(* Title: HOL/Infnite_Set.thy
ID: $Id$
Author: Stephan Merz
*)
header {* Infinite Sets and Related Concepts *}
theory Infinite_Set
imports ATP_Linkup
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" where
"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
lemma int_infinite [simp]:
shows "infinite (UNIV::int set)"
proof -
from inj_int have "infinite (range int)" by (rule range_inj_infinite)
moreover
have "range int \<subseteq> (UNIV::int set)" by simp
ultimately show "infinite (UNIV::int set)" by (simp add: infinite_super)
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 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 = infinite {x. P x}"
definition
Alm_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "MOST " 10) where
"Alm_all P = (\<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 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" where
"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