src/HOL/Hilbert_Choice.thy
 author wenzelm Thu May 24 17:25:53 2012 +0200 (2012-05-24) changeset 47988 e4b69e10b990 parent 46950 d0181abdbdac child 48891 c0eafbd55de3 permissions -rw-r--r--
tuned proofs;
```     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 uses ("Tools/choice_specification.ML")
```
```    12 begin
```
```    13
```
```    14 subsection {* Hilbert's epsilon *}
```
```    15
```
```    16 axiomatization Eps :: "('a => bool) => 'a" where
```
```    17   someI: "P x ==> P (Eps P)"
```
```    18
```
```    19 syntax (epsilon)
```
```    20   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3\<some>_./ _)" [0, 10] 10)
```
```    21 syntax (HOL)
```
```    22   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3@ _./ _)" [0, 10] 10)
```
```    23 syntax
```
```    24   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3SOME _./ _)" [0, 10] 10)
```
```    25 translations
```
```    26   "SOME x. P" == "CONST Eps (%x. P)"
```
```    27
```
```    28 print_translation {*
```
```    29   [(@{const_syntax Eps}, fn [Abs abs] =>
```
```    30       let val (x, t) = Syntax_Trans.atomic_abs_tr' abs
```
```    31       in Syntax.const @{syntax_const "_Eps"} \$ x \$ t end)]
```
```    32 *} -- {* to avoid eta-contraction of body *}
```
```    33
```
```    34 definition inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
```
```    35 "inv_into A f == %x. SOME y. y : A & f y = x"
```
```    36
```
```    37 abbreviation inv :: "('a => 'b) => ('b => 'a)" where
```
```    38 "inv == inv_into UNIV"
```
```    39
```
```    40
```
```    41 subsection {*Hilbert's Epsilon-operator*}
```
```    42
```
```    43 text{*Easier to apply than @{text someI} if the witness comes from an
```
```    44 existential formula*}
```
```    45 lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)"
```
```    46 apply (erule exE)
```
```    47 apply (erule someI)
```
```    48 done
```
```    49
```
```    50 text{*Easier to apply than @{text someI} because the conclusion has only one
```
```    51 occurrence of @{term P}.*}
```
```    52 lemma someI2: "[| P a;  !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
```
```    53 by (blast intro: someI)
```
```    54
```
```    55 text{*Easier to apply than @{text someI2} if the witness comes from an
```
```    56 existential formula*}
```
```    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 some_equality [intro]:
```
```    61      "[| P a;  !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
```
```    62 by (blast intro: someI2)
```
```    63
```
```    64 lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
```
```    65 by blast
```
```    66
```
```    67 lemma some_eq_ex: "P (SOME x. P x) =  (\<exists>x. P x)"
```
```    68 by (blast intro: someI)
```
```    69
```
```    70 lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
```
```    71 apply (rule some_equality)
```
```    72 apply (rule refl, assumption)
```
```    73 done
```
```    74
```
```    75 lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"
```
```    76 apply (rule some_equality)
```
```    77 apply (rule refl)
```
```    78 apply (erule sym)
```
```    79 done
```
```    80
```
```    81
```
```    82 subsection{*Axiom of Choice, Proved Using the Description Operator*}
```
```    83
```
```    84 lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)"
```
```    85 by (fast elim: someI)
```
```    86
```
```    87 lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)"
```
```    88 by (fast elim: someI)
```
```    89
```
```    90
```
```    91 subsection {*Function Inverse*}
```
```    92
```
```    93 lemma inv_def: "inv f = (%y. SOME x. f x = y)"
```
```    94 by(simp add: inv_into_def)
```
```    95
```
```    96 lemma inv_into_into: "x : f ` A ==> inv_into A f x : A"
```
```    97 apply (simp add: inv_into_def)
```
```    98 apply (fast intro: someI2)
```
```    99 done
```
```   100
```
```   101 lemma inv_id [simp]: "inv id = id"
```
```   102 by (simp add: inv_into_def id_def)
```
```   103
```
```   104 lemma inv_into_f_f [simp]:
```
```   105   "[| inj_on f A;  x : A |] ==> inv_into A f (f x) = x"
```
```   106 apply (simp add: inv_into_def inj_on_def)
```
```   107 apply (blast intro: someI2)
```
```   108 done
```
```   109
```
```   110 lemma inv_f_f: "inj f ==> inv f (f x) = x"
```
```   111 by simp
```
```   112
```
```   113 lemma f_inv_into_f: "y : f`A  ==> f (inv_into A f y) = y"
```
```   114 apply (simp add: inv_into_def)
```
```   115 apply (fast intro: someI2)
```
```   116 done
```
```   117
```
```   118 lemma inv_into_f_eq: "[| inj_on f A; x : A; f x = y |] ==> inv_into A f y = x"
```
```   119 apply (erule subst)
```
```   120 apply (fast intro: inv_into_f_f)
```
```   121 done
```
```   122
```
```   123 lemma inv_f_eq: "[| inj f; f x = y |] ==> inv f y = x"
```
```   124 by (simp add:inv_into_f_eq)
```
```   125
```
```   126 lemma inj_imp_inv_eq: "[| inj f; ALL x. f(g x) = x |] ==> inv f = g"
```
```   127   by (blast intro: inv_into_f_eq)
```
```   128
```
```   129 text{*But is it useful?*}
```
```   130 lemma inj_transfer:
```
```   131   assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)"
```
```   132   shows "P x"
```
```   133 proof -
```
```   134   have "f x \<in> range f" by auto
```
```   135   hence "P(inv f (f x))" by (rule minor)
```
```   136   thus "P x" by (simp add: inv_into_f_f [OF injf])
```
```   137 qed
```
```   138
```
```   139 lemma inj_iff: "(inj f) = (inv f o f = id)"
```
```   140 apply (simp add: o_def fun_eq_iff)
```
```   141 apply (blast intro: inj_on_inverseI inv_into_f_f)
```
```   142 done
```
```   143
```
```   144 lemma inv_o_cancel[simp]: "inj f ==> inv f o f = id"
```
```   145 by (simp add: inj_iff)
```
```   146
```
```   147 lemma o_inv_o_cancel[simp]: "inj f ==> g o inv f o f = g"
```
```   148 by (simp add: o_assoc[symmetric])
```
```   149
```
```   150 lemma inv_into_image_cancel[simp]:
```
```   151   "inj_on f A ==> S <= A ==> inv_into A f ` f ` S = S"
```
```   152 by(fastforce simp: image_def)
```
```   153
```
```   154 lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
```
```   155 by (blast intro!: surjI inv_into_f_f)
```
```   156
```
```   157 lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
```
```   158 by (simp add: f_inv_into_f)
```
```   159
```
```   160 lemma inv_into_injective:
```
```   161   assumes eq: "inv_into A f x = inv_into A f y"
```
```   162       and x: "x: f`A"
```
```   163       and y: "y: f`A"
```
```   164   shows "x=y"
```
```   165 proof -
```
```   166   have "f (inv_into A f x) = f (inv_into A f y)" using eq by simp
```
```   167   thus ?thesis by (simp add: f_inv_into_f x y)
```
```   168 qed
```
```   169
```
```   170 lemma inj_on_inv_into: "B <= f`A ==> inj_on (inv_into A f) B"
```
```   171 by (blast intro: inj_onI dest: inv_into_injective injD)
```
```   172
```
```   173 lemma bij_betw_inv_into: "bij_betw f A B ==> bij_betw (inv_into A f) B A"
```
```   174 by (auto simp add: bij_betw_def inj_on_inv_into)
```
```   175
```
```   176 lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
```
```   177 by (simp add: inj_on_inv_into)
```
```   178
```
```   179 lemma surj_iff: "(surj f) = (f o inv f = id)"
```
```   180 by (auto intro!: surjI simp: surj_f_inv_f fun_eq_iff[where 'b='a])
```
```   181
```
```   182 lemma surj_iff_all: "surj f \<longleftrightarrow> (\<forall>x. f (inv f x) = x)"
```
```   183   unfolding surj_iff by (simp add: o_def fun_eq_iff)
```
```   184
```
```   185 lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
```
```   186 apply (rule ext)
```
```   187 apply (drule_tac x = "inv f x" in spec)
```
```   188 apply (simp add: surj_f_inv_f)
```
```   189 done
```
```   190
```
```   191 lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
```
```   192 by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
```
```   193
```
```   194 lemma inv_equality: "[| !!x. g (f x) = x;  !!y. f (g y) = y |] ==> inv f = g"
```
```   195 apply (rule ext)
```
```   196 apply (auto simp add: inv_into_def)
```
```   197 done
```
```   198
```
```   199 lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
```
```   200 apply (rule inv_equality)
```
```   201 apply (auto simp add: bij_def surj_f_inv_f)
```
```   202 done
```
```   203
```
```   204 (** bij(inv f) implies little about f.  Consider f::bool=>bool such that
```
```   205     f(True)=f(False)=True.  Then it's consistent with axiom someI that
```
```   206     inv f could be any function at all, including the identity function.
```
```   207     If inv f=id then inv f is a bijection, but inj f, surj(f) and
```
```   208     inv(inv f)=f all fail.
```
```   209 **)
```
```   210
```
```   211 lemma inv_into_comp:
```
```   212   "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
```
```   213   inv_into A (f o g) x = (inv_into A g o inv_into (g ` A) f) x"
```
```   214 apply (rule inv_into_f_eq)
```
```   215   apply (fast intro: comp_inj_on)
```
```   216  apply (simp add: inv_into_into)
```
```   217 apply (simp add: f_inv_into_f inv_into_into)
```
```   218 done
```
```   219
```
```   220 lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
```
```   221 apply (rule inv_equality)
```
```   222 apply (auto simp add: bij_def surj_f_inv_f)
```
```   223 done
```
```   224
```
```   225 lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
```
```   226 by (simp add: image_eq_UN surj_f_inv_f)
```
```   227
```
```   228 lemma image_inv_f_f: "inj f ==> (inv f) ` (f ` A) = A"
```
```   229 by (simp add: image_eq_UN)
```
```   230
```
```   231 lemma inv_image_comp: "inj f ==> inv f ` (f`X) = X"
```
```   232 by (auto simp add: image_def)
```
```   233
```
```   234 lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
```
```   235 apply auto
```
```   236 apply (force simp add: bij_is_inj)
```
```   237 apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
```
```   238 done
```
```   239
```
```   240 lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A"
```
```   241 apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
```
```   242 apply (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric])
```
```   243 done
```
```   244
```
```   245 lemma finite_fun_UNIVD1:
```
```   246   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
```
```   247   and card: "card (UNIV :: 'b set) \<noteq> Suc 0"
```
```   248   shows "finite (UNIV :: 'a set)"
```
```   249 proof -
```
```   250   from fin have finb: "finite (UNIV :: 'b set)" by (rule finite_fun_UNIVD2)
```
```   251   with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)"
```
```   252     by (cases "card (UNIV :: 'b set)") (auto simp add: card_eq_0_iff)
```
```   253   then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - Suc (Suc 0)" by auto
```
```   254   then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by (auto simp add: card_Suc_eq)
```
```   255   from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by (rule finite_imageI)
```
```   256   moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)"
```
```   257   proof (rule UNIV_eq_I)
```
```   258     fix x :: 'a
```
```   259     from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_into_def)
```
```   260     thus "x \<in> range (\<lambda>f\<Colon>'a \<Rightarrow> 'b. inv f b1)" by blast
```
```   261   qed
```
```   262   ultimately show "finite (UNIV :: 'a set)" by simp
```
```   263 qed
```
```   264
```
```   265 lemma image_inv_into_cancel:
```
```   266   assumes SURJ: "f`A=A'" and SUB: "B' \<le> A'"
```
```   267   shows "f `((inv_into A f)`B') = B'"
```
```   268   using assms
```
```   269 proof (auto simp add: f_inv_into_f)
```
```   270   let ?f' = "(inv_into A f)"
```
```   271   fix a' assume *: "a' \<in> B'"
```
```   272   then have "a' \<in> A'" using SUB by auto
```
```   273   then have "a' = f (?f' a')"
```
```   274     using SURJ by (auto simp add: f_inv_into_f)
```
```   275   then show "a' \<in> f ` (?f' ` B')" using * by blast
```
```   276 qed
```
```   277
```
```   278 lemma inv_into_inv_into_eq:
```
```   279   assumes "bij_betw f A A'" "a \<in> A"
```
```   280   shows "inv_into A' (inv_into A f) a = f a"
```
```   281 proof -
```
```   282   let ?f' = "inv_into A f"   let ?f'' = "inv_into A' ?f'"
```
```   283   have 1: "bij_betw ?f' A' A" using assms
```
```   284   by (auto simp add: bij_betw_inv_into)
```
```   285   obtain a' where 2: "a' \<in> A'" and 3: "?f' a' = a"
```
```   286     using 1 `a \<in> A` unfolding bij_betw_def by force
```
```   287   hence "?f'' a = a'"
```
```   288     using `a \<in> A` 1 3 by (auto simp add: f_inv_into_f bij_betw_def)
```
```   289   moreover have "f a = a'" using assms 2 3
```
```   290     by (auto simp add: bij_betw_def)
```
```   291   ultimately show "?f'' a = f a" by simp
```
```   292 qed
```
```   293
```
```   294 lemma inj_on_iff_surj:
```
```   295   assumes "A \<noteq> {}"
```
```   296   shows "(\<exists>f. inj_on f A \<and> f ` A \<le> A') \<longleftrightarrow> (\<exists>g. g ` A' = A)"
```
```   297 proof safe
```
```   298   fix f assume INJ: "inj_on f A" and INCL: "f ` A \<le> A'"
```
```   299   let ?phi = "\<lambda>a' a. a \<in> A \<and> f a = a'"  let ?csi = "\<lambda>a. a \<in> A"
```
```   300   let ?g = "\<lambda>a'. if a' \<in> f ` A then (SOME a. ?phi a' a) else (SOME a. ?csi a)"
```
```   301   have "?g ` A' = A"
```
```   302   proof
```
```   303     show "?g ` A' \<le> A"
```
```   304     proof clarify
```
```   305       fix a' assume *: "a' \<in> A'"
```
```   306       show "?g a' \<in> A"
```
```   307       proof cases
```
```   308         assume Case1: "a' \<in> f ` A"
```
```   309         then obtain a where "?phi a' a" by blast
```
```   310         hence "?phi a' (SOME a. ?phi a' a)" using someI[of "?phi a'" a] by blast
```
```   311         with Case1 show ?thesis by auto
```
```   312       next
```
```   313         assume Case2: "a' \<notin> f ` A"
```
```   314         hence "?csi (SOME a. ?csi a)" using assms someI_ex[of ?csi] by blast
```
```   315         with Case2 show ?thesis by auto
```
```   316       qed
```
```   317     qed
```
```   318   next
```
```   319     show "A \<le> ?g ` A'"
```
```   320     proof-
```
```   321       {fix a assume *: "a \<in> A"
```
```   322        let ?b = "SOME aa. ?phi (f a) aa"
```
```   323        have "?phi (f a) a" using * by auto
```
```   324        hence 1: "?phi (f a) ?b" using someI[of "?phi(f a)" a] by blast
```
```   325        hence "?g(f a) = ?b" using * by auto
```
```   326        moreover have "a = ?b" using 1 INJ * by (auto simp add: inj_on_def)
```
```   327        ultimately have "?g(f a) = a" by simp
```
```   328        with INCL * have "?g(f a) = a \<and> f a \<in> A'" by auto
```
```   329       }
```
```   330       thus ?thesis by force
```
```   331     qed
```
```   332   qed
```
```   333   thus "\<exists>g. g ` A' = A" by blast
```
```   334 next
```
```   335   fix g  let ?f = "inv_into A' g"
```
```   336   have "inj_on ?f (g ` A')"
```
```   337     by (auto simp add: inj_on_inv_into)
```
```   338   moreover
```
```   339   {fix a' assume *: "a' \<in> A'"
```
```   340    let ?phi = "\<lambda> b'. b' \<in> A' \<and> g b' = g a'"
```
```   341    have "?phi a'" using * by auto
```
```   342    hence "?phi(SOME b'. ?phi b')" using someI[of ?phi] by blast
```
```   343    hence "?f(g a') \<in> A'" unfolding inv_into_def by auto
```
```   344   }
```
```   345   ultimately show "\<exists>f. inj_on f (g ` A') \<and> f ` g ` A' \<subseteq> A'" by auto
```
```   346 qed
```
```   347
```
```   348 lemma Ex_inj_on_UNION_Sigma:
```
```   349   "\<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))"
```
```   350 proof
```
```   351   let ?phi = "\<lambda> a i. i \<in> I \<and> a \<in> A i"
```
```   352   let ?sm = "\<lambda> a. SOME i. ?phi a i"
```
```   353   let ?f = "\<lambda>a. (?sm a, a)"
```
```   354   have "inj_on ?f (\<Union> i \<in> I. A i)" unfolding inj_on_def by auto
```
```   355   moreover
```
```   356   { { fix i a assume "i \<in> I" and "a \<in> A i"
```
```   357       hence "?sm a \<in> I \<and> a \<in> A(?sm a)" using someI[of "?phi a" i] by auto
```
```   358     }
```
```   359     hence "?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)" by auto
```
```   360   }
```
```   361   ultimately
```
```   362   show "inj_on ?f (\<Union> i \<in> I. A i) \<and> ?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)"
```
```   363   by auto
```
```   364 qed
```
```   365
```
```   366 subsection {* The Cantor-Bernstein Theorem *}
```
```   367
```
```   368 lemma Cantor_Bernstein_aux:
```
```   369   shows "\<exists>A' h. A' \<le> A \<and>
```
```   370                 (\<forall>a \<in> A'. a \<notin> g`(B - f ` A')) \<and>
```
```   371                 (\<forall>a \<in> A'. h a = f a) \<and>
```
```   372                 (\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a))"
```
```   373 proof-
```
```   374   obtain H where H_def: "H = (\<lambda> A'. A - (g`(B - (f ` A'))))" by blast
```
```   375   have 0: "mono H" unfolding mono_def H_def by blast
```
```   376   then obtain A' where 1: "H A' = A'" using lfp_unfold by blast
```
```   377   hence 2: "A' = A - (g`(B - (f ` A')))" unfolding H_def by simp
```
```   378   hence 3: "A' \<le> A" by blast
```
```   379   have 4: "\<forall>a \<in> A'.  a \<notin> g`(B - f ` A')"
```
```   380   using 2 by blast
```
```   381   have 5: "\<forall>a \<in> A - A'. \<exists>b \<in> B - (f ` A'). a = g b"
```
```   382   using 2 by blast
```
```   383   (*  *)
```
```   384   obtain h where h_def:
```
```   385   "h = (\<lambda> a. if a \<in> A' then f a else (SOME b. b \<in> B - (f ` A') \<and> a = g b))" by blast
```
```   386   hence "\<forall>a \<in> A'. h a = f a" by auto
```
```   387   moreover
```
```   388   have "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
```
```   389   proof
```
```   390     fix a assume *: "a \<in> A - A'"
```
```   391     let ?phi = "\<lambda> b. b \<in> B - (f ` A') \<and> a = g b"
```
```   392     have "h a = (SOME b. ?phi b)" using h_def * by auto
```
```   393     moreover have "\<exists>b. ?phi b" using 5 *  by auto
```
```   394     ultimately show  "?phi (h a)" using someI_ex[of ?phi] by auto
```
```   395   qed
```
```   396   ultimately show ?thesis using 3 4 by blast
```
```   397 qed
```
```   398
```
```   399 theorem Cantor_Bernstein:
```
```   400   assumes INJ1: "inj_on f A" and SUB1: "f ` A \<le> B" and
```
```   401           INJ2: "inj_on g B" and SUB2: "g ` B \<le> A"
```
```   402   shows "\<exists>h. bij_betw h A B"
```
```   403 proof-
```
```   404   obtain A' and h where 0: "A' \<le> A" and
```
```   405   1: "\<forall>a \<in> A'. a \<notin> g`(B - f ` A')" and
```
```   406   2: "\<forall>a \<in> A'. h a = f a" and
```
```   407   3: "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
```
```   408   using Cantor_Bernstein_aux[of A g B f] by blast
```
```   409   have "inj_on h A"
```
```   410   proof (intro inj_onI)
```
```   411     fix a1 a2
```
```   412     assume 4: "a1 \<in> A" and 5: "a2 \<in> A" and 6: "h a1 = h a2"
```
```   413     show "a1 = a2"
```
```   414     proof(cases "a1 \<in> A'")
```
```   415       assume Case1: "a1 \<in> A'"
```
```   416       show ?thesis
```
```   417       proof(cases "a2 \<in> A'")
```
```   418         assume Case11: "a2 \<in> A'"
```
```   419         hence "f a1 = f a2" using Case1 2 6 by auto
```
```   420         thus ?thesis using INJ1 Case1 Case11 0
```
```   421         unfolding inj_on_def by blast
```
```   422       next
```
```   423         assume Case12: "a2 \<notin> A'"
```
```   424         hence False using 3 5 2 6 Case1 by force
```
```   425         thus ?thesis by simp
```
```   426       qed
```
```   427     next
```
```   428     assume Case2: "a1 \<notin> A'"
```
```   429       show ?thesis
```
```   430       proof(cases "a2 \<in> A'")
```
```   431         assume Case21: "a2 \<in> A'"
```
```   432         hence False using 3 4 2 6 Case2 by auto
```
```   433         thus ?thesis by simp
```
```   434       next
```
```   435         assume Case22: "a2 \<notin> A'"
```
```   436         hence "a1 = g(h a1) \<and> a2 = g(h a2)" using Case2 4 5 3 by auto
```
```   437         thus ?thesis using 6 by simp
```
```   438       qed
```
```   439     qed
```
```   440   qed
```
```   441   (*  *)
```
```   442   moreover
```
```   443   have "h ` A = B"
```
```   444   proof safe
```
```   445     fix a assume "a \<in> A"
```
```   446     thus "h a \<in> B" using SUB1 2 3 by (cases "a \<in> A'") auto
```
```   447   next
```
```   448     fix b assume *: "b \<in> B"
```
```   449     show "b \<in> h ` A"
```
```   450     proof(cases "b \<in> f ` A'")
```
```   451       assume Case1: "b \<in> f ` A'"
```
```   452       then obtain a where "a \<in> A' \<and> b = f a" by blast
```
```   453       thus ?thesis using 2 0 by force
```
```   454     next
```
```   455       assume Case2: "b \<notin> f ` A'"
```
```   456       hence "g b \<notin> A'" using 1 * by auto
```
```   457       hence 4: "g b \<in> A - A'" using * SUB2 by auto
```
```   458       hence "h(g b) \<in> B \<and> g(h(g b)) = g b"
```
```   459       using 3 by auto
```
```   460       hence "h(g b) = b" using * INJ2 unfolding inj_on_def by auto
```
```   461       thus ?thesis using 4 by force
```
```   462     qed
```
```   463   qed
```
```   464   (*  *)
```
```   465   ultimately show ?thesis unfolding bij_betw_def by auto
```
```   466 qed
```
```   467
```
```   468 subsection {*Other Consequences of Hilbert's Epsilon*}
```
```   469
```
```   470 text {*Hilbert's Epsilon and the @{term split} Operator*}
```
```   471
```
```   472 text{*Looping simprule*}
```
```   473 lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
```
```   474   by simp
```
```   475
```
```   476 lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"
```
```   477   by (simp add: split_def)
```
```   478
```
```   479 lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
```
```   480   by blast
```
```   481
```
```   482
```
```   483 text{*A relation is wellfounded iff it has no infinite descending chain*}
```
```   484 lemma wf_iff_no_infinite_down_chain:
```
```   485   "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
```
```   486 apply (simp only: wf_eq_minimal)
```
```   487 apply (rule iffI)
```
```   488  apply (rule notI)
```
```   489  apply (erule exE)
```
```   490  apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
```
```   491 apply (erule contrapos_np, simp, clarify)
```
```   492 apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
```
```   493  apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI)
```
```   494  apply (rule allI, simp)
```
```   495  apply (rule someI2_ex, blast, blast)
```
```   496 apply (rule allI)
```
```   497 apply (induct_tac "n", simp_all)
```
```   498 apply (rule someI2_ex, blast+)
```
```   499 done
```
```   500
```
```   501 lemma wf_no_infinite_down_chainE:
```
```   502   assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r"
```
```   503 using `wf r` wf_iff_no_infinite_down_chain[of r] by blast
```
```   504
```
```   505
```
```   506 text{*A dynamically-scoped fact for TFL *}
```
```   507 lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
```
```   508   by (blast intro: someI)
```
```   509
```
```   510
```
```   511 subsection {* Least value operator *}
```
```   512
```
```   513 definition
```
```   514   LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where
```
```   515   "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
```
```   516
```
```   517 syntax
```
```   518   "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
```
```   519 translations
```
```   520   "LEAST x WRT m. P" == "CONST LeastM m (%x. P)"
```
```   521
```
```   522 lemma LeastMI2:
```
```   523   "P x ==> (!!y. P y ==> m x <= m y)
```
```   524     ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
```
```   525     ==> Q (LeastM m P)"
```
```   526   apply (simp add: LeastM_def)
```
```   527   apply (rule someI2_ex, blast, blast)
```
```   528   done
```
```   529
```
```   530 lemma LeastM_equality:
```
```   531   "P k ==> (!!x. P x ==> m k <= m x)
```
```   532     ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
```
```   533   apply (rule LeastMI2, assumption, blast)
```
```   534   apply (blast intro!: order_antisym)
```
```   535   done
```
```   536
```
```   537 lemma wf_linord_ex_has_least:
```
```   538   "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
```
```   539     ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
```
```   540   apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
```
```   541   apply (drule_tac x = "m`Collect P" in spec, force)
```
```   542   done
```
```   543
```
```   544 lemma ex_has_least_nat:
```
```   545     "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
```
```   546   apply (simp only: pred_nat_trancl_eq_le [symmetric])
```
```   547   apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
```
```   548    apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
```
```   549   done
```
```   550
```
```   551 lemma LeastM_nat_lemma:
```
```   552     "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
```
```   553   apply (simp add: LeastM_def)
```
```   554   apply (rule someI_ex)
```
```   555   apply (erule ex_has_least_nat)
```
```   556   done
```
```   557
```
```   558 lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1]
```
```   559
```
```   560 lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
```
```   561 by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
```
```   562
```
```   563
```
```   564 subsection {* Greatest value operator *}
```
```   565
```
```   566 definition
```
```   567   GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where
```
```   568   "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
```
```   569
```
```   570 definition
```
```   571   Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where
```
```   572   "Greatest == GreatestM (%x. x)"
```
```   573
```
```   574 syntax
```
```   575   "_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"
```
```   576       ("GREATEST _ WRT _. _" [0, 4, 10] 10)
```
```   577 translations
```
```   578   "GREATEST x WRT m. P" == "CONST GreatestM m (%x. P)"
```
```   579
```
```   580 lemma GreatestMI2:
```
```   581   "P x ==> (!!y. P y ==> m y <= m x)
```
```   582     ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
```
```   583     ==> Q (GreatestM m P)"
```
```   584   apply (simp add: GreatestM_def)
```
```   585   apply (rule someI2_ex, blast, blast)
```
```   586   done
```
```   587
```
```   588 lemma GreatestM_equality:
```
```   589  "P k ==> (!!x. P x ==> m x <= m k)
```
```   590     ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
```
```   591   apply (rule_tac m = m in GreatestMI2, assumption, blast)
```
```   592   apply (blast intro!: order_antisym)
```
```   593   done
```
```   594
```
```   595 lemma Greatest_equality:
```
```   596   "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
```
```   597   apply (simp add: Greatest_def)
```
```   598   apply (erule GreatestM_equality, blast)
```
```   599   done
```
```   600
```
```   601 lemma ex_has_greatest_nat_lemma:
```
```   602   "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
```
```   603     ==> \<exists>y. P y & ~ (m y < m k + n)"
```
```   604   apply (induct n, force)
```
```   605   apply (force simp add: le_Suc_eq)
```
```   606   done
```
```   607
```
```   608 lemma ex_has_greatest_nat:
```
```   609   "P k ==> \<forall>y. P y --> m y < b
```
```   610     ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
```
```   611   apply (rule ccontr)
```
```   612   apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
```
```   613     apply (subgoal_tac  "m k <= b", auto)
```
```   614   done
```
```   615
```
```   616 lemma GreatestM_nat_lemma:
```
```   617   "P k ==> \<forall>y. P y --> m y < b
```
```   618     ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
```
```   619   apply (simp add: GreatestM_def)
```
```   620   apply (rule someI_ex)
```
```   621   apply (erule ex_has_greatest_nat, assumption)
```
```   622   done
```
```   623
```
```   624 lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1]
```
```   625
```
```   626 lemma GreatestM_nat_le:
```
```   627   "P x ==> \<forall>y. P y --> m y < b
```
```   628     ==> (m x::nat) <= m (GreatestM m P)"
```
```   629   apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
```
```   630   done
```
```   631
```
```   632
```
```   633 text {* \medskip Specialization to @{text GREATEST}. *}
```
```   634
```
```   635 lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
```
```   636   apply (simp add: Greatest_def)
```
```   637   apply (rule GreatestM_natI, auto)
```
```   638   done
```
```   639
```
```   640 lemma Greatest_le:
```
```   641     "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
```
```   642   apply (simp add: Greatest_def)
```
```   643   apply (rule GreatestM_nat_le, auto)
```
```   644   done
```
```   645
```
```   646
```
```   647 subsection {* Specification package -- Hilbertized version *}
```
```   648
```
```   649 lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
```
```   650   by (simp only: someI_ex)
```
```   651
```
```   652 use "Tools/choice_specification.ML"
```
```   653
```
```   654 end
```