(* 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 Stephan 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 auto lemma infinite_remove: "infinite S ==> 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 ==> infinite (S ∪ T)" by simp lemma infinite_Un: "infinite (S ∪ T) <-> infinite S ∨ infinite T" by simp lemma 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 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 "∃k. S ⊆ {..<k}" (is "∃k. ?bounded S k") using S proof 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 qed qed lemma 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) simp qed lemma 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) simp qed lemma 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) qed next 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 qed qed lemma 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" .. qed next 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" .. 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: "∀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 lemma nat_not_finite: "finite (UNIV::nat set) ==> R" by simp 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 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 "∃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) qed lemma 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_cofinite lemma 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 simp lemma MOST_const [simp]: "(MOST x::'a. P) <-> P ∨ finite (UNIV::'a set)" unfolding Alm_all_def by simp lemma INFM_EX: "(∃⇩_{∞}x. P x) ==> (∃x. P x)" apply (erule contrapos_pp) apply simp done lemma ALL_MOST: "∀x. P x ==> ∀⇩_{∞}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 "!!x. P x" shows "MOST x. P x" using assms by simp lemma 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) qed lemma 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) done lemma 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) simp qed lemma 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) simp qed lemma 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 apply clarify apply (drule (1) finite_vimageI) apply simp done lemma MOST_inj: "MOST x. P x ==> inj f ==> 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]: "¬ (INFM x. x = a)" "¬ (INFM x. a = x)" unfolding INFM_iff_infinite by simp_all lemma MOST_neq [simp]: "MOST x. x ≠ a" "MOST x. a ≠ x" unfolding MOST_iff_cofinite by simp_all lemma 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_all lemma 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_all lemma MOST_eq_imp: "MOST x. x = a --> P x" "MOST x. a = x --> P x" unfolding MOST_iff_cofinite by simp_all text {* 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 simp lemma enumerate_in_set: "infinite S ==> 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 ==> 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 ==> infinite S ==> 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 ≤ enumerate S n" using S proof (induct n) case 0 then show ?case by simp next 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 simp qed lemma enumerate_Suc'': fixes S :: "'a::wellorder set" assumes "infinite S" shows "enumerate S (Suc n) = (LEAST s. s ∈ S ∧ enumerate S n < s)" using assms 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) 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 ∈ 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) qed qed lemma 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) 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 => 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