src/HOL/Library/Infinite_Set.thy
 author Andreas Lochbihler Fri Sep 20 10:09:16 2013 +0200 (2013-09-20) changeset 53745 788730ab7da4 parent 53239 2f21813cf2f0 child 54557 d71c2737ee21 permissions -rw-r--r--
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```     1 (*  Title:      HOL/Library/Infinite_Set.thy
```
```     2     Author:     Stephan Merz
```
```     3 *)
```
```     4
```
```     5 header {* Infinite Sets and Related Concepts *}
```
```     6
```
```     7 theory Infinite_Set
```
```     8 imports Main
```
```     9 begin
```
```    10
```
```    11 subsection "Infinite Sets"
```
```    12
```
```    13 text {*
```
```    14   Some elementary facts about infinite sets, mostly by Stefan Merz.
```
```    15   Beware! Because "infinite" merely abbreviates a negation, these
```
```    16   lemmas may not work well with @{text "blast"}.
```
```    17 *}
```
```    18
```
```    19 abbreviation infinite :: "'a set \<Rightarrow> bool"
```
```    20   where "infinite S \<equiv> \<not> finite S"
```
```    21
```
```    22 text {*
```
```    23   Infinite sets are non-empty, and if we remove some elements from an
```
```    24   infinite set, the result is still infinite.
```
```    25 *}
```
```    26
```
```    27 lemma infinite_imp_nonempty: "infinite S \<Longrightarrow> S \<noteq> {}"
```
```    28   by auto
```
```    29
```
```    30 lemma infinite_remove: "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: "infinite S \<Longrightarrow> infinite (S \<union> T)"
```
```    51   by simp
```
```    52
```
```    53 lemma infinite_Un: "infinite (S \<union> T) \<longleftrightarrow> infinite S \<or> infinite T"
```
```    54   by simp
```
```    55
```
```    56 lemma infinite_super:
```
```    57   assumes T: "S \<subseteq> T" and S: "infinite S"
```
```    58   shows "infinite T"
```
```    59 proof
```
```    60   assume "finite T"
```
```    61   with T have "finite S" by (simp add: finite_subset)
```
```    62   with S show False by simp
```
```    63 qed
```
```    64
```
```    65 text {*
```
```    66   As a concrete example, we prove that the set of natural numbers is
```
```    67   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   then show "\<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     then show ?thesis by auto
```
```    89   qed
```
```    90 qed
```
```    91
```
```    92 lemma finite_nat_iff_bounded:
```
```    93   "finite (S::nat set) \<longleftrightarrow> (\<exists>k. S \<subseteq> {..<k})"  (is "?lhs \<longleftrightarrow> ?rhs")
```
```    94 proof
```
```    95   assume ?lhs
```
```    96   then show ?rhs by (rule finite_nat_bounded)
```
```    97 next
```
```    98   assume ?rhs
```
```    99   then obtain k where "S \<subseteq> {..<k}" ..
```
```   100   then show "finite S"
```
```   101     by (rule finite_subset) simp
```
```   102 qed
```
```   103
```
```   104 lemma finite_nat_iff_bounded_le:
```
```   105   "finite (S::nat set) \<longleftrightarrow> (\<exists>k. S \<subseteq> {..k})"  (is "?lhs \<longleftrightarrow> ?rhs")
```
```   106 proof
```
```   107   assume ?lhs
```
```   108   then obtain k where "S \<subseteq> {..<k}"
```
```   109     by (blast dest: finite_nat_bounded)
```
```   110   then have "S \<subseteq> {..k}" by auto
```
```   111   then show ?rhs ..
```
```   112 next
```
```   113   assume ?rhs
```
```   114   then obtain k where "S \<subseteq> {..k}" ..
```
```   115   then show "finite S"
```
```   116     by (rule finite_subset) simp
```
```   117 qed
```
```   118
```
```   119 lemma infinite_nat_iff_unbounded:
```
```   120   "infinite (S::nat set) \<longleftrightarrow> (\<forall>m. \<exists>n. m < n \<and> n \<in> S)"
```
```   121   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   122 proof
```
```   123   assume ?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     then have "S \<subseteq> {..m}"
```
```   129       by (auto simp add: sym [OF linorder_not_less])
```
```   130     with `?lhs` show False
```
```   131       by (simp add: finite_nat_iff_bounded_le)
```
```   132   qed
```
```   133 next
```
```   134   assume ?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     then have "\<forall>n. m < n \<longrightarrow> n \<notin> S" by auto
```
```   141     with `?rhs` show False by blast
```
```   142   qed
```
```   143 qed
```
```   144
```
```   145 lemma infinite_nat_iff_unbounded_le:
```
```   146   "infinite (S::nat set) \<longleftrightarrow> (\<forall>m. \<exists>n. m \<le> n \<and> n \<in> S)"
```
```   147   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   148 proof
```
```   149   assume ?lhs
```
```   150   show ?rhs
```
```   151   proof
```
```   152     fix m
```
```   153     from `?lhs` obtain n where "m < n \<and> n \<in> S"
```
```   154       by (auto simp add: infinite_nat_iff_unbounded)
```
```   155     then have "m \<le> n \<and> n \<in> S" by simp
```
```   156     then show "\<exists>n. m \<le> n \<and> n \<in> S" ..
```
```   157   qed
```
```   158 next
```
```   159   assume ?rhs
```
```   160   show ?lhs
```
```   161   proof (auto simp add: infinite_nat_iff_unbounded)
```
```   162     fix m
```
```   163     from `?rhs` obtain n where "Suc m \<le> n \<and> n \<in> S"
```
```   164       by blast
```
```   165     then have "m < n \<and> n \<in> S" by simp
```
```   166     then show "\<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 to know
```
```   172   that for any number larger than some @{text k}, there is some larger
```
```   173   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 -
```
```   180   {
```
```   181     fix m have "\<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       then show ?thesis ..
```
```   190     qed
```
```   191   }
```
```   192   then show ?thesis
```
```   193     by (auto simp add: infinite_nat_iff_unbounded)
```
```   194 qed
```
```   195
```
```   196 (* duplicates Finite_Set.infinite_UNIV_nat *)
```
```   197 lemma nat_infinite: "infinite (UNIV :: nat set)"
```
```   198   by (auto simp add: infinite_nat_iff_unbounded)
```
```   199
```
```   200 lemma nat_not_finite: "finite (UNIV::nat set) \<Longrightarrow> R"
```
```   201   by simp
```
```   202
```
```   203 text {*
```
```   204   Every infinite set contains a countable subset. More precisely we
```
```   205   show that a set @{text S} is infinite if and only if there exists an
```
```   206   injective function from the naturals into @{text S}.
```
```   207 *}
```
```   208
```
```   209 lemma range_inj_infinite:
```
```   210   "inj (f::nat \<Rightarrow> 'a) \<Longrightarrow> infinite (range f)"
```
```   211 proof
```
```   212   assume "finite (range f)" and "inj f"
```
```   213   then have "finite (UNIV::nat set)"
```
```   214     by (rule finite_imageD)
```
```   215   then show False by simp
```
```   216 qed
```
```   217
```
```   218 lemma int_infinite [simp]: "infinite (UNIV::int set)"
```
```   219 proof -
```
```   220   from inj_int have "infinite (range int)"
```
```   221     by (rule range_inj_infinite)
```
```   222   moreover
```
```   223   have "range int \<subseteq> (UNIV::int set)" by simp
```
```   224   ultimately show "infinite (UNIV::int set)"
```
```   225     by (simp add: infinite_super)
```
```   226 qed
```
```   227
```
```   228 text {*
```
```   229   The ``only if'' direction is harder because it requires the
```
```   230   construction of a sequence of pairwise different elements of an
```
```   231   infinite set @{text S}. The idea is to construct a sequence of
```
```   232   non-empty and infinite subsets of @{text S} obtained by successively
```
```   233   removing elements of @{text S}.
```
```   234 *}
```
```   235
```
```   236 lemma linorder_injI:
```
```   237   assumes hyp: "\<And>x y. x < (y::'a::linorder) \<Longrightarrow> f x \<noteq> f y"
```
```   238   shows "inj f"
```
```   239 proof (rule inj_onI)
```
```   240   fix x y
```
```   241   assume f_eq: "f x = f y"
```
```   242   show "x = y"
```
```   243   proof (rule linorder_cases)
```
```   244     assume "x < y"
```
```   245     with hyp have "f x \<noteq> f y" by blast
```
```   246     with f_eq show ?thesis by simp
```
```   247   next
```
```   248     assume "x = y"
```
```   249     then show ?thesis .
```
```   250   next
```
```   251     assume "y < x"
```
```   252     with hyp have "f y \<noteq> f x" by blast
```
```   253     with f_eq show ?thesis by simp
```
```   254   qed
```
```   255 qed
```
```   256
```
```   257 lemma infinite_countable_subset:
```
```   258   assumes inf: "infinite (S::'a set)"
```
```   259   shows "\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S"
```
```   260 proof -
```
```   261   def Sseq \<equiv> "nat_rec S (\<lambda>n T. T - {SOME e. e \<in> T})"
```
```   262   def pick \<equiv> "\<lambda>n. (SOME e. e \<in> Sseq n)"
```
```   263   have Sseq_inf: "\<And>n. infinite (Sseq n)"
```
```   264   proof -
```
```   265     fix n
```
```   266     show "infinite (Sseq n)"
```
```   267     proof (induct n)
```
```   268       from inf show "infinite (Sseq 0)"
```
```   269         by (simp add: Sseq_def)
```
```   270     next
```
```   271       fix n
```
```   272       assume "infinite (Sseq n)" then show "infinite (Sseq (Suc n))"
```
```   273         by (simp add: Sseq_def infinite_remove)
```
```   274     qed
```
```   275   qed
```
```   276   have Sseq_S: "\<And>n. Sseq n \<subseteq> S"
```
```   277   proof -
```
```   278     fix n
```
```   279     show "Sseq n \<subseteq> S"
```
```   280       by (induct n) (auto simp add: Sseq_def)
```
```   281   qed
```
```   282   have Sseq_pick: "\<And>n. pick n \<in> Sseq n"
```
```   283   proof -
```
```   284     fix n
```
```   285     show "pick n \<in> Sseq n"
```
```   286       unfolding pick_def
```
```   287     proof (rule someI_ex)
```
```   288       from Sseq_inf have "infinite (Sseq n)" .
```
```   289       then have "Sseq n \<noteq> {}" by auto
```
```   290       then show "\<exists>x. x \<in> Sseq n" by auto
```
```   291     qed
```
```   292   qed
```
```   293   with Sseq_S have rng: "range pick \<subseteq> S"
```
```   294     by auto
```
```   295   have pick_Sseq_gt: "\<And>n m. pick n \<notin> Sseq (n + Suc m)"
```
```   296   proof -
```
```   297     fix n m
```
```   298     show "pick n \<notin> Sseq (n + Suc m)"
```
```   299       by (induct m) (auto simp add: Sseq_def pick_def)
```
```   300   qed
```
```   301   have pick_pick: "\<And>n m. pick n \<noteq> pick (n + Suc m)"
```
```   302   proof -
```
```   303     fix n m
```
```   304     from Sseq_pick have "pick (n + Suc m) \<in> Sseq (n + Suc m)" .
```
```   305     moreover from pick_Sseq_gt
```
```   306     have "pick n \<notin> Sseq (n + Suc m)" .
```
```   307     ultimately show "pick n \<noteq> pick (n + Suc m)"
```
```   308       by auto
```
```   309   qed
```
```   310   have inj: "inj pick"
```
```   311   proof (rule linorder_injI)
```
```   312     fix i j :: nat
```
```   313     assume "i < j"
```
```   314     show "pick i \<noteq> pick j"
```
```   315     proof
```
```   316       assume eq: "pick i = pick j"
```
```   317       from `i < j` obtain k where "j = i + Suc k"
```
```   318         by (auto simp add: less_iff_Suc_add)
```
```   319       with pick_pick have "pick i \<noteq> pick j" by simp
```
```   320       with eq show False by simp
```
```   321     qed
```
```   322   qed
```
```   323   from rng inj show ?thesis by auto
```
```   324 qed
```
```   325
```
```   326 lemma infinite_iff_countable_subset:
```
```   327     "infinite S \<longleftrightarrow> (\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S)"
```
```   328   by (auto simp add: infinite_countable_subset range_inj_infinite infinite_super)
```
```   329
```
```   330 text {*
```
```   331   For any function with infinite domain and finite range there is some
```
```   332   element that is the image of infinitely many domain elements.  In
```
```   333   particular, any infinite sequence of elements from a finite set
```
```   334   contains some element that occurs infinitely often.
```
```   335 *}
```
```   336
```
```   337 lemma inf_img_fin_dom:
```
```   338   assumes img: "finite (f`A)" and dom: "infinite A"
```
```   339   shows "\<exists>y \<in> f`A. infinite (f -` {y})"
```
```   340 proof (rule ccontr)
```
```   341   assume "\<not> ?thesis"
```
```   342   with img have "finite (UN y:f`A. f -` {y})" by blast
```
```   343   moreover have "A \<subseteq> (UN y:f`A. f -` {y})" by auto
```
```   344   moreover note dom
```
```   345   ultimately show False by (simp add: infinite_super)
```
```   346 qed
```
```   347
```
```   348 lemma inf_img_fin_domE:
```
```   349   assumes "finite (f`A)" and "infinite A"
```
```   350   obtains y where "y \<in> f`A" and "infinite (f -` {y})"
```
```   351   using assms by (blast dest: inf_img_fin_dom)
```
```   352
```
```   353
```
```   354 subsection "Infinitely Many and Almost All"
```
```   355
```
```   356 text {*
```
```   357   We often need to reason about the existence of infinitely many
```
```   358   (resp., all but finitely many) objects satisfying some predicate, so
```
```   359   we introduce corresponding binders and their proof rules.
```
```   360 *}
```
```   361
```
```   362 definition Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "INFM " 10)
```
```   363   where "Inf_many P \<longleftrightarrow> infinite {x. P x}"
```
```   364
```
```   365 definition Alm_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "MOST " 10)
```
```   366   where "Alm_all P \<longleftrightarrow> \<not> (INFM x. \<not> P x)"
```
```   367
```
```   368 notation (xsymbols)
```
```   369   Inf_many  (binder "\<exists>\<^sub>\<infinity>" 10) and
```
```   370   Alm_all  (binder "\<forall>\<^sub>\<infinity>" 10)
```
```   371
```
```   372 notation (HTML output)
```
```   373   Inf_many  (binder "\<exists>\<^sub>\<infinity>" 10) and
```
```   374   Alm_all  (binder "\<forall>\<^sub>\<infinity>" 10)
```
```   375
```
```   376 lemma INFM_iff_infinite: "(INFM x. P x) \<longleftrightarrow> infinite {x. P x}"
```
```   377   unfolding Inf_many_def ..
```
```   378
```
```   379 lemma MOST_iff_cofinite: "(MOST x. P x) \<longleftrightarrow> finite {x. \<not> P x}"
```
```   380   unfolding Alm_all_def Inf_many_def by simp
```
```   381
```
```   382 (* legacy name *)
```
```   383 lemmas MOST_iff_finiteNeg = MOST_iff_cofinite
```
```   384
```
```   385 lemma not_INFM [simp]: "\<not> (INFM x. P x) \<longleftrightarrow> (MOST x. \<not> P x)"
```
```   386   unfolding Alm_all_def not_not ..
```
```   387
```
```   388 lemma not_MOST [simp]: "\<not> (MOST x. P x) \<longleftrightarrow> (INFM x. \<not> P x)"
```
```   389   unfolding Alm_all_def not_not ..
```
```   390
```
```   391 lemma INFM_const [simp]: "(INFM x::'a. P) \<longleftrightarrow> P \<and> infinite (UNIV::'a set)"
```
```   392   unfolding Inf_many_def by simp
```
```   393
```
```   394 lemma MOST_const [simp]: "(MOST x::'a. P) \<longleftrightarrow> P \<or> finite (UNIV::'a set)"
```
```   395   unfolding Alm_all_def by simp
```
```   396
```
```   397 lemma INFM_EX: "(\<exists>\<^sub>\<infinity>x. P x) \<Longrightarrow> (\<exists>x. P x)"
```
```   398   apply (erule contrapos_pp)
```
```   399   apply simp
```
```   400   done
```
```   401
```
```   402 lemma ALL_MOST: "\<forall>x. P x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. P x"
```
```   403   by simp
```
```   404
```
```   405 lemma INFM_E:
```
```   406   assumes "INFM x. P x"
```
```   407   obtains x where "P x"
```
```   408   using INFM_EX [OF assms] by (rule exE)
```
```   409
```
```   410 lemma MOST_I:
```
```   411   assumes "\<And>x. P x"
```
```   412   shows "MOST x. P x"
```
```   413   using assms by simp
```
```   414
```
```   415 lemma INFM_mono:
```
```   416   assumes inf: "\<exists>\<^sub>\<infinity>x. P x" and q: "\<And>x. P x \<Longrightarrow> Q x"
```
```   417   shows "\<exists>\<^sub>\<infinity>x. Q x"
```
```   418 proof -
```
```   419   from inf have "infinite {x. P x}" unfolding Inf_many_def .
```
```   420   moreover from q have "{x. P x} \<subseteq> {x. Q x}" by auto
```
```   421   ultimately show ?thesis
```
```   422     by (simp add: Inf_many_def infinite_super)
```
```   423 qed
```
```   424
```
```   425 lemma MOST_mono: "\<forall>\<^sub>\<infinity>x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> \<forall>\<^sub>\<infinity>x. Q x"
```
```   426   unfolding Alm_all_def by (blast intro: INFM_mono)
```
```   427
```
```   428 lemma INFM_disj_distrib:
```
```   429   "(\<exists>\<^sub>\<infinity>x. P x \<or> Q x) \<longleftrightarrow> (\<exists>\<^sub>\<infinity>x. P x) \<or> (\<exists>\<^sub>\<infinity>x. Q x)"
```
```   430   unfolding Inf_many_def by (simp add: Collect_disj_eq)
```
```   431
```
```   432 lemma INFM_imp_distrib:
```
```   433   "(INFM x. P x \<longrightarrow> Q x) \<longleftrightarrow> ((MOST x. P x) \<longrightarrow> (INFM x. Q x))"
```
```   434   by (simp only: imp_conv_disj INFM_disj_distrib not_MOST)
```
```   435
```
```   436 lemma MOST_conj_distrib:
```
```   437   "(\<forall>\<^sub>\<infinity>x. P x \<and> Q x) \<longleftrightarrow> (\<forall>\<^sub>\<infinity>x. P x) \<and> (\<forall>\<^sub>\<infinity>x. Q x)"
```
```   438   unfolding Alm_all_def by (simp add: INFM_disj_distrib del: disj_not1)
```
```   439
```
```   440 lemma MOST_conjI:
```
```   441   "MOST x. P x \<Longrightarrow> MOST x. Q x \<Longrightarrow> MOST x. P x \<and> Q x"
```
```   442   by (simp add: MOST_conj_distrib)
```
```   443
```
```   444 lemma INFM_conjI:
```
```   445   "INFM x. P x \<Longrightarrow> MOST x. Q x \<Longrightarrow> INFM x. P x \<and> Q x"
```
```   446   unfolding MOST_iff_cofinite INFM_iff_infinite
```
```   447   apply (drule (1) Diff_infinite_finite)
```
```   448   apply (simp add: Collect_conj_eq Collect_neg_eq)
```
```   449   done
```
```   450
```
```   451 lemma MOST_rev_mp:
```
```   452   assumes "\<forall>\<^sub>\<infinity>x. P x" and "\<forall>\<^sub>\<infinity>x. P x \<longrightarrow> Q x"
```
```   453   shows "\<forall>\<^sub>\<infinity>x. Q x"
```
```   454 proof -
```
```   455   have "\<forall>\<^sub>\<infinity>x. P x \<and> (P x \<longrightarrow> Q x)"
```
```   456     using assms by (rule MOST_conjI)
```
```   457   thus ?thesis by (rule MOST_mono) simp
```
```   458 qed
```
```   459
```
```   460 lemma MOST_imp_iff:
```
```   461   assumes "MOST x. P x"
```
```   462   shows "(MOST x. P x \<longrightarrow> Q x) \<longleftrightarrow> (MOST x. Q x)"
```
```   463 proof
```
```   464   assume "MOST x. P x \<longrightarrow> Q x"
```
```   465   with assms show "MOST x. Q x" by (rule MOST_rev_mp)
```
```   466 next
```
```   467   assume "MOST x. Q x"
```
```   468   then show "MOST x. P x \<longrightarrow> Q x" by (rule MOST_mono) simp
```
```   469 qed
```
```   470
```
```   471 lemma INFM_MOST_simps [simp]:
```
```   472   "\<And>P Q. (INFM x. P x \<and> Q) \<longleftrightarrow> (INFM x. P x) \<and> Q"
```
```   473   "\<And>P Q. (INFM x. P \<and> Q x) \<longleftrightarrow> P \<and> (INFM x. Q x)"
```
```   474   "\<And>P Q. (MOST x. P x \<or> Q) \<longleftrightarrow> (MOST x. P x) \<or> Q"
```
```   475   "\<And>P Q. (MOST x. P \<or> Q x) \<longleftrightarrow> P \<or> (MOST x. Q x)"
```
```   476   "\<And>P Q. (MOST x. P x \<longrightarrow> Q) \<longleftrightarrow> ((INFM x. P x) \<longrightarrow> Q)"
```
```   477   "\<And>P Q. (MOST x. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (MOST x. Q x))"
```
```   478   unfolding Alm_all_def Inf_many_def
```
```   479   by (simp_all add: Collect_conj_eq)
```
```   480
```
```   481 text {* Properties of quantifiers with injective functions. *}
```
```   482
```
```   483 lemma INFM_inj: "INFM x. P (f x) \<Longrightarrow> inj f \<Longrightarrow> INFM x. P x"
```
```   484   unfolding INFM_iff_infinite
```
```   485   apply clarify
```
```   486   apply (drule (1) finite_vimageI)
```
```   487   apply simp
```
```   488   done
```
```   489
```
```   490 lemma MOST_inj: "MOST x. P x \<Longrightarrow> inj f \<Longrightarrow> MOST x. P (f x)"
```
```   491   unfolding MOST_iff_cofinite
```
```   492   apply (drule (1) finite_vimageI)
```
```   493   apply simp
```
```   494   done
```
```   495
```
```   496 text {* Properties of quantifiers with singletons. *}
```
```   497
```
```   498 lemma not_INFM_eq [simp]:
```
```   499   "\<not> (INFM x. x = a)"
```
```   500   "\<not> (INFM x. a = x)"
```
```   501   unfolding INFM_iff_infinite by simp_all
```
```   502
```
```   503 lemma MOST_neq [simp]:
```
```   504   "MOST x. x \<noteq> a"
```
```   505   "MOST x. a \<noteq> x"
```
```   506   unfolding MOST_iff_cofinite by simp_all
```
```   507
```
```   508 lemma INFM_neq [simp]:
```
```   509   "(INFM x::'a. x \<noteq> a) \<longleftrightarrow> infinite (UNIV::'a set)"
```
```   510   "(INFM x::'a. a \<noteq> x) \<longleftrightarrow> infinite (UNIV::'a set)"
```
```   511   unfolding INFM_iff_infinite by simp_all
```
```   512
```
```   513 lemma MOST_eq [simp]:
```
```   514   "(MOST x::'a. x = a) \<longleftrightarrow> finite (UNIV::'a set)"
```
```   515   "(MOST x::'a. a = x) \<longleftrightarrow> finite (UNIV::'a set)"
```
```   516   unfolding MOST_iff_cofinite by simp_all
```
```   517
```
```   518 lemma MOST_eq_imp:
```
```   519   "MOST x. x = a \<longrightarrow> P x"
```
```   520   "MOST x. a = x \<longrightarrow> P x"
```
```   521   unfolding MOST_iff_cofinite by simp_all
```
```   522
```
```   523 text {* Properties of quantifiers over the naturals. *}
```
```   524
```
```   525 lemma INFM_nat: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) \<longleftrightarrow> (\<forall>m. \<exists>n. m < n \<and> P n)"
```
```   526   by (simp add: Inf_many_def infinite_nat_iff_unbounded)
```
```   527
```
```   528 lemma INFM_nat_le: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) \<longleftrightarrow> (\<forall>m. \<exists>n. m \<le> n \<and> P n)"
```
```   529   by (simp add: Inf_many_def infinite_nat_iff_unbounded_le)
```
```   530
```
```   531 lemma MOST_nat: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) \<longleftrightarrow> (\<exists>m. \<forall>n. m < n \<longrightarrow> P n)"
```
```   532   by (simp add: Alm_all_def INFM_nat)
```
```   533
```
```   534 lemma MOST_nat_le: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) \<longleftrightarrow> (\<exists>m. \<forall>n. m \<le> n \<longrightarrow> P n)"
```
```   535   by (simp add: Alm_all_def INFM_nat_le)
```
```   536
```
```   537
```
```   538 subsection "Enumeration of an Infinite Set"
```
```   539
```
```   540 text {*
```
```   541   The set's element type must be wellordered (e.g. the natural numbers).
```
```   542 *}
```
```   543
```
```   544 primrec (in wellorder) enumerate :: "'a set \<Rightarrow> nat \<Rightarrow> 'a"
```
```   545 where
```
```   546   enumerate_0: "enumerate S 0 = (LEAST n. n \<in> S)"
```
```   547 | enumerate_Suc: "enumerate S (Suc n) = enumerate (S - {LEAST n. n \<in> S}) n"
```
```   548
```
```   549 lemma enumerate_Suc': "enumerate S (Suc n) = enumerate (S - {enumerate S 0}) n"
```
```   550   by simp
```
```   551
```
```   552 lemma enumerate_in_set: "infinite S \<Longrightarrow> enumerate S n : S"
```
```   553   apply (induct n arbitrary: S)
```
```   554    apply (fastforce intro: LeastI dest!: infinite_imp_nonempty)
```
```   555   apply simp
```
```   556   apply (metis DiffE infinite_remove)
```
```   557   done
```
```   558
```
```   559 declare enumerate_0 [simp del] enumerate_Suc [simp del]
```
```   560
```
```   561 lemma enumerate_step: "infinite S \<Longrightarrow> enumerate S n < enumerate S (Suc n)"
```
```   562   apply (induct n arbitrary: S)
```
```   563    apply (rule order_le_neq_trans)
```
```   564     apply (simp add: enumerate_0 Least_le enumerate_in_set)
```
```   565    apply (simp only: enumerate_Suc')
```
```   566    apply (subgoal_tac "enumerate (S - {enumerate S 0}) 0 : S - {enumerate S 0}")
```
```   567     apply (blast intro: sym)
```
```   568    apply (simp add: enumerate_in_set del: Diff_iff)
```
```   569   apply (simp add: enumerate_Suc')
```
```   570   done
```
```   571
```
```   572 lemma enumerate_mono: "m<n \<Longrightarrow> infinite S \<Longrightarrow> enumerate S m < enumerate S n"
```
```   573   apply (erule less_Suc_induct)
```
```   574   apply (auto intro: enumerate_step)
```
```   575   done
```
```   576
```
```   577
```
```   578 lemma le_enumerate:
```
```   579   assumes S: "infinite S"
```
```   580   shows "n \<le> enumerate S n"
```
```   581   using S
```
```   582 proof (induct n)
```
```   583   case 0
```
```   584   then show ?case by simp
```
```   585 next
```
```   586   case (Suc n)
```
```   587   then have "n \<le> enumerate S n" by simp
```
```   588   also note enumerate_mono[of n "Suc n", OF _ `infinite S`]
```
```   589   finally show ?case by simp
```
```   590 qed
```
```   591
```
```   592 lemma enumerate_Suc'':
```
```   593   fixes S :: "'a::wellorder set"
```
```   594   assumes "infinite S"
```
```   595   shows "enumerate S (Suc n) = (LEAST s. s \<in> S \<and> enumerate S n < s)"
```
```   596   using assms
```
```   597 proof (induct n arbitrary: S)
```
```   598   case 0
```
```   599   then have "\<forall>s \<in> S. enumerate S 0 \<le> s"
```
```   600     by (auto simp: enumerate.simps intro: Least_le)
```
```   601   then show ?case
```
```   602     unfolding enumerate_Suc' enumerate_0[of "S - {enumerate S 0}"]
```
```   603     by (intro arg_cong[where f = Least] ext) auto
```
```   604 next
```
```   605   case (Suc n S)
```
```   606   show ?case
```
```   607     using enumerate_mono[OF zero_less_Suc `infinite S`, of n] `infinite S`
```
```   608     apply (subst (1 2) enumerate_Suc')
```
```   609     apply (subst Suc)
```
```   610     using `infinite S`
```
```   611     apply simp
```
```   612     apply (intro arg_cong[where f = Least] ext)
```
```   613     apply (auto simp: enumerate_Suc'[symmetric])
```
```   614     done
```
```   615 qed
```
```   616
```
```   617 lemma enumerate_Ex:
```
```   618   assumes S: "infinite (S::nat set)"
```
```   619   shows "s \<in> S \<Longrightarrow> \<exists>n. enumerate S n = s"
```
```   620 proof (induct s rule: less_induct)
```
```   621   case (less s)
```
```   622   show ?case
```
```   623   proof cases
```
```   624     let ?y = "Max {s'\<in>S. s' < s}"
```
```   625     assume "\<exists>y\<in>S. y < s"
```
```   626     then have y: "\<And>x. ?y < x \<longleftrightarrow> (\<forall>s'\<in>S. s' < s \<longrightarrow> s' < x)"
```
```   627       by (subst Max_less_iff) auto
```
```   628     then have y_in: "?y \<in> {s'\<in>S. s' < s}"
```
```   629       by (intro Max_in) auto
```
```   630     with less.hyps[of ?y] obtain n where "enumerate S n = ?y"
```
```   631       by auto
```
```   632     with S have "enumerate S (Suc n) = s"
```
```   633       by (auto simp: y less enumerate_Suc'' intro!: Least_equality)
```
```   634     then show ?case by auto
```
```   635   next
```
```   636     assume *: "\<not> (\<exists>y\<in>S. y < s)"
```
```   637     then have "\<forall>t\<in>S. s \<le> t" by auto
```
```   638     with `s \<in> S` show ?thesis
```
```   639       by (auto intro!: exI[of _ 0] Least_equality simp: enumerate_0)
```
```   640   qed
```
```   641 qed
```
```   642
```
```   643 lemma bij_enumerate:
```
```   644   fixes S :: "nat set"
```
```   645   assumes S: "infinite S"
```
```   646   shows "bij_betw (enumerate S) UNIV S"
```
```   647 proof -
```
```   648   have "\<And>n m. n \<noteq> m \<Longrightarrow> enumerate S n \<noteq> enumerate S m"
```
```   649     using enumerate_mono[OF _ `infinite S`] by (auto simp: neq_iff)
```
```   650   then have "inj (enumerate S)"
```
```   651     by (auto simp: inj_on_def)
```
```   652   moreover have "\<forall>s \<in> S. \<exists>i. enumerate S i = s"
```
```   653     using enumerate_Ex[OF S] by auto
```
```   654   moreover note `infinite S`
```
```   655   ultimately show ?thesis
```
```   656     unfolding bij_betw_def by (auto intro: enumerate_in_set)
```
```   657 qed
```
```   658
```
```   659 subsection "Miscellaneous"
```
```   660
```
```   661 text {*
```
```   662   A few trivial lemmas about sets that contain at most one element.
```
```   663   These simplify the reasoning about deterministic automata.
```
```   664 *}
```
```   665
```
```   666 definition atmost_one :: "'a set \<Rightarrow> bool"
```
```   667   where "atmost_one S \<longleftrightarrow> (\<forall>x y. x\<in>S \<and> y\<in>S \<longrightarrow> x = y)"
```
```   668
```
```   669 lemma atmost_one_empty: "S = {} \<Longrightarrow> atmost_one S"
```
```   670   by (simp add: atmost_one_def)
```
```   671
```
```   672 lemma atmost_one_singleton: "S = {x} \<Longrightarrow> atmost_one S"
```
```   673   by (simp add: atmost_one_def)
```
```   674
```
```   675 lemma atmost_one_unique [elim]: "atmost_one S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> y = x"
```
```   676   by (simp add: atmost_one_def)
```
```   677
```
```   678 end
```
```   679
```