# Theory Infinite_Set

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theory Infinite_Set
imports Main
`(*  Title:      HOL/Library/Infinite_Set.thy    Author:     Stephan Merz*)header {* Infinite Sets and Related Concepts *}theory Infinite_Setimports Mainbeginsubsection "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 => bool" where  "infinite S == ¬ 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 ≠ {}"  by autolemma infinite_remove:  "infinite S ==> infinite (S - {a})"  by simplemma Diff_infinite_finite:  assumes T: "finite T" and S: "infinite S"  shows "infinite (S - T)"  using Tproof induct  from S  show "infinite (S - {})" by autonext  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)qedlemma Un_infinite: "infinite S ==> infinite (S ∪ T)"  by simplemma infinite_Un: "infinite (S ∪ T) <-> infinite S ∨ infinite T"  by simplemma infinite_super:  assumes T: "S ⊆ 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 simpqedtext {*  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 "∃k. S ⊆ {..<k}"  (is "∃k. ?bounded S k")using Sproof induct  have "?bounded {} 0" by simp  then show "∃k. ?bounded {} k" ..next  fix S x  assume "∃k. ?bounded S k"  then obtain k where k: "?bounded S k" ..  show "∃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  qedqedlemma finite_nat_iff_bounded:  "finite (S::nat set) = (∃k. S ⊆ {..<k})"  (is "?lhs = ?rhs")proof  assume ?lhs  then show ?rhs by (rule finite_nat_bounded)next  assume ?rhs  then obtain k where "S ⊆ {..<k}" ..  then show "finite S"    by (rule finite_subset) simpqedlemma finite_nat_iff_bounded_le:  "finite (S::nat set) = (∃k. S ⊆ {..k})"  (is "?lhs = ?rhs")proof  assume ?lhs  then obtain k where "S ⊆ {..<k}"    by (blast dest: finite_nat_bounded)  then have "S ⊆ {..k}" by auto  then show ?rhs ..next  assume ?rhs  then obtain k where "S ⊆ {..k}" ..  then show "finite S"    by (rule finite_subset) simpqedlemma infinite_nat_iff_unbounded:  "infinite (S::nat set) = (∀m. ∃n. m<n ∧ n∈S)"  (is "?lhs = ?rhs")proof  assume ?lhs  show ?rhs  proof (rule ccontr)    assume "¬ ?rhs"    then obtain m where m: "∀n. m<n --> n∉S" by blast    then have "S ⊆ {..m}"      by (auto simp add: sym [OF linorder_not_less])    with `?lhs` show False      by (simp add: finite_nat_iff_bounded_le)  qednext  assume ?rhs  show ?lhs  proof    assume "finite S"    then obtain m where "S ⊆ {..m}"      by (auto simp add: finite_nat_iff_bounded_le)    then have "∀n. m<n --> n∉S" by auto    with `?rhs` show False by blast  qedqedlemma infinite_nat_iff_unbounded_le:  "infinite (S::nat set) = (∀m. ∃n. m≤n ∧ n∈S)"  (is "?lhs = ?rhs")proof  assume ?lhs  show ?rhs  proof    fix m    from `?lhs` obtain n where "m<n ∧ n∈S"      by (auto simp add: infinite_nat_iff_unbounded)    then have "m≤n ∧ n∈S" by simp    then show "∃n. m ≤ n ∧ n ∈ S" ..  qednext  assume ?rhs  show ?lhs  proof (auto simp add: infinite_nat_iff_unbounded)    fix m    from `?rhs` obtain n where "Suc m ≤ n ∧ n∈S"      by blast    then have "m<n ∧ n∈S" by simp    then show "∃n. m < n ∧ n ∈ S" ..  qedqedtext {*  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: "∀m. k<m --> (∃n. m<n ∧ n∈S)"  shows "infinite (S::nat set)"proof -  {    fix m have "∃n. m<n ∧ n∈S"    proof (cases "k<m")      case True      with k show ?thesis by blast    next      case False      from k obtain n where "Suc k < n ∧ n∈S" by auto      with False have "m<n ∧ n∈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) ==> R"  by simptext {*  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 => 'a) ==> 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 simpqedlemma 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 ⊆ (UNIV::int set)" by simp  ultimately show "infinite (UNIV::int set)" by (simp add: infinite_super)qedtext {*  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 ≠ 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 ≠ 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 ≠ f x" by blast    with f_eq show ?thesis by simp  qedqedlemma infinite_countable_subset:  assumes inf: "infinite (S::'a set)"  shows "∃f. inj (f::nat => 'a) ∧ range f ⊆ S"proof -  def Sseq ≡ "nat_rec S (λn T. T - {SOME e. e ∈ T})"  def pick ≡ "λn. (SOME e. e ∈ Sseq n)"  have Sseq_inf: "!!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: "!!n. Sseq n ⊆ S"  proof -    fix n    show "Sseq n ⊆ S"      by (induct n) (auto simp add: Sseq_def)  qed  have Sseq_pick: "!!n. pick n ∈ Sseq n"  proof -    fix n    show "pick n ∈ Sseq n"    proof (unfold pick_def, rule someI_ex)      from Sseq_inf have "infinite (Sseq n)" .      then have "Sseq n ≠ {}" by auto      then show "∃x. x ∈ Sseq n" by auto    qed  qed  with Sseq_S have rng: "range pick ⊆ S"    by auto  have pick_Sseq_gt: "!!n m. pick n ∉ Sseq (n + Suc m)"  proof -    fix n m    show "pick n ∉ Sseq (n + Suc m)"      by (induct m) (auto simp add: Sseq_def pick_def)  qed  have pick_pick: "!!n m. pick n ≠ pick (n + Suc m)"  proof -    fix n m    from Sseq_pick have "pick (n + Suc m) ∈ Sseq (n + Suc m)" .    moreover from pick_Sseq_gt    have "pick n ∉ Sseq (n + Suc m)" .    ultimately show "pick n ≠ 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 ≠ 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 ≠ pick j" by simp      with eq show False by simp    qed  qed  from rng inj show ?thesis by autoqedlemma infinite_iff_countable_subset:    "infinite S = (∃f. inj (f::nat => 'a) ∧ range f ⊆ 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 "∃y ∈ f`A. infinite (f -` {y})"proof (rule ccontr)  assume "¬ ?thesis"  with img have "finite (UN y:f`A. f -` {y})" by blast  moreover have "A ⊆ (UN y:f`A. f -` {y})" by auto  moreover note dom  ultimately show False by (simp add: infinite_super)qedlemma inf_img_fin_domE:  assumes "finite (f`A)" and "infinite A"  obtains y where "y ∈ 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 => bool) => bool"  (binder "INFM " 10) where  "Inf_many P = infinite {x. P x}"definition  Alm_all :: "('a => bool) => bool"  (binder "MOST " 10) where  "Alm_all P = (¬ (INFM x. ¬ P x))"notation (xsymbols)  Inf_many  (binder "∃⇩∞" 10) and  Alm_all  (binder "∀⇩∞" 10)notation (HTML output)  Inf_many  (binder "∃⇩∞" 10) and  Alm_all  (binder "∀⇩∞" 10)lemma INFM_iff_infinite: "(INFM x. P x) <-> infinite {x. P x}"  unfolding Inf_many_def ..lemma MOST_iff_cofinite: "(MOST x. P x) <-> finite {x. ¬ P x}"  unfolding Alm_all_def Inf_many_def by simp(* legacy name *)lemmas MOST_iff_finiteNeg = MOST_iff_cofinitelemma not_INFM [simp]: "¬ (INFM x. P x) <-> (MOST x. ¬ P x)"  unfolding Alm_all_def not_not ..lemma not_MOST [simp]: "¬ (MOST x. P x) <-> (INFM x. ¬ P x)"  unfolding Alm_all_def not_not ..lemma INFM_const [simp]: "(INFM x::'a. P) <-> P ∧ infinite (UNIV::'a set)"  unfolding Inf_many_def by simplemma MOST_const [simp]: "(MOST x::'a. P) <-> P ∨ finite (UNIV::'a set)"  unfolding Alm_all_def by simplemma INFM_EX: "(∃⇩∞x. P x) ==> (∃x. P x)"  by (erule contrapos_pp, simp)lemma ALL_MOST: "∀x. P x ==> ∀⇩∞x. P x"  by simplemma INFM_E: assumes "INFM x. P x" obtains x where "P x"  using INFM_EX [OF assms] by (rule exE)lemma MOST_I: assumes "!!x. P x" shows "MOST x. P x"  using assms by simplemma INFM_mono:  assumes inf: "∃⇩∞x. P x" and q: "!!x. P x ==> Q x"  shows "∃⇩∞x. Q x"proof -  from inf have "infinite {x. P x}" unfolding Inf_many_def .  moreover from q have "{x. P x} ⊆ {x. Q x}" by auto  ultimately show ?thesis    by (simp add: Inf_many_def infinite_super)qedlemma MOST_mono: "∀⇩∞x. P x ==> (!!x. P x ==> Q x) ==> ∀⇩∞x. Q x"  unfolding Alm_all_def by (blast intro: INFM_mono)lemma INFM_disj_distrib:  "(∃⇩∞x. P x ∨ Q x) <-> (∃⇩∞x. P x) ∨ (∃⇩∞x. Q x)"  unfolding Inf_many_def by (simp add: Collect_disj_eq)lemma INFM_imp_distrib:  "(INFM x. P x --> Q x) <-> ((MOST x. P x) --> (INFM x. Q x))"  by (simp only: imp_conv_disj INFM_disj_distrib not_MOST)lemma MOST_conj_distrib:  "(∀⇩∞x. P x ∧ Q x) <-> (∀⇩∞x. P x) ∧ (∀⇩∞x. Q x)"  unfolding Alm_all_def by (simp add: INFM_disj_distrib del: disj_not1)lemma MOST_conjI:  "MOST x. P x ==> MOST x. Q x ==> MOST x. P x ∧ Q x"  by (simp add: MOST_conj_distrib)lemma INFM_conjI:  "INFM x. P x ==> MOST x. Q x ==> INFM x. P x ∧ Q x"  unfolding MOST_iff_cofinite INFM_iff_infinite  apply (drule (1) Diff_infinite_finite)  apply (simp add: Collect_conj_eq Collect_neg_eq)  donelemma MOST_rev_mp:  assumes "∀⇩∞x. P x" and "∀⇩∞x. P x --> Q x"  shows "∀⇩∞x. Q x"proof -  have "∀⇩∞x. P x ∧ (P x --> Q x)"    using assms by (rule MOST_conjI)  thus ?thesis by (rule MOST_mono) simpqedlemma MOST_imp_iff:  assumes "MOST x. P x"  shows "(MOST x. P x --> Q x) <-> (MOST x. Q x)"proof  assume "MOST x. P x --> 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 --> Q x" by (rule MOST_mono) simpqedlemma INFM_MOST_simps [simp]:  "!!P Q. (INFM x. P x ∧ Q) <-> (INFM x. P x) ∧ Q"  "!!P Q. (INFM x. P ∧ Q x) <-> P ∧ (INFM x. Q x)"  "!!P Q. (MOST x. P x ∨ Q) <-> (MOST x. P x) ∨ Q"  "!!P Q. (MOST x. P ∨ Q x) <-> P ∨ (MOST x. Q x)"  "!!P Q. (MOST x. P x --> Q) <-> ((INFM x. P x) --> Q)"  "!!P Q. (MOST x. P --> Q x) <-> (P --> (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) ==> inj f ==> INFM x. P x"  unfolding INFM_iff_infinite  by (clarify, drule (1) finite_vimageI, simp)lemma MOST_inj:  "MOST x. P x ==> inj f ==> 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]:  "¬ (INFM x. x = a)"  "¬ (INFM x. a = x)"  unfolding INFM_iff_infinite by simp_alllemma MOST_neq [simp]:  "MOST x. x ≠ a"  "MOST x. a ≠ x"  unfolding MOST_iff_cofinite by simp_alllemma INFM_neq [simp]:  "(INFM x::'a. x ≠ a) <-> infinite (UNIV::'a set)"  "(INFM x::'a. a ≠ x) <-> infinite (UNIV::'a set)"  unfolding INFM_iff_infinite by simp_alllemma MOST_eq [simp]:  "(MOST x::'a. x = a) <-> finite (UNIV::'a set)"  "(MOST x::'a. a = x) <-> finite (UNIV::'a set)"  unfolding MOST_iff_cofinite by simp_alllemma MOST_eq_imp:  "MOST x. x = a --> P x"  "MOST x. a = x --> P x"  unfolding MOST_iff_cofinite by simp_alltext {* Properties of quantifiers over the naturals. *}lemma INFM_nat: "(∃⇩∞n. P (n::nat)) = (∀m. ∃n. m<n ∧ P n)"  by (simp add: Inf_many_def infinite_nat_iff_unbounded)lemma INFM_nat_le: "(∃⇩∞n. P (n::nat)) = (∀m. ∃n. m≤n ∧ P n)"  by (simp add: Inf_many_def infinite_nat_iff_unbounded_le)lemma MOST_nat: "(∀⇩∞n. P (n::nat)) = (∃m. ∀n. m<n --> P n)"  by (simp add: Alm_all_def INFM_nat)lemma MOST_nat_le: "(∀⇩∞n. P (n::nat)) = (∃m. ∀n. m≤n --> 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 => nat => 'a" where    enumerate_0:   "enumerate S 0       = (LEAST n. n ∈ S)"  | enumerate_Suc: "enumerate S (Suc n) = enumerate (S - {LEAST n. n ∈ S}) n"lemma enumerate_Suc':    "enumerate S (Suc n) = enumerate (S - {enumerate S 0}) n"  by simplemma enumerate_in_set: "infinite S ==> enumerate S n : S"apply (induct n arbitrary: S) apply (fastforce intro: LeastI dest!: infinite_imp_nonempty)apply simpapply (metis DiffE infinite_remove)donedeclare enumerate_0 [simp del] enumerate_Suc [simp del]lemma enumerate_step: "infinite S ==> 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')  donelemma enumerate_mono: "m<n ==> infinite S ==> enumerate S m < enumerate S n"  apply (erule less_Suc_induct)  apply (auto intro: enumerate_step)  donelemma le_enumerate:  assumes S: "infinite S"  shows "n ≤ enumerate S n"  using S proof (induct n)  case (Suc n)  then have "n ≤ enumerate S n" by simp  also note enumerate_mono[of n "Suc n", OF _ `infinite S`]  finally show ?case by simpqed simplemma enumerate_Suc'':  fixes S :: "'a::wellorder set"  shows "infinite S  ==> enumerate S (Suc n) = (LEAST s. s ∈ S ∧ enumerate S n < s)"proof (induct n arbitrary: S)  case 0  then have "∀s∈S. enumerate S 0 ≤ 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) autonext  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)    apply (insert `infinite S`, simp)    by (intro arg_cong[where f=Least] ext)       (auto simp: enumerate_Suc'[symmetric])qedlemma enumerate_Ex:  assumes S: "infinite (S::nat set)"  shows "s ∈ S ==> ∃n. enumerate S n = s"proof (induct s rule: less_induct)  case (less s)  show ?case  proof cases    let ?y = "Max {s'∈S. s' < s}"    assume "∃y∈S. y < s"    then have y: "!!x. ?y < x <-> (∀s'∈S. s' < s --> s' < x)" by (subst Max_less_iff) auto    then have y_in: "?y ∈ {s'∈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 *: "¬ (∃y∈S. y < s)"    then have "∀t∈S. s ≤ t" by auto    with `s ∈ S` show ?thesis      by (auto intro!: exI[of _ 0] Least_equality simp: enumerate_0)  qedqedlemma bij_enumerate:  fixes S :: "nat set"  assumes S: "infinite S"  shows "bij_betw (enumerate S) UNIV S"proof -  have "!!n m. n ≠ m ==> enumerate S n ≠ 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 "∀s∈S. ∃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)qedsubsection "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 => bool" where  "atmost_one S = (∀x y. x∈S ∧ y∈S --> x=y)"lemma atmost_one_empty: "S = {} ==> atmost_one S"  by (simp add: atmost_one_def)lemma atmost_one_singleton: "S = {x} ==> atmost_one S"  by (simp add: atmost_one_def)lemma atmost_one_unique [elim]: "atmost_one S ==> x ∈ S ==> y ∈ S ==> y = x"  by (simp add: atmost_one_def)end`