src/HOL/Library/Infinite_Set.thy
 author wenzelm Mon Dec 28 17:43:30 2015 +0100 (2015-12-28) changeset 61952 546958347e05 parent 61945 1135b8de26c3 child 63492 a662e8139804 permissions -rw-r--r--
prefer symbols for "Union", "Inter";
```     1 (*  Title:      HOL/Library/Infinite_Set.thy
```
```     2     Author:     Stephan Merz
```
```     3 *)
```
```     4
```
```     5 section \<open>Infinite Sets and Related Concepts\<close>
```
```     6
```
```     7 theory Infinite_Set
```
```     8 imports Main
```
```     9 begin
```
```    10
```
```    11 text \<open>The set of natural numbers is infinite.\<close>
```
```    12
```
```    13 lemma infinite_nat_iff_unbounded_le: "infinite (S::nat set) \<longleftrightarrow> (\<forall>m. \<exists>n\<ge>m. n \<in> S)"
```
```    14   using frequently_cofinite[of "\<lambda>x. x \<in> S"]
```
```    15   by (simp add: cofinite_eq_sequentially frequently_def eventually_sequentially)
```
```    16
```
```    17 lemma infinite_nat_iff_unbounded: "infinite (S::nat set) \<longleftrightarrow> (\<forall>m. \<exists>n>m. n \<in> S)"
```
```    18   using frequently_cofinite[of "\<lambda>x. x \<in> S"]
```
```    19   by (simp add: cofinite_eq_sequentially frequently_def eventually_at_top_dense)
```
```    20
```
```    21 lemma finite_nat_iff_bounded: "finite (S::nat set) \<longleftrightarrow> (\<exists>k. S \<subseteq> {..<k})"
```
```    22   using infinite_nat_iff_unbounded_le[of S] by (simp add: subset_eq) (metis not_le)
```
```    23
```
```    24 lemma finite_nat_iff_bounded_le: "finite (S::nat set) \<longleftrightarrow> (\<exists>k. S \<subseteq> {.. k})"
```
```    25   using infinite_nat_iff_unbounded[of S] by (simp add: subset_eq) (metis not_le)
```
```    26
```
```    27 lemma finite_nat_bounded: "finite (S::nat set) \<Longrightarrow> \<exists>k. S \<subseteq> {..<k}"
```
```    28   by (simp add: finite_nat_iff_bounded)
```
```    29
```
```    30
```
```    31 text \<open>
```
```    32   For a set of natural numbers to be infinite, it is enough to know
```
```    33   that for any number larger than some \<open>k\<close>, there is some larger
```
```    34   number that is an element of the set.
```
```    35 \<close>
```
```    36
```
```    37 lemma unbounded_k_infinite: "\<forall>m>k. \<exists>n>m. n \<in> S \<Longrightarrow> infinite (S::nat set)"
```
```    38 apply (clarsimp simp add: finite_nat_set_iff_bounded)
```
```    39 apply (drule_tac x="Suc (max m k)" in spec)
```
```    40 using less_Suc_eq by fastforce
```
```    41
```
```    42 lemma nat_not_finite: "finite (UNIV::nat set) \<Longrightarrow> R"
```
```    43   by simp
```
```    44
```
```    45 lemma range_inj_infinite:
```
```    46   "inj (f::nat \<Rightarrow> 'a) \<Longrightarrow> infinite (range f)"
```
```    47 proof
```
```    48   assume "finite (range f)" and "inj f"
```
```    49   then have "finite (UNIV::nat set)"
```
```    50     by (rule finite_imageD)
```
```    51   then show False by simp
```
```    52 qed
```
```    53
```
```    54 text \<open>The set of integers is also infinite.\<close>
```
```    55
```
```    56 lemma infinite_int_iff_infinite_nat_abs: "infinite (S::int set) \<longleftrightarrow> infinite ((nat o abs) ` S)"
```
```    57   by (auto simp: transfer_nat_int_set_relations o_def image_comp dest: finite_image_absD)
```
```    58
```
```    59 proposition infinite_int_iff_unbounded_le: "infinite (S::int set) \<longleftrightarrow> (\<forall>m. \<exists>n. \<bar>n\<bar> \<ge> m \<and> n \<in> S)"
```
```    60   apply (simp add: infinite_int_iff_infinite_nat_abs infinite_nat_iff_unbounded_le o_def image_def)
```
```    61   apply (metis abs_ge_zero nat_le_eq_zle le_nat_iff)
```
```    62   done
```
```    63
```
```    64 proposition infinite_int_iff_unbounded: "infinite (S::int set) \<longleftrightarrow> (\<forall>m. \<exists>n. \<bar>n\<bar> > m \<and> n \<in> S)"
```
```    65   apply (simp add: infinite_int_iff_infinite_nat_abs infinite_nat_iff_unbounded o_def image_def)
```
```    66   apply (metis (full_types) nat_le_iff nat_mono not_le)
```
```    67   done
```
```    68
```
```    69 proposition finite_int_iff_bounded: "finite (S::int set) \<longleftrightarrow> (\<exists>k. abs ` S \<subseteq> {..<k})"
```
```    70   using infinite_int_iff_unbounded_le[of S] by (simp add: subset_eq) (metis not_le)
```
```    71
```
```    72 proposition finite_int_iff_bounded_le: "finite (S::int set) \<longleftrightarrow> (\<exists>k. abs ` S \<subseteq> {.. k})"
```
```    73   using infinite_int_iff_unbounded[of S] by (simp add: subset_eq) (metis not_le)
```
```    74
```
```    75 subsection "Infinitely Many and Almost All"
```
```    76
```
```    77 text \<open>
```
```    78   We often need to reason about the existence of infinitely many
```
```    79   (resp., all but finitely many) objects satisfying some predicate, so
```
```    80   we introduce corresponding binders and their proof rules.
```
```    81 \<close>
```
```    82
```
```    83 (* The following two lemmas are available as filter-rules, but not in the simp-set *)
```
```    84 lemma not_INFM [simp]: "\<not> (INFM x. P x) \<longleftrightarrow> (MOST x. \<not> P x)" by (fact not_frequently)
```
```    85 lemma not_MOST [simp]: "\<not> (MOST x. P x) \<longleftrightarrow> (INFM x. \<not> P x)" by (fact not_eventually)
```
```    86
```
```    87 lemma INFM_const [simp]: "(INFM x::'a. P) \<longleftrightarrow> P \<and> infinite (UNIV::'a set)"
```
```    88   by (simp add: frequently_const_iff)
```
```    89
```
```    90 lemma MOST_const [simp]: "(MOST x::'a. P) \<longleftrightarrow> P \<or> finite (UNIV::'a set)"
```
```    91   by (simp add: eventually_const_iff)
```
```    92
```
```    93 lemma INFM_imp_distrib: "(INFM x. P x \<longrightarrow> Q x) \<longleftrightarrow> ((MOST x. P x) \<longrightarrow> (INFM x. Q x))"
```
```    94   by (simp only: imp_conv_disj frequently_disj_iff not_eventually)
```
```    95
```
```    96 lemma MOST_imp_iff: "MOST x. P x \<Longrightarrow> (MOST x. P x \<longrightarrow> Q x) \<longleftrightarrow> (MOST x. Q x)"
```
```    97   by (auto intro: eventually_rev_mp eventually_mono)
```
```    98
```
```    99 lemma INFM_conjI: "INFM x. P x \<Longrightarrow> MOST x. Q x \<Longrightarrow> INFM x. P x \<and> Q x"
```
```   100   by (rule frequently_rev_mp[of P]) (auto elim: eventually_mono)
```
```   101
```
```   102 text \<open>Properties of quantifiers with injective functions.\<close>
```
```   103
```
```   104 lemma INFM_inj: "INFM x. P (f x) \<Longrightarrow> inj f \<Longrightarrow> INFM x. P x"
```
```   105   using finite_vimageI[of "{x. P x}" f] by (auto simp: frequently_cofinite)
```
```   106
```
```   107 lemma MOST_inj: "MOST x. P x \<Longrightarrow> inj f \<Longrightarrow> MOST x. P (f x)"
```
```   108   using finite_vimageI[of "{x. \<not> P x}" f] by (auto simp: eventually_cofinite)
```
```   109
```
```   110 text \<open>Properties of quantifiers with singletons.\<close>
```
```   111
```
```   112 lemma not_INFM_eq [simp]:
```
```   113   "\<not> (INFM x. x = a)"
```
```   114   "\<not> (INFM x. a = x)"
```
```   115   unfolding frequently_cofinite by simp_all
```
```   116
```
```   117 lemma MOST_neq [simp]:
```
```   118   "MOST x. x \<noteq> a"
```
```   119   "MOST x. a \<noteq> x"
```
```   120   unfolding eventually_cofinite by simp_all
```
```   121
```
```   122 lemma INFM_neq [simp]:
```
```   123   "(INFM x::'a. x \<noteq> a) \<longleftrightarrow> infinite (UNIV::'a set)"
```
```   124   "(INFM x::'a. a \<noteq> x) \<longleftrightarrow> infinite (UNIV::'a set)"
```
```   125   unfolding frequently_cofinite by simp_all
```
```   126
```
```   127 lemma MOST_eq [simp]:
```
```   128   "(MOST x::'a. x = a) \<longleftrightarrow> finite (UNIV::'a set)"
```
```   129   "(MOST x::'a. a = x) \<longleftrightarrow> finite (UNIV::'a set)"
```
```   130   unfolding eventually_cofinite by simp_all
```
```   131
```
```   132 lemma MOST_eq_imp:
```
```   133   "MOST x. x = a \<longrightarrow> P x"
```
```   134   "MOST x. a = x \<longrightarrow> P x"
```
```   135   unfolding eventually_cofinite by simp_all
```
```   136
```
```   137 text \<open>Properties of quantifiers over the naturals.\<close>
```
```   138
```
```   139 lemma MOST_nat: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) \<longleftrightarrow> (\<exists>m. \<forall>n>m. P n)"
```
```   140   by (auto simp add: eventually_cofinite finite_nat_iff_bounded_le subset_eq not_le[symmetric])
```
```   141
```
```   142 lemma MOST_nat_le: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) \<longleftrightarrow> (\<exists>m. \<forall>n\<ge>m. P n)"
```
```   143   by (auto simp add: eventually_cofinite finite_nat_iff_bounded subset_eq not_le[symmetric])
```
```   144
```
```   145 lemma INFM_nat: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) \<longleftrightarrow> (\<forall>m. \<exists>n>m. P n)"
```
```   146   by (simp add: frequently_cofinite infinite_nat_iff_unbounded)
```
```   147
```
```   148 lemma INFM_nat_le: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) \<longleftrightarrow> (\<forall>m. \<exists>n\<ge>m. P n)"
```
```   149   by (simp add: frequently_cofinite infinite_nat_iff_unbounded_le)
```
```   150
```
```   151 lemma MOST_INFM: "infinite (UNIV::'a set) \<Longrightarrow> MOST x::'a. P x \<Longrightarrow> INFM x::'a. P x"
```
```   152   by (simp add: eventually_frequently)
```
```   153
```
```   154 lemma MOST_Suc_iff: "(MOST n. P (Suc n)) \<longleftrightarrow> (MOST n. P n)"
```
```   155   by (simp add: cofinite_eq_sequentially eventually_sequentially_Suc)
```
```   156
```
```   157 lemma
```
```   158   shows MOST_SucI: "MOST n. P n \<Longrightarrow> MOST n. P (Suc n)"
```
```   159     and MOST_SucD: "MOST n. P (Suc n) \<Longrightarrow> MOST n. P n"
```
```   160   by (simp_all add: MOST_Suc_iff)
```
```   161
```
```   162 lemma MOST_ge_nat: "MOST n::nat. m \<le> n"
```
```   163   by (simp add: cofinite_eq_sequentially eventually_ge_at_top)
```
```   164
```
```   165 (* legacy names *)
```
```   166 lemma Inf_many_def: "Inf_many P \<longleftrightarrow> infinite {x. P x}" by (fact frequently_cofinite)
```
```   167 lemma Alm_all_def: "Alm_all P \<longleftrightarrow> \<not> (INFM x. \<not> P x)" by simp
```
```   168 lemma INFM_iff_infinite: "(INFM x. P x) \<longleftrightarrow> infinite {x. P x}" by (fact frequently_cofinite)
```
```   169 lemma MOST_iff_cofinite: "(MOST x. P x) \<longleftrightarrow> finite {x. \<not> P x}" by (fact eventually_cofinite)
```
```   170 lemma INFM_EX: "(\<exists>\<^sub>\<infinity>x. P x) \<Longrightarrow> (\<exists>x. P x)" by (fact frequently_ex)
```
```   171 lemma ALL_MOST: "\<forall>x. P x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. P x" by (fact always_eventually)
```
```   172 lemma INFM_mono: "\<exists>\<^sub>\<infinity>x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> \<exists>\<^sub>\<infinity>x. Q x" by (fact frequently_elim1)
```
```   173 lemma MOST_mono: "\<forall>\<^sub>\<infinity>x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> \<forall>\<^sub>\<infinity>x. Q x" by (fact eventually_mono)
```
```   174 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)" by (fact frequently_disj_iff)
```
```   175 lemma MOST_rev_mp: "\<forall>\<^sub>\<infinity>x. P x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. P x \<longrightarrow> Q x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. Q x" by (fact eventually_rev_mp)
```
```   176 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)" by (fact eventually_conj_iff)
```
```   177 lemma MOST_conjI: "MOST x. P x \<Longrightarrow> MOST x. Q x \<Longrightarrow> MOST x. P x \<and> Q x" by (fact eventually_conj)
```
```   178 lemma INFM_finite_Bex_distrib: "finite A \<Longrightarrow> (INFM y. \<exists>x\<in>A. P x y) \<longleftrightarrow> (\<exists>x\<in>A. INFM y. P x y)" by (fact frequently_bex_finite_distrib)
```
```   179 lemma MOST_finite_Ball_distrib: "finite A \<Longrightarrow> (MOST y. \<forall>x\<in>A. P x y) \<longleftrightarrow> (\<forall>x\<in>A. MOST y. P x y)" by (fact eventually_ball_finite_distrib)
```
```   180 lemma INFM_E: "INFM x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> thesis) \<Longrightarrow> thesis" by (fact frequentlyE)
```
```   181 lemma MOST_I: "(\<And>x. P x) \<Longrightarrow> MOST x. P x" by (rule eventuallyI)
```
```   182 lemmas MOST_iff_finiteNeg = MOST_iff_cofinite
```
```   183
```
```   184 subsection "Enumeration of an Infinite Set"
```
```   185
```
```   186 text \<open>
```
```   187   The set's element type must be wellordered (e.g. the natural numbers).
```
```   188 \<close>
```
```   189
```
```   190 text \<open>
```
```   191   Could be generalized to
```
```   192     @{term "enumerate' S n = (SOME t. t \<in> s \<and> finite {s\<in>S. s < t} \<and> card {s\<in>S. s < t} = n)"}.
```
```   193 \<close>
```
```   194
```
```   195 primrec (in wellorder) enumerate :: "'a set \<Rightarrow> nat \<Rightarrow> 'a"
```
```   196 where
```
```   197   enumerate_0: "enumerate S 0 = (LEAST n. n \<in> S)"
```
```   198 | enumerate_Suc: "enumerate S (Suc n) = enumerate (S - {LEAST n. n \<in> S}) n"
```
```   199
```
```   200 lemma enumerate_Suc': "enumerate S (Suc n) = enumerate (S - {enumerate S 0}) n"
```
```   201   by simp
```
```   202
```
```   203 lemma enumerate_in_set: "infinite S \<Longrightarrow> enumerate S n \<in> S"
```
```   204   apply (induct n arbitrary: S)
```
```   205    apply (fastforce intro: LeastI dest!: infinite_imp_nonempty)
```
```   206   apply simp
```
```   207   apply (metis DiffE infinite_remove)
```
```   208   done
```
```   209
```
```   210 declare enumerate_0 [simp del] enumerate_Suc [simp del]
```
```   211
```
```   212 lemma enumerate_step: "infinite S \<Longrightarrow> enumerate S n < enumerate S (Suc n)"
```
```   213   apply (induct n arbitrary: S)
```
```   214    apply (rule order_le_neq_trans)
```
```   215     apply (simp add: enumerate_0 Least_le enumerate_in_set)
```
```   216    apply (simp only: enumerate_Suc')
```
```   217    apply (subgoal_tac "enumerate (S - {enumerate S 0}) 0 \<in> S - {enumerate S 0}")
```
```   218     apply (blast intro: sym)
```
```   219    apply (simp add: enumerate_in_set del: Diff_iff)
```
```   220   apply (simp add: enumerate_Suc')
```
```   221   done
```
```   222
```
```   223 lemma enumerate_mono: "m < n \<Longrightarrow> infinite S \<Longrightarrow> enumerate S m < enumerate S n"
```
```   224   apply (erule less_Suc_induct)
```
```   225   apply (auto intro: enumerate_step)
```
```   226   done
```
```   227
```
```   228
```
```   229 lemma le_enumerate:
```
```   230   assumes S: "infinite S"
```
```   231   shows "n \<le> enumerate S n"
```
```   232   using S
```
```   233 proof (induct n)
```
```   234   case 0
```
```   235   then show ?case by simp
```
```   236 next
```
```   237   case (Suc n)
```
```   238   then have "n \<le> enumerate S n" by simp
```
```   239   also note enumerate_mono[of n "Suc n", OF _ \<open>infinite S\<close>]
```
```   240   finally show ?case by simp
```
```   241 qed
```
```   242
```
```   243 lemma enumerate_Suc'':
```
```   244   fixes S :: "'a::wellorder set"
```
```   245   assumes "infinite S"
```
```   246   shows "enumerate S (Suc n) = (LEAST s. s \<in> S \<and> enumerate S n < s)"
```
```   247   using assms
```
```   248 proof (induct n arbitrary: S)
```
```   249   case 0
```
```   250   then have "\<forall>s \<in> S. enumerate S 0 \<le> s"
```
```   251     by (auto simp: enumerate.simps intro: Least_le)
```
```   252   then show ?case
```
```   253     unfolding enumerate_Suc' enumerate_0[of "S - {enumerate S 0}"]
```
```   254     by (intro arg_cong[where f = Least] ext) auto
```
```   255 next
```
```   256   case (Suc n S)
```
```   257   show ?case
```
```   258     using enumerate_mono[OF zero_less_Suc \<open>infinite S\<close>, of n] \<open>infinite S\<close>
```
```   259     apply (subst (1 2) enumerate_Suc')
```
```   260     apply (subst Suc)
```
```   261     using \<open>infinite S\<close>
```
```   262     apply simp
```
```   263     apply (intro arg_cong[where f = Least] ext)
```
```   264     apply (auto simp: enumerate_Suc'[symmetric])
```
```   265     done
```
```   266 qed
```
```   267
```
```   268 lemma enumerate_Ex:
```
```   269   assumes S: "infinite (S::nat set)"
```
```   270   shows "s \<in> S \<Longrightarrow> \<exists>n. enumerate S n = s"
```
```   271 proof (induct s rule: less_induct)
```
```   272   case (less s)
```
```   273   show ?case
```
```   274   proof cases
```
```   275     let ?y = "Max {s'\<in>S. s' < s}"
```
```   276     assume "\<exists>y\<in>S. y < s"
```
```   277     then have y: "\<And>x. ?y < x \<longleftrightarrow> (\<forall>s'\<in>S. s' < s \<longrightarrow> s' < x)"
```
```   278       by (subst Max_less_iff) auto
```
```   279     then have y_in: "?y \<in> {s'\<in>S. s' < s}"
```
```   280       by (intro Max_in) auto
```
```   281     with less.hyps[of ?y] obtain n where "enumerate S n = ?y"
```
```   282       by auto
```
```   283     with S have "enumerate S (Suc n) = s"
```
```   284       by (auto simp: y less enumerate_Suc'' intro!: Least_equality)
```
```   285     then show ?case by auto
```
```   286   next
```
```   287     assume *: "\<not> (\<exists>y\<in>S. y < s)"
```
```   288     then have "\<forall>t\<in>S. s \<le> t" by auto
```
```   289     with \<open>s \<in> S\<close> show ?thesis
```
```   290       by (auto intro!: exI[of _ 0] Least_equality simp: enumerate_0)
```
```   291   qed
```
```   292 qed
```
```   293
```
```   294 lemma bij_enumerate:
```
```   295   fixes S :: "nat set"
```
```   296   assumes S: "infinite S"
```
```   297   shows "bij_betw (enumerate S) UNIV S"
```
```   298 proof -
```
```   299   have "\<And>n m. n \<noteq> m \<Longrightarrow> enumerate S n \<noteq> enumerate S m"
```
```   300     using enumerate_mono[OF _ \<open>infinite S\<close>] by (auto simp: neq_iff)
```
```   301   then have "inj (enumerate S)"
```
```   302     by (auto simp: inj_on_def)
```
```   303   moreover have "\<forall>s \<in> S. \<exists>i. enumerate S i = s"
```
```   304     using enumerate_Ex[OF S] by auto
```
```   305   moreover note \<open>infinite S\<close>
```
```   306   ultimately show ?thesis
```
```   307     unfolding bij_betw_def by (auto intro: enumerate_in_set)
```
```   308 qed
```
```   309
```
```   310 end
```
```   311
```