src/HOL/Library/Infinite_Set.thy
 author wenzelm Thu Oct 16 22:44:24 2008 +0200 (2008-10-16) changeset 28615 4c8fa015ec7f parent 27487 c8a6ce181805 child 29839 018ac1fa1ed3 permissions -rw-r--r--
explicit SORT_CONSTRAINT for proofs depending implicitly on certain sorts;
```     1 (*  Title:      HOL/Library/Infinite_Set.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Stephan Merz
```
```     4 *)
```
```     5
```
```     6 header {* Infinite Sets and Related Concepts *}
```
```     7
```
```     8 theory Infinite_Set
```
```     9 imports Plain "~~/src/HOL/SetInterval" "~~/src/HOL/Hilbert_Choice"
```
```    10 begin
```
```    11
```
```    12
```
```    13 subsection "Infinite Sets"
```
```    14
```
```    15 text {*
```
```    16   Some elementary facts about infinite sets, mostly by Stefan Merz.
```
```    17   Beware! Because "infinite" merely abbreviates a negation, these
```
```    18   lemmas may not work well with @{text "blast"}.
```
```    19 *}
```
```    20
```
```    21 abbreviation
```
```    22   infinite :: "'a set \<Rightarrow> bool" where
```
```    23   "infinite S == \<not> finite S"
```
```    24
```
```    25 text {*
```
```    26   Infinite sets are non-empty, and if we remove some elements from an
```
```    27   infinite set, the result is still infinite.
```
```    28 *}
```
```    29
```
```    30 lemma infinite_imp_nonempty: "infinite S ==> S \<noteq> {}"
```
```    31   by auto
```
```    32
```
```    33 lemma infinite_remove:
```
```    34   "infinite S \<Longrightarrow> infinite (S - {a})"
```
```    35   by simp
```
```    36
```
```    37 lemma Diff_infinite_finite:
```
```    38   assumes T: "finite T" and S: "infinite S"
```
```    39   shows "infinite (S - T)"
```
```    40   using T
```
```    41 proof induct
```
```    42   from S
```
```    43   show "infinite (S - {})" by auto
```
```    44 next
```
```    45   fix T x
```
```    46   assume ih: "infinite (S - T)"
```
```    47   have "S - (insert x T) = (S - T) - {x}"
```
```    48     by (rule Diff_insert)
```
```    49   with ih
```
```    50   show "infinite (S - (insert x T))"
```
```    51     by (simp add: infinite_remove)
```
```    52 qed
```
```    53
```
```    54 lemma Un_infinite: "infinite S \<Longrightarrow> infinite (S \<union> T)"
```
```    55   by simp
```
```    56
```
```    57 lemma infinite_super:
```
```    58   assumes T: "S \<subseteq> T" and S: "infinite S"
```
```    59   shows "infinite T"
```
```    60 proof
```
```    61   assume "finite T"
```
```    62   with T have "finite S" by (simp add: finite_subset)
```
```    63   with S show False by simp
```
```    64 qed
```
```    65
```
```    66 text {*
```
```    67   As a concrete example, we prove that the set of natural numbers is
```
```    68   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   then show "\<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     then show ?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   then show ?rhs by (rule finite_nat_bounded)
```
```    98 next
```
```    99   assume ?rhs
```
```   100   then obtain k where "S \<subseteq> {..<k}" ..
```
```   101   then show "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   then have "S \<subseteq> {..k}" by auto
```
```   112   then show ?rhs ..
```
```   113 next
```
```   114   assume ?rhs
```
```   115   then obtain k where "S \<subseteq> {..k}" ..
```
```   116   then show "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 ?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     then have "S \<subseteq> {..m}"
```
```   130       by (auto simp add: sym [OF linorder_not_less])
```
```   131     with `?lhs` show False
```
```   132       by (simp add: finite_nat_iff_bounded_le)
```
```   133   qed
```
```   134 next
```
```   135   assume ?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     then have "\<forall>n. m<n \<longrightarrow> n\<notin>S" by auto
```
```   142     with `?rhs` 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 ?lhs
```
```   151   show ?rhs
```
```   152   proof
```
```   153     fix m
```
```   154     from `?lhs` obtain n where "m<n \<and> n\<in>S"
```
```   155       by (auto simp add: infinite_nat_iff_unbounded)
```
```   156     then have "m\<le>n \<and> n\<in>S" by simp
```
```   157     then show "\<exists>n. m \<le> n \<and> n \<in> S" ..
```
```   158   qed
```
```   159 next
```
```   160   assume ?rhs
```
```   161   show ?lhs
```
```   162   proof (auto simp add: infinite_nat_iff_unbounded)
```
```   163     fix m
```
```   164     from `?rhs` obtain n where "Suc m \<le> n \<and> n\<in>S"
```
```   165       by blast
```
```   166     then have "m<n \<and> n\<in>S" by simp
```
```   167     then show "\<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 to know
```
```   173   that for any number larger than some @{text k}, there is some larger
```
```   174   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 -
```
```   181   {
```
```   182     fix m have "\<exists>n. m<n \<and> n\<in>S"
```
```   183     proof (cases "k<m")
```
```   184       case True
```
```   185       with k show ?thesis by blast
```
```   186     next
```
```   187       case False
```
```   188       from k obtain n where "Suc k < n \<and> n\<in>S" by auto
```
```   189       with False have "m<n \<and> n\<in>S" by auto
```
```   190       then show ?thesis ..
```
```   191     qed
```
```   192   }
```
```   193   then show ?thesis
```
```   194     by (auto simp add: infinite_nat_iff_unbounded)
```
```   195 qed
```
```   196
```
```   197 lemma nat_infinite [simp]: "infinite (UNIV :: nat set)"
```
```   198   by (auto simp add: infinite_nat_iff_unbounded)
```
```   199
```
```   200 lemma nat_not_finite [elim]: "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]:
```
```   219   shows "infinite (UNIV::int set)"
```
```   220 proof -
```
```   221   from inj_int have "infinite (range int)" by (rule range_inj_infinite)
```
```   222   moreover
```
```   223   have "range int \<subseteq> (UNIV::int set)" by simp
```
```   224   ultimately show "infinite (UNIV::int set)" by (simp add: infinite_super)
```
```   225 qed
```
```   226
```
```   227 text {*
```
```   228   The ``only if'' direction is harder because it requires the
```
```   229   construction of a sequence of pairwise different elements of an
```
```   230   infinite set @{text S}. The idea is to construct a sequence of
```
```   231   non-empty and infinite subsets of @{text S} obtained by successively
```
```   232   removing elements of @{text S}.
```
```   233 *}
```
```   234
```
```   235 lemma linorder_injI:
```
```   236   assumes hyp: "!!x y. x < (y::'a::linorder) ==> f x \<noteq> f y"
```
```   237   shows "inj f"
```
```   238 proof (rule inj_onI)
```
```   239   fix x y
```
```   240   assume f_eq: "f x = f y"
```
```   241   show "x = y"
```
```   242   proof (rule linorder_cases)
```
```   243     assume "x < y"
```
```   244     with hyp have "f x \<noteq> f y" by blast
```
```   245     with f_eq show ?thesis by simp
```
```   246   next
```
```   247     assume "x = y"
```
```   248     then show ?thesis .
```
```   249   next
```
```   250     assume "y < x"
```
```   251     with hyp have "f y \<noteq> f x" by blast
```
```   252     with f_eq show ?thesis by simp
```
```   253   qed
```
```   254 qed
```
```   255
```
```   256 lemma infinite_countable_subset:
```
```   257   assumes inf: "infinite (S::'a set)"
```
```   258   shows "\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S"
```
```   259 proof -
```
```   260   def Sseq \<equiv> "nat_rec S (\<lambda>n T. T - {SOME e. e \<in> T})"
```
```   261   def pick \<equiv> "\<lambda>n. (SOME e. e \<in> Sseq n)"
```
```   262   have Sseq_inf: "\<And>n. infinite (Sseq n)"
```
```   263   proof -
```
```   264     fix n
```
```   265     show "infinite (Sseq n)"
```
```   266     proof (induct n)
```
```   267       from inf show "infinite (Sseq 0)"
```
```   268         by (simp add: Sseq_def)
```
```   269     next
```
```   270       fix n
```
```   271       assume "infinite (Sseq n)" then show "infinite (Sseq (Suc n))"
```
```   272         by (simp add: Sseq_def infinite_remove)
```
```   273     qed
```
```   274   qed
```
```   275   have Sseq_S: "\<And>n. Sseq n \<subseteq> S"
```
```   276   proof -
```
```   277     fix n
```
```   278     show "Sseq n \<subseteq> S"
```
```   279       by (induct n) (auto simp add: Sseq_def)
```
```   280   qed
```
```   281   have Sseq_pick: "\<And>n. pick n \<in> Sseq n"
```
```   282   proof -
```
```   283     fix n
```
```   284     show "pick n \<in> Sseq n"
```
```   285     proof (unfold pick_def, rule someI_ex)
```
```   286       from Sseq_inf have "infinite (Sseq n)" .
```
```   287       then have "Sseq n \<noteq> {}" by auto
```
```   288       then show "\<exists>x. x \<in> Sseq n" by auto
```
```   289     qed
```
```   290   qed
```
```   291   with Sseq_S have rng: "range pick \<subseteq> S"
```
```   292     by auto
```
```   293   have pick_Sseq_gt: "\<And>n m. pick n \<notin> Sseq (n + Suc m)"
```
```   294   proof -
```
```   295     fix n m
```
```   296     show "pick n \<notin> Sseq (n + Suc m)"
```
```   297       by (induct m) (auto simp add: Sseq_def pick_def)
```
```   298   qed
```
```   299   have pick_pick: "\<And>n m. pick n \<noteq> pick (n + Suc m)"
```
```   300   proof -
```
```   301     fix n m
```
```   302     from Sseq_pick have "pick (n + Suc m) \<in> Sseq (n + Suc m)" .
```
```   303     moreover from pick_Sseq_gt
```
```   304     have "pick n \<notin> Sseq (n + Suc m)" .
```
```   305     ultimately show "pick n \<noteq> pick (n + Suc m)"
```
```   306       by auto
```
```   307   qed
```
```   308   have inj: "inj pick"
```
```   309   proof (rule linorder_injI)
```
```   310     fix i j :: nat
```
```   311     assume "i < j"
```
```   312     show "pick i \<noteq> pick j"
```
```   313     proof
```
```   314       assume eq: "pick i = pick j"
```
```   315       from `i < j` obtain k where "j = i + Suc k"
```
```   316         by (auto simp add: less_iff_Suc_add)
```
```   317       with pick_pick have "pick i \<noteq> pick j" by simp
```
```   318       with eq show False by simp
```
```   319     qed
```
```   320   qed
```
```   321   from rng inj show ?thesis by auto
```
```   322 qed
```
```   323
```
```   324 lemma infinite_iff_countable_subset:
```
```   325     "infinite S = (\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S)"
```
```   326   by (auto simp add: infinite_countable_subset range_inj_infinite infinite_super)
```
```   327
```
```   328 text {*
```
```   329   For any function with infinite domain and finite range there is some
```
```   330   element that is the image of infinitely many domain elements.  In
```
```   331   particular, any infinite sequence of elements from a finite set
```
```   332   contains some element that occurs infinitely often.
```
```   333 *}
```
```   334
```
```   335 lemma inf_img_fin_dom:
```
```   336   assumes img: "finite (f`A)" and dom: "infinite A"
```
```   337   shows "\<exists>y \<in> f`A. infinite (f -` {y})"
```
```   338 proof (rule ccontr)
```
```   339   assume "\<not> ?thesis"
```
```   340   with img have "finite (UN y:f`A. f -` {y})" by (blast intro: finite_UN_I)
```
```   341   moreover have "A \<subseteq> (UN y:f`A. f -` {y})" by auto
```
```   342   moreover note dom
```
```   343   ultimately show False by (simp add: infinite_super)
```
```   344 qed
```
```   345
```
```   346 lemma inf_img_fin_domE:
```
```   347   assumes "finite (f`A)" and "infinite A"
```
```   348   obtains y where "y \<in> f`A" and "infinite (f -` {y})"
```
```   349   using assms by (blast dest: inf_img_fin_dom)
```
```   350
```
```   351
```
```   352 subsection "Infinitely Many and Almost All"
```
```   353
```
```   354 text {*
```
```   355   We often need to reason about the existence of infinitely many
```
```   356   (resp., all but finitely many) objects satisfying some predicate, so
```
```   357   we introduce corresponding binders and their proof rules.
```
```   358 *}
```
```   359
```
```   360 definition
```
```   361   Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "INFM " 10) where
```
```   362   "Inf_many P = infinite {x. P x}"
```
```   363
```
```   364 definition
```
```   365   Alm_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "MOST " 10) where
```
```   366   "Alm_all P = (\<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_EX:
```
```   377   "(\<exists>\<^sub>\<infinity>x. P x) \<Longrightarrow> (\<exists>x. P x)"
```
```   378   unfolding Inf_many_def
```
```   379 proof (rule ccontr)
```
```   380   assume inf: "infinite {x. P x}"
```
```   381   assume "\<not> ?thesis" then have "{x. P x} = {}" by simp
```
```   382   then have "finite {x. P x}" by simp
```
```   383   with inf show False by simp
```
```   384 qed
```
```   385
```
```   386 lemma MOST_iff_finiteNeg: "(\<forall>\<^sub>\<infinity>x. P x) = finite {x. \<not> P x}"
```
```   387   by (simp add: Alm_all_def Inf_many_def)
```
```   388
```
```   389 lemma ALL_MOST: "\<forall>x. P x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. P x"
```
```   390   by (simp add: MOST_iff_finiteNeg)
```
```   391
```
```   392 lemma INFM_mono:
```
```   393   assumes inf: "\<exists>\<^sub>\<infinity>x. P x" and q: "\<And>x. P x \<Longrightarrow> Q x"
```
```   394   shows "\<exists>\<^sub>\<infinity>x. Q x"
```
```   395 proof -
```
```   396   from inf have "infinite {x. P x}" unfolding Inf_many_def .
```
```   397   moreover from q have "{x. P x} \<subseteq> {x. Q x}" by auto
```
```   398   ultimately show ?thesis
```
```   399     by (simp add: Inf_many_def infinite_super)
```
```   400 qed
```
```   401
```
```   402 lemma MOST_mono: "\<forall>\<^sub>\<infinity>x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> \<forall>\<^sub>\<infinity>x. Q x"
```
```   403   unfolding Alm_all_def by (blast intro: INFM_mono)
```
```   404
```
```   405 lemma INFM_disj_distrib:
```
```   406   "(\<exists>\<^sub>\<infinity>x. P x \<or> Q x) \<longleftrightarrow> (\<exists>\<^sub>\<infinity>x. P x) \<or> (\<exists>\<^sub>\<infinity>x. Q x)"
```
```   407   unfolding Inf_many_def by (simp add: Collect_disj_eq)
```
```   408
```
```   409 lemma MOST_conj_distrib:
```
```   410   "(\<forall>\<^sub>\<infinity>x. P x \<and> Q x) \<longleftrightarrow> (\<forall>\<^sub>\<infinity>x. P x) \<and> (\<forall>\<^sub>\<infinity>x. Q x)"
```
```   411   unfolding Alm_all_def by (simp add: INFM_disj_distrib del: disj_not1)
```
```   412
```
```   413 lemma MOST_rev_mp:
```
```   414   assumes "\<forall>\<^sub>\<infinity>x. P x" and "\<forall>\<^sub>\<infinity>x. P x \<longrightarrow> Q x"
```
```   415   shows "\<forall>\<^sub>\<infinity>x. Q x"
```
```   416 proof -
```
```   417   have "\<forall>\<^sub>\<infinity>x. P x \<and> (P x \<longrightarrow> Q x)"
```
```   418     using prems by (simp add: MOST_conj_distrib)
```
```   419   thus ?thesis by (rule MOST_mono) simp
```
```   420 qed
```
```   421
```
```   422 lemma not_INFM [simp]: "\<not> (INFM x. P x) \<longleftrightarrow> (MOST x. \<not> P x)"
```
```   423 unfolding Alm_all_def not_not ..
```
```   424
```
```   425 lemma not_MOST [simp]: "\<not> (MOST x. P x) \<longleftrightarrow> (INFM x. \<not> P x)"
```
```   426 unfolding Alm_all_def not_not ..
```
```   427
```
```   428 lemma INFM_const [simp]: "(INFM x::'a. P) \<longleftrightarrow> P \<and> infinite (UNIV::'a set)"
```
```   429   unfolding Inf_many_def by simp
```
```   430
```
```   431 lemma MOST_const [simp]: "(MOST x::'a. P) \<longleftrightarrow> P \<or> finite (UNIV::'a set)"
```
```   432   unfolding Alm_all_def by simp
```
```   433
```
```   434 lemma INFM_nat: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) = (\<forall>m. \<exists>n. m<n \<and> P n)"
```
```   435   by (simp add: Inf_many_def infinite_nat_iff_unbounded)
```
```   436
```
```   437 lemma INFM_nat_le: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) = (\<forall>m. \<exists>n. m\<le>n \<and> P n)"
```
```   438   by (simp add: Inf_many_def infinite_nat_iff_unbounded_le)
```
```   439
```
```   440 lemma MOST_nat: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) = (\<exists>m. \<forall>n. m<n \<longrightarrow> P n)"
```
```   441   by (simp add: Alm_all_def INFM_nat)
```
```   442
```
```   443 lemma MOST_nat_le: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) = (\<exists>m. \<forall>n. m\<le>n \<longrightarrow> P n)"
```
```   444   by (simp add: Alm_all_def INFM_nat_le)
```
```   445
```
```   446
```
```   447 subsection "Enumeration of an Infinite Set"
```
```   448
```
```   449 text {*
```
```   450   The set's element type must be wellordered (e.g. the natural numbers).
```
```   451 *}
```
```   452
```
```   453 consts
```
```   454   enumerate   :: "'a::wellorder set => (nat => 'a::wellorder)"
```
```   455 primrec
```
```   456   enumerate_0:   "enumerate S 0       = (LEAST n. n \<in> S)"
```
```   457   enumerate_Suc: "enumerate S (Suc n) = enumerate (S - {LEAST n. n \<in> S}) n"
```
```   458
```
```   459 lemma enumerate_Suc':
```
```   460     "enumerate S (Suc n) = enumerate (S - {enumerate S 0}) n"
```
```   461   by simp
```
```   462
```
```   463 lemma enumerate_in_set: "infinite S \<Longrightarrow> enumerate S n : S"
```
```   464   apply (induct n arbitrary: S)
```
```   465    apply (fastsimp intro: LeastI dest!: infinite_imp_nonempty)
```
```   466   apply (fastsimp iff: finite_Diff_singleton)
```
```   467   done
```
```   468
```
```   469 declare enumerate_0 [simp del] enumerate_Suc [simp del]
```
```   470
```
```   471 lemma enumerate_step: "infinite S \<Longrightarrow> enumerate S n < enumerate S (Suc n)"
```
```   472   apply (induct n arbitrary: S)
```
```   473    apply (rule order_le_neq_trans)
```
```   474     apply (simp add: enumerate_0 Least_le enumerate_in_set)
```
```   475    apply (simp only: enumerate_Suc')
```
```   476    apply (subgoal_tac "enumerate (S - {enumerate S 0}) 0 : S - {enumerate S 0}")
```
```   477     apply (blast intro: sym)
```
```   478    apply (simp add: enumerate_in_set del: Diff_iff)
```
```   479   apply (simp add: enumerate_Suc')
```
```   480   done
```
```   481
```
```   482 lemma enumerate_mono: "m<n \<Longrightarrow> infinite S \<Longrightarrow> enumerate S m < enumerate S n"
```
```   483   apply (erule less_Suc_induct)
```
```   484   apply (auto intro: enumerate_step)
```
```   485   done
```
```   486
```
```   487
```
```   488 subsection "Miscellaneous"
```
```   489
```
```   490 text {*
```
```   491   A few trivial lemmas about sets that contain at most one element.
```
```   492   These simplify the reasoning about deterministic automata.
```
```   493 *}
```
```   494
```
```   495 definition
```
```   496   atmost_one :: "'a set \<Rightarrow> bool" where
```
```   497   "atmost_one S = (\<forall>x y. x\<in>S \<and> y\<in>S \<longrightarrow> x=y)"
```
```   498
```
```   499 lemma atmost_one_empty: "S = {} \<Longrightarrow> atmost_one S"
```
```   500   by (simp add: atmost_one_def)
```
```   501
```
```   502 lemma atmost_one_singleton: "S = {x} \<Longrightarrow> atmost_one S"
```
```   503   by (simp add: atmost_one_def)
```
```   504
```
```   505 lemma atmost_one_unique [elim]: "atmost_one S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> y = x"
```
```   506   by (simp add: atmost_one_def)
```
```   507
```
```   508 end
```