src/HOL/Hilbert_Choice.thy
 author Christian Sternagel Thu Aug 30 15:44:03 2012 +0900 (2012-08-30) changeset 49093 fdc301f592c4 parent 48891 c0eafbd55de3 child 49739 13aa6d8268ec permissions -rw-r--r--
```     1 (*  Title:      HOL/Hilbert_Choice.thy
```
```     2     Author:     Lawrence C Paulson, Tobias Nipkow
```
```     3     Copyright   2001  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 header {* Hilbert's Epsilon-Operator and the Axiom of Choice *}
```
```     7
```
```     8 theory Hilbert_Choice
```
```     9 imports Nat Wellfounded Plain
```
```    10 keywords "specification" "ax_specification" :: thy_goal
```
```    11 begin
```
```    12
```
```    13 subsection {* Hilbert's epsilon *}
```
```    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 (HOL)
```
```    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 {*
```
```    28   [(@{const_syntax Eps}, 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 *} -- {* to avoid eta-contraction of body *}
```
```    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 {*Hilbert's Epsilon-operator*}
```
```    41
```
```    42 text{*Easier to apply than @{text someI} if the witness comes from an
```
```    43 existential formula*}
```
```    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{*Easier to apply than @{text someI} because the conclusion has only one
```
```    50 occurrence of @{term P}.*}
```
```    51 lemma someI2: "[| P a;  !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
```
```    52 by (blast intro: someI)
```
```    53
```
```    54 text{*Easier to apply than @{text someI2} if the witness comes from an
```
```    55 existential formula*}
```
```    56 lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
```
```    57 by (blast intro: someI2)
```
```    58
```
```    59 lemma some_equality [intro]:
```
```    60      "[| P a;  !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
```
```    61 by (blast intro: someI2)
```
```    62
```
```    63 lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
```
```    64 by blast
```
```    65
```
```    66 lemma some_eq_ex: "P (SOME x. P x) =  (\<exists>x. P x)"
```
```    67 by (blast intro: someI)
```
```    68
```
```    69 lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
```
```    70 apply (rule some_equality)
```
```    71 apply (rule refl, assumption)
```
```    72 done
```
```    73
```
```    74 lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"
```
```    75 apply (rule some_equality)
```
```    76 apply (rule refl)
```
```    77 apply (erule sym)
```
```    78 done
```
```    79
```
```    80
```
```    81 subsection{*Axiom of Choice, Proved Using the Description Operator*}
```
```    82
```
```    83 lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)"
```
```    84 by (fast elim: someI)
```
```    85
```
```    86 lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)"
```
```    87 by (fast elim: someI)
```
```    88
```
```    89
```
```    90 subsection {*Function Inverse*}
```
```    91
```
```    92 lemma inv_def: "inv f = (%y. SOME x. f x = y)"
```
```    93 by(simp add: inv_into_def)
```
```    94
```
```    95 lemma inv_into_into: "x : f ` A ==> inv_into A f x : A"
```
```    96 apply (simp add: inv_into_def)
```
```    97 apply (fast intro: someI2)
```
```    98 done
```
```    99
```
```   100 lemma inv_id [simp]: "inv id = id"
```
```   101 by (simp add: inv_into_def id_def)
```
```   102
```
```   103 lemma inv_into_f_f [simp]:
```
```   104   "[| inj_on f A;  x : A |] ==> inv_into A f (f x) = x"
```
```   105 apply (simp add: inv_into_def inj_on_def)
```
```   106 apply (blast intro: someI2)
```
```   107 done
```
```   108
```
```   109 lemma inv_f_f: "inj f ==> inv f (f x) = x"
```
```   110 by simp
```
```   111
```
```   112 lemma f_inv_into_f: "y : f`A  ==> f (inv_into A f y) = y"
```
```   113 apply (simp add: inv_into_def)
```
```   114 apply (fast intro: someI2)
```
```   115 done
```
```   116
```
```   117 lemma inv_into_f_eq: "[| inj_on f A; x : A; f x = y |] ==> inv_into A f y = x"
```
```   118 apply (erule subst)
```
```   119 apply (fast intro: inv_into_f_f)
```
```   120 done
```
```   121
```
```   122 lemma inv_f_eq: "[| inj f; f x = y |] ==> inv f y = x"
```
```   123 by (simp add:inv_into_f_eq)
```
```   124
```
```   125 lemma inj_imp_inv_eq: "[| inj f; ALL x. f(g x) = x |] ==> inv f = g"
```
```   126   by (blast intro: inv_into_f_eq)
```
```   127
```
```   128 text{*But is it useful?*}
```
```   129 lemma inj_transfer:
```
```   130   assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)"
```
```   131   shows "P x"
```
```   132 proof -
```
```   133   have "f x \<in> range f" by auto
```
```   134   hence "P(inv f (f x))" by (rule minor)
```
```   135   thus "P x" by (simp add: inv_into_f_f [OF injf])
```
```   136 qed
```
```   137
```
```   138 lemma inj_iff: "(inj f) = (inv f o f = id)"
```
```   139 apply (simp add: o_def fun_eq_iff)
```
```   140 apply (blast intro: inj_on_inverseI inv_into_f_f)
```
```   141 done
```
```   142
```
```   143 lemma inv_o_cancel[simp]: "inj f ==> inv f o f = id"
```
```   144 by (simp add: inj_iff)
```
```   145
```
```   146 lemma o_inv_o_cancel[simp]: "inj f ==> g o inv f o f = g"
```
```   147 by (simp add: o_assoc[symmetric])
```
```   148
```
```   149 lemma inv_into_image_cancel[simp]:
```
```   150   "inj_on f A ==> S <= A ==> inv_into A f ` f ` S = S"
```
```   151 by(fastforce simp: image_def)
```
```   152
```
```   153 lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
```
```   154 by (blast intro!: surjI inv_into_f_f)
```
```   155
```
```   156 lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
```
```   157 by (simp add: f_inv_into_f)
```
```   158
```
```   159 lemma inv_into_injective:
```
```   160   assumes eq: "inv_into A f x = inv_into A f y"
```
```   161       and x: "x: f`A"
```
```   162       and y: "y: f`A"
```
```   163   shows "x=y"
```
```   164 proof -
```
```   165   have "f (inv_into A f x) = f (inv_into A f y)" using eq by simp
```
```   166   thus ?thesis by (simp add: f_inv_into_f x y)
```
```   167 qed
```
```   168
```
```   169 lemma inj_on_inv_into: "B <= f`A ==> inj_on (inv_into A f) B"
```
```   170 by (blast intro: inj_onI dest: inv_into_injective injD)
```
```   171
```
```   172 lemma bij_betw_inv_into: "bij_betw f A B ==> bij_betw (inv_into A f) B A"
```
```   173 by (auto simp add: bij_betw_def inj_on_inv_into)
```
```   174
```
```   175 lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
```
```   176 by (simp add: inj_on_inv_into)
```
```   177
```
```   178 lemma surj_iff: "(surj f) = (f o inv f = id)"
```
```   179 by (auto intro!: surjI simp: surj_f_inv_f fun_eq_iff[where 'b='a])
```
```   180
```
```   181 lemma surj_iff_all: "surj f \<longleftrightarrow> (\<forall>x. f (inv f x) = x)"
```
```   182   unfolding surj_iff by (simp add: o_def fun_eq_iff)
```
```   183
```
```   184 lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
```
```   185 apply (rule ext)
```
```   186 apply (drule_tac x = "inv f x" in spec)
```
```   187 apply (simp add: surj_f_inv_f)
```
```   188 done
```
```   189
```
```   190 lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
```
```   191 by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
```
```   192
```
```   193 lemma inv_equality: "[| !!x. g (f x) = x;  !!y. f (g y) = y |] ==> inv f = g"
```
```   194 apply (rule ext)
```
```   195 apply (auto simp add: inv_into_def)
```
```   196 done
```
```   197
```
```   198 lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
```
```   199 apply (rule inv_equality)
```
```   200 apply (auto simp add: bij_def surj_f_inv_f)
```
```   201 done
```
```   202
```
```   203 (** bij(inv f) implies little about f.  Consider f::bool=>bool such that
```
```   204     f(True)=f(False)=True.  Then it's consistent with axiom someI that
```
```   205     inv f could be any function at all, including the identity function.
```
```   206     If inv f=id then inv f is a bijection, but inj f, surj(f) and
```
```   207     inv(inv f)=f all fail.
```
```   208 **)
```
```   209
```
```   210 lemma inv_into_comp:
```
```   211   "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
```
```   212   inv_into A (f o g) x = (inv_into A g o inv_into (g ` A) f) x"
```
```   213 apply (rule inv_into_f_eq)
```
```   214   apply (fast intro: comp_inj_on)
```
```   215  apply (simp add: inv_into_into)
```
```   216 apply (simp add: f_inv_into_f inv_into_into)
```
```   217 done
```
```   218
```
```   219 lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
```
```   220 apply (rule inv_equality)
```
```   221 apply (auto simp add: bij_def surj_f_inv_f)
```
```   222 done
```
```   223
```
```   224 lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
```
```   225 by (simp add: image_eq_UN surj_f_inv_f)
```
```   226
```
```   227 lemma image_inv_f_f: "inj f ==> (inv f) ` (f ` A) = A"
```
```   228 by (simp add: image_eq_UN)
```
```   229
```
```   230 lemma inv_image_comp: "inj f ==> inv f ` (f`X) = X"
```
```   231 by (auto simp add: image_def)
```
```   232
```
```   233 lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
```
```   234 apply auto
```
```   235 apply (force simp add: bij_is_inj)
```
```   236 apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
```
```   237 done
```
```   238
```
```   239 lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A"
```
```   240 apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
```
```   241 apply (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric])
```
```   242 done
```
```   243
```
```   244 lemma finite_fun_UNIVD1:
```
```   245   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
```
```   246   and card: "card (UNIV :: 'b set) \<noteq> Suc 0"
```
```   247   shows "finite (UNIV :: 'a set)"
```
```   248 proof -
```
```   249   from fin have finb: "finite (UNIV :: 'b set)" by (rule finite_fun_UNIVD2)
```
```   250   with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)"
```
```   251     by (cases "card (UNIV :: 'b set)") (auto simp add: card_eq_0_iff)
```
```   252   then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - Suc (Suc 0)" by auto
```
```   253   then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by (auto simp add: card_Suc_eq)
```
```   254   from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by (rule finite_imageI)
```
```   255   moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)"
```
```   256   proof (rule UNIV_eq_I)
```
```   257     fix x :: 'a
```
```   258     from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_into_def)
```
```   259     thus "x \<in> range (\<lambda>f\<Colon>'a \<Rightarrow> 'b. inv f b1)" by blast
```
```   260   qed
```
```   261   ultimately show "finite (UNIV :: 'a set)" by simp
```
```   262 qed
```
```   263
```
```   264 lemma image_inv_into_cancel:
```
```   265   assumes SURJ: "f`A=A'" and SUB: "B' \<le> A'"
```
```   266   shows "f `((inv_into A f)`B') = B'"
```
```   267   using assms
```
```   268 proof (auto simp add: f_inv_into_f)
```
```   269   let ?f' = "(inv_into A f)"
```
```   270   fix a' assume *: "a' \<in> B'"
```
```   271   then have "a' \<in> A'" using SUB by auto
```
```   272   then have "a' = f (?f' a')"
```
```   273     using SURJ by (auto simp add: f_inv_into_f)
```
```   274   then show "a' \<in> f ` (?f' ` B')" using * by blast
```
```   275 qed
```
```   276
```
```   277 lemma inv_into_inv_into_eq:
```
```   278   assumes "bij_betw f A A'" "a \<in> A"
```
```   279   shows "inv_into A' (inv_into A f) a = f a"
```
```   280 proof -
```
```   281   let ?f' = "inv_into A f"   let ?f'' = "inv_into A' ?f'"
```
```   282   have 1: "bij_betw ?f' A' A" using assms
```
```   283   by (auto simp add: bij_betw_inv_into)
```
```   284   obtain a' where 2: "a' \<in> A'" and 3: "?f' a' = a"
```
```   285     using 1 `a \<in> A` unfolding bij_betw_def by force
```
```   286   hence "?f'' a = a'"
```
```   287     using `a \<in> A` 1 3 by (auto simp add: f_inv_into_f bij_betw_def)
```
```   288   moreover have "f a = a'" using assms 2 3
```
```   289     by (auto simp add: bij_betw_def)
```
```   290   ultimately show "?f'' a = f a" by simp
```
```   291 qed
```
```   292
```
```   293 lemma inj_on_iff_surj:
```
```   294   assumes "A \<noteq> {}"
```
```   295   shows "(\<exists>f. inj_on f A \<and> f ` A \<le> A') \<longleftrightarrow> (\<exists>g. g ` A' = A)"
```
```   296 proof safe
```
```   297   fix f assume INJ: "inj_on f A" and INCL: "f ` A \<le> A'"
```
```   298   let ?phi = "\<lambda>a' a. a \<in> A \<and> f a = a'"  let ?csi = "\<lambda>a. a \<in> A"
```
```   299   let ?g = "\<lambda>a'. if a' \<in> f ` A then (SOME a. ?phi a' a) else (SOME a. ?csi a)"
```
```   300   have "?g ` A' = A"
```
```   301   proof
```
```   302     show "?g ` A' \<le> A"
```
```   303     proof clarify
```
```   304       fix a' assume *: "a' \<in> A'"
```
```   305       show "?g a' \<in> A"
```
```   306       proof cases
```
```   307         assume Case1: "a' \<in> f ` A"
```
```   308         then obtain a where "?phi a' a" by blast
```
```   309         hence "?phi a' (SOME a. ?phi a' a)" using someI[of "?phi a'" a] by blast
```
```   310         with Case1 show ?thesis by auto
```
```   311       next
```
```   312         assume Case2: "a' \<notin> f ` A"
```
```   313         hence "?csi (SOME a. ?csi a)" using assms someI_ex[of ?csi] by blast
```
```   314         with Case2 show ?thesis by auto
```
```   315       qed
```
```   316     qed
```
```   317   next
```
```   318     show "A \<le> ?g ` A'"
```
```   319     proof-
```
```   320       {fix a assume *: "a \<in> A"
```
```   321        let ?b = "SOME aa. ?phi (f a) aa"
```
```   322        have "?phi (f a) a" using * by auto
```
```   323        hence 1: "?phi (f a) ?b" using someI[of "?phi(f a)" a] by blast
```
```   324        hence "?g(f a) = ?b" using * by auto
```
```   325        moreover have "a = ?b" using 1 INJ * by (auto simp add: inj_on_def)
```
```   326        ultimately have "?g(f a) = a" by simp
```
```   327        with INCL * have "?g(f a) = a \<and> f a \<in> A'" by auto
```
```   328       }
```
```   329       thus ?thesis by force
```
```   330     qed
```
```   331   qed
```
```   332   thus "\<exists>g. g ` A' = A" by blast
```
```   333 next
```
```   334   fix g  let ?f = "inv_into A' g"
```
```   335   have "inj_on ?f (g ` A')"
```
```   336     by (auto simp add: inj_on_inv_into)
```
```   337   moreover
```
```   338   {fix a' assume *: "a' \<in> A'"
```
```   339    let ?phi = "\<lambda> b'. b' \<in> A' \<and> g b' = g a'"
```
```   340    have "?phi a'" using * by auto
```
```   341    hence "?phi(SOME b'. ?phi b')" using someI[of ?phi] by blast
```
```   342    hence "?f(g a') \<in> A'" unfolding inv_into_def by auto
```
```   343   }
```
```   344   ultimately show "\<exists>f. inj_on f (g ` A') \<and> f ` g ` A' \<subseteq> A'" by auto
```
```   345 qed
```
```   346
```
```   347 lemma Ex_inj_on_UNION_Sigma:
```
```   348   "\<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))"
```
```   349 proof
```
```   350   let ?phi = "\<lambda> a i. i \<in> I \<and> a \<in> A i"
```
```   351   let ?sm = "\<lambda> a. SOME i. ?phi a i"
```
```   352   let ?f = "\<lambda>a. (?sm a, a)"
```
```   353   have "inj_on ?f (\<Union> i \<in> I. A i)" unfolding inj_on_def by auto
```
```   354   moreover
```
```   355   { { fix i a assume "i \<in> I" and "a \<in> A i"
```
```   356       hence "?sm a \<in> I \<and> a \<in> A(?sm a)" using someI[of "?phi a" i] by auto
```
```   357     }
```
```   358     hence "?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)" by auto
```
```   359   }
```
```   360   ultimately
```
```   361   show "inj_on ?f (\<Union> i \<in> I. A i) \<and> ?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)"
```
```   362   by auto
```
```   363 qed
```
```   364
```
```   365 subsection {* The Cantor-Bernstein Theorem *}
```
```   366
```
```   367 lemma Cantor_Bernstein_aux:
```
```   368   shows "\<exists>A' h. A' \<le> A \<and>
```
```   369                 (\<forall>a \<in> A'. a \<notin> g`(B - f ` A')) \<and>
```
```   370                 (\<forall>a \<in> A'. h a = f a) \<and>
```
```   371                 (\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a))"
```
```   372 proof-
```
```   373   obtain H where H_def: "H = (\<lambda> A'. A - (g`(B - (f ` A'))))" by blast
```
```   374   have 0: "mono H" unfolding mono_def H_def by blast
```
```   375   then obtain A' where 1: "H A' = A'" using lfp_unfold by blast
```
```   376   hence 2: "A' = A - (g`(B - (f ` A')))" unfolding H_def by simp
```
```   377   hence 3: "A' \<le> A" by blast
```
```   378   have 4: "\<forall>a \<in> A'.  a \<notin> g`(B - f ` A')"
```
```   379   using 2 by blast
```
```   380   have 5: "\<forall>a \<in> A - A'. \<exists>b \<in> B - (f ` A'). a = g b"
```
```   381   using 2 by blast
```
```   382   (*  *)
```
```   383   obtain h where h_def:
```
```   384   "h = (\<lambda> a. if a \<in> A' then f a else (SOME b. b \<in> B - (f ` A') \<and> a = g b))" by blast
```
```   385   hence "\<forall>a \<in> A'. h a = f a" by auto
```
```   386   moreover
```
```   387   have "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
```
```   388   proof
```
```   389     fix a assume *: "a \<in> A - A'"
```
```   390     let ?phi = "\<lambda> b. b \<in> B - (f ` A') \<and> a = g b"
```
```   391     have "h a = (SOME b. ?phi b)" using h_def * by auto
```
```   392     moreover have "\<exists>b. ?phi b" using 5 *  by auto
```
```   393     ultimately show  "?phi (h a)" using someI_ex[of ?phi] by auto
```
```   394   qed
```
```   395   ultimately show ?thesis using 3 4 by blast
```
```   396 qed
```
```   397
```
```   398 theorem Cantor_Bernstein:
```
```   399   assumes INJ1: "inj_on f A" and SUB1: "f ` A \<le> B" and
```
```   400           INJ2: "inj_on g B" and SUB2: "g ` B \<le> A"
```
```   401   shows "\<exists>h. bij_betw h A B"
```
```   402 proof-
```
```   403   obtain A' and h where 0: "A' \<le> A" and
```
```   404   1: "\<forall>a \<in> A'. a \<notin> g`(B - f ` A')" and
```
```   405   2: "\<forall>a \<in> A'. h a = f a" and
```
```   406   3: "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
```
```   407   using Cantor_Bernstein_aux[of A g B f] by blast
```
```   408   have "inj_on h A"
```
```   409   proof (intro inj_onI)
```
```   410     fix a1 a2
```
```   411     assume 4: "a1 \<in> A" and 5: "a2 \<in> A" and 6: "h a1 = h a2"
```
```   412     show "a1 = a2"
```
```   413     proof(cases "a1 \<in> A'")
```
```   414       assume Case1: "a1 \<in> A'"
```
```   415       show ?thesis
```
```   416       proof(cases "a2 \<in> A'")
```
```   417         assume Case11: "a2 \<in> A'"
```
```   418         hence "f a1 = f a2" using Case1 2 6 by auto
```
```   419         thus ?thesis using INJ1 Case1 Case11 0
```
```   420         unfolding inj_on_def by blast
```
```   421       next
```
```   422         assume Case12: "a2 \<notin> A'"
```
```   423         hence False using 3 5 2 6 Case1 by force
```
```   424         thus ?thesis by simp
```
```   425       qed
```
```   426     next
```
```   427     assume Case2: "a1 \<notin> A'"
```
```   428       show ?thesis
```
```   429       proof(cases "a2 \<in> A'")
```
```   430         assume Case21: "a2 \<in> A'"
```
```   431         hence False using 3 4 2 6 Case2 by auto
```
```   432         thus ?thesis by simp
```
```   433       next
```
```   434         assume Case22: "a2 \<notin> A'"
```
```   435         hence "a1 = g(h a1) \<and> a2 = g(h a2)" using Case2 4 5 3 by auto
```
```   436         thus ?thesis using 6 by simp
```
```   437       qed
```
```   438     qed
```
```   439   qed
```
```   440   (*  *)
```
```   441   moreover
```
```   442   have "h ` A = B"
```
```   443   proof safe
```
```   444     fix a assume "a \<in> A"
```
```   445     thus "h a \<in> B" using SUB1 2 3 by (cases "a \<in> A'") auto
```
```   446   next
```
```   447     fix b assume *: "b \<in> B"
```
```   448     show "b \<in> h ` A"
```
```   449     proof(cases "b \<in> f ` A'")
```
```   450       assume Case1: "b \<in> f ` A'"
```
```   451       then obtain a where "a \<in> A' \<and> b = f a" by blast
```
```   452       thus ?thesis using 2 0 by force
```
```   453     next
```
```   454       assume Case2: "b \<notin> f ` A'"
```
```   455       hence "g b \<notin> A'" using 1 * by auto
```
```   456       hence 4: "g b \<in> A - A'" using * SUB2 by auto
```
```   457       hence "h(g b) \<in> B \<and> g(h(g b)) = g b"
```
```   458       using 3 by auto
```
```   459       hence "h(g b) = b" using * INJ2 unfolding inj_on_def by auto
```
```   460       thus ?thesis using 4 by force
```
```   461     qed
```
```   462   qed
```
```   463   (*  *)
```
```   464   ultimately show ?thesis unfolding bij_betw_def by auto
```
```   465 qed
```
```   466
```
```   467 subsection {*Other Consequences of Hilbert's Epsilon*}
```
```   468
```
```   469 text {*Hilbert's Epsilon and the @{term split} Operator*}
```
```   470
```
```   471 text{*Looping simprule*}
```
```   472 lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
```
```   473   by simp
```
```   474
```
```   475 lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"
```
```   476   by (simp add: split_def)
```
```   477
```
```   478 lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
```
```   479   by blast
```
```   480
```
```   481
```
```   482 text{*A relation is wellfounded iff it has no infinite descending chain*}
```
```   483 lemma wf_iff_no_infinite_down_chain:
```
```   484   "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
```
```   485 apply (simp only: wf_eq_minimal)
```
```   486 apply (rule iffI)
```
```   487  apply (rule notI)
```
```   488  apply (erule exE)
```
```   489  apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
```
```   490 apply (erule contrapos_np, simp, clarify)
```
```   491 apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
```
```   492  apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI)
```
```   493  apply (rule allI, simp)
```
```   494  apply (rule someI2_ex, blast, blast)
```
```   495 apply (rule allI)
```
```   496 apply (induct_tac "n", simp_all)
```
```   497 apply (rule someI2_ex, blast+)
```
```   498 done
```
```   499
```
```   500 lemma wf_no_infinite_down_chainE:
```
```   501   assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r"
```
```   502 using `wf r` wf_iff_no_infinite_down_chain[of r] by blast
```
```   503
```
```   504
```
```   505 text{*A dynamically-scoped fact for TFL *}
```
```   506 lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
```
```   507   by (blast intro: someI)
```
```   508
```
```   509
```
```   510 subsection {* Least value operator *}
```
```   511
```
```   512 definition
```
```   513   LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where
```
```   514   "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
```
```   515
```
```   516 syntax
```
```   517   "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
```
```   518 translations
```
```   519   "LEAST x WRT m. P" == "CONST LeastM m (%x. P)"
```
```   520
```
```   521 lemma LeastMI2:
```
```   522   "P x ==> (!!y. P y ==> m x <= m y)
```
```   523     ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
```
```   524     ==> Q (LeastM m P)"
```
```   525   apply (simp add: LeastM_def)
```
```   526   apply (rule someI2_ex, blast, blast)
```
```   527   done
```
```   528
```
```   529 lemma LeastM_equality:
```
```   530   "P k ==> (!!x. P x ==> m k <= m x)
```
```   531     ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
```
```   532   apply (rule LeastMI2, assumption, blast)
```
```   533   apply (blast intro!: order_antisym)
```
```   534   done
```
```   535
```
```   536 lemma wf_linord_ex_has_least:
```
```   537   "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
```
```   538     ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
```
```   539   apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
```
```   540   apply (drule_tac x = "m`Collect P" in spec, force)
```
```   541   done
```
```   542
```
```   543 lemma ex_has_least_nat:
```
```   544     "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
```
```   545   apply (simp only: pred_nat_trancl_eq_le [symmetric])
```
```   546   apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
```
```   547    apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
```
```   548   done
```
```   549
```
```   550 lemma LeastM_nat_lemma:
```
```   551     "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
```
```   552   apply (simp add: LeastM_def)
```
```   553   apply (rule someI_ex)
```
```   554   apply (erule ex_has_least_nat)
```
```   555   done
```
```   556
```
```   557 lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1]
```
```   558
```
```   559 lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
```
```   560 by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
```
```   561
```
```   562
```
```   563 subsection {* Greatest value operator *}
```
```   564
```
```   565 definition
```
```   566   GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where
```
```   567   "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
```
```   568
```
```   569 definition
```
```   570   Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where
```
```   571   "Greatest == GreatestM (%x. x)"
```
```   572
```
```   573 syntax
```
```   574   "_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"
```
```   575       ("GREATEST _ WRT _. _" [0, 4, 10] 10)
```
```   576 translations
```
```   577   "GREATEST x WRT m. P" == "CONST GreatestM m (%x. P)"
```
```   578
```
```   579 lemma GreatestMI2:
```
```   580   "P x ==> (!!y. P y ==> m y <= m x)
```
```   581     ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
```
```   582     ==> Q (GreatestM m P)"
```
```   583   apply (simp add: GreatestM_def)
```
```   584   apply (rule someI2_ex, blast, blast)
```
```   585   done
```
```   586
```
```   587 lemma GreatestM_equality:
```
```   588  "P k ==> (!!x. P x ==> m x <= m k)
```
```   589     ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
```
```   590   apply (rule_tac m = m in GreatestMI2, assumption, blast)
```
```   591   apply (blast intro!: order_antisym)
```
```   592   done
```
```   593
```
```   594 lemma Greatest_equality:
```
```   595   "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
```
```   596   apply (simp add: Greatest_def)
```
```   597   apply (erule GreatestM_equality, blast)
```
```   598   done
```
```   599
```
```   600 lemma ex_has_greatest_nat_lemma:
```
```   601   "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
```
```   602     ==> \<exists>y. P y & ~ (m y < m k + n)"
```
```   603   apply (induct n, force)
```
```   604   apply (force simp add: le_Suc_eq)
```
```   605   done
```
```   606
```
```   607 lemma ex_has_greatest_nat:
```
```   608   "P k ==> \<forall>y. P y --> m y < b
```
```   609     ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
```
```   610   apply (rule ccontr)
```
```   611   apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
```
```   612     apply (subgoal_tac  "m k <= b", auto)
```
```   613   done
```
```   614
```
```   615 lemma GreatestM_nat_lemma:
```
```   616   "P k ==> \<forall>y. P y --> m y < b
```
```   617     ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
```
```   618   apply (simp add: GreatestM_def)
```
```   619   apply (rule someI_ex)
```
```   620   apply (erule ex_has_greatest_nat, assumption)
```
```   621   done
```
```   622
```
```   623 lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1]
```
```   624
```
```   625 lemma GreatestM_nat_le:
```
```   626   "P x ==> \<forall>y. P y --> m y < b
```
```   627     ==> (m x::nat) <= m (GreatestM m P)"
```
```   628   apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
```
```   629   done
```
```   630
```
```   631
```
```   632 text {* \medskip Specialization to @{text GREATEST}. *}
```
```   633
```
```   634 lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
```
```   635   apply (simp add: Greatest_def)
```
```   636   apply (rule GreatestM_natI, auto)
```
```   637   done
```
```   638
```
```   639 lemma Greatest_le:
```
```   640     "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
```
```   641   apply (simp add: Greatest_def)
```
```   642   apply (rule GreatestM_nat_le, auto)
```
```   643   done
```
```   644
```
```   645
```
```   646 subsection {* Specification package -- Hilbertized version *}
```
```   647
```
```   648 lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
```
```   649   by (simp only: someI_ex)
```
```   650
```
```   651 ML_file "Tools/choice_specification.ML"
```
```   652
```
```   653 end
```