(*  Title:      HOL/Library/Infinite_Set.thy

    Author:     Stephan Merz

*)

header {* Infinite Sets and Related Concepts *}

theory Infinite_Set

imports Main

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_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}"  (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

(* duplicates Finite_Set.infinite_UNIV_nat *)

lemma nat_infinite: "infinite (UNIV :: nat set)"

  by (auto simp add: infinite_nat_iff_unbounded)

lemma nat_not_finite: "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 "finite (range f)" and "inj f"

  then have "finite (UNIV::nat set)"

    by (rule finite_imageD)

  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

  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 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)"

  by (erule contrapos_pp, simp)

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

  by (clarify, drule (1) finite_vimageI, simp)

lemma MOST_inj:

  "MOST x. P x \<Longrightarrow> inj f \<Longrightarrow> MOST x. P (f x)"

  unfolding MOST_iff_cofinite

  by (drule (1) finite_vimageI, simp)

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)) = (\<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)) = (\<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 INFM_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 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 (fastsimp intro: LeastI dest!: infinite_imp_nonempty)

apply simp

apply (metis Collect_def Collect_mem_eq 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

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