# HG changeset patch
# User wenzelm
# Date 1159720166 -7200
# Node ID 6c4fd0b4b63ad927b392f95791457a5fddbd0688
# Parent  96d413f78870a0046e83541da209ecd76e8aa3d9
moved theory Infinite_Set to Library;

diff -r 96d413f78870 -r 6c4fd0b4b63a src/HOL/Complex/ROOT.ML
--- a/src/HOL/Complex/ROOT.ML	Sun Oct 01 18:29:25 2006 +0200
+++ b/src/HOL/Complex/ROOT.ML	Sun Oct 01 18:29:26 2006 +0200
@@ -5,6 +5,7 @@
 The Complex Numbers
 *)
 
+no_document use_thy "Infinite_Set";
 with_path "../Real"      use_thy "Float";
 with_path "../Hyperreal" use_thy "Hyperreal";
 time_use_thy "Complex_Main";
diff -r 96d413f78870 -r 6c4fd0b4b63a src/HOL/Hyperreal/Filter.thy
--- a/src/HOL/Hyperreal/Filter.thy	Sun Oct 01 18:29:25 2006 +0200
+++ b/src/HOL/Hyperreal/Filter.thy	Sun Oct 01 18:29:26 2006 +0200
@@ -9,7 +9,7 @@
 header {* Filters and Ultrafilters *}
 
 theory Filter
-imports Zorn
+imports Zorn Infinite_Set
 begin
 
 subsection {* Definitions and basic properties *}
diff -r 96d413f78870 -r 6c4fd0b4b63a src/HOL/Infinite_Set.thy
--- a/src/HOL/Infinite_Set.thy	Sun Oct 01 18:29:25 2006 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,470 +0,0 @@
-(*  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 "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 (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 linorder_not_less])
-    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 @{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 (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 @{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)"
-  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 @{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"
-    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 - {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)" 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 "!!i j. i<(j::nat) ==> pick i \<noteq> pick j"
-    proof
-      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.
-*}
-
-definition
-  Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool"      (binder "INF " 10)
-  INF_def:  "Inf_many P \<equiv> infinite {x. P x}"
-  Alm_all  :: "('a \<Rightarrow> bool) \<Rightarrow> bool"      (binder "MOST " 10)
-  MOST_def: "Alm_all P \<equiv> \<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_def
-proof (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 "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 [rule_format]: "\<forall>S. infinite S --> enumerate S n : S"
-apply (induct n) 
- apply (force intro: LeastI dest!:infinite_imp_nonempty)
-apply (auto iff: finite_Diff_singleton) 
-done
-
-declare enumerate_0 [simp del] enumerate_Suc [simp del]
-
-lemma enumerate_step [rule_format]:
-     "\<forall>S. infinite S --> enumerate S n < enumerate S (Suc n)"
-apply (induct n, clarify) 
- 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; infinite S|] ==> 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 \<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
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
diff -r 96d413f78870 -r 6c4fd0b4b63a src/HOL/Library/Library.thy
--- a/src/HOL/Library/Library.thy	Sun Oct 01 18:29:25 2006 +0200
+++ b/src/HOL/Library/Library.thy	Sun Oct 01 18:29:26 2006 +0200
@@ -1,3 +1,4 @@
+(* $Id$ *)
 (*<*)
 theory Library
 imports
@@ -23,6 +24,7 @@
   Commutative_Ring
   Coinductive_List
   AssocList
+  Infinite_Set
 begin
 end
 (*>*)
diff -r 96d413f78870 -r 6c4fd0b4b63a src/HOL/Nominal/Nominal.thy
--- a/src/HOL/Nominal/Nominal.thy	Sun Oct 01 18:29:25 2006 +0200
+++ b/src/HOL/Nominal/Nominal.thy	Sun Oct 01 18:29:26 2006 +0200
@@ -1,7 +1,7 @@
 (* $Id$ *)
 
 theory Nominal 
-imports Main
+imports Main Infinite_Set
 uses
   ("nominal_atoms.ML")
   ("nominal_package.ML")
diff -r 96d413f78870 -r 6c4fd0b4b63a src/HOL/Nominal/ROOT.ML
--- a/src/HOL/Nominal/ROOT.ML	Sun Oct 01 18:29:25 2006 +0200
+++ b/src/HOL/Nominal/ROOT.ML	Sun Oct 01 18:29:26 2006 +0200
@@ -5,4 +5,5 @@
 The nominal datatype package.
 *)
 
+no_document use_thy "Infinite_Set";
 use_thy "Nominal";
diff -r 96d413f78870 -r 6c4fd0b4b63a src/HOL/NumberTheory/Finite2.thy
--- a/src/HOL/NumberTheory/Finite2.thy	Sun Oct 01 18:29:25 2006 +0200
+++ b/src/HOL/NumberTheory/Finite2.thy	Sun Oct 01 18:29:26 2006 +0200
@@ -6,7 +6,7 @@
 header {*Finite Sets and Finite Sums*}
 
 theory Finite2
-imports IntFact
+imports IntFact Infinite_Set
 begin
 
 text{*
diff -r 96d413f78870 -r 6c4fd0b4b63a src/HOL/NumberTheory/ROOT.ML
--- a/src/HOL/NumberTheory/ROOT.ML	Sun Oct 01 18:29:25 2006 +0200
+++ b/src/HOL/NumberTheory/ROOT.ML	Sun Oct 01 18:29:26 2006 +0200
@@ -1,5 +1,6 @@
 (* $Id$ *)
 
+no_document use_thy "Infinite_Set";
 no_document use_thy "Permutation";
 no_document use_thy "Primes";
 
diff -r 96d413f78870 -r 6c4fd0b4b63a src/HOL/Reconstruction.thy
--- a/src/HOL/Reconstruction.thy	Sun Oct 01 18:29:25 2006 +0200
+++ b/src/HOL/Reconstruction.thy	Sun Oct 01 18:29:26 2006 +0200
@@ -7,7 +7,7 @@
 header{* Reconstructing external resolution proofs *}
 
 theory Reconstruction
-imports Hilbert_Choice Map Infinite_Set Extraction
+imports Hilbert_Choice Map Extraction
 uses 	 "Tools/polyhash.ML"
 	 "Tools/ATP/AtpCommunication.ML"
 	 "Tools/ATP/watcher.ML"