src/HOL/Infinite_Set.thy
 author nipkow Mon Aug 16 14:22:27 2004 +0200 (2004-08-16) changeset 15131 c69542757a4d parent 15045 d59f7e2e18d3 child 15140 322485b816ac permissions -rw-r--r--
```     1 (*  Title:      HOL/Infnite_Set.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Stephan Merz
```
```     4 *)
```
```     5
```
```     6 header {* Infnite Sets and Related Concepts*}
```
```     7
```
```     8 theory Infinite_Set
```
```     9 import Hilbert_Choice Finite_Set SetInterval
```
```    10 begin
```
```    11
```
```    12 subsection "Infinite Sets"
```
```    13
```
```    14 text {* Some elementary facts about infinite sets, by Stefan Merz. *}
```
```    15
```
```    16 syntax
```
```    17   infinite :: "'a set \<Rightarrow> bool"
```
```    18 translations
```
```    19   "infinite S" == "S \<notin> Finites"
```
```    20
```
```    21 text {*
```
```    22   Infinite sets are non-empty, and if we remove some elements
```
```    23   from an infinite set, the result is still infinite.
```
```    24 *}
```
```    25
```
```    26 lemma infinite_nonempty:
```
```    27   "\<not> (infinite {})"
```
```    28 by simp
```
```    29
```
```    30 lemma infinite_remove:
```
```    31   "infinite S \<Longrightarrow> infinite (S - {a})"
```
```    32 by simp
```
```    33
```
```    34 lemma Diff_infinite_finite:
```
```    35   assumes T: "finite T" and S: "infinite S"
```
```    36   shows "infinite (S-T)"
```
```    37 using T
```
```    38 proof (induct)
```
```    39   from S
```
```    40   show "infinite (S - {})" by auto
```
```    41 next
```
```    42   fix T x
```
```    43   assume ih: "infinite (S-T)"
```
```    44   have "S - (insert x T) = (S-T) - {x}"
```
```    45     by (rule Diff_insert)
```
```    46   with ih
```
```    47   show "infinite (S - (insert x T))"
```
```    48     by (simp add: infinite_remove)
```
```    49 qed
```
```    50
```
```    51 lemma Un_infinite:
```
```    52   "infinite S \<Longrightarrow> infinite (S \<union> T)"
```
```    53 by simp
```
```    54
```
```    55 lemma infinite_super:
```
```    56   assumes T: "S \<subseteq> T" and S: "infinite S"
```
```    57   shows "infinite T"
```
```    58 proof (rule ccontr)
```
```    59   assume "\<not>(infinite T)"
```
```    60   with T
```
```    61   have "finite S" by (simp add: finite_subset)
```
```    62   with S
```
```    63   show False by simp
```
```    64 qed
```
```    65
```
```    66 text {*
```
```    67   As a concrete example, we prove that the set of natural
```
```    68   numbers is infinite.
```
```    69 *}
```
```    70
```
```    71 lemma finite_nat_bounded:
```
```    72   assumes S: "finite (S::nat set)"
```
```    73   shows "\<exists>k. S \<subseteq> {..<k}" (is "\<exists>k. ?bounded S k")
```
```    74 using S
```
```    75 proof (induct)
```
```    76   have "?bounded {} 0" by simp
```
```    77   thus "\<exists>k. ?bounded {} k" ..
```
```    78 next
```
```    79   fix S x
```
```    80   assume "\<exists>k. ?bounded S k"
```
```    81   then obtain k where k: "?bounded S k" ..
```
```    82   show "\<exists>k. ?bounded (insert x S) k"
```
```    83   proof (cases "x<k")
```
```    84     case True
```
```    85     with k show ?thesis by auto
```
```    86   next
```
```    87     case False
```
```    88     with k have "?bounded S (Suc x)" by auto
```
```    89     thus ?thesis by auto
```
```    90   qed
```
```    91 qed
```
```    92
```
```    93 lemma finite_nat_iff_bounded:
```
```    94   "finite (S::nat set) = (\<exists>k. S \<subseteq> {..<k})" (is "?lhs = ?rhs")
```
```    95 proof
```
```    96   assume ?lhs
```
```    97   thus ?rhs by (rule finite_nat_bounded)
```
```    98 next
```
```    99   assume ?rhs
```
```   100   then obtain k where "S \<subseteq> {..<k}" ..
```
```   101   thus "finite S"
```
```   102     by (rule finite_subset, simp)
```
```   103 qed
```
```   104
```
```   105 lemma finite_nat_iff_bounded_le:
```
```   106   "finite (S::nat set) = (\<exists>k. S \<subseteq> {..k})" (is "?lhs = ?rhs")
```
```   107 proof
```
```   108   assume ?lhs
```
```   109   then obtain k where "S \<subseteq> {..<k}"
```
```   110     by (blast dest: finite_nat_bounded)
```
```   111   hence "S \<subseteq> {..k}" by auto
```
```   112   thus ?rhs ..
```
```   113 next
```
```   114   assume ?rhs
```
```   115   then obtain k where "S \<subseteq> {..k}" ..
```
```   116   thus "finite S"
```
```   117     by (rule finite_subset, simp)
```
```   118 qed
```
```   119
```
```   120 lemma infinite_nat_iff_unbounded:
```
```   121   "infinite (S::nat set) = (\<forall>m. \<exists>n. m<n \<and> n\<in>S)"
```
```   122   (is "?lhs = ?rhs")
```
```   123 proof
```
```   124   assume inf: ?lhs
```
```   125   show ?rhs
```
```   126   proof (rule ccontr)
```
```   127     assume "\<not> ?rhs"
```
```   128     then obtain m where m: "\<forall>n. m<n \<longrightarrow> n\<notin>S" by blast
```
```   129     hence "S \<subseteq> {..m}"
```
```   130       by (auto simp add: sym[OF not_less_iff_le])
```
```   131     with inf show "False"
```
```   132       by (simp add: finite_nat_iff_bounded_le)
```
```   133   qed
```
```   134 next
```
```   135   assume unbounded: ?rhs
```
```   136   show ?lhs
```
```   137   proof
```
```   138     assume "finite S"
```
```   139     then obtain m where "S \<subseteq> {..m}"
```
```   140       by (auto simp add: finite_nat_iff_bounded_le)
```
```   141     hence "\<forall>n. m<n \<longrightarrow> n\<notin>S" by auto
```
```   142     with unbounded show "False" by blast
```
```   143   qed
```
```   144 qed
```
```   145
```
```   146 lemma infinite_nat_iff_unbounded_le:
```
```   147   "infinite (S::nat set) = (\<forall>m. \<exists>n. m\<le>n \<and> n\<in>S)"
```
```   148   (is "?lhs = ?rhs")
```
```   149 proof
```
```   150   assume inf: ?lhs
```
```   151   show ?rhs
```
```   152   proof
```
```   153     fix m
```
```   154     from inf obtain n where "m<n \<and> n\<in>S"
```
```   155       by (auto simp add: infinite_nat_iff_unbounded)
```
```   156     hence "m\<le>n \<and> n\<in>S" by auto
```
```   157     thus "\<exists>n. m \<le> n \<and> n \<in> S" ..
```
```   158   qed
```
```   159 next
```
```   160   assume unbounded: ?rhs
```
```   161   show ?lhs
```
```   162   proof (auto simp add: infinite_nat_iff_unbounded)
```
```   163     fix m
```
```   164     from unbounded obtain n where "(Suc m)\<le>n \<and> n\<in>S"
```
```   165       by blast
```
```   166     hence "m<n \<and> n\<in>S" by auto
```
```   167     thus "\<exists>n. m < n \<and> n \<in> S" ..
```
```   168   qed
```
```   169 qed
```
```   170
```
```   171 text {*
```
```   172   For a set of natural numbers to be infinite, it is enough
```
```   173   to know that for any number larger than some @{text k}, there
```
```   174   is some larger number that is an element of the set.
```
```   175 *}
```
```   176
```
```   177 lemma unbounded_k_infinite:
```
```   178   assumes k: "\<forall>m. k<m \<longrightarrow> (\<exists>n. m<n \<and> n\<in>S)"
```
```   179   shows "infinite (S::nat set)"
```
```   180 proof (auto simp add: infinite_nat_iff_unbounded)
```
```   181   fix m show "\<exists>n. m<n \<and> n\<in>S"
```
```   182   proof (cases "k<m")
```
```   183     case True
```
```   184     with k show ?thesis by blast
```
```   185   next
```
```   186     case False
```
```   187     from k obtain n where "Suc k < n \<and> n\<in>S" by auto
```
```   188     with False have "m<n \<and> n\<in>S" by auto
```
```   189     thus ?thesis ..
```
```   190   qed
```
```   191 qed
```
```   192
```
```   193 theorem nat_infinite [simp]:
```
```   194   "infinite (UNIV :: nat set)"
```
```   195 by (auto simp add: infinite_nat_iff_unbounded)
```
```   196
```
```   197 theorem nat_not_finite [elim]:
```
```   198   "finite (UNIV::nat set) \<Longrightarrow> R"
```
```   199 by simp
```
```   200
```
```   201 text {*
```
```   202   Every infinite set contains a countable subset. More precisely
```
```   203   we show that a set @{text S} is infinite if and only if there exists
```
```   204   an injective function from the naturals into @{text S}.
```
```   205 *}
```
```   206
```
```   207 lemma range_inj_infinite:
```
```   208   "inj (f::nat \<Rightarrow> 'a) \<Longrightarrow> infinite (range f)"
```
```   209 proof
```
```   210   assume "inj f"
```
```   211     and  "finite (range f)"
```
```   212   hence "finite (UNIV::nat set)"
```
```   213     by (auto intro: finite_imageD simp del: nat_infinite)
```
```   214   thus "False" by simp
```
```   215 qed
```
```   216
```
```   217 text {*
```
```   218   The ``only if'' direction is harder because it requires the
```
```   219   construction of a sequence of pairwise different elements of
```
```   220   an infinite set @{text S}. The idea is to construct a sequence of
```
```   221   non-empty and infinite subsets of @{text S} obtained by successively
```
```   222   removing elements of @{text S}.
```
```   223 *}
```
```   224
```
```   225 lemma linorder_injI:
```
```   226   assumes hyp: "\<forall>x y. x < (y::'a::linorder) \<longrightarrow> f x \<noteq> f y"
```
```   227   shows "inj f"
```
```   228 proof (rule inj_onI)
```
```   229   fix x y
```
```   230   assume f_eq: "f x = f y"
```
```   231   show "x = y"
```
```   232   proof (rule linorder_cases)
```
```   233     assume "x < y"
```
```   234     with hyp have "f x \<noteq> f y" by blast
```
```   235     with f_eq show ?thesis by simp
```
```   236   next
```
```   237     assume "x = y"
```
```   238     thus ?thesis .
```
```   239   next
```
```   240     assume "y < x"
```
```   241     with hyp have "f y \<noteq> f x" by blast
```
```   242     with f_eq show ?thesis by simp
```
```   243   qed
```
```   244 qed
```
```   245
```
```   246 lemma infinite_countable_subset:
```
```   247   assumes inf: "infinite (S::'a set)"
```
```   248   shows "\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S"
```
```   249 proof -
```
```   250   def Sseq \<equiv> "nat_rec S (\<lambda>n T. T - {SOME e. e \<in> T})"
```
```   251   def pick \<equiv> "\<lambda>n. (SOME e. e \<in> Sseq n)"
```
```   252   have Sseq_inf: "\<And>n. infinite (Sseq n)"
```
```   253   proof -
```
```   254     fix n
```
```   255     show "infinite (Sseq n)"
```
```   256     proof (induct n)
```
```   257       from inf show "infinite (Sseq 0)"
```
```   258 	by (simp add: Sseq_def)
```
```   259     next
```
```   260       fix n
```
```   261       assume "infinite (Sseq n)" thus "infinite (Sseq (Suc n))"
```
```   262 	by (simp add: Sseq_def infinite_remove)
```
```   263     qed
```
```   264   qed
```
```   265   have Sseq_S: "\<And>n. Sseq n \<subseteq> S"
```
```   266   proof -
```
```   267     fix n
```
```   268     show "Sseq n \<subseteq> S"
```
```   269       by (induct n, auto simp add: Sseq_def)
```
```   270   qed
```
```   271   have Sseq_pick: "\<And>n. pick n \<in> Sseq n"
```
```   272   proof -
```
```   273     fix n
```
```   274     show "pick n \<in> Sseq n"
```
```   275     proof (unfold pick_def, rule someI_ex)
```
```   276       from Sseq_inf have "infinite (Sseq n)" .
```
```   277       hence "Sseq n \<noteq> {}" by auto
```
```   278       thus "\<exists>x. x \<in> Sseq n" by auto
```
```   279     qed
```
```   280   qed
```
```   281   with Sseq_S have rng: "range pick \<subseteq> S"
```
```   282     by auto
```
```   283   have pick_Sseq_gt: "\<And>n m. pick n \<notin> Sseq (n + Suc m)"
```
```   284   proof -
```
```   285     fix n m
```
```   286     show "pick n \<notin> Sseq (n + Suc m)"
```
```   287       by (induct m, auto simp add: Sseq_def pick_def)
```
```   288   qed
```
```   289   have pick_pick: "\<And>n m. pick n \<noteq> pick (n + Suc m)"
```
```   290   proof -
```
```   291     fix n m
```
```   292     from Sseq_pick have "pick (n + Suc m) \<in> Sseq (n + Suc m)" .
```
```   293     moreover from pick_Sseq_gt
```
```   294     have "pick n \<notin> Sseq (n + Suc m)" .
```
```   295     ultimately show "pick n \<noteq> pick (n + Suc m)"
```
```   296       by auto
```
```   297   qed
```
```   298   have inj: "inj pick"
```
```   299   proof (rule linorder_injI)
```
```   300     show "\<forall>i j. i<(j::nat) \<longrightarrow> pick i \<noteq> pick j"
```
```   301     proof (clarify)
```
```   302       fix i j
```
```   303       assume ij: "i<(j::nat)"
```
```   304 	and eq: "pick i = pick j"
```
```   305       from ij obtain k where "j = i + (Suc k)"
```
```   306 	by (auto simp add: less_iff_Suc_add)
```
```   307       with pick_pick have "pick i \<noteq> pick j" by simp
```
```   308       with eq show "False" by simp
```
```   309     qed
```
```   310   qed
```
```   311   from rng inj show ?thesis by auto
```
```   312 qed
```
```   313
```
```   314 theorem infinite_iff_countable_subset:
```
```   315   "infinite S = (\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S)"
```
```   316   (is "?lhs = ?rhs")
```
```   317 by (auto simp add: infinite_countable_subset
```
```   318                    range_inj_infinite infinite_super)
```
```   319
```
```   320 text {*
```
```   321   For any function with infinite domain and finite range
```
```   322   there is some element that is the image of infinitely
```
```   323   many domain elements. In particular, any infinite sequence
```
```   324   of elements from a finite set contains some element that
```
```   325   occurs infinitely often.
```
```   326 *}
```
```   327
```
```   328 theorem inf_img_fin_dom:
```
```   329   assumes img: "finite (f`A)" and dom: "infinite A"
```
```   330   shows "\<exists>y \<in> f`A. infinite (f -` {y})"
```
```   331 proof (rule ccontr)
```
```   332   assume "\<not> (\<exists>y\<in>f ` A. infinite (f -` {y}))"
```
```   333   with img have "finite (UN y:f`A. f -` {y})"
```
```   334     by (blast intro: finite_UN_I)
```
```   335   moreover have "A \<subseteq> (UN y:f`A. f -` {y})" by auto
```
```   336   moreover note dom
```
```   337   ultimately show "False"
```
```   338     by (simp add: infinite_super)
```
```   339 qed
```
```   340
```
```   341 theorems inf_img_fin_domE = inf_img_fin_dom[THEN bexE]
```
```   342
```
```   343
```
```   344 subsection "Infinitely Many and Almost All"
```
```   345
```
```   346 text {*
```
```   347   We often need to reason about the existence of infinitely many
```
```   348   (resp., all but finitely many) objects satisfying some predicate,
```
```   349   so we introduce corresponding binders and their proof rules.
```
```   350 *}
```
```   351
```
```   352 consts
```
```   353   Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool"      (binder "INF " 10)
```
```   354   Alm_all  :: "('a \<Rightarrow> bool) \<Rightarrow> bool"      (binder "MOST " 10)
```
```   355
```
```   356 defs
```
```   357   INF_def:  "Inf_many P \<equiv> infinite {x. P x}"
```
```   358   MOST_def: "Alm_all P \<equiv> \<not>(INF x. \<not> P x)"
```
```   359
```
```   360 syntax (xsymbols)
```
```   361   "MOST " :: "[idts, bool] \<Rightarrow> bool"       ("(3\<forall>\<^sub>\<infinity>_./ _)" [0,10] 10)
```
```   362   "INF "    :: "[idts, bool] \<Rightarrow> bool"     ("(3\<exists>\<^sub>\<infinity>_./ _)" [0,10] 10)
```
```   363
```
```   364 syntax (HTML output)
```
```   365   "MOST " :: "[idts, bool] \<Rightarrow> bool"       ("(3\<forall>\<^sub>\<infinity>_./ _)" [0,10] 10)
```
```   366   "INF "    :: "[idts, bool] \<Rightarrow> bool"     ("(3\<exists>\<^sub>\<infinity>_./ _)" [0,10] 10)
```
```   367
```
```   368 lemma INF_EX:
```
```   369   "(\<exists>\<^sub>\<infinity>x. P x) \<Longrightarrow> (\<exists>x. P x)"
```
```   370 proof (unfold INF_def, rule ccontr)
```
```   371   assume inf: "infinite {x. P x}"
```
```   372     and notP: "\<not>(\<exists>x. P x)"
```
```   373   from notP have "{x. P x} = {}" by simp
```
```   374   hence "finite {x. P x}" by simp
```
```   375   with inf show "False" by simp
```
```   376 qed
```
```   377
```
```   378 lemma MOST_iff_finiteNeg:
```
```   379   "(\<forall>\<^sub>\<infinity>x. P x) = finite {x. \<not> P x}"
```
```   380 by (simp add: MOST_def INF_def)
```
```   381
```
```   382 lemma ALL_MOST:
```
```   383   "\<forall>x. P x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. P x"
```
```   384 by (simp add: MOST_iff_finiteNeg)
```
```   385
```
```   386 lemma INF_mono:
```
```   387   assumes inf: "\<exists>\<^sub>\<infinity>x. P x" and q: "\<And>x. P x \<Longrightarrow> Q x"
```
```   388   shows "\<exists>\<^sub>\<infinity>x. Q x"
```
```   389 proof -
```
```   390   from inf have "infinite {x. P x}" by (unfold INF_def)
```
```   391   moreover from q have "{x. P x} \<subseteq> {x. Q x}" by auto
```
```   392   ultimately show ?thesis
```
```   393     by (simp add: INF_def infinite_super)
```
```   394 qed
```
```   395
```
```   396 lemma MOST_mono:
```
```   397   "\<lbrakk> \<forall>\<^sub>\<infinity>x. P x; \<And>x. P x \<Longrightarrow> Q x \<rbrakk> \<Longrightarrow> \<forall>\<^sub>\<infinity>x. Q x"
```
```   398 by (unfold MOST_def, blast intro: INF_mono)
```
```   399
```
```   400 lemma INF_nat: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) = (\<forall>m. \<exists>n. m<n \<and> P n)"
```
```   401 by (simp add: INF_def infinite_nat_iff_unbounded)
```
```   402
```
```   403 lemma INF_nat_le: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) = (\<forall>m. \<exists>n. m\<le>n \<and> P n)"
```
```   404 by (simp add: INF_def infinite_nat_iff_unbounded_le)
```
```   405
```
```   406 lemma MOST_nat: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) = (\<exists>m. \<forall>n. m<n \<longrightarrow> P n)"
```
```   407 by (simp add: MOST_def INF_nat)
```
```   408
```
```   409 lemma MOST_nat_le: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) = (\<exists>m. \<forall>n. m\<le>n \<longrightarrow> P n)"
```
```   410 by (simp add: MOST_def INF_nat_le)
```
```   411
```
```   412
```
```   413 subsection "Miscellaneous"
```
```   414
```
```   415 text {*
```
```   416   A few trivial lemmas about sets that contain at most one element.
```
```   417   These simplify the reasoning about deterministic automata.
```
```   418 *}
```
```   419
```
```   420 constdefs
```
```   421   atmost_one :: "'a set \<Rightarrow> bool"
```
```   422   "atmost_one S \<equiv> \<forall>x y. x\<in>S \<and> y\<in>S \<longrightarrow> x=y"
```
```   423
```
```   424 lemma atmost_one_empty: "S={} \<Longrightarrow> atmost_one S"
```
```   425 by (simp add: atmost_one_def)
```
```   426
```
```   427 lemma atmost_one_singleton: "S = {x} \<Longrightarrow> atmost_one S"
```
```   428 by (simp add: atmost_one_def)
```
```   429
```
```   430 lemma atmost_one_unique [elim]: "\<lbrakk> atmost_one S; x \<in> S; y \<in> S \<rbrakk> \<Longrightarrow> y=x"
```
```   431 by (simp add: atmost_one_def)
```
```   432
```
```   433 end
```