src/HOL/Hilbert_Choice.thy
author blanchet
Mon Jan 20 23:07:23 2014 +0100 (2014-01-20)
changeset 55088 57c82e01022b
parent 55020 96b05fd2aee4
child 55415 05f5fdb8d093
permissions -rw-r--r--
moved 'bacc' back to 'Enum' (cf. 744934b818c7) -- reduces baggage loaded by 'Hilbert_Choice'
     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 Metis
    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 _ => 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 lemma choice_iff: "(\<forall>x. \<exists>y. Q x y) \<longleftrightarrow> (\<exists>f. \<forall>x. Q x (f x))"
    90 by (fast elim: someI)
    91 
    92 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))"
    93 by (fast elim: someI)
    94 
    95 lemma bchoice_iff: "(\<forall>x\<in>S. \<exists>y. Q x y) \<longleftrightarrow> (\<exists>f. \<forall>x\<in>S. Q x (f x))"
    96 by (fast elim: someI)
    97 
    98 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))"
    99 by (fast elim: someI)
   100 
   101 subsection {*Function Inverse*}
   102 
   103 lemma inv_def: "inv f = (%y. SOME x. f x = y)"
   104 by(simp add: inv_into_def)
   105 
   106 lemma inv_into_into: "x : f ` A ==> inv_into A f x : A"
   107 apply (simp add: inv_into_def)
   108 apply (fast intro: someI2)
   109 done
   110 
   111 lemma inv_id [simp]: "inv id = id"
   112 by (simp add: inv_into_def id_def)
   113 
   114 lemma inv_into_f_f [simp]:
   115   "[| inj_on f A;  x : A |] ==> inv_into A f (f x) = x"
   116 apply (simp add: inv_into_def inj_on_def)
   117 apply (blast intro: someI2)
   118 done
   119 
   120 lemma inv_f_f: "inj f ==> inv f (f x) = x"
   121 by simp
   122 
   123 lemma f_inv_into_f: "y : f`A  ==> f (inv_into A f y) = y"
   124 apply (simp add: inv_into_def)
   125 apply (fast intro: someI2)
   126 done
   127 
   128 lemma inv_into_f_eq: "[| inj_on f A; x : A; f x = y |] ==> inv_into A f y = x"
   129 apply (erule subst)
   130 apply (fast intro: inv_into_f_f)
   131 done
   132 
   133 lemma inv_f_eq: "[| inj f; f x = y |] ==> inv f y = x"
   134 by (simp add:inv_into_f_eq)
   135 
   136 lemma inj_imp_inv_eq: "[| inj f; ALL x. f(g x) = x |] ==> inv f = g"
   137   by (blast intro: inv_into_f_eq)
   138 
   139 text{*But is it useful?*}
   140 lemma inj_transfer:
   141   assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)"
   142   shows "P x"
   143 proof -
   144   have "f x \<in> range f" by auto
   145   hence "P(inv f (f x))" by (rule minor)
   146   thus "P x" by (simp add: inv_into_f_f [OF injf])
   147 qed
   148 
   149 lemma inj_iff: "(inj f) = (inv f o f = id)"
   150 apply (simp add: o_def fun_eq_iff)
   151 apply (blast intro: inj_on_inverseI inv_into_f_f)
   152 done
   153 
   154 lemma inv_o_cancel[simp]: "inj f ==> inv f o f = id"
   155 by (simp add: inj_iff)
   156 
   157 lemma o_inv_o_cancel[simp]: "inj f ==> g o inv f o f = g"
   158 by (simp add: comp_assoc)
   159 
   160 lemma inv_into_image_cancel[simp]:
   161   "inj_on f A ==> S <= A ==> inv_into A f ` f ` S = S"
   162 by(fastforce simp: image_def)
   163 
   164 lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
   165 by (blast intro!: surjI inv_into_f_f)
   166 
   167 lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
   168 by (simp add: f_inv_into_f)
   169 
   170 lemma inv_into_injective:
   171   assumes eq: "inv_into A f x = inv_into A f y"
   172       and x: "x: f`A"
   173       and y: "y: f`A"
   174   shows "x=y"
   175 proof -
   176   have "f (inv_into A f x) = f (inv_into A f y)" using eq by simp
   177   thus ?thesis by (simp add: f_inv_into_f x y)
   178 qed
   179 
   180 lemma inj_on_inv_into: "B <= f`A ==> inj_on (inv_into A f) B"
   181 by (blast intro: inj_onI dest: inv_into_injective injD)
   182 
   183 lemma bij_betw_inv_into: "bij_betw f A B ==> bij_betw (inv_into A f) B A"
   184 by (auto simp add: bij_betw_def inj_on_inv_into)
   185 
   186 lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
   187 by (simp add: inj_on_inv_into)
   188 
   189 lemma surj_iff: "(surj f) = (f o inv f = id)"
   190 by (auto intro!: surjI simp: surj_f_inv_f fun_eq_iff[where 'b='a])
   191 
   192 lemma surj_iff_all: "surj f \<longleftrightarrow> (\<forall>x. f (inv f x) = x)"
   193   unfolding surj_iff by (simp add: o_def fun_eq_iff)
   194 
   195 lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
   196 apply (rule ext)
   197 apply (drule_tac x = "inv f x" in spec)
   198 apply (simp add: surj_f_inv_f)
   199 done
   200 
   201 lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
   202 by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
   203 
   204 lemma inv_equality: "[| !!x. g (f x) = x;  !!y. f (g y) = y |] ==> inv f = g"
   205 apply (rule ext)
   206 apply (auto simp add: inv_into_def)
   207 done
   208 
   209 lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
   210 apply (rule inv_equality)
   211 apply (auto simp add: bij_def surj_f_inv_f)
   212 done
   213 
   214 (** bij(inv f) implies little about f.  Consider f::bool=>bool such that
   215     f(True)=f(False)=True.  Then it's consistent with axiom someI that
   216     inv f could be any function at all, including the identity function.
   217     If inv f=id then inv f is a bijection, but inj f, surj(f) and
   218     inv(inv f)=f all fail.
   219 **)
   220 
   221 lemma inv_into_comp:
   222   "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
   223   inv_into A (f o g) x = (inv_into A g o inv_into (g ` A) f) x"
   224 apply (rule inv_into_f_eq)
   225   apply (fast intro: comp_inj_on)
   226  apply (simp add: inv_into_into)
   227 apply (simp add: f_inv_into_f inv_into_into)
   228 done
   229 
   230 lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
   231 apply (rule inv_equality)
   232 apply (auto simp add: bij_def surj_f_inv_f)
   233 done
   234 
   235 lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
   236 by (simp add: image_eq_UN surj_f_inv_f)
   237 
   238 lemma image_inv_f_f: "inj f ==> (inv f) ` (f ` A) = A"
   239 by (simp add: image_eq_UN)
   240 
   241 lemma inv_image_comp: "inj f ==> inv f ` (f`X) = X"
   242 by (auto simp add: image_def)
   243 
   244 lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
   245 apply auto
   246 apply (force simp add: bij_is_inj)
   247 apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
   248 done
   249 
   250 lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A" 
   251 apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
   252 apply (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric])
   253 done
   254 
   255 lemma finite_fun_UNIVD1:
   256   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
   257   and card: "card (UNIV :: 'b set) \<noteq> Suc 0"
   258   shows "finite (UNIV :: 'a set)"
   259 proof -
   260   from fin have finb: "finite (UNIV :: 'b set)" by (rule finite_fun_UNIVD2)
   261   with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)"
   262     by (cases "card (UNIV :: 'b set)") (auto simp add: card_eq_0_iff)
   263   then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - Suc (Suc 0)" by auto
   264   then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by (auto simp add: card_Suc_eq)
   265   from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by (rule finite_imageI)
   266   moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)"
   267   proof (rule UNIV_eq_I)
   268     fix x :: 'a
   269     from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_into_def)
   270     thus "x \<in> range (\<lambda>f\<Colon>'a \<Rightarrow> 'b. inv f b1)" by blast
   271   qed
   272   ultimately show "finite (UNIV :: 'a set)" by simp
   273 qed
   274 
   275 text {*
   276   Every infinite set contains a countable subset. More precisely we
   277   show that a set @{text S} is infinite if and only if there exists an
   278   injective function from the naturals into @{text S}.
   279 
   280   The ``only if'' direction is harder because it requires the
   281   construction of a sequence of pairwise different elements of an
   282   infinite set @{text S}. The idea is to construct a sequence of
   283   non-empty and infinite subsets of @{text S} obtained by successively
   284   removing elements of @{text S}.
   285 *}
   286 
   287 lemma infinite_countable_subset:
   288   assumes inf: "\<not> finite (S::'a set)"
   289   shows "\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S"
   290   -- {* Courtesy of Stephan Merz *}
   291 proof -
   292   def Sseq \<equiv> "nat_rec S (\<lambda>n T. T - {SOME e. e \<in> T})"
   293   def pick \<equiv> "\<lambda>n. (SOME e. e \<in> Sseq n)"
   294   { fix n have "Sseq n \<subseteq> S" "\<not> finite (Sseq n)" by (induct n) (auto simp add: Sseq_def inf) }
   295   moreover then have *: "\<And>n. pick n \<in> Sseq n" by (metis someI_ex pick_def ex_in_conv finite.simps)
   296   ultimately have "range pick \<subseteq> S" by auto
   297   moreover
   298   { fix n m                 
   299     have "pick n \<notin> Sseq (n + Suc m)" by (induct m) (auto simp add: Sseq_def pick_def)
   300     hence "pick n \<noteq> pick (n + Suc m)" by (metis *)
   301   }
   302   then have "inj pick" by (intro linorder_injI) (auto simp add: less_iff_Suc_add)
   303   ultimately show ?thesis by blast
   304 qed
   305 
   306 lemma infinite_iff_countable_subset: "\<not> finite S \<longleftrightarrow> (\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S)"
   307   -- {* Courtesy of Stephan Merz *}
   308   by (metis finite_imageD finite_subset infinite_UNIV_char_0 infinite_countable_subset)
   309 
   310 lemma image_inv_into_cancel:
   311   assumes SURJ: "f`A=A'" and SUB: "B' \<le> A'"
   312   shows "f `((inv_into A f)`B') = B'"
   313   using assms
   314 proof (auto simp add: f_inv_into_f)
   315   let ?f' = "(inv_into A f)"
   316   fix a' assume *: "a' \<in> B'"
   317   then have "a' \<in> A'" using SUB by auto
   318   then have "a' = f (?f' a')"
   319     using SURJ by (auto simp add: f_inv_into_f)
   320   then show "a' \<in> f ` (?f' ` B')" using * by blast
   321 qed
   322 
   323 lemma inv_into_inv_into_eq:
   324   assumes "bij_betw f A A'" "a \<in> A"
   325   shows "inv_into A' (inv_into A f) a = f a"
   326 proof -
   327   let ?f' = "inv_into A f"   let ?f'' = "inv_into A' ?f'"
   328   have 1: "bij_betw ?f' A' A" using assms
   329   by (auto simp add: bij_betw_inv_into)
   330   obtain a' where 2: "a' \<in> A'" and 3: "?f' a' = a"
   331     using 1 `a \<in> A` unfolding bij_betw_def by force
   332   hence "?f'' a = a'"
   333     using `a \<in> A` 1 3 by (auto simp add: f_inv_into_f bij_betw_def)
   334   moreover have "f a = a'" using assms 2 3
   335     by (auto simp add: bij_betw_def)
   336   ultimately show "?f'' a = f a" by simp
   337 qed
   338 
   339 lemma inj_on_iff_surj:
   340   assumes "A \<noteq> {}"
   341   shows "(\<exists>f. inj_on f A \<and> f ` A \<le> A') \<longleftrightarrow> (\<exists>g. g ` A' = A)"
   342 proof safe
   343   fix f assume INJ: "inj_on f A" and INCL: "f ` A \<le> A'"
   344   let ?phi = "\<lambda>a' a. a \<in> A \<and> f a = a'"  let ?csi = "\<lambda>a. a \<in> A"
   345   let ?g = "\<lambda>a'. if a' \<in> f ` A then (SOME a. ?phi a' a) else (SOME a. ?csi a)"
   346   have "?g ` A' = A"
   347   proof
   348     show "?g ` A' \<le> A"
   349     proof clarify
   350       fix a' assume *: "a' \<in> A'"
   351       show "?g a' \<in> A"
   352       proof cases
   353         assume Case1: "a' \<in> f ` A"
   354         then obtain a where "?phi a' a" by blast
   355         hence "?phi a' (SOME a. ?phi a' a)" using someI[of "?phi a'" a] by blast
   356         with Case1 show ?thesis by auto
   357       next
   358         assume Case2: "a' \<notin> f ` A"
   359         hence "?csi (SOME a. ?csi a)" using assms someI_ex[of ?csi] by blast
   360         with Case2 show ?thesis by auto
   361       qed
   362     qed
   363   next
   364     show "A \<le> ?g ` A'"
   365     proof-
   366       {fix a assume *: "a \<in> A"
   367        let ?b = "SOME aa. ?phi (f a) aa"
   368        have "?phi (f a) a" using * by auto
   369        hence 1: "?phi (f a) ?b" using someI[of "?phi(f a)" a] by blast
   370        hence "?g(f a) = ?b" using * by auto
   371        moreover have "a = ?b" using 1 INJ * by (auto simp add: inj_on_def)
   372        ultimately have "?g(f a) = a" by simp
   373        with INCL * have "?g(f a) = a \<and> f a \<in> A'" by auto
   374       }
   375       thus ?thesis by force
   376     qed
   377   qed
   378   thus "\<exists>g. g ` A' = A" by blast
   379 next
   380   fix g  let ?f = "inv_into A' g"
   381   have "inj_on ?f (g ` A')"
   382     by (auto simp add: inj_on_inv_into)
   383   moreover
   384   {fix a' assume *: "a' \<in> A'"
   385    let ?phi = "\<lambda> b'. b' \<in> A' \<and> g b' = g a'"
   386    have "?phi a'" using * by auto
   387    hence "?phi(SOME b'. ?phi b')" using someI[of ?phi] by blast
   388    hence "?f(g a') \<in> A'" unfolding inv_into_def by auto
   389   }
   390   ultimately show "\<exists>f. inj_on f (g ` A') \<and> f ` g ` A' \<subseteq> A'" by auto
   391 qed
   392 
   393 lemma Ex_inj_on_UNION_Sigma:
   394   "\<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))"
   395 proof
   396   let ?phi = "\<lambda> a i. i \<in> I \<and> a \<in> A i"
   397   let ?sm = "\<lambda> a. SOME i. ?phi a i"
   398   let ?f = "\<lambda>a. (?sm a, a)"
   399   have "inj_on ?f (\<Union> i \<in> I. A i)" unfolding inj_on_def by auto
   400   moreover
   401   { { fix i a assume "i \<in> I" and "a \<in> A i"
   402       hence "?sm a \<in> I \<and> a \<in> A(?sm a)" using someI[of "?phi a" i] by auto
   403     }
   404     hence "?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)" by auto
   405   }
   406   ultimately
   407   show "inj_on ?f (\<Union> i \<in> I. A i) \<and> ?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)"
   408   by auto
   409 qed
   410 
   411 subsection {* The Cantor-Bernstein Theorem *}
   412 
   413 lemma Cantor_Bernstein_aux:
   414   shows "\<exists>A' h. A' \<le> A \<and>
   415                 (\<forall>a \<in> A'. a \<notin> g`(B - f ` A')) \<and>
   416                 (\<forall>a \<in> A'. h a = f a) \<and>
   417                 (\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a))"
   418 proof-
   419   obtain H where H_def: "H = (\<lambda> A'. A - (g`(B - (f ` A'))))" by blast
   420   have 0: "mono H" unfolding mono_def H_def by blast
   421   then obtain A' where 1: "H A' = A'" using lfp_unfold by blast
   422   hence 2: "A' = A - (g`(B - (f ` A')))" unfolding H_def by simp
   423   hence 3: "A' \<le> A" by blast
   424   have 4: "\<forall>a \<in> A'.  a \<notin> g`(B - f ` A')"
   425   using 2 by blast
   426   have 5: "\<forall>a \<in> A - A'. \<exists>b \<in> B - (f ` A'). a = g b"
   427   using 2 by blast
   428   (*  *)
   429   obtain h where h_def:
   430   "h = (\<lambda> a. if a \<in> A' then f a else (SOME b. b \<in> B - (f ` A') \<and> a = g b))" by blast
   431   hence "\<forall>a \<in> A'. h a = f a" by auto
   432   moreover
   433   have "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
   434   proof
   435     fix a assume *: "a \<in> A - A'"
   436     let ?phi = "\<lambda> b. b \<in> B - (f ` A') \<and> a = g b"
   437     have "h a = (SOME b. ?phi b)" using h_def * by auto
   438     moreover have "\<exists>b. ?phi b" using 5 *  by auto
   439     ultimately show  "?phi (h a)" using someI_ex[of ?phi] by auto
   440   qed
   441   ultimately show ?thesis using 3 4 by blast
   442 qed
   443 
   444 theorem Cantor_Bernstein:
   445   assumes INJ1: "inj_on f A" and SUB1: "f ` A \<le> B" and
   446           INJ2: "inj_on g B" and SUB2: "g ` B \<le> A"
   447   shows "\<exists>h. bij_betw h A B"
   448 proof-
   449   obtain A' and h where 0: "A' \<le> A" and
   450   1: "\<forall>a \<in> A'. a \<notin> g`(B - f ` A')" and
   451   2: "\<forall>a \<in> A'. h a = f a" and
   452   3: "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
   453   using Cantor_Bernstein_aux[of A g B f] by blast
   454   have "inj_on h A"
   455   proof (intro inj_onI)
   456     fix a1 a2
   457     assume 4: "a1 \<in> A" and 5: "a2 \<in> A" and 6: "h a1 = h a2"
   458     show "a1 = a2"
   459     proof(cases "a1 \<in> A'")
   460       assume Case1: "a1 \<in> A'"
   461       show ?thesis
   462       proof(cases "a2 \<in> A'")
   463         assume Case11: "a2 \<in> A'"
   464         hence "f a1 = f a2" using Case1 2 6 by auto
   465         thus ?thesis using INJ1 Case1 Case11 0
   466         unfolding inj_on_def by blast
   467       next
   468         assume Case12: "a2 \<notin> A'"
   469         hence False using 3 5 2 6 Case1 by force
   470         thus ?thesis by simp
   471       qed
   472     next
   473     assume Case2: "a1 \<notin> A'"
   474       show ?thesis
   475       proof(cases "a2 \<in> A'")
   476         assume Case21: "a2 \<in> A'"
   477         hence False using 3 4 2 6 Case2 by auto
   478         thus ?thesis by simp
   479       next
   480         assume Case22: "a2 \<notin> A'"
   481         hence "a1 = g(h a1) \<and> a2 = g(h a2)" using Case2 4 5 3 by auto
   482         thus ?thesis using 6 by simp
   483       qed
   484     qed
   485   qed
   486   (*  *)
   487   moreover
   488   have "h ` A = B"
   489   proof safe
   490     fix a assume "a \<in> A"
   491     thus "h a \<in> B" using SUB1 2 3 by (cases "a \<in> A'") auto
   492   next
   493     fix b assume *: "b \<in> B"
   494     show "b \<in> h ` A"
   495     proof(cases "b \<in> f ` A'")
   496       assume Case1: "b \<in> f ` A'"
   497       then obtain a where "a \<in> A' \<and> b = f a" by blast
   498       thus ?thesis using 2 0 by force
   499     next
   500       assume Case2: "b \<notin> f ` A'"
   501       hence "g b \<notin> A'" using 1 * by auto
   502       hence 4: "g b \<in> A - A'" using * SUB2 by auto
   503       hence "h(g b) \<in> B \<and> g(h(g b)) = g b"
   504       using 3 by auto
   505       hence "h(g b) = b" using * INJ2 unfolding inj_on_def by auto
   506       thus ?thesis using 4 by force
   507     qed
   508   qed
   509   (*  *)
   510   ultimately show ?thesis unfolding bij_betw_def by auto
   511 qed
   512 
   513 subsection {*Other Consequences of Hilbert's Epsilon*}
   514 
   515 text {*Hilbert's Epsilon and the @{term split} Operator*}
   516 
   517 text{*Looping simprule*}
   518 lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
   519   by simp
   520 
   521 lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"
   522   by (simp add: split_def)
   523 
   524 lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
   525   by blast
   526 
   527 
   528 text{*A relation is wellfounded iff it has no infinite descending chain*}
   529 lemma wf_iff_no_infinite_down_chain:
   530   "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
   531 apply (simp only: wf_eq_minimal)
   532 apply (rule iffI)
   533  apply (rule notI)
   534  apply (erule exE)
   535  apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
   536 apply (erule contrapos_np, simp, clarify)
   537 apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
   538  apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI)
   539  apply (rule allI, simp)
   540  apply (rule someI2_ex, blast, blast)
   541 apply (rule allI)
   542 apply (induct_tac "n", simp_all)
   543 apply (rule someI2_ex, blast+)
   544 done
   545 
   546 lemma wf_no_infinite_down_chainE:
   547   assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r"
   548 using `wf r` wf_iff_no_infinite_down_chain[of r] by blast
   549 
   550 
   551 text{*A dynamically-scoped fact for TFL *}
   552 lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
   553   by (blast intro: someI)
   554 
   555 
   556 subsection {* Least value operator *}
   557 
   558 definition
   559   LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where
   560   "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
   561 
   562 syntax
   563   "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
   564 translations
   565   "LEAST x WRT m. P" == "CONST LeastM m (%x. P)"
   566 
   567 lemma LeastMI2:
   568   "P x ==> (!!y. P y ==> m x <= m y)
   569     ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
   570     ==> Q (LeastM m P)"
   571   apply (simp add: LeastM_def)
   572   apply (rule someI2_ex, blast, blast)
   573   done
   574 
   575 lemma LeastM_equality:
   576   "P k ==> (!!x. P x ==> m k <= m x)
   577     ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
   578   apply (rule LeastMI2, assumption, blast)
   579   apply (blast intro!: order_antisym)
   580   done
   581 
   582 lemma wf_linord_ex_has_least:
   583   "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
   584     ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
   585   apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
   586   apply (drule_tac x = "m`Collect P" in spec, force)
   587   done
   588 
   589 lemma ex_has_least_nat:
   590     "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
   591   apply (simp only: pred_nat_trancl_eq_le [symmetric])
   592   apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
   593    apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
   594   done
   595 
   596 lemma LeastM_nat_lemma:
   597     "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
   598   apply (simp add: LeastM_def)
   599   apply (rule someI_ex)
   600   apply (erule ex_has_least_nat)
   601   done
   602 
   603 lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1]
   604 
   605 lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
   606 by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
   607 
   608 
   609 subsection {* Greatest value operator *}
   610 
   611 definition
   612   GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where
   613   "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
   614 
   615 definition
   616   Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where
   617   "Greatest == GreatestM (%x. x)"
   618 
   619 syntax
   620   "_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"
   621       ("GREATEST _ WRT _. _" [0, 4, 10] 10)
   622 translations
   623   "GREATEST x WRT m. P" == "CONST GreatestM m (%x. P)"
   624 
   625 lemma GreatestMI2:
   626   "P x ==> (!!y. P y ==> m y <= m x)
   627     ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
   628     ==> Q (GreatestM m P)"
   629   apply (simp add: GreatestM_def)
   630   apply (rule someI2_ex, blast, blast)
   631   done
   632 
   633 lemma GreatestM_equality:
   634  "P k ==> (!!x. P x ==> m x <= m k)
   635     ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
   636   apply (rule_tac m = m in GreatestMI2, assumption, blast)
   637   apply (blast intro!: order_antisym)
   638   done
   639 
   640 lemma Greatest_equality:
   641   "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
   642   apply (simp add: Greatest_def)
   643   apply (erule GreatestM_equality, blast)
   644   done
   645 
   646 lemma ex_has_greatest_nat_lemma:
   647   "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
   648     ==> \<exists>y. P y & ~ (m y < m k + n)"
   649   apply (induct n, force)
   650   apply (force simp add: le_Suc_eq)
   651   done
   652 
   653 lemma ex_has_greatest_nat:
   654   "P k ==> \<forall>y. P y --> m y < b
   655     ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
   656   apply (rule ccontr)
   657   apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
   658     apply (subgoal_tac [3] "m k <= b", auto)
   659   done
   660 
   661 lemma GreatestM_nat_lemma:
   662   "P k ==> \<forall>y. P y --> m y < b
   663     ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
   664   apply (simp add: GreatestM_def)
   665   apply (rule someI_ex)
   666   apply (erule ex_has_greatest_nat, assumption)
   667   done
   668 
   669 lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1]
   670 
   671 lemma GreatestM_nat_le:
   672   "P x ==> \<forall>y. P y --> m y < b
   673     ==> (m x::nat) <= m (GreatestM m P)"
   674   apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
   675   done
   676 
   677 
   678 text {* \medskip Specialization to @{text GREATEST}. *}
   679 
   680 lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
   681   apply (simp add: Greatest_def)
   682   apply (rule GreatestM_natI, auto)
   683   done
   684 
   685 lemma Greatest_le:
   686     "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
   687   apply (simp add: Greatest_def)
   688   apply (rule GreatestM_nat_le, auto)
   689   done
   690 
   691 
   692 subsection {* An aside: bounded accessible part *}
   693 
   694 text {* Finite monotone eventually stable sequences *}
   695 
   696 lemma finite_mono_remains_stable_implies_strict_prefix:
   697   fixes f :: "nat \<Rightarrow> 'a::order"
   698   assumes S: "finite (range f)" "mono f" and eq: "\<forall>n. f n = f (Suc n) \<longrightarrow> f (Suc n) = f (Suc (Suc n))"
   699   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)"
   700   using assms
   701 proof -
   702   have "\<exists>n. f n = f (Suc n)"
   703   proof (rule ccontr)
   704     assume "\<not> ?thesis"
   705     then have "\<And>n. f n \<noteq> f (Suc n)" by auto
   706     then have "\<And>n. f n < f (Suc n)"
   707       using  `mono f` by (auto simp: le_less mono_iff_le_Suc)
   708     with lift_Suc_mono_less_iff[of f]
   709     have "\<And>n m. n < m \<Longrightarrow> f n < f m" by auto
   710     then have "inj f"
   711       by (auto simp: inj_on_def) (metis linorder_less_linear order_less_imp_not_eq)
   712     with `finite (range f)` have "finite (UNIV::nat set)"
   713       by (rule finite_imageD)
   714     then show False by simp
   715   qed
   716   then obtain n where n: "f n = f (Suc n)" ..
   717   def N \<equiv> "LEAST n. f n = f (Suc n)"
   718   have N: "f N = f (Suc N)"
   719     unfolding N_def using n by (rule LeastI)
   720   show ?thesis
   721   proof (intro exI[of _ N] conjI allI impI)
   722     fix n assume "N \<le> n"
   723     then have "\<And>m. N \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m = f N"
   724     proof (induct rule: dec_induct)
   725       case (step n) then show ?case
   726         using eq[rule_format, of "n - 1"] N
   727         by (cases n) (auto simp add: le_Suc_eq)
   728     qed simp
   729     from this[of n] `N \<le> n` show "f N = f n" by auto
   730   next
   731     fix n m :: nat assume "m < n" "n \<le> N"
   732     then show "f m < f n"
   733     proof (induct rule: less_Suc_induct[consumes 1])
   734       case (1 i)
   735       then have "i < N" by simp
   736       then have "f i \<noteq> f (Suc i)"
   737         unfolding N_def by (rule not_less_Least)
   738       with `mono f` show ?case by (simp add: mono_iff_le_Suc less_le)
   739     qed auto
   740   qed
   741 qed
   742 
   743 lemma finite_mono_strict_prefix_implies_finite_fixpoint:
   744   fixes f :: "nat \<Rightarrow> 'a set"
   745   assumes S: "\<And>i. f i \<subseteq> S" "finite S"
   746     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)"
   747   shows "f (card S) = (\<Union>n. f n)"
   748 proof -
   749   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
   750 
   751   { fix i have "i \<le> N \<Longrightarrow> i \<le> card (f i)"
   752     proof (induct i)
   753       case 0 then show ?case by simp
   754     next
   755       case (Suc i)
   756       with inj[rule_format, of "Suc i" i]
   757       have "(f i) \<subset> (f (Suc i))" by auto
   758       moreover have "finite (f (Suc i))" using S by (rule finite_subset)
   759       ultimately have "card (f i) < card (f (Suc i))" by (intro psubset_card_mono)
   760       with Suc show ?case using inj by auto
   761     qed
   762   }
   763   then have "N \<le> card (f N)" by simp
   764   also have "\<dots> \<le> card S" using S by (intro card_mono)
   765   finally have "f (card S) = f N" using eq by auto
   766   then show ?thesis using eq inj[rule_format, of N]
   767     apply auto
   768     apply (case_tac "n < N")
   769     apply (auto simp: not_less)
   770     done
   771 qed
   772 
   773 
   774 subsection {* More on injections, bijections, and inverses *}
   775 
   776 lemma infinite_imp_bij_betw:
   777 assumes INF: "\<not> finite A"
   778 shows "\<exists>h. bij_betw h A (A - {a})"
   779 proof(cases "a \<in> A")
   780   assume Case1: "a \<notin> A"  hence "A - {a} = A" by blast
   781   thus ?thesis using bij_betw_id[of A] by auto
   782 next
   783   assume Case2: "a \<in> A"
   784 find_theorems "\<not> finite _"
   785   have "\<not> finite (A - {a})" using INF by auto
   786   with infinite_iff_countable_subset[of "A - {a}"] obtain f::"nat \<Rightarrow> 'a"
   787   where 1: "inj f" and 2: "f ` UNIV \<le> A - {a}" by blast
   788   obtain g where g_def: "g = (\<lambda> n. if n = 0 then a else f (Suc n))" by blast
   789   obtain A' where A'_def: "A' = g ` UNIV" by blast
   790   have temp: "\<forall>y. f y \<noteq> a" using 2 by blast
   791   have 3: "inj_on g UNIV \<and> g ` UNIV \<le> A \<and> a \<in> g ` UNIV"
   792   proof(auto simp add: Case2 g_def, unfold inj_on_def, intro ballI impI,
   793         case_tac "x = 0", auto simp add: 2)
   794     fix y  assume "a = (if y = 0 then a else f (Suc y))"
   795     thus "y = 0" using temp by (case_tac "y = 0", auto)
   796   next
   797     fix x y
   798     assume "f (Suc x) = (if y = 0 then a else f (Suc y))"
   799     thus "x = y" using 1 temp unfolding inj_on_def by (case_tac "y = 0", auto)
   800   next
   801     fix n show "f (Suc n) \<in> A" using 2 by blast
   802   qed
   803   hence 4: "bij_betw g UNIV A' \<and> a \<in> A' \<and> A' \<le> A"
   804   using inj_on_imp_bij_betw[of g] unfolding A'_def by auto
   805   hence 5: "bij_betw (inv g) A' UNIV"
   806   by (auto simp add: bij_betw_inv_into)
   807   (*  *)
   808   obtain n where "g n = a" using 3 by auto
   809   hence 6: "bij_betw g (UNIV - {n}) (A' - {a})"
   810   using 3 4 unfolding A'_def
   811   by clarify (rule bij_betw_subset, auto simp: image_set_diff)
   812   (*  *)
   813   obtain v where v_def: "v = (\<lambda> m. if m < n then m else Suc m)" by blast
   814   have 7: "bij_betw v UNIV (UNIV - {n})"
   815   proof(unfold bij_betw_def inj_on_def, intro conjI, clarify)
   816     fix m1 m2 assume "v m1 = v m2"
   817     thus "m1 = m2"
   818     by(case_tac "m1 < n", case_tac "m2 < n",
   819        auto simp add: inj_on_def v_def, case_tac "m2 < n", auto)
   820   next
   821     show "v ` UNIV = UNIV - {n}"
   822     proof(auto simp add: v_def)
   823       fix m assume *: "m \<noteq> n" and **: "m \<notin> Suc ` {m'. \<not> m' < n}"
   824       {assume "n \<le> m" with * have 71: "Suc n \<le> m" by auto
   825        then obtain m' where 72: "m = Suc m'" using Suc_le_D by auto
   826        with 71 have "n \<le> m'" by auto
   827        with 72 ** have False by auto
   828       }
   829       thus "m < n" by force
   830     qed
   831   qed
   832   (*  *)
   833   obtain h' where h'_def: "h' = g o v o (inv g)" by blast
   834   hence 8: "bij_betw h' A' (A' - {a})" using 5 7 6
   835   by (auto simp add: bij_betw_trans)
   836   (*  *)
   837   obtain h where h_def: "h = (\<lambda> b. if b \<in> A' then h' b else b)" by blast
   838   have "\<forall>b \<in> A'. h b = h' b" unfolding h_def by auto
   839   hence "bij_betw h  A' (A' - {a})" using 8 bij_betw_cong[of A' h] by auto
   840   moreover
   841   {have "\<forall>b \<in> A - A'. h b = b" unfolding h_def by auto
   842    hence "bij_betw h  (A - A') (A - A')"
   843    using bij_betw_cong[of "A - A'" h id] bij_betw_id[of "A - A'"] by auto
   844   }
   845   moreover
   846   have "(A' Int (A - A') = {} \<and> A' \<union> (A - A') = A) \<and>
   847         ((A' - {a}) Int (A - A') = {} \<and> (A' - {a}) \<union> (A - A') = A - {a})"
   848   using 4 by blast
   849   ultimately have "bij_betw h A (A - {a})"
   850   using bij_betw_combine[of h A' "A' - {a}" "A - A'" "A - A'"] by simp
   851   thus ?thesis by blast
   852 qed
   853 
   854 lemma infinite_imp_bij_betw2:
   855 assumes INF: "\<not> finite A"
   856 shows "\<exists>h. bij_betw h A (A \<union> {a})"
   857 proof(cases "a \<in> A")
   858   assume Case1: "a \<in> A"  hence "A \<union> {a} = A" by blast
   859   thus ?thesis using bij_betw_id[of A] by auto
   860 next
   861   let ?A' = "A \<union> {a}"
   862   assume Case2: "a \<notin> A" hence "A = ?A' - {a}" by blast
   863   moreover have "\<not> finite ?A'" using INF by auto
   864   ultimately obtain f where "bij_betw f ?A' A"
   865   using infinite_imp_bij_betw[of ?A' a] by auto
   866   hence "bij_betw(inv_into ?A' f) A ?A'" using bij_betw_inv_into by blast
   867   thus ?thesis by auto
   868 qed
   869 
   870 lemma bij_betw_inv_into_left:
   871 assumes BIJ: "bij_betw f A A'" and IN: "a \<in> A"
   872 shows "(inv_into A f) (f a) = a"
   873 using assms unfolding bij_betw_def
   874 by clarify (rule inv_into_f_f)
   875 
   876 lemma bij_betw_inv_into_right:
   877 assumes "bij_betw f A A'" "a' \<in> A'"
   878 shows "f(inv_into A f a') = a'"
   879 using assms unfolding bij_betw_def using f_inv_into_f by force
   880 
   881 lemma bij_betw_inv_into_subset:
   882 assumes BIJ: "bij_betw f A A'" and
   883         SUB: "B \<le> A" and IM: "f ` B = B'"
   884 shows "bij_betw (inv_into A f) B' B"
   885 using assms unfolding bij_betw_def
   886 by (auto intro: inj_on_inv_into)
   887 
   888 
   889 subsection {* Specification package -- Hilbertized version *}
   890 
   891 lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
   892   by (simp only: someI_ex)
   893 
   894 ML_file "Tools/choice_specification.ML"
   895 
   896 end
   897