(* Title: HOL/Library/Infinite_Set.thy
Author: Stephan Merz
*)
section {* Infinite Sets and Related Concepts *}
theory Infinite_Set
imports Main
begin
subsection "Infinite Sets"
text {*
Some elementary facts about infinite sets, mostly by Stephan Merz.
Beware! Because "infinite" merely abbreviates a negation, these
lemmas may not work well with @{text "blast"}.
*}
abbreviation infinite :: "'a set \<Rightarrow> bool"
where "infinite S \<equiv> \<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 \<Longrightarrow> 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_Un: "infinite (S \<union> T) \<longleftrightarrow> infinite S \<or> infinite 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}"
proof cases
assume "S \<noteq> {}" with Max_less_iff[OF S this] show ?thesis
by auto
qed simp
lemma finite_nat_iff_bounded: "finite (S::nat set) \<longleftrightarrow> (\<exists>k. S \<subseteq> {..<k})"
by (auto intro: finite_nat_bounded dest: finite_subset)
lemma finite_nat_iff_bounded_le:
"finite (S::nat set) \<longleftrightarrow> (\<exists>k. S \<subseteq> {..k})" (is "?lhs \<longleftrightarrow> ?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) \<longleftrightarrow> (\<forall>m. \<exists>n. m < n \<and> n \<in> S)"
unfolding finite_nat_iff_bounded_le by (auto simp: subset_eq not_le)
lemma infinite_nat_iff_unbounded_le:
"infinite (S::nat set) \<longleftrightarrow> (\<forall>m. \<exists>n. m \<le> n \<and> n \<in> S)"
unfolding finite_nat_iff_bounded by (auto simp: subset_eq not_less)
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_not_finite: "finite (UNIV::nat set) \<Longrightarrow> R"
by simp
lemma range_inj_infinite:
"inj (f::nat \<Rightarrow> 'a) \<Longrightarrow> infinite (range f)"
proof
assume "finite (range f)" and "inj f"
then have "finite (UNIV::nat set)"
by (rule finite_imageD)
then show False by simp
qed
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} \<inter> A)"
proof (rule ccontr)
have "A \<subseteq> (\<Union>y\<in>f ` A. f -` {y} \<inter> A)" by auto
moreover
assume "\<not> ?thesis"
with img have "finite (\<Union>y\<in>f ` A. f -` {y} \<inter> A)" by blast
ultimately have "finite A" by(rule finite_subset)
with dom show False by contradiction
qed
lemma inf_img_fin_domE':
assumes "finite (f ` A)" and "infinite A"
obtains y where "y \<in> f`A" and "infinite (f -` {y} \<inter> A)"
using assms by (blast dest: inf_img_fin_dom')
lemma inf_img_fin_dom:
assumes img: "finite (f`A)" and dom: "infinite A"
shows "\<exists>y \<in> f`A. infinite (f -` {y})"
using inf_img_fin_dom'[OF assms] by auto
lemma inf_img_fin_domE:
assumes "finite (f`A)" and "infinite A"
obtains y where "y \<in> f`A" and "infinite (f -` {y})"
using assms 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 "INFM " 10)
where "Inf_many P \<longleftrightarrow> infinite {x. P x}"
definition Alm_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "MOST " 10)
where "Alm_all P \<longleftrightarrow> \<not> (INFM x. \<not> P x)"
notation (xsymbols)
Inf_many (binder "\<exists>\<^sub>\<infinity>" 10) and
Alm_all (binder "\<forall>\<^sub>\<infinity>" 10)
notation (HTML output)
Inf_many (binder "\<exists>\<^sub>\<infinity>" 10) and
Alm_all (binder "\<forall>\<^sub>\<infinity>" 10)
lemma INFM_iff_infinite: "(INFM x. P x) \<longleftrightarrow> infinite {x. P x}"
unfolding Inf_many_def ..
lemma MOST_iff_cofinite: "(MOST x. P x) \<longleftrightarrow> finite {x. \<not> P x}"
unfolding Alm_all_def Inf_many_def by simp
(* legacy name *)
lemmas MOST_iff_finiteNeg = MOST_iff_cofinite
lemma not_INFM [simp]: "\<not> (INFM x. P x) \<longleftrightarrow> (MOST x. \<not> P x)"
unfolding Alm_all_def not_not ..
lemma not_MOST [simp]: "\<not> (MOST x. P x) \<longleftrightarrow> (INFM x. \<not> P x)"
unfolding Alm_all_def not_not ..
lemma INFM_const [simp]: "(INFM x::'a. P) \<longleftrightarrow> P \<and> infinite (UNIV::'a set)"
unfolding Inf_many_def by simp
lemma MOST_const [simp]: "(MOST x::'a. P) \<longleftrightarrow> P \<or> finite (UNIV::'a set)"
unfolding Alm_all_def by simp
lemma INFM_EX: "(\<exists>\<^sub>\<infinity>x. P x) \<Longrightarrow> (\<exists>x. P x)"
apply (erule contrapos_pp)
apply simp
done
lemma ALL_MOST: "\<forall>x. P x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. P x"
by simp
lemma INFM_E:
assumes "INFM x. P x"
obtains x where "P x"
using INFM_EX [OF assms] by (rule exE)
lemma MOST_I:
assumes "\<And>x. P x"
shows "MOST x. P x"
using assms by simp
lemma INFM_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: INFM_mono)
lemma INFM_disj_distrib:
"(\<exists>\<^sub>\<infinity>x. P x \<or> Q x) \<longleftrightarrow> (\<exists>\<^sub>\<infinity>x. P x) \<or> (\<exists>\<^sub>\<infinity>x. Q x)"
unfolding Inf_many_def by (simp add: Collect_disj_eq)
lemma INFM_imp_distrib:
"(INFM x. P x \<longrightarrow> Q x) \<longleftrightarrow> ((MOST x. P x) \<longrightarrow> (INFM x. Q x))"
by (simp only: imp_conv_disj INFM_disj_distrib not_MOST)
lemma MOST_conj_distrib:
"(\<forall>\<^sub>\<infinity>x. P x \<and> Q x) \<longleftrightarrow> (\<forall>\<^sub>\<infinity>x. P x) \<and> (\<forall>\<^sub>\<infinity>x. Q x)"
unfolding Alm_all_def by (simp add: INFM_disj_distrib del: disj_not1)
lemma MOST_conjI:
"MOST x. P x \<Longrightarrow> MOST x. Q x \<Longrightarrow> MOST x. P x \<and> Q x"
by (simp add: MOST_conj_distrib)
lemma INFM_conjI:
"INFM x. P x \<Longrightarrow> MOST x. Q x \<Longrightarrow> INFM x. P x \<and> Q x"
unfolding MOST_iff_cofinite INFM_iff_infinite
apply (drule (1) Diff_infinite_finite)
apply (simp add: Collect_conj_eq Collect_neg_eq)
done
lemma MOST_rev_mp:
assumes "\<forall>\<^sub>\<infinity>x. P x" and "\<forall>\<^sub>\<infinity>x. P x \<longrightarrow> Q x"
shows "\<forall>\<^sub>\<infinity>x. Q x"
proof -
have "\<forall>\<^sub>\<infinity>x. P x \<and> (P x \<longrightarrow> Q x)"
using assms by (rule MOST_conjI)
thus ?thesis by (rule MOST_mono) simp
qed
lemma MOST_imp_iff:
assumes "MOST x. P x"
shows "(MOST x. P x \<longrightarrow> Q x) \<longleftrightarrow> (MOST x. Q x)"
proof
assume "MOST x. P x \<longrightarrow> Q x"
with assms show "MOST x. Q x" by (rule MOST_rev_mp)
next
assume "MOST x. Q x"
then show "MOST x. P x \<longrightarrow> Q x" by (rule MOST_mono) simp
qed
lemma INFM_MOST_simps [simp]:
"\<And>P Q. (INFM x. P x \<and> Q) \<longleftrightarrow> (INFM x. P x) \<and> Q"
"\<And>P Q. (INFM x. P \<and> Q x) \<longleftrightarrow> P \<and> (INFM x. Q x)"
"\<And>P Q. (MOST x. P x \<or> Q) \<longleftrightarrow> (MOST x. P x) \<or> Q"
"\<And>P Q. (MOST x. P \<or> Q x) \<longleftrightarrow> P \<or> (MOST x. Q x)"
"\<And>P Q. (MOST x. P x \<longrightarrow> Q) \<longleftrightarrow> ((INFM x. P x) \<longrightarrow> Q)"
"\<And>P Q. (MOST x. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (MOST x. Q x))"
unfolding Alm_all_def Inf_many_def
by (simp_all add: Collect_conj_eq)
text {* Properties of quantifiers with injective functions. *}
lemma INFM_inj: "INFM x. P (f x) \<Longrightarrow> inj f \<Longrightarrow> INFM x. P x"
unfolding INFM_iff_infinite
apply clarify
apply (drule (1) finite_vimageI)
apply simp
done
lemma MOST_inj: "MOST x. P x \<Longrightarrow> inj f \<Longrightarrow> MOST x. P (f x)"
unfolding MOST_iff_cofinite
apply (drule (1) finite_vimageI)
apply simp
done
text {* Properties of quantifiers with singletons. *}
lemma not_INFM_eq [simp]:
"\<not> (INFM x. x = a)"
"\<not> (INFM x. a = x)"
unfolding INFM_iff_infinite by simp_all
lemma MOST_neq [simp]:
"MOST x. x \<noteq> a"
"MOST x. a \<noteq> x"
unfolding MOST_iff_cofinite by simp_all
lemma INFM_neq [simp]:
"(INFM x::'a. x \<noteq> a) \<longleftrightarrow> infinite (UNIV::'a set)"
"(INFM x::'a. a \<noteq> x) \<longleftrightarrow> infinite (UNIV::'a set)"
unfolding INFM_iff_infinite by simp_all
lemma MOST_eq [simp]:
"(MOST x::'a. x = a) \<longleftrightarrow> finite (UNIV::'a set)"
"(MOST x::'a. a = x) \<longleftrightarrow> finite (UNIV::'a set)"
unfolding MOST_iff_cofinite by simp_all
lemma MOST_eq_imp:
"MOST x. x = a \<longrightarrow> P x"
"MOST x. a = x \<longrightarrow> P x"
unfolding MOST_iff_cofinite by simp_all
text {* Properties of quantifiers over the naturals. *}
lemma INFM_nat: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) \<longleftrightarrow> (\<forall>m. \<exists>n. m < n \<and> P n)"
by (simp add: Inf_many_def infinite_nat_iff_unbounded)
lemma INFM_nat_le: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) \<longleftrightarrow> (\<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)) \<longleftrightarrow> (\<exists>m. \<forall>n. m < n \<longrightarrow> P n)"
by (simp add: Alm_all_def INFM_nat)
lemma MOST_nat_le: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) \<longleftrightarrow> (\<exists>m. \<forall>n. m \<le> n \<longrightarrow> P n)"
by (simp add: Alm_all_def INFM_nat_le)
subsection "Enumeration of an Infinite Set"
text {*
The set's element type must be wellordered (e.g. the natural numbers).
*}
primrec (in wellorder) enumerate :: "'a set \<Rightarrow> nat \<Rightarrow> 'a"
where
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 (fastforce intro: LeastI dest!: infinite_imp_nonempty)
apply simp
apply (metis DiffE infinite_remove)
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
lemma le_enumerate:
assumes S: "infinite S"
shows "n \<le> enumerate S n"
using S
proof (induct n)
case 0
then show ?case by simp
next
case (Suc n)
then have "n \<le> enumerate S n" by simp
also note enumerate_mono[of n "Suc n", OF _ `infinite S`]
finally show ?case by simp
qed
lemma enumerate_Suc'':
fixes S :: "'a::wellorder set"
assumes "infinite S"
shows "enumerate S (Suc n) = (LEAST s. s \<in> S \<and> enumerate S n < s)"
using assms
proof (induct n arbitrary: S)
case 0
then have "\<forall>s \<in> S. enumerate S 0 \<le> s"
by (auto simp: enumerate.simps intro: Least_le)
then show ?case
unfolding enumerate_Suc' enumerate_0[of "S - {enumerate S 0}"]
by (intro arg_cong[where f = Least] ext) auto
next
case (Suc n S)
show ?case
using enumerate_mono[OF zero_less_Suc `infinite S`, of n] `infinite S`
apply (subst (1 2) enumerate_Suc')
apply (subst Suc)
using `infinite S`
apply simp
apply (intro arg_cong[where f = Least] ext)
apply (auto simp: enumerate_Suc'[symmetric])
done
qed
lemma enumerate_Ex:
assumes S: "infinite (S::nat set)"
shows "s \<in> S \<Longrightarrow> \<exists>n. enumerate S n = s"
proof (induct s rule: less_induct)
case (less s)
show ?case
proof cases
let ?y = "Max {s'\<in>S. s' < s}"
assume "\<exists>y\<in>S. y < s"
then have y: "\<And>x. ?y < x \<longleftrightarrow> (\<forall>s'\<in>S. s' < s \<longrightarrow> s' < x)"
by (subst Max_less_iff) auto
then have y_in: "?y \<in> {s'\<in>S. s' < s}"
by (intro Max_in) auto
with less.hyps[of ?y] obtain n where "enumerate S n = ?y"
by auto
with S have "enumerate S (Suc n) = s"
by (auto simp: y less enumerate_Suc'' intro!: Least_equality)
then show ?case by auto
next
assume *: "\<not> (\<exists>y\<in>S. y < s)"
then have "\<forall>t\<in>S. s \<le> t" by auto
with `s \<in> S` show ?thesis
by (auto intro!: exI[of _ 0] Least_equality simp: enumerate_0)
qed
qed
lemma bij_enumerate:
fixes S :: "nat set"
assumes S: "infinite S"
shows "bij_betw (enumerate S) UNIV S"
proof -
have "\<And>n m. n \<noteq> m \<Longrightarrow> enumerate S n \<noteq> enumerate S m"
using enumerate_mono[OF _ `infinite S`] by (auto simp: neq_iff)
then have "inj (enumerate S)"
by (auto simp: inj_on_def)
moreover have "\<forall>s \<in> S. \<exists>i. enumerate S i = s"
using enumerate_Ex[OF S] by auto
moreover note `infinite S`
ultimately show ?thesis
unfolding bij_betw_def by (auto intro: enumerate_in_set)
qed
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"
where "atmost_one S \<longleftrightarrow> (\<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