--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Infinite_Set.thy Mon Mar 08 11:11:58 2004 +0100
@@ -0,0 +1,427 @@
+(* 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
--- a/src/HOL/IsaMakefile Sat Mar 06 19:32:21 2004 +0100
+++ b/src/HOL/IsaMakefile Mon Mar 08 11:11:58 2004 +0100
@@ -84,7 +84,7 @@
Divides.thy Extraction.thy Finite_Set.ML Finite_Set.thy \
Fun.thy Gfp.ML Gfp.thy \
Hilbert_Choice.thy Hilbert_Choice_lemmas.ML HOL.ML \
- HOL.thy HOL_lemmas.ML Inductive.thy Integ/Bin.thy \
+ HOL.thy HOL_lemmas.ML Inductive.thy Infinite_Set.thy Integ/Bin.thy \
Integ/cooper_dec.ML Integ/cooper_proof.ML \
Integ/Equiv.thy Integ/IntArith.thy Integ/IntDef.thy \
Integ/IntDiv.thy Integ/NatBin.thy Integ/NatSimprocs.thy Integ/Parity.thy \