteq> {}" by auto
+      then show "\<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)
+    fix i j :: nat
+    assume "i < j"
+    show "pick i \<noteq> pick j"
+    proof
+      assume eq: "pick i = pick j"
+      from `i < j` 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
+
+lemma infinite_iff_countable_subset:
+    "infinite S = (\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S)"
+  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.
+*}
+
+lemma 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> ?thesis"
+  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
+
+lemma inf_img_fin_domE:
+  assumes "finite (f`A)" and "infinite A"
+  obtains y where "y \<in> f`A" and "infinite (f -` {y})"
+  using prems by (blast dest: inf_img_fin_dom)
+
+
+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.
+*}
+
+definition
+  Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool"      (binder "INF " 10)
+  "Inf_many P = infinite {x. P x}"
+  Alm_all  :: "('a \<Rightarrow> bool) \<Rightarrow> bool"      (binder "MOST " 10)
+  "Alm_all P = (\<not> (INF x. \<not> P x))"
+
+const_syntax (xsymbols)
+  Inf_many  (binder "\<exists>\<^sub>\<infinity>" 10)
+  Alm_all  (binder "\<forall>\<^sub>\<infinity>" 10)
+
+const_syntax (HTML output)
+  Inf_many  (binder "\<exists>\<^sub>\<infinity>" 10)
+  Alm_all  (binder "\<forall>\<^sub>\<infinity>" 10)
+
+lemma INF_EX:
+  "(\<exists>\<^sub>\<infinity>x. P x) \<Longrightarrow> (\<exists>x. P x)"
+  unfolding Inf_many_def
+proof (rule ccontr)
+  assume inf: "infinite {x. P x}"
+  assume "\<not> ?thesis" then have "{x. P x} = {}" by simp
+  then have "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: Alm_all_def Inf_many_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}" unfolding Inf_many_def .
+  moreover from q have "{x. P x} \<subseteq> {x. Q x}" by auto
+  ultimately show ?thesis
+    by (simp add: Inf_many_def infinite_super)
+qed
+
+lemma MOST_mono: "\<forall>\<^sub>\<infinity>x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> \<forall>\<^sub>\<infinity>x. Q x"
+  unfolding Alm_all_def by (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_many_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_many_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: Alm_all_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: Alm_all_def INF_nat_le)
+
+
+subsection "Enumeration of an Infinite Set"
+
+text {*
+  The set's element type must be wellordered (e.g. the natural numbers).
+*}
+
+consts
+  enumerate   :: "'a::wellorder set => (nat => 'a::wellorder)"
+primrec
+  enumerate_0:   "enumerate S 0       = (LEAST n. n \<in> S)"
+  enumerate_Suc: "enumerate S (Suc n) = enumerate (S - {LEAST n. n \<in> S}) n"
+
+lemma enumerate_Suc':
+    "enumerate S (Suc n) = enumerate (S - {enumerate S 0}) n"
+  by simp
+
+lemma enumerate_in_set: "infinite S \<Longrightarrow> enumerate S n : S"
+  apply (induct n arbitrary: S)
+   apply (fastsimp intro: LeastI dest!: infinite_imp_nonempty)
+  apply (fastsimp iff: finite_Diff_singleton)
+  done
+
+declare enumerate_0 [simp del] enumerate_Suc [simp del]
+
+lemma enumerate_step: "infinite S \<Longrightarrow> enumerate S n < enumerate S (Suc n)"
+  apply (induct n arbitrary: S)
+   apply (rule order_le_neq_trans)
+    apply (simp add: enumerate_0 Least_le enumerate_in_set)
+   apply (simp only: enumerate_Suc')
+   apply (subgoal_tac "enumerate (S - {enumerate S 0}) 0 : S - {enumerate S 0}")
+    apply (blast intro: sym)
+   apply (simp add: enumerate_in_set del: Diff_iff)
+  apply (simp add: enumerate_Suc')
+  done
+
+lemma enumerate_mono: "m<n \<Longrightarrow> infinite S \<Longrightarrow> enumerate S m < enumerate S n"
+  apply (erule less_Suc_induct)
+  apply (auto intro: enumerate_step)
+  done
+
+
+subsection "Miscellaneous"
+
+text {*
+  A few trivial lemmas about sets that contain at most one element.
+  These simplify the reasoning about deterministic automata.
+*}
+
+definition
+  atmost_one :: "'a set \<Rightarrow> bool"
+  "atmost_one S = (\<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]: "atmost_one S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> y = x"
+  by (simp add: atmost_one_def)
+
+end