src/HOL/Hilbert_Choice.thy
 author paulson Mon May 23 15:33:24 2016 +0100 (2016-05-23) changeset 63114 27afe7af7379 parent 63040 eb4ddd18d635 child 63365 5340fb6633d0 permissions -rw-r--r--
Lots of new material for multivariate analysis
```     1 (*  Title:      HOL/Hilbert_Choice.thy
```
```     2     Author:     Lawrence C Paulson, Tobias Nipkow
```
```     3     Copyright   2001  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 section \<open>Hilbert's Epsilon-Operator and the Axiom of Choice\<close>
```
```     7
```
```     8 theory Hilbert_Choice
```
```     9 imports Wellfounded
```
```    10 keywords "specification" :: thy_goal
```
```    11 begin
```
```    12
```
```    13 subsection \<open>Hilbert's epsilon\<close>
```
```    14
```
```    15 axiomatization Eps :: "('a => bool) => 'a" where
```
```    16   someI: "P x ==> P (Eps P)"
```
```    17
```
```    18 syntax (epsilon)
```
```    19   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3\<some>_./ _)" [0, 10] 10)
```
```    20 syntax (input)
```
```    21   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3@ _./ _)" [0, 10] 10)
```
```    22 syntax
```
```    23   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3SOME _./ _)" [0, 10] 10)
```
```    24 translations
```
```    25   "SOME x. P" == "CONST Eps (%x. P)"
```
```    26
```
```    27 print_translation \<open>
```
```    28   [(@{const_syntax Eps}, fn _ => fn [Abs abs] =>
```
```    29       let val (x, t) = Syntax_Trans.atomic_abs_tr' abs
```
```    30       in Syntax.const @{syntax_const "_Eps"} \$ x \$ t end)]
```
```    31 \<close> \<comment> \<open>to avoid eta-contraction of body\<close>
```
```    32
```
```    33 definition inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
```
```    34 "inv_into A f == %x. SOME y. y : A & f y = x"
```
```    35
```
```    36 abbreviation inv :: "('a => 'b) => ('b => 'a)" where
```
```    37 "inv == inv_into UNIV"
```
```    38
```
```    39
```
```    40 subsection \<open>Hilbert's Epsilon-operator\<close>
```
```    41
```
```    42 text\<open>Easier to apply than \<open>someI\<close> if the witness comes from an
```
```    43 existential formula\<close>
```
```    44 lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)"
```
```    45 apply (erule exE)
```
```    46 apply (erule someI)
```
```    47 done
```
```    48
```
```    49 text\<open>Easier to apply than \<open>someI\<close> because the conclusion has only one
```
```    50 occurrence of @{term P}.\<close>
```
```    51 lemma someI2: "[| P a;  !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
```
```    52   by (blast intro: someI)
```
```    53
```
```    54 text\<open>Easier to apply than \<open>someI2\<close> if the witness comes from an
```
```    55 existential formula\<close>
```
```    56
```
```    57 lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
```
```    58   by (blast intro: someI2)
```
```    59
```
```    60 lemma someI2_bex: "[| \<exists>a\<in>A. P a; !!x. x \<in> A \<and> P x ==> Q x |] ==> Q (SOME x. x \<in> A \<and> P x)"
```
```    61   by (blast intro: someI2)
```
```    62
```
```    63 lemma some_equality [intro]:
```
```    64      "[| P a;  !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
```
```    65 by (blast intro: someI2)
```
```    66
```
```    67 lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
```
```    68 by blast
```
```    69
```
```    70 lemma some_eq_ex: "P (SOME x. P x) =  (\<exists>x. P x)"
```
```    71 by (blast intro: someI)
```
```    72
```
```    73 lemma some_in_eq: "(SOME x. x \<in> A) \<in> A \<longleftrightarrow> A \<noteq> {}"
```
```    74   unfolding ex_in_conv[symmetric] by (rule some_eq_ex)
```
```    75
```
```    76 lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
```
```    77 apply (rule some_equality)
```
```    78 apply (rule refl, assumption)
```
```    79 done
```
```    80
```
```    81 lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"
```
```    82 apply (rule some_equality)
```
```    83 apply (rule refl)
```
```    84 apply (erule sym)
```
```    85 done
```
```    86
```
```    87
```
```    88 subsection\<open>Axiom of Choice, Proved Using the Description Operator\<close>
```
```    89
```
```    90 lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)"
```
```    91 by (fast elim: someI)
```
```    92
```
```    93 lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)"
```
```    94 by (fast elim: someI)
```
```    95
```
```    96 lemma choice_iff: "(\<forall>x. \<exists>y. Q x y) \<longleftrightarrow> (\<exists>f. \<forall>x. Q x (f x))"
```
```    97 by (fast elim: someI)
```
```    98
```
```    99 lemma choice_iff': "(\<forall>x. P x \<longrightarrow> (\<exists>y. Q x y)) \<longleftrightarrow> (\<exists>f. \<forall>x. P x \<longrightarrow> Q x (f x))"
```
```   100 by (fast elim: someI)
```
```   101
```
```   102 lemma bchoice_iff: "(\<forall>x\<in>S. \<exists>y. Q x y) \<longleftrightarrow> (\<exists>f. \<forall>x\<in>S. Q x (f x))"
```
```   103 by (fast elim: someI)
```
```   104
```
```   105 lemma bchoice_iff': "(\<forall>x\<in>S. P x \<longrightarrow> (\<exists>y. Q x y)) \<longleftrightarrow> (\<exists>f. \<forall>x\<in>S. P x \<longrightarrow> Q x (f x))"
```
```   106 by (fast elim: someI)
```
```   107
```
```   108 lemma dependent_nat_choice:
```
```   109   assumes  1: "\<exists>x. P 0 x" and
```
```   110            2: "\<And>x n. P n x \<Longrightarrow> \<exists>y. P (Suc n) y \<and> Q n x y"
```
```   111   shows "\<exists>f. \<forall>n. P n (f n) \<and> Q n (f n) (f (Suc n))"
```
```   112 proof (intro exI allI conjI)
```
```   113   fix n
```
```   114   define f where "f = rec_nat (SOME x. P 0 x) (\<lambda>n x. SOME y. P (Suc n) y \<and> Q n x y)"
```
```   115   have "P 0 (f 0)" "\<And>n. P n (f n) \<Longrightarrow> P (Suc n) (f (Suc n)) \<and> Q n (f n) (f (Suc n))"
```
```   116     using someI_ex[OF 1] someI_ex[OF 2] by (simp_all add: f_def)
```
```   117   then show "P n (f n)" "Q n (f n) (f (Suc n))"
```
```   118     by (induct n) auto
```
```   119 qed
```
```   120
```
```   121
```
```   122 subsection \<open>Function Inverse\<close>
```
```   123
```
```   124 lemma inv_def: "inv f = (%y. SOME x. f x = y)"
```
```   125 by(simp add: inv_into_def)
```
```   126
```
```   127 lemma inv_into_into: "x : f ` A ==> inv_into A f x : A"
```
```   128 apply (simp add: inv_into_def)
```
```   129 apply (fast intro: someI2)
```
```   130 done
```
```   131
```
```   132 lemma inv_id [simp]: "inv id = id"
```
```   133 by (simp add: inv_into_def id_def)
```
```   134
```
```   135 lemma inv_into_f_f [simp]:
```
```   136   "[| inj_on f A;  x : A |] ==> inv_into A f (f x) = x"
```
```   137 apply (simp add: inv_into_def inj_on_def)
```
```   138 apply (blast intro: someI2)
```
```   139 done
```
```   140
```
```   141 lemma inv_f_f: "inj f ==> inv f (f x) = x"
```
```   142 by simp
```
```   143
```
```   144 lemma f_inv_into_f: "y : f`A  ==> f (inv_into A f y) = y"
```
```   145 apply (simp add: inv_into_def)
```
```   146 apply (fast intro: someI2)
```
```   147 done
```
```   148
```
```   149 lemma inv_into_f_eq: "[| inj_on f A; x : A; f x = y |] ==> inv_into A f y = x"
```
```   150 apply (erule subst)
```
```   151 apply (fast intro: inv_into_f_f)
```
```   152 done
```
```   153
```
```   154 lemma inv_f_eq: "[| inj f; f x = y |] ==> inv f y = x"
```
```   155 by (simp add:inv_into_f_eq)
```
```   156
```
```   157 lemma inj_imp_inv_eq: "[| inj f; ALL x. f(g x) = x |] ==> inv f = g"
```
```   158   by (blast intro: inv_into_f_eq)
```
```   159
```
```   160 text\<open>But is it useful?\<close>
```
```   161 lemma inj_transfer:
```
```   162   assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)"
```
```   163   shows "P x"
```
```   164 proof -
```
```   165   have "f x \<in> range f" by auto
```
```   166   hence "P(inv f (f x))" by (rule minor)
```
```   167   thus "P x" by (simp add: inv_into_f_f [OF injf])
```
```   168 qed
```
```   169
```
```   170 lemma inj_iff: "(inj f) = (inv f o f = id)"
```
```   171 apply (simp add: o_def fun_eq_iff)
```
```   172 apply (blast intro: inj_on_inverseI inv_into_f_f)
```
```   173 done
```
```   174
```
```   175 lemma inv_o_cancel[simp]: "inj f ==> inv f o f = id"
```
```   176 by (simp add: inj_iff)
```
```   177
```
```   178 lemma o_inv_o_cancel[simp]: "inj f ==> g o inv f o f = g"
```
```   179 by (simp add: comp_assoc)
```
```   180
```
```   181 lemma inv_into_image_cancel[simp]:
```
```   182   "inj_on f A ==> S <= A ==> inv_into A f ` f ` S = S"
```
```   183 by(fastforce simp: image_def)
```
```   184
```
```   185 lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
```
```   186 by (blast intro!: surjI inv_into_f_f)
```
```   187
```
```   188 lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
```
```   189 by (simp add: f_inv_into_f)
```
```   190
```
```   191 lemma inv_into_injective:
```
```   192   assumes eq: "inv_into A f x = inv_into A f y"
```
```   193       and x: "x: f`A"
```
```   194       and y: "y: f`A"
```
```   195   shows "x=y"
```
```   196 proof -
```
```   197   have "f (inv_into A f x) = f (inv_into A f y)" using eq by simp
```
```   198   thus ?thesis by (simp add: f_inv_into_f x y)
```
```   199 qed
```
```   200
```
```   201 lemma inj_on_inv_into: "B <= f`A ==> inj_on (inv_into A f) B"
```
```   202 by (blast intro: inj_onI dest: inv_into_injective injD)
```
```   203
```
```   204 lemma bij_betw_inv_into: "bij_betw f A B ==> bij_betw (inv_into A f) B A"
```
```   205 by (auto simp add: bij_betw_def inj_on_inv_into)
```
```   206
```
```   207 lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
```
```   208 by (simp add: inj_on_inv_into)
```
```   209
```
```   210 lemma surj_iff: "(surj f) = (f o inv f = id)"
```
```   211 by (auto intro!: surjI simp: surj_f_inv_f fun_eq_iff[where 'b='a])
```
```   212
```
```   213 lemma surj_iff_all: "surj f \<longleftrightarrow> (\<forall>x. f (inv f x) = x)"
```
```   214   unfolding surj_iff by (simp add: o_def fun_eq_iff)
```
```   215
```
```   216 lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
```
```   217 apply (rule ext)
```
```   218 apply (drule_tac x = "inv f x" in spec)
```
```   219 apply (simp add: surj_f_inv_f)
```
```   220 done
```
```   221
```
```   222 lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
```
```   223 by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
```
```   224
```
```   225 lemma inv_equality: "[| !!x. g (f x) = x;  !!y. f (g y) = y |] ==> inv f = g"
```
```   226 apply (rule ext)
```
```   227 apply (auto simp add: inv_into_def)
```
```   228 done
```
```   229
```
```   230 lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
```
```   231 apply (rule inv_equality)
```
```   232 apply (auto simp add: bij_def surj_f_inv_f)
```
```   233 done
```
```   234
```
```   235 (** bij(inv f) implies little about f.  Consider f::bool=>bool such that
```
```   236     f(True)=f(False)=True.  Then it's consistent with axiom someI that
```
```   237     inv f could be any function at all, including the identity function.
```
```   238     If inv f=id then inv f is a bijection, but inj f, surj(f) and
```
```   239     inv(inv f)=f all fail.
```
```   240 **)
```
```   241
```
```   242 lemma inv_into_comp:
```
```   243   "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
```
```   244   inv_into A (f o g) x = (inv_into A g o inv_into (g ` A) f) x"
```
```   245 apply (rule inv_into_f_eq)
```
```   246   apply (fast intro: comp_inj_on)
```
```   247  apply (simp add: inv_into_into)
```
```   248 apply (simp add: f_inv_into_f inv_into_into)
```
```   249 done
```
```   250
```
```   251 lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
```
```   252 apply (rule inv_equality)
```
```   253 apply (auto simp add: bij_def surj_f_inv_f)
```
```   254 done
```
```   255
```
```   256 lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
```
```   257   by (simp add: surj_f_inv_f image_comp comp_def)
```
```   258
```
```   259 lemma image_inv_f_f: "inj f ==> inv f ` (f ` A) = A"
```
```   260   by simp
```
```   261
```
```   262 lemma inv_image_comp: "inj f ==> inv f ` (f ` X) = X"
```
```   263   by (fact image_inv_f_f)
```
```   264
```
```   265 lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
```
```   266 apply auto
```
```   267 apply (force simp add: bij_is_inj)
```
```   268 apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
```
```   269 done
```
```   270
```
```   271 lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A"
```
```   272 apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
```
```   273 apply (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric])
```
```   274 done
```
```   275
```
```   276 lemma finite_fun_UNIVD1:
```
```   277   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
```
```   278   and card: "card (UNIV :: 'b set) \<noteq> Suc 0"
```
```   279   shows "finite (UNIV :: 'a set)"
```
```   280 proof -
```
```   281   from fin have finb: "finite (UNIV :: 'b set)" by (rule finite_fun_UNIVD2)
```
```   282   with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)"
```
```   283     by (cases "card (UNIV :: 'b set)") (auto simp add: card_eq_0_iff)
```
```   284   then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - Suc (Suc 0)" by auto
```
```   285   then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by (auto simp add: card_Suc_eq)
```
```   286   from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by (rule finite_imageI)
```
```   287   moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)"
```
```   288   proof (rule UNIV_eq_I)
```
```   289     fix x :: 'a
```
```   290     from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_into_def)
```
```   291     thus "x \<in> range (\<lambda>f::'a \<Rightarrow> 'b. inv f b1)" by blast
```
```   292   qed
```
```   293   ultimately show "finite (UNIV :: 'a set)" by simp
```
```   294 qed
```
```   295
```
```   296 text \<open>
```
```   297   Every infinite set contains a countable subset. More precisely we
```
```   298   show that a set \<open>S\<close> is infinite if and only if there exists an
```
```   299   injective function from the naturals into \<open>S\<close>.
```
```   300
```
```   301   The ``only if'' direction is harder because it requires the
```
```   302   construction of a sequence of pairwise different elements of an
```
```   303   infinite set \<open>S\<close>. The idea is to construct a sequence of
```
```   304   non-empty and infinite subsets of \<open>S\<close> obtained by successively
```
```   305   removing elements of \<open>S\<close>.
```
```   306 \<close>
```
```   307
```
```   308 lemma infinite_countable_subset:
```
```   309   assumes inf: "\<not> finite (S::'a set)"
```
```   310   shows "\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S"
```
```   311   \<comment> \<open>Courtesy of Stephan Merz\<close>
```
```   312 proof -
```
```   313   define Sseq where "Sseq = rec_nat S (\<lambda>n T. T - {SOME e. e \<in> T})"
```
```   314   define pick where "pick n = (SOME e. e \<in> Sseq n)" for n
```
```   315   { fix n have "Sseq n \<subseteq> S" "\<not> finite (Sseq n)" by (induct n) (auto simp add: Sseq_def inf) }
```
```   316   moreover then have *: "\<And>n. pick n \<in> Sseq n"
```
```   317     unfolding pick_def by (subst (asm) finite.simps) (auto simp add: ex_in_conv intro: someI_ex)
```
```   318   ultimately have "range pick \<subseteq> S" by auto
```
```   319   moreover
```
```   320   { fix n m
```
```   321     have "pick n \<notin> Sseq (n + Suc m)" by (induct m) (auto simp add: Sseq_def pick_def)
```
```   322     with * have "pick n \<noteq> pick (n + Suc m)" by auto
```
```   323   }
```
```   324   then have "inj pick" by (intro linorder_injI) (auto simp add: less_iff_Suc_add)
```
```   325   ultimately show ?thesis by blast
```
```   326 qed
```
```   327
```
```   328 lemma infinite_iff_countable_subset: "\<not> finite S \<longleftrightarrow> (\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S)"
```
```   329   \<comment> \<open>Courtesy of Stephan Merz\<close>
```
```   330   using finite_imageD finite_subset infinite_UNIV_char_0 infinite_countable_subset by auto
```
```   331
```
```   332 lemma image_inv_into_cancel:
```
```   333   assumes SURJ: "f`A=A'" and SUB: "B' \<le> A'"
```
```   334   shows "f `((inv_into A f)`B') = B'"
```
```   335   using assms
```
```   336 proof (auto simp add: f_inv_into_f)
```
```   337   let ?f' = "(inv_into A f)"
```
```   338   fix a' assume *: "a' \<in> B'"
```
```   339   then have "a' \<in> A'" using SUB by auto
```
```   340   then have "a' = f (?f' a')"
```
```   341     using SURJ by (auto simp add: f_inv_into_f)
```
```   342   then show "a' \<in> f ` (?f' ` B')" using * by blast
```
```   343 qed
```
```   344
```
```   345 lemma inv_into_inv_into_eq:
```
```   346   assumes "bij_betw f A A'" "a \<in> A"
```
```   347   shows "inv_into A' (inv_into A f) a = f a"
```
```   348 proof -
```
```   349   let ?f' = "inv_into A f"   let ?f'' = "inv_into A' ?f'"
```
```   350   have 1: "bij_betw ?f' A' A" using assms
```
```   351   by (auto simp add: bij_betw_inv_into)
```
```   352   obtain a' where 2: "a' \<in> A'" and 3: "?f' a' = a"
```
```   353     using 1 \<open>a \<in> A\<close> unfolding bij_betw_def by force
```
```   354   hence "?f'' a = a'"
```
```   355     using \<open>a \<in> A\<close> 1 3 by (auto simp add: f_inv_into_f bij_betw_def)
```
```   356   moreover have "f a = a'" using assms 2 3
```
```   357     by (auto simp add: bij_betw_def)
```
```   358   ultimately show "?f'' a = f a" by simp
```
```   359 qed
```
```   360
```
```   361 lemma inj_on_iff_surj:
```
```   362   assumes "A \<noteq> {}"
```
```   363   shows "(\<exists>f. inj_on f A \<and> f ` A \<le> A') \<longleftrightarrow> (\<exists>g. g ` A' = A)"
```
```   364 proof safe
```
```   365   fix f assume INJ: "inj_on f A" and INCL: "f ` A \<le> A'"
```
```   366   let ?phi = "\<lambda>a' a. a \<in> A \<and> f a = a'"  let ?csi = "\<lambda>a. a \<in> A"
```
```   367   let ?g = "\<lambda>a'. if a' \<in> f ` A then (SOME a. ?phi a' a) else (SOME a. ?csi a)"
```
```   368   have "?g ` A' = A"
```
```   369   proof
```
```   370     show "?g ` A' \<le> A"
```
```   371     proof clarify
```
```   372       fix a' assume *: "a' \<in> A'"
```
```   373       show "?g a' \<in> A"
```
```   374       proof cases
```
```   375         assume Case1: "a' \<in> f ` A"
```
```   376         then obtain a where "?phi a' a" by blast
```
```   377         hence "?phi a' (SOME a. ?phi a' a)" using someI[of "?phi a'" a] by blast
```
```   378         with Case1 show ?thesis by auto
```
```   379       next
```
```   380         assume Case2: "a' \<notin> f ` A"
```
```   381         hence "?csi (SOME a. ?csi a)" using assms someI_ex[of ?csi] by blast
```
```   382         with Case2 show ?thesis by auto
```
```   383       qed
```
```   384     qed
```
```   385   next
```
```   386     show "A \<le> ?g ` A'"
```
```   387     proof-
```
```   388       {fix a assume *: "a \<in> A"
```
```   389        let ?b = "SOME aa. ?phi (f a) aa"
```
```   390        have "?phi (f a) a" using * by auto
```
```   391        hence 1: "?phi (f a) ?b" using someI[of "?phi(f a)" a] by blast
```
```   392        hence "?g(f a) = ?b" using * by auto
```
```   393        moreover have "a = ?b" using 1 INJ * by (auto simp add: inj_on_def)
```
```   394        ultimately have "?g(f a) = a" by simp
```
```   395        with INCL * have "?g(f a) = a \<and> f a \<in> A'" by auto
```
```   396       }
```
```   397       thus ?thesis by force
```
```   398     qed
```
```   399   qed
```
```   400   thus "\<exists>g. g ` A' = A" by blast
```
```   401 next
```
```   402   fix g  let ?f = "inv_into A' g"
```
```   403   have "inj_on ?f (g ` A')"
```
```   404     by (auto simp add: inj_on_inv_into)
```
```   405   moreover
```
```   406   {fix a' assume *: "a' \<in> A'"
```
```   407    let ?phi = "\<lambda> b'. b' \<in> A' \<and> g b' = g a'"
```
```   408    have "?phi a'" using * by auto
```
```   409    hence "?phi(SOME b'. ?phi b')" using someI[of ?phi] by blast
```
```   410    hence "?f(g a') \<in> A'" unfolding inv_into_def by auto
```
```   411   }
```
```   412   ultimately show "\<exists>f. inj_on f (g ` A') \<and> f ` g ` A' \<subseteq> A'" by auto
```
```   413 qed
```
```   414
```
```   415 lemma Ex_inj_on_UNION_Sigma:
```
```   416   "\<exists>f. (inj_on f (\<Union>i \<in> I. A i) \<and> f ` (\<Union>i \<in> I. A i) \<le> (SIGMA i : I. A i))"
```
```   417 proof
```
```   418   let ?phi = "\<lambda> a i. i \<in> I \<and> a \<in> A i"
```
```   419   let ?sm = "\<lambda> a. SOME i. ?phi a i"
```
```   420   let ?f = "\<lambda>a. (?sm a, a)"
```
```   421   have "inj_on ?f (\<Union>i \<in> I. A i)" unfolding inj_on_def by auto
```
```   422   moreover
```
```   423   { { fix i a assume "i \<in> I" and "a \<in> A i"
```
```   424       hence "?sm a \<in> I \<and> a \<in> A(?sm a)" using someI[of "?phi a" i] by auto
```
```   425     }
```
```   426     hence "?f ` (\<Union>i \<in> I. A i) \<le> (SIGMA i : I. A i)" by auto
```
```   427   }
```
```   428   ultimately
```
```   429   show "inj_on ?f (\<Union>i \<in> I. A i) \<and> ?f ` (\<Union>i \<in> I. A i) \<le> (SIGMA i : I. A i)"
```
```   430   by auto
```
```   431 qed
```
```   432
```
```   433 lemma inv_unique_comp:
```
```   434   assumes fg: "f \<circ> g = id"
```
```   435     and gf: "g \<circ> f = id"
```
```   436   shows "inv f = g"
```
```   437   using fg gf inv_equality[of g f] by (auto simp add: fun_eq_iff)
```
```   438
```
```   439
```
```   440 subsection \<open>The Cantor-Bernstein Theorem\<close>
```
```   441
```
```   442 lemma Cantor_Bernstein_aux:
```
```   443   shows "\<exists>A' h. A' \<le> A \<and>
```
```   444                 (\<forall>a \<in> A'. a \<notin> g`(B - f ` A')) \<and>
```
```   445                 (\<forall>a \<in> A'. h a = f a) \<and>
```
```   446                 (\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a))"
```
```   447 proof-
```
```   448   obtain H where H_def: "H = (\<lambda> A'. A - (g`(B - (f ` A'))))" by blast
```
```   449   have 0: "mono H" unfolding mono_def H_def by blast
```
```   450   then obtain A' where 1: "H A' = A'" using lfp_unfold by blast
```
```   451   hence 2: "A' = A - (g`(B - (f ` A')))" unfolding H_def by simp
```
```   452   hence 3: "A' \<le> A" by blast
```
```   453   have 4: "\<forall>a \<in> A'.  a \<notin> g`(B - f ` A')"
```
```   454   using 2 by blast
```
```   455   have 5: "\<forall>a \<in> A - A'. \<exists>b \<in> B - (f ` A'). a = g b"
```
```   456   using 2 by blast
```
```   457   (*  *)
```
```   458   obtain h where h_def:
```
```   459   "h = (\<lambda> a. if a \<in> A' then f a else (SOME b. b \<in> B - (f ` A') \<and> a = g b))" by blast
```
```   460   hence "\<forall>a \<in> A'. h a = f a" by auto
```
```   461   moreover
```
```   462   have "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
```
```   463   proof
```
```   464     fix a assume *: "a \<in> A - A'"
```
```   465     let ?phi = "\<lambda> b. b \<in> B - (f ` A') \<and> a = g b"
```
```   466     have "h a = (SOME b. ?phi b)" using h_def * by auto
```
```   467     moreover have "\<exists>b. ?phi b" using 5 *  by auto
```
```   468     ultimately show  "?phi (h a)" using someI_ex[of ?phi] by auto
```
```   469   qed
```
```   470   ultimately show ?thesis using 3 4 by blast
```
```   471 qed
```
```   472
```
```   473 theorem Cantor_Bernstein:
```
```   474   assumes INJ1: "inj_on f A" and SUB1: "f ` A \<le> B" and
```
```   475           INJ2: "inj_on g B" and SUB2: "g ` B \<le> A"
```
```   476   shows "\<exists>h. bij_betw h A B"
```
```   477 proof-
```
```   478   obtain A' and h where 0: "A' \<le> A" and
```
```   479   1: "\<forall>a \<in> A'. a \<notin> g`(B - f ` A')" and
```
```   480   2: "\<forall>a \<in> A'. h a = f a" and
```
```   481   3: "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
```
```   482   using Cantor_Bernstein_aux[of A g B f] by blast
```
```   483   have "inj_on h A"
```
```   484   proof (intro inj_onI)
```
```   485     fix a1 a2
```
```   486     assume 4: "a1 \<in> A" and 5: "a2 \<in> A" and 6: "h a1 = h a2"
```
```   487     show "a1 = a2"
```
```   488     proof(cases "a1 \<in> A'")
```
```   489       assume Case1: "a1 \<in> A'"
```
```   490       show ?thesis
```
```   491       proof(cases "a2 \<in> A'")
```
```   492         assume Case11: "a2 \<in> A'"
```
```   493         hence "f a1 = f a2" using Case1 2 6 by auto
```
```   494         thus ?thesis using INJ1 Case1 Case11 0
```
```   495         unfolding inj_on_def by blast
```
```   496       next
```
```   497         assume Case12: "a2 \<notin> A'"
```
```   498         hence False using 3 5 2 6 Case1 by force
```
```   499         thus ?thesis by simp
```
```   500       qed
```
```   501     next
```
```   502     assume Case2: "a1 \<notin> A'"
```
```   503       show ?thesis
```
```   504       proof(cases "a2 \<in> A'")
```
```   505         assume Case21: "a2 \<in> A'"
```
```   506         hence False using 3 4 2 6 Case2 by auto
```
```   507         thus ?thesis by simp
```
```   508       next
```
```   509         assume Case22: "a2 \<notin> A'"
```
```   510         hence "a1 = g(h a1) \<and> a2 = g(h a2)" using Case2 4 5 3 by auto
```
```   511         thus ?thesis using 6 by simp
```
```   512       qed
```
```   513     qed
```
```   514   qed
```
```   515   (*  *)
```
```   516   moreover
```
```   517   have "h ` A = B"
```
```   518   proof safe
```
```   519     fix a assume "a \<in> A"
```
```   520     thus "h a \<in> B" using SUB1 2 3 by (cases "a \<in> A'") auto
```
```   521   next
```
```   522     fix b assume *: "b \<in> B"
```
```   523     show "b \<in> h ` A"
```
```   524     proof(cases "b \<in> f ` A'")
```
```   525       assume Case1: "b \<in> f ` A'"
```
```   526       then obtain a where "a \<in> A' \<and> b = f a" by blast
```
```   527       thus ?thesis using 2 0 by force
```
```   528     next
```
```   529       assume Case2: "b \<notin> f ` A'"
```
```   530       hence "g b \<notin> A'" using 1 * by auto
```
```   531       hence 4: "g b \<in> A - A'" using * SUB2 by auto
```
```   532       hence "h(g b) \<in> B \<and> g(h(g b)) = g b"
```
```   533       using 3 by auto
```
```   534       hence "h(g b) = b" using * INJ2 unfolding inj_on_def by auto
```
```   535       thus ?thesis using 4 by force
```
```   536     qed
```
```   537   qed
```
```   538   (*  *)
```
```   539   ultimately show ?thesis unfolding bij_betw_def by auto
```
```   540 qed
```
```   541
```
```   542 subsection \<open>Other Consequences of Hilbert's Epsilon\<close>
```
```   543
```
```   544 text \<open>Hilbert's Epsilon and the @{term split} Operator\<close>
```
```   545
```
```   546 text\<open>Looping simprule\<close>
```
```   547 lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
```
```   548   by simp
```
```   549
```
```   550 lemma Eps_case_prod: "Eps (case_prod P) = (SOME xy. P (fst xy) (snd xy))"
```
```   551   by (simp add: split_def)
```
```   552
```
```   553 lemma Eps_case_prod_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
```
```   554   by blast
```
```   555
```
```   556
```
```   557 text\<open>A relation is wellfounded iff it has no infinite descending chain\<close>
```
```   558 lemma wf_iff_no_infinite_down_chain:
```
```   559   "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
```
```   560 apply (simp only: wf_eq_minimal)
```
```   561 apply (rule iffI)
```
```   562  apply (rule notI)
```
```   563  apply (erule exE)
```
```   564  apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
```
```   565 apply (erule contrapos_np, simp, clarify)
```
```   566 apply (subgoal_tac "\<forall>n. rec_nat x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
```
```   567  apply (rule_tac x = "rec_nat x (%i y. @z. z:Q & (z,y) :r)" in exI)
```
```   568  apply (rule allI, simp)
```
```   569  apply (rule someI2_ex, blast, blast)
```
```   570 apply (rule allI)
```
```   571 apply (induct_tac "n", simp_all)
```
```   572 apply (rule someI2_ex, blast+)
```
```   573 done
```
```   574
```
```   575 lemma wf_no_infinite_down_chainE:
```
```   576   assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r"
```
```   577 using \<open>wf r\<close> wf_iff_no_infinite_down_chain[of r] by blast
```
```   578
```
```   579
```
```   580 text\<open>A dynamically-scoped fact for TFL\<close>
```
```   581 lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
```
```   582   by (blast intro: someI)
```
```   583
```
```   584
```
```   585 subsection \<open>Least value operator\<close>
```
```   586
```
```   587 definition
```
```   588   LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where
```
```   589   "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
```
```   590
```
```   591 syntax
```
```   592   "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
```
```   593 translations
```
```   594   "LEAST x WRT m. P" == "CONST LeastM m (%x. P)"
```
```   595
```
```   596 lemma LeastMI2:
```
```   597   "P x ==> (!!y. P y ==> m x <= m y)
```
```   598     ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
```
```   599     ==> Q (LeastM m P)"
```
```   600   apply (simp add: LeastM_def)
```
```   601   apply (rule someI2_ex, blast, blast)
```
```   602   done
```
```   603
```
```   604 lemma LeastM_equality:
```
```   605   "P k ==> (!!x. P x ==> m k <= m x)
```
```   606     ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
```
```   607   apply (rule LeastMI2, assumption, blast)
```
```   608   apply (blast intro!: order_antisym)
```
```   609   done
```
```   610
```
```   611 lemma wf_linord_ex_has_least:
```
```   612   "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
```
```   613     ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
```
```   614   apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
```
```   615   apply (drule_tac x = "m`Collect P" in spec, force)
```
```   616   done
```
```   617
```
```   618 lemma ex_has_least_nat:
```
```   619     "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
```
```   620   apply (simp only: pred_nat_trancl_eq_le [symmetric])
```
```   621   apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
```
```   622    apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
```
```   623   done
```
```   624
```
```   625 lemma LeastM_nat_lemma:
```
```   626     "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
```
```   627   apply (simp add: LeastM_def)
```
```   628   apply (rule someI_ex)
```
```   629   apply (erule ex_has_least_nat)
```
```   630   done
```
```   631
```
```   632 lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1]
```
```   633
```
```   634 lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
```
```   635 by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
```
```   636
```
```   637
```
```   638 subsection \<open>Greatest value operator\<close>
```
```   639
```
```   640 definition
```
```   641   GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where
```
```   642   "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
```
```   643
```
```   644 definition
```
```   645   Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where
```
```   646   "Greatest == GreatestM (%x. x)"
```
```   647
```
```   648 syntax
```
```   649   "_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"
```
```   650       ("GREATEST _ WRT _. _" [0, 4, 10] 10)
```
```   651 translations
```
```   652   "GREATEST x WRT m. P" == "CONST GreatestM m (%x. P)"
```
```   653
```
```   654 lemma GreatestMI2:
```
```   655   "P x ==> (!!y. P y ==> m y <= m x)
```
```   656     ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
```
```   657     ==> Q (GreatestM m P)"
```
```   658   apply (simp add: GreatestM_def)
```
```   659   apply (rule someI2_ex, blast, blast)
```
```   660   done
```
```   661
```
```   662 lemma GreatestM_equality:
```
```   663  "P k ==> (!!x. P x ==> m x <= m k)
```
```   664     ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
```
```   665   apply (rule_tac m = m in GreatestMI2, assumption, blast)
```
```   666   apply (blast intro!: order_antisym)
```
```   667   done
```
```   668
```
```   669 lemma Greatest_equality:
```
```   670   "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
```
```   671   apply (simp add: Greatest_def)
```
```   672   apply (erule GreatestM_equality, blast)
```
```   673   done
```
```   674
```
```   675 lemma ex_has_greatest_nat_lemma:
```
```   676   "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
```
```   677     ==> \<exists>y. P y & ~ (m y < m k + n)"
```
```   678   apply (induct n, force)
```
```   679   apply (force simp add: le_Suc_eq)
```
```   680   done
```
```   681
```
```   682 lemma ex_has_greatest_nat:
```
```   683   "P k ==> \<forall>y. P y --> m y < b
```
```   684     ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
```
```   685   apply (rule ccontr)
```
```   686   apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
```
```   687     apply (subgoal_tac  "m k <= b", auto)
```
```   688   done
```
```   689
```
```   690 lemma GreatestM_nat_lemma:
```
```   691   "P k ==> \<forall>y. P y --> m y < b
```
```   692     ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
```
```   693   apply (simp add: GreatestM_def)
```
```   694   apply (rule someI_ex)
```
```   695   apply (erule ex_has_greatest_nat, assumption)
```
```   696   done
```
```   697
```
```   698 lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1]
```
```   699
```
```   700 lemma GreatestM_nat_le:
```
```   701   "P x ==> \<forall>y. P y --> m y < b
```
```   702     ==> (m x::nat) <= m (GreatestM m P)"
```
```   703   apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
```
```   704   done
```
```   705
```
```   706
```
```   707 text \<open>\medskip Specialization to \<open>GREATEST\<close>.\<close>
```
```   708
```
```   709 lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
```
```   710   apply (simp add: Greatest_def)
```
```   711   apply (rule GreatestM_natI, auto)
```
```   712   done
```
```   713
```
```   714 lemma Greatest_le:
```
```   715     "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
```
```   716   apply (simp add: Greatest_def)
```
```   717   apply (rule GreatestM_nat_le, auto)
```
```   718   done
```
```   719
```
```   720
```
```   721 subsection \<open>An aside: bounded accessible part\<close>
```
```   722
```
```   723 text \<open>Finite monotone eventually stable sequences\<close>
```
```   724
```
```   725 lemma finite_mono_remains_stable_implies_strict_prefix:
```
```   726   fixes f :: "nat \<Rightarrow> 'a::order"
```
```   727   assumes S: "finite (range f)" "mono f" and eq: "\<forall>n. f n = f (Suc n) \<longrightarrow> f (Suc n) = f (Suc (Suc n))"
```
```   728   shows "\<exists>N. (\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m < f n) \<and> (\<forall>n\<ge>N. f N = f n)"
```
```   729   using assms
```
```   730 proof -
```
```   731   have "\<exists>n. f n = f (Suc n)"
```
```   732   proof (rule ccontr)
```
```   733     assume "\<not> ?thesis"
```
```   734     then have "\<And>n. f n \<noteq> f (Suc n)" by auto
```
```   735     then have "\<And>n. f n < f (Suc n)"
```
```   736       using  \<open>mono f\<close> by (auto simp: le_less mono_iff_le_Suc)
```
```   737     with lift_Suc_mono_less_iff[of f]
```
```   738     have *: "\<And>n m. n < m \<Longrightarrow> f n < f m" by auto
```
```   739     have "inj f"
```
```   740     proof (intro injI)
```
```   741       fix x y
```
```   742       assume "f x = f y"
```
```   743       then show "x = y" by (cases x y rule: linorder_cases) (auto dest: *)
```
```   744     qed
```
```   745     with \<open>finite (range f)\<close> have "finite (UNIV::nat set)"
```
```   746       by (rule finite_imageD)
```
```   747     then show False by simp
```
```   748   qed
```
```   749   then obtain n where n: "f n = f (Suc n)" ..
```
```   750   define N where "N = (LEAST n. f n = f (Suc n))"
```
```   751   have N: "f N = f (Suc N)"
```
```   752     unfolding N_def using n by (rule LeastI)
```
```   753   show ?thesis
```
```   754   proof (intro exI[of _ N] conjI allI impI)
```
```   755     fix n assume "N \<le> n"
```
```   756     then have "\<And>m. N \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m = f N"
```
```   757     proof (induct rule: dec_induct)
```
```   758       case (step n) then show ?case
```
```   759         using eq[rule_format, of "n - 1"] N
```
```   760         by (cases n) (auto simp add: le_Suc_eq)
```
```   761     qed simp
```
```   762     from this[of n] \<open>N \<le> n\<close> show "f N = f n" by auto
```
```   763   next
```
```   764     fix n m :: nat assume "m < n" "n \<le> N"
```
```   765     then show "f m < f n"
```
```   766     proof (induct rule: less_Suc_induct)
```
```   767       case (1 i)
```
```   768       then have "i < N" by simp
```
```   769       then have "f i \<noteq> f (Suc i)"
```
```   770         unfolding N_def by (rule not_less_Least)
```
```   771       with \<open>mono f\<close> show ?case by (simp add: mono_iff_le_Suc less_le)
```
```   772     qed auto
```
```   773   qed
```
```   774 qed
```
```   775
```
```   776 lemma finite_mono_strict_prefix_implies_finite_fixpoint:
```
```   777   fixes f :: "nat \<Rightarrow> 'a set"
```
```   778   assumes S: "\<And>i. f i \<subseteq> S" "finite S"
```
```   779     and inj: "\<exists>N. (\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m \<subset> f n) \<and> (\<forall>n\<ge>N. f N = f n)"
```
```   780   shows "f (card S) = (\<Union>n. f n)"
```
```   781 proof -
```
```   782   from inj obtain N where inj: "(\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m \<subset> f n)" and eq: "(\<forall>n\<ge>N. f N = f n)" by auto
```
```   783
```
```   784   { fix i have "i \<le> N \<Longrightarrow> i \<le> card (f i)"
```
```   785     proof (induct i)
```
```   786       case 0 then show ?case by simp
```
```   787     next
```
```   788       case (Suc i)
```
```   789       with inj[rule_format, of "Suc i" i]
```
```   790       have "(f i) \<subset> (f (Suc i))" by auto
```
```   791       moreover have "finite (f (Suc i))" using S by (rule finite_subset)
```
```   792       ultimately have "card (f i) < card (f (Suc i))" by (intro psubset_card_mono)
```
```   793       with Suc show ?case using inj by auto
```
```   794     qed
```
```   795   }
```
```   796   then have "N \<le> card (f N)" by simp
```
```   797   also have "\<dots> \<le> card S" using S by (intro card_mono)
```
```   798   finally have "f (card S) = f N" using eq by auto
```
```   799   then show ?thesis using eq inj[rule_format, of N]
```
```   800     apply auto
```
```   801     apply (case_tac "n < N")
```
```   802     apply (auto simp: not_less)
```
```   803     done
```
```   804 qed
```
```   805
```
```   806
```
```   807 subsection \<open>More on injections, bijections, and inverses\<close>
```
```   808
```
```   809 lemma infinite_imp_bij_betw:
```
```   810 assumes INF: "\<not> finite A"
```
```   811 shows "\<exists>h. bij_betw h A (A - {a})"
```
```   812 proof(cases "a \<in> A")
```
```   813   assume Case1: "a \<notin> A"  hence "A - {a} = A" by blast
```
```   814   thus ?thesis using bij_betw_id[of A] by auto
```
```   815 next
```
```   816   assume Case2: "a \<in> A"
```
```   817   have "\<not> finite (A - {a})" using INF by auto
```
```   818   with infinite_iff_countable_subset[of "A - {a}"] obtain f::"nat \<Rightarrow> 'a"
```
```   819   where 1: "inj f" and 2: "f ` UNIV \<le> A - {a}" by blast
```
```   820   obtain g where g_def: "g = (\<lambda> n. if n = 0 then a else f (Suc n))" by blast
```
```   821   obtain A' where A'_def: "A' = g ` UNIV" by blast
```
```   822   have temp: "\<forall>y. f y \<noteq> a" using 2 by blast
```
```   823   have 3: "inj_on g UNIV \<and> g ` UNIV \<le> A \<and> a \<in> g ` UNIV"
```
```   824   proof(auto simp add: Case2 g_def, unfold inj_on_def, intro ballI impI,
```
```   825         case_tac "x = 0", auto simp add: 2)
```
```   826     fix y  assume "a = (if y = 0 then a else f (Suc y))"
```
```   827     thus "y = 0" using temp by (case_tac "y = 0", auto)
```
```   828   next
```
```   829     fix x y
```
```   830     assume "f (Suc x) = (if y = 0 then a else f (Suc y))"
```
```   831     thus "x = y" using 1 temp unfolding inj_on_def by (case_tac "y = 0", auto)
```
```   832   next
```
```   833     fix n show "f (Suc n) \<in> A" using 2 by blast
```
```   834   qed
```
```   835   hence 4: "bij_betw g UNIV A' \<and> a \<in> A' \<and> A' \<le> A"
```
```   836   using inj_on_imp_bij_betw[of g] unfolding A'_def by auto
```
```   837   hence 5: "bij_betw (inv g) A' UNIV"
```
```   838   by (auto simp add: bij_betw_inv_into)
```
```   839   (*  *)
```
```   840   obtain n where "g n = a" using 3 by auto
```
```   841   hence 6: "bij_betw g (UNIV - {n}) (A' - {a})"
```
```   842   using 3 4 unfolding A'_def
```
```   843   by clarify (rule bij_betw_subset, auto simp: image_set_diff)
```
```   844   (*  *)
```
```   845   obtain v where v_def: "v = (\<lambda> m. if m < n then m else Suc m)" by blast
```
```   846   have 7: "bij_betw v UNIV (UNIV - {n})"
```
```   847   proof(unfold bij_betw_def inj_on_def, intro conjI, clarify)
```
```   848     fix m1 m2 assume "v m1 = v m2"
```
```   849     thus "m1 = m2"
```
```   850     by(case_tac "m1 < n", case_tac "m2 < n",
```
```   851        auto simp add: inj_on_def v_def, case_tac "m2 < n", auto)
```
```   852   next
```
```   853     show "v ` UNIV = UNIV - {n}"
```
```   854     proof(auto simp add: v_def)
```
```   855       fix m assume *: "m \<noteq> n" and **: "m \<notin> Suc ` {m'. \<not> m' < n}"
```
```   856       {assume "n \<le> m" with * have 71: "Suc n \<le> m" by auto
```
```   857        then obtain m' where 72: "m = Suc m'" using Suc_le_D by auto
```
```   858        with 71 have "n \<le> m'" by auto
```
```   859        with 72 ** have False by auto
```
```   860       }
```
```   861       thus "m < n" by force
```
```   862     qed
```
```   863   qed
```
```   864   (*  *)
```
```   865   obtain h' where h'_def: "h' = g o v o (inv g)" by blast
```
```   866   hence 8: "bij_betw h' A' (A' - {a})" using 5 7 6
```
```   867   by (auto simp add: bij_betw_trans)
```
```   868   (*  *)
```
```   869   obtain h where h_def: "h = (\<lambda> b. if b \<in> A' then h' b else b)" by blast
```
```   870   have "\<forall>b \<in> A'. h b = h' b" unfolding h_def by auto
```
```   871   hence "bij_betw h  A' (A' - {a})" using 8 bij_betw_cong[of A' h] by auto
```
```   872   moreover
```
```   873   {have "\<forall>b \<in> A - A'. h b = b" unfolding h_def by auto
```
```   874    hence "bij_betw h  (A - A') (A - A')"
```
```   875    using bij_betw_cong[of "A - A'" h id] bij_betw_id[of "A - A'"] by auto
```
```   876   }
```
```   877   moreover
```
```   878   have "(A' Int (A - A') = {} \<and> A' \<union> (A - A') = A) \<and>
```
```   879         ((A' - {a}) Int (A - A') = {} \<and> (A' - {a}) \<union> (A - A') = A - {a})"
```
```   880   using 4 by blast
```
```   881   ultimately have "bij_betw h A (A - {a})"
```
```   882   using bij_betw_combine[of h A' "A' - {a}" "A - A'" "A - A'"] by simp
```
```   883   thus ?thesis by blast
```
```   884 qed
```
```   885
```
```   886 lemma infinite_imp_bij_betw2:
```
```   887 assumes INF: "\<not> finite A"
```
```   888 shows "\<exists>h. bij_betw h A (A \<union> {a})"
```
```   889 proof(cases "a \<in> A")
```
```   890   assume Case1: "a \<in> A"  hence "A \<union> {a} = A" by blast
```
```   891   thus ?thesis using bij_betw_id[of A] by auto
```
```   892 next
```
```   893   let ?A' = "A \<union> {a}"
```
```   894   assume Case2: "a \<notin> A" hence "A = ?A' - {a}" by blast
```
```   895   moreover have "\<not> finite ?A'" using INF by auto
```
```   896   ultimately obtain f where "bij_betw f ?A' A"
```
```   897   using infinite_imp_bij_betw[of ?A' a] by auto
```
```   898   hence "bij_betw(inv_into ?A' f) A ?A'" using bij_betw_inv_into by blast
```
```   899   thus ?thesis by auto
```
```   900 qed
```
```   901
```
```   902 lemma bij_betw_inv_into_left:
```
```   903 assumes BIJ: "bij_betw f A A'" and IN: "a \<in> A"
```
```   904 shows "(inv_into A f) (f a) = a"
```
```   905 using assms unfolding bij_betw_def
```
```   906 by clarify (rule inv_into_f_f)
```
```   907
```
```   908 lemma bij_betw_inv_into_right:
```
```   909 assumes "bij_betw f A A'" "a' \<in> A'"
```
```   910 shows "f(inv_into A f a') = a'"
```
```   911 using assms unfolding bij_betw_def using f_inv_into_f by force
```
```   912
```
```   913 lemma bij_betw_inv_into_subset:
```
```   914 assumes BIJ: "bij_betw f A A'" and
```
```   915         SUB: "B \<le> A" and IM: "f ` B = B'"
```
```   916 shows "bij_betw (inv_into A f) B' B"
```
```   917 using assms unfolding bij_betw_def
```
```   918 by (auto intro: inj_on_inv_into)
```
```   919
```
```   920
```
```   921 subsection \<open>Specification package -- Hilbertized version\<close>
```
```   922
```
```   923 lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
```
```   924   by (simp only: someI_ex)
```
```   925
```
```   926 ML_file "Tools/choice_specification.ML"
```
```   927
```
```   928 end
```