src/HOL/Infinite_Set.thy
 author obua Mon Apr 10 16:00:34 2006 +0200 (2006-04-10) changeset 19404 9bf2cdc9e8e8 parent 19363 667b5ea637dd child 19457 b6eb4b4546fa permissions -rw-r--r--
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```     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 imports Hilbert_Choice Binomial
```
```    10 begin
```
```    11
```
```    12 subsection "Infinite Sets"
```
```    13
```
```    14 text {* Some elementary facts about infinite sets, by Stefan Merz. *}
```
```    15
```
```    16 abbreviation
```
```    17   infinite :: "'a set \<Rightarrow> bool"
```
```    18   "infinite S == \<not> finite S"
```
```    19
```
```    20 text {*
```
```    21   Infinite sets are non-empty, and if we remove some elements
```
```    22   from an infinite set, the result is still infinite.
```
```    23 *}
```
```    24
```
```    25 lemma infinite_nonempty:
```
```    26   "\<not> (infinite {})"
```
```    27 by simp
```
```    28
```
```    29 lemma infinite_remove:
```
```    30   "infinite S \<Longrightarrow> infinite (S - {a})"
```
```    31 by simp
```
```    32
```
```    33 lemma Diff_infinite_finite:
```
```    34   assumes T: "finite T" and S: "infinite S"
```
```    35   shows "infinite (S-T)"
```
```    36 using T
```
```    37 proof (induct)
```
```    38   from S
```
```    39   show "infinite (S - {})" by auto
```
```    40 next
```
```    41   fix T x
```
```    42   assume ih: "infinite (S-T)"
```
```    43   have "S - (insert x T) = (S-T) - {x}"
```
```    44     by (rule Diff_insert)
```
```    45   with ih
```
```    46   show "infinite (S - (insert x T))"
```
```    47     by (simp add: infinite_remove)
```
```    48 qed
```
```    49
```
```    50 lemma Un_infinite:
```
```    51   "infinite S \<Longrightarrow> infinite (S \<union> T)"
```
```    52 by simp
```
```    53
```
```    54 lemma infinite_super:
```
```    55   assumes T: "S \<subseteq> T" and S: "infinite S"
```
```    56   shows "infinite T"
```
```    57 proof (rule ccontr)
```
```    58   assume "\<not>(infinite T)"
```
```    59   with T
```
```    60   have "finite S" by (simp add: finite_subset)
```
```    61   with S
```
```    62   show False by simp
```
```    63 qed
```
```    64
```
```    65 text {*
```
```    66   As a concrete example, we prove that the set of natural
```
```    67   numbers is infinite.
```
```    68 *}
```
```    69
```
```    70 lemma finite_nat_bounded:
```
```    71   assumes S: "finite (S::nat set)"
```
```    72   shows "\<exists>k. S \<subseteq> {..<k}" (is "\<exists>k. ?bounded S k")
```
```    73 using S
```
```    74 proof (induct)
```
```    75   have "?bounded {} 0" by simp
```
```    76   thus "\<exists>k. ?bounded {} k" ..
```
```    77 next
```
```    78   fix S x
```
```    79   assume "\<exists>k. ?bounded S k"
```
```    80   then obtain k where k: "?bounded S k" ..
```
```    81   show "\<exists>k. ?bounded (insert x S) k"
```
```    82   proof (cases "x<k")
```
```    83     case True
```
```    84     with k show ?thesis by auto
```
```    85   next
```
```    86     case False
```
```    87     with k have "?bounded S (Suc x)" by auto
```
```    88     thus ?thesis by auto
```
```    89   qed
```
```    90 qed
```
```    91
```
```    92 lemma finite_nat_iff_bounded:
```
```    93   "finite (S::nat set) = (\<exists>k. S \<subseteq> {..<k})" (is "?lhs = ?rhs")
```
```    94 proof
```
```    95   assume ?lhs
```
```    96   thus ?rhs by (rule finite_nat_bounded)
```
```    97 next
```
```    98   assume ?rhs
```
```    99   then obtain k where "S \<subseteq> {..<k}" ..
```
```   100   thus "finite S"
```
```   101     by (rule finite_subset, simp)
```
```   102 qed
```
```   103
```
```   104 lemma finite_nat_iff_bounded_le:
```
```   105   "finite (S::nat set) = (\<exists>k. S \<subseteq> {..k})" (is "?lhs = ?rhs")
```
```   106 proof
```
```   107   assume ?lhs
```
```   108   then obtain k where "S \<subseteq> {..<k}"
```
```   109     by (blast dest: finite_nat_bounded)
```
```   110   hence "S \<subseteq> {..k}" by auto
```
```   111   thus ?rhs ..
```
```   112 next
```
```   113   assume ?rhs
```
```   114   then obtain k where "S \<subseteq> {..k}" ..
```
```   115   thus "finite S"
```
```   116     by (rule finite_subset, simp)
```
```   117 qed
```
```   118
```
```   119 lemma infinite_nat_iff_unbounded:
```
```   120   "infinite (S::nat set) = (\<forall>m. \<exists>n. m<n \<and> n\<in>S)"
```
```   121   (is "?lhs = ?rhs")
```
```   122 proof
```
```   123   assume inf: ?lhs
```
```   124   show ?rhs
```
```   125   proof (rule ccontr)
```
```   126     assume "\<not> ?rhs"
```
```   127     then obtain m where m: "\<forall>n. m<n \<longrightarrow> n\<notin>S" by blast
```
```   128     hence "S \<subseteq> {..m}"
```
```   129       by (auto simp add: sym[OF linorder_not_less])
```
```   130     with inf show "False"
```
```   131       by (simp add: finite_nat_iff_bounded_le)
```
```   132   qed
```
```   133 next
```
```   134   assume unbounded: ?rhs
```
```   135   show ?lhs
```
```   136   proof
```
```   137     assume "finite S"
```
```   138     then obtain m where "S \<subseteq> {..m}"
```
```   139       by (auto simp add: finite_nat_iff_bounded_le)
```
```   140     hence "\<forall>n. m<n \<longrightarrow> n\<notin>S" by auto
```
```   141     with unbounded show "False" by blast
```
```   142   qed
```
```   143 qed
```
```   144
```
```   145 lemma infinite_nat_iff_unbounded_le:
```
```   146   "infinite (S::nat set) = (\<forall>m. \<exists>n. m\<le>n \<and> n\<in>S)"
```
```   147   (is "?lhs = ?rhs")
```
```   148 proof
```
```   149   assume inf: ?lhs
```
```   150   show ?rhs
```
```   151   proof
```
```   152     fix m
```
```   153     from inf obtain n where "m<n \<and> n\<in>S"
```
```   154       by (auto simp add: infinite_nat_iff_unbounded)
```
```   155     hence "m\<le>n \<and> n\<in>S" by auto
```
```   156     thus "\<exists>n. m \<le> n \<and> n \<in> S" ..
```
```   157   qed
```
```   158 next
```
```   159   assume unbounded: ?rhs
```
```   160   show ?lhs
```
```   161   proof (auto simp add: infinite_nat_iff_unbounded)
```
```   162     fix m
```
```   163     from unbounded obtain n where "(Suc m)\<le>n \<and> n\<in>S"
```
```   164       by blast
```
```   165     hence "m<n \<and> n\<in>S" by auto
```
```   166     thus "\<exists>n. m < n \<and> n \<in> S" ..
```
```   167   qed
```
```   168 qed
```
```   169
```
```   170 text {*
```
```   171   For a set of natural numbers to be infinite, it is enough
```
```   172   to know that for any number larger than some @{text k}, there
```
```   173   is some larger number that is an element of the set.
```
```   174 *}
```
```   175
```
```   176 lemma unbounded_k_infinite:
```
```   177   assumes k: "\<forall>m. k<m \<longrightarrow> (\<exists>n. m<n \<and> n\<in>S)"
```
```   178   shows "infinite (S::nat set)"
```
```   179 proof (auto simp add: infinite_nat_iff_unbounded)
```
```   180   fix m show "\<exists>n. m<n \<and> n\<in>S"
```
```   181   proof (cases "k<m")
```
```   182     case True
```
```   183     with k show ?thesis by blast
```
```   184   next
```
```   185     case False
```
```   186     from k obtain n where "Suc k < n \<and> n\<in>S" by auto
```
```   187     with False have "m<n \<and> n\<in>S" by auto
```
```   188     thus ?thesis ..
```
```   189   qed
```
```   190 qed
```
```   191
```
```   192 theorem nat_infinite [simp]:
```
```   193   "infinite (UNIV :: nat set)"
```
```   194 by (auto simp add: infinite_nat_iff_unbounded)
```
```   195
```
```   196 theorem nat_not_finite [elim]:
```
```   197   "finite (UNIV::nat set) \<Longrightarrow> R"
```
```   198 by simp
```
```   199
```
```   200 text {*
```
```   201   Every infinite set contains a countable subset. More precisely
```
```   202   we show that a set @{text S} is infinite if and only if there exists
```
```   203   an injective function from the naturals into @{text S}.
```
```   204 *}
```
```   205
```
```   206 lemma range_inj_infinite:
```
```   207   "inj (f::nat \<Rightarrow> 'a) \<Longrightarrow> infinite (range f)"
```
```   208 proof
```
```   209   assume "inj f"
```
```   210     and  "finite (range f)"
```
```   211   hence "finite (UNIV::nat set)"
```
```   212     by (auto intro: finite_imageD simp del: nat_infinite)
```
```   213   thus "False" by simp
```
```   214 qed
```
```   215
```
```   216 text {*
```
```   217   The ``only if'' direction is harder because it requires the
```
```   218   construction of a sequence of pairwise different elements of
```
```   219   an infinite set @{text S}. The idea is to construct a sequence of
```
```   220   non-empty and infinite subsets of @{text S} obtained by successively
```
```   221   removing elements of @{text S}.
```
```   222 *}
```
```   223
```
```   224 lemma linorder_injI:
```
```   225   assumes hyp: "\<forall>x y. x < (y::'a::linorder) \<longrightarrow> f x \<noteq> f y"
```
```   226   shows "inj f"
```
```   227 proof (rule inj_onI)
```
```   228   fix x y
```
```   229   assume f_eq: "f x = f y"
```
```   230   show "x = y"
```
```   231   proof (rule linorder_cases)
```
```   232     assume "x < y"
```
```   233     with hyp have "f x \<noteq> f y" by blast
```
```   234     with f_eq show ?thesis by simp
```
```   235   next
```
```   236     assume "x = y"
```
```   237     thus ?thesis .
```
```   238   next
```
```   239     assume "y < x"
```
```   240     with hyp have "f y \<noteq> f x" by blast
```
```   241     with f_eq show ?thesis by simp
```
```   242   qed
```
```   243 qed
```
```   244
```
```   245 lemma infinite_countable_subset:
```
```   246   assumes inf: "infinite (S::'a set)"
```
```   247   shows "\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S"
```
```   248 proof -
```
```   249   def Sseq \<equiv> "nat_rec S (\<lambda>n T. T - {SOME e. e \<in> T})"
```
```   250   def pick \<equiv> "\<lambda>n. (SOME e. e \<in> Sseq n)"
```
```   251   have Sseq_inf: "\<And>n. infinite (Sseq n)"
```
```   252   proof -
```
```   253     fix n
```
```   254     show "infinite (Sseq n)"
```
```   255     proof (induct n)
```
```   256       from inf show "infinite (Sseq 0)"
```
```   257 	by (simp add: Sseq_def)
```
```   258     next
```
```   259       fix n
```
```   260       assume "infinite (Sseq n)" thus "infinite (Sseq (Suc n))"
```
```   261 	by (simp add: Sseq_def infinite_remove)
```
```   262     qed
```
```   263   qed
```
```   264   have Sseq_S: "\<And>n. Sseq n \<subseteq> S"
```
```   265   proof -
```
```   266     fix n
```
```   267     show "Sseq n \<subseteq> S"
```
```   268       by (induct n, auto simp add: Sseq_def)
```
```   269   qed
```
```   270   have Sseq_pick: "\<And>n. pick n \<in> Sseq n"
```
```   271   proof -
```
```   272     fix n
```
```   273     show "pick n \<in> Sseq n"
```
```   274     proof (unfold pick_def, rule someI_ex)
```
```   275       from Sseq_inf have "infinite (Sseq n)" .
```
```   276       hence "Sseq n \<noteq> {}" by auto
```
```   277       thus "\<exists>x. x \<in> Sseq n" by auto
```
```   278     qed
```
```   279   qed
```
```   280   with Sseq_S have rng: "range pick \<subseteq> S"
```
```   281     by auto
```
```   282   have pick_Sseq_gt: "\<And>n m. pick n \<notin> Sseq (n + Suc m)"
```
```   283   proof -
```
```   284     fix n m
```
```   285     show "pick n \<notin> Sseq (n + Suc m)"
```
```   286       by (induct m, auto simp add: Sseq_def pick_def)
```
```   287   qed
```
```   288   have pick_pick: "\<And>n m. pick n \<noteq> pick (n + Suc m)"
```
```   289   proof -
```
```   290     fix n m
```
```   291     from Sseq_pick have "pick (n + Suc m) \<in> Sseq (n + Suc m)" .
```
```   292     moreover from pick_Sseq_gt
```
```   293     have "pick n \<notin> Sseq (n + Suc m)" .
```
```   294     ultimately show "pick n \<noteq> pick (n + Suc m)"
```
```   295       by auto
```
```   296   qed
```
```   297   have inj: "inj pick"
```
```   298   proof (rule linorder_injI)
```
```   299     show "\<forall>i j. i<(j::nat) \<longrightarrow> pick i \<noteq> pick j"
```
```   300     proof (clarify)
```
```   301       fix i j
```
```   302       assume ij: "i<(j::nat)"
```
```   303 	and eq: "pick i = pick j"
```
```   304       from ij obtain k where "j = i + (Suc k)"
```
```   305 	by (auto simp add: less_iff_Suc_add)
```
```   306       with pick_pick have "pick i \<noteq> pick j" by simp
```
```   307       with eq show "False" by simp
```
```   308     qed
```
```   309   qed
```
```   310   from rng inj show ?thesis by auto
```
```   311 qed
```
```   312
```
```   313 theorem infinite_iff_countable_subset:
```
```   314   "infinite S = (\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S)"
```
```   315   (is "?lhs = ?rhs")
```
```   316 by (auto simp add: infinite_countable_subset
```
```   317                    range_inj_infinite infinite_super)
```
```   318
```
```   319 text {*
```
```   320   For any function with infinite domain and finite range
```
```   321   there is some element that is the image of infinitely
```
```   322   many domain elements. In particular, any infinite sequence
```
```   323   of elements from a finite set contains some element that
```
```   324   occurs infinitely often.
```
```   325 *}
```
```   326
```
```   327 theorem inf_img_fin_dom:
```
```   328   assumes img: "finite (f`A)" and dom: "infinite A"
```
```   329   shows "\<exists>y \<in> f`A. infinite (f -` {y})"
```
```   330 proof (rule ccontr)
```
```   331   assume "\<not> (\<exists>y\<in>f ` A. infinite (f -` {y}))"
```
```   332   with img have "finite (UN y:f`A. f -` {y})"
```
```   333     by (blast intro: finite_UN_I)
```
```   334   moreover have "A \<subseteq> (UN y:f`A. f -` {y})" by auto
```
```   335   moreover note dom
```
```   336   ultimately show "False"
```
```   337     by (simp add: infinite_super)
```
```   338 qed
```
```   339
```
```   340 theorems inf_img_fin_domE = inf_img_fin_dom[THEN bexE]
```
```   341
```
```   342
```
```   343 subsection "Infinitely Many and Almost All"
```
```   344
```
```   345 text {*
```
```   346   We often need to reason about the existence of infinitely many
```
```   347   (resp., all but finitely many) objects satisfying some predicate,
```
```   348   so we introduce corresponding binders and their proof rules.
```
```   349 *}
```
```   350
```
```   351 consts
```
```   352   Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool"      (binder "INF " 10)
```
```   353   Alm_all  :: "('a \<Rightarrow> bool) \<Rightarrow> bool"      (binder "MOST " 10)
```
```   354
```
```   355 defs
```
```   356   INF_def:  "Inf_many P \<equiv> infinite {x. P x}"
```
```   357   MOST_def: "Alm_all P \<equiv> \<not>(INF x. \<not> P x)"
```
```   358
```
```   359 syntax (xsymbols)
```
```   360   "MOST " :: "[idts, bool] \<Rightarrow> bool"       ("(3\<forall>\<^sub>\<infinity>_./ _)" [0,10] 10)
```
```   361   "INF "    :: "[idts, bool] \<Rightarrow> bool"     ("(3\<exists>\<^sub>\<infinity>_./ _)" [0,10] 10)
```
```   362
```
```   363 syntax (HTML output)
```
```   364   "MOST " :: "[idts, bool] \<Rightarrow> bool"       ("(3\<forall>\<^sub>\<infinity>_./ _)" [0,10] 10)
```
```   365   "INF "    :: "[idts, bool] \<Rightarrow> bool"     ("(3\<exists>\<^sub>\<infinity>_./ _)" [0,10] 10)
```
```   366
```
```   367 lemma INF_EX:
```
```   368   "(\<exists>\<^sub>\<infinity>x. P x) \<Longrightarrow> (\<exists>x. P x)"
```
```   369 proof (unfold INF_def, rule ccontr)
```
```   370   assume inf: "infinite {x. P x}"
```
```   371     and notP: "\<not>(\<exists>x. P x)"
```
```   372   from notP have "{x. P x} = {}" by simp
```
```   373   hence "finite {x. P x}" by simp
```
```   374   with inf show "False" by simp
```
```   375 qed
```
```   376
```
```   377 lemma MOST_iff_finiteNeg:
```
```   378   "(\<forall>\<^sub>\<infinity>x. P x) = finite {x. \<not> P x}"
```
```   379 by (simp add: MOST_def INF_def)
```
```   380
```
```   381 lemma ALL_MOST:
```
```   382   "\<forall>x. P x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. P x"
```
```   383 by (simp add: MOST_iff_finiteNeg)
```
```   384
```
```   385 lemma INF_mono:
```
```   386   assumes inf: "\<exists>\<^sub>\<infinity>x. P x" and q: "\<And>x. P x \<Longrightarrow> Q x"
```
```   387   shows "\<exists>\<^sub>\<infinity>x. Q x"
```
```   388 proof -
```
```   389   from inf have "infinite {x. P x}" by (unfold INF_def)
```
```   390   moreover from q have "{x. P x} \<subseteq> {x. Q x}" by auto
```
```   391   ultimately show ?thesis
```
```   392     by (simp add: INF_def infinite_super)
```
```   393 qed
```
```   394
```
```   395 lemma MOST_mono:
```
```   396   "\<lbrakk> \<forall>\<^sub>\<infinity>x. P x; \<And>x. P x \<Longrightarrow> Q x \<rbrakk> \<Longrightarrow> \<forall>\<^sub>\<infinity>x. Q x"
```
```   397 by (unfold MOST_def, blast intro: INF_mono)
```
```   398
```
```   399 lemma INF_nat: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) = (\<forall>m. \<exists>n. m<n \<and> P n)"
```
```   400 by (simp add: INF_def infinite_nat_iff_unbounded)
```
```   401
```
```   402 lemma INF_nat_le: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) = (\<forall>m. \<exists>n. m\<le>n \<and> P n)"
```
```   403 by (simp add: INF_def infinite_nat_iff_unbounded_le)
```
```   404
```
```   405 lemma MOST_nat: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) = (\<exists>m. \<forall>n. m<n \<longrightarrow> P n)"
```
```   406 by (simp add: MOST_def INF_nat)
```
```   407
```
```   408 lemma MOST_nat_le: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) = (\<exists>m. \<forall>n. m\<le>n \<longrightarrow> P n)"
```
```   409 by (simp add: MOST_def INF_nat_le)
```
```   410
```
```   411
```
```   412 subsection "Miscellaneous"
```
```   413
```
```   414 text {*
```
```   415   A few trivial lemmas about sets that contain at most one element.
```
```   416   These simplify the reasoning about deterministic automata.
```
```   417 *}
```
```   418
```
```   419 constdefs
```
```   420   atmost_one :: "'a set \<Rightarrow> bool"
```
```   421   "atmost_one S \<equiv> \<forall>x y. x\<in>S \<and> y\<in>S \<longrightarrow> x=y"
```
```   422
```
```   423 lemma atmost_one_empty: "S={} \<Longrightarrow> atmost_one S"
```
```   424 by (simp add: atmost_one_def)
```
```   425
```
```   426 lemma atmost_one_singleton: "S = {x} \<Longrightarrow> atmost_one S"
```
```   427 by (simp add: atmost_one_def)
```
```   428
```
```   429 lemma atmost_one_unique [elim]: "\<lbrakk> atmost_one S; x \<in> S; y \<in> S \<rbrakk> \<Longrightarrow> y=x"
```
```   430 by (simp add: atmost_one_def)
```
```   431
```
```   432 end
```