src/HOL/Hilbert_Choice.thy

basic facts about almost everywhere fix bijections

(*  Title:      HOL/Hilbert_Choice.thy
    Author:     Lawrence C Paulson, Tobias Nipkow
    Copyright   2001  University of Cambridge
*)

section \<open>Hilbert's Epsilon-Operator and the Axiom of Choice\<close>

theory Hilbert_Choice
imports Wellfounded
keywords "specification" :: thy_goal
begin

subsection \<open>Hilbert's epsilon\<close>

axiomatization Eps :: "('a => bool) => 'a" where
  someI: "P x ==> P (Eps P)"

syntax (epsilon)
  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3\<some>_./ _)" [0, 10] 10)
syntax (input)
  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3@ _./ _)" [0, 10] 10)
syntax
  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3SOME _./ _)" [0, 10] 10)
translations
  "SOME x. P" == "CONST Eps (%x. P)"

print_translation \<open>
  [(@{const_syntax Eps}, fn _ => fn [Abs abs] =>
      let val (x, t) = Syntax_Trans.atomic_abs_tr' abs
      in Syntax.const @{syntax_const "_Eps"} $ x $ t end)]
\<close> \<comment> \<open>to avoid eta-contraction of body\<close>

definition inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
"inv_into A f == %x. SOME y. y : A & f y = x"

abbreviation inv :: "('a => 'b) => ('b => 'a)" where
"inv == inv_into UNIV"


subsection \<open>Hilbert's Epsilon-operator\<close>

text\<open>Easier to apply than \<open>someI\<close> if the witness comes from an
existential formula\<close>
lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)"
apply (erule exE)
apply (erule someI)
done

text\<open>Easier to apply than \<open>someI\<close> because the conclusion has only one
occurrence of @{term P}.\<close>
lemma someI2: "[| P a;  !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
  by (blast intro: someI)

text\<open>Easier to apply than \<open>someI2\<close> if the witness comes from an
existential formula\<close>

lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
  by (blast intro: someI2)

lemma someI2_bex: "[| \<exists>a\<in>A. P a; !!x. x \<in> A \<and> P x ==> Q x |] ==> Q (SOME x. x \<in> A \<and> P x)"
  by (blast intro: someI2)

lemma some_equality [intro]:
     "[| P a;  !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
by (blast intro: someI2)

lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
by blast

lemma some_eq_ex: "P (SOME x. P x) =  (\<exists>x. P x)"
by (blast intro: someI)

lemma some_in_eq: "(SOME x. x \<in> A) \<in> A \<longleftrightarrow> A \<noteq> {}"
  unfolding ex_in_conv[symmetric] by (rule some_eq_ex)

lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
apply (rule some_equality)
apply (rule refl, assumption)
done

lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"
apply (rule some_equality)
apply (rule refl)
apply (erule sym)
done


subsection\<open>Axiom of Choice, Proved Using the Description Operator\<close>

lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)"
by (fast elim: someI)

lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)"
by (fast elim: someI)

lemma choice_iff: "(\<forall>x. \<exists>y. Q x y) \<longleftrightarrow> (\<exists>f. \<forall>x. Q x (f x))"
by (fast elim: someI)

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))"
by (fast elim: someI)

lemma bchoice_iff: "(\<forall>x\<in>S. \<exists>y. Q x y) \<longleftrightarrow> (\<exists>f. \<forall>x\<in>S. Q x (f x))"
by (fast elim: someI)

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))"
by (fast elim: someI)

lemma dependent_nat_choice:
  assumes  1: "\<exists>x. P 0 x" and
           2: "\<And>x n. P n x \<Longrightarrow> \<exists>y. P (Suc n) y \<and> Q n x y"
  shows "\<exists>f. \<forall>n. P n (f n) \<and> Q n (f n) (f (Suc n))"
proof (intro exI allI conjI)
  fix n
  define f where "f = rec_nat (SOME x. P 0 x) (\<lambda>n x. SOME y. P (Suc n) y \<and> Q n x y)"
  have "P 0 (f 0)" "\<And>n. P n (f n) \<Longrightarrow> P (Suc n) (f (Suc n)) \<and> Q n (f n) (f (Suc n))"
    using someI_ex[OF 1] someI_ex[OF 2] by (simp_all add: f_def)
  then show "P n (f n)" "Q n (f n) (f (Suc n))"
    by (induct n) auto
qed


subsection \<open>Function Inverse\<close>

lemma inv_def: "inv f = (%y. SOME x. f x = y)"
by(simp add: inv_into_def)

lemma inv_into_into: "x : f ` A ==> inv_into A f x : A"
apply (simp add: inv_into_def)
apply (fast intro: someI2)
done

lemma inv_identity [simp]:
  "inv (\<lambda>a. a) = (\<lambda>a. a)"
  by (simp add: inv_def)

lemma inv_id [simp]:
  "inv id = id"
  by (simp add: id_def)

lemma inv_into_f_f [simp]:
  "[| inj_on f A;  x : A |] ==> inv_into A f (f x) = x"
apply (simp add: inv_into_def inj_on_def)
apply (blast intro: someI2)
done

lemma inv_f_f: "inj f ==> inv f (f x) = x"
by simp

lemma f_inv_into_f: "y : f`A  ==> f (inv_into A f y) = y"
apply (simp add: inv_into_def)
apply (fast intro: someI2)
done

lemma inv_into_f_eq: "[| inj_on f A; x : A; f x = y |] ==> inv_into A f y = x"
apply (erule subst)
apply (fast intro: inv_into_f_f)
done

lemma inv_f_eq: "[| inj f; f x = y |] ==> inv f y = x"
by (simp add:inv_into_f_eq)

lemma inj_imp_inv_eq: "[| inj f; ALL x. f(g x) = x |] ==> inv f = g"
  by (blast intro: inv_into_f_eq)

text\<open>But is it useful?\<close>
lemma inj_transfer:
  assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)"
  shows "P x"
proof -
  have "f x \<in> range f" by auto
  hence "P(inv f (f x))" by (rule minor)
  thus "P x" by (simp add: inv_into_f_f [OF injf])
qed

lemma inj_iff: "(inj f) = (inv f o f = id)"
apply (simp add: o_def fun_eq_iff)
apply (blast intro: inj_on_inverseI inv_into_f_f)
done

lemma inv_o_cancel[simp]: "inj f ==> inv f o f = id"
by (simp add: inj_iff)

lemma o_inv_o_cancel[simp]: "inj f ==> g o inv f o f = g"
by (simp add: comp_assoc)

lemma inv_into_image_cancel[simp]:
  "inj_on f A ==> S <= A ==> inv_into A f ` f ` S = S"
by(fastforce simp: image_def)

lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
by (blast intro!: surjI inv_into_f_f)

lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
by (simp add: f_inv_into_f)

lemma inv_into_injective:
  assumes eq: "inv_into A f x = inv_into A f y"
      and x: "x: f`A"
      and y: "y: f`A"
  shows "x=y"
proof -
  have "f (inv_into A f x) = f (inv_into A f y)" using eq by simp
  thus ?thesis by (simp add: f_inv_into_f x y)
qed

lemma inj_on_inv_into: "B <= f`A ==> inj_on (inv_into A f) B"
by (blast intro: inj_onI dest: inv_into_injective injD)

lemma bij_betw_inv_into: "bij_betw f A B ==> bij_betw (inv_into A f) B A"
by (auto simp add: bij_betw_def inj_on_inv_into)

lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
by (simp add: inj_on_inv_into)

lemma surj_iff: "(surj f) = (f o inv f = id)"
by (auto intro!: surjI simp: surj_f_inv_f fun_eq_iff[where 'b='a])

lemma surj_iff_all: "surj f \<longleftrightarrow> (\<forall>x. f (inv f x) = x)"
  unfolding surj_iff by (simp add: o_def fun_eq_iff)

lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
apply (rule ext)
apply (drule_tac x = "inv f x" in spec)
apply (simp add: surj_f_inv_f)
done

lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)

lemma inv_equality: "[| !!x. g (f x) = x;  !!y. f (g y) = y |] ==> inv f = g"
apply (rule ext)
apply (auto simp add: inv_into_def)
done

lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
apply (rule inv_equality)
apply (auto simp add: bij_def surj_f_inv_f)
done

(** bij(inv f) implies little about f.  Consider f::bool=>bool such that
    f(True)=f(False)=True.  Then it's consistent with axiom someI that
    inv f could be any function at all, including the identity function.
    If inv f=id then inv f is a bijection, but inj f, surj(f) and
    inv(inv f)=f all fail.
**)

lemma inv_into_comp:
  "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
  inv_into A (f o g) x = (inv_into A g o inv_into (g ` A) f) x"
apply (rule inv_into_f_eq)
  apply (fast intro: comp_inj_on)
 apply (simp add: inv_into_into)
apply (simp add: f_inv_into_f inv_into_into)
done

lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
apply (rule inv_equality)
apply (auto simp add: bij_def surj_f_inv_f)
done

lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
  by (simp add: surj_f_inv_f image_comp comp_def)

lemma image_inv_f_f: "inj f ==> inv f ` (f ` A) = A"
  by simp

lemma inv_image_comp: "inj f ==> inv f ` (f ` X) = X"
  by (fact image_inv_f_f)

lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
apply auto
apply (force simp add: bij_is_inj)
apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
done

lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A"
apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
apply (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric])
done

lemma finite_fun_UNIVD1:
  assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
  and card: "card (UNIV :: 'b set) \<noteq> Suc 0"
  shows "finite (UNIV :: 'a set)"
proof -
  from fin have finb: "finite (UNIV :: 'b set)" by (rule finite_fun_UNIVD2)
  with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)"
    by (cases "card (UNIV :: 'b set)") (auto simp add: card_eq_0_iff)
  then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - Suc (Suc 0)" by auto
  then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by (auto simp add: card_Suc_eq)
  from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by (rule finite_imageI)
  moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)"
  proof (rule UNIV_eq_I)
    fix x :: 'a
    from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_into_def)
    thus "x \<in> range (\<lambda>f::'a \<Rightarrow> 'b. inv f b1)" by blast
  qed
  ultimately show "finite (UNIV :: 'a set)" by simp
qed

text \<open>
  Every infinite set contains a countable subset. More precisely we
  show that a set \<open>S\<close> is infinite if and only if there exists an
  injective function from the naturals into \<open>S\<close>.

  The ``only if'' direction is harder because it requires the
  construction of a sequence of pairwise different elements of an
  infinite set \<open>S\<close>. The idea is to construct a sequence of
  non-empty and infinite subsets of \<open>S\<close> obtained by successively
  removing elements of \<open>S\<close>.
\<close>

lemma infinite_countable_subset:
  assumes inf: "\<not> finite (S::'a set)"
  shows "\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S"
  \<comment> \<open>Courtesy of Stephan Merz\<close>
proof -
  define Sseq where "Sseq = rec_nat S (\<lambda>n T. T - {SOME e. e \<in> T})"
  define pick where "pick n = (SOME e. e \<in> Sseq n)" for n
  { fix n have "Sseq n \<subseteq> S" "\<not> finite (Sseq n)" by (induct n) (auto simp add: Sseq_def inf) }
  moreover then have *: "\<And>n. pick n \<in> Sseq n"
    unfolding pick_def by (subst (asm) finite.simps) (auto simp add: ex_in_conv intro: someI_ex)
  ultimately have "range pick \<subseteq> S" by auto
  moreover
  { fix n m
    have "pick n \<notin> Sseq (n + Suc m)" by (induct m) (auto simp add: Sseq_def pick_def)
    with * have "pick n \<noteq> pick (n + Suc m)" by auto
  }
  then have "inj pick" by (intro linorder_injI) (auto simp add: less_iff_Suc_add)
  ultimately show ?thesis by blast
qed

lemma infinite_iff_countable_subset: "\<not> finite S \<longleftrightarrow> (\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S)"
  \<comment> \<open>Courtesy of Stephan Merz\<close>
  using finite_imageD finite_subset infinite_UNIV_char_0 infinite_countable_subset by auto

lemma image_inv_into_cancel:
  assumes SURJ: "f`A=A'" and SUB: "B' \<le> A'"
  shows "f `((inv_into A f)`B') = B'"
  using assms
proof (auto simp add: f_inv_into_f)
  let ?f' = "(inv_into A f)"
  fix a' assume *: "a' \<in> B'"
  then have "a' \<in> A'" using SUB by auto
  then have "a' = f (?f' a')"
    using SURJ by (auto simp add: f_inv_into_f)
  then show "a' \<in> f ` (?f' ` B')" using * by blast
qed

lemma inv_into_inv_into_eq:
  assumes "bij_betw f A A'" "a \<in> A"
  shows "inv_into A' (inv_into A f) a = f a"
proof -
  let ?f' = "inv_into A f"   let ?f'' = "inv_into A' ?f'"
  have 1: "bij_betw ?f' A' A" using assms
  by (auto simp add: bij_betw_inv_into)
  obtain a' where 2: "a' \<in> A'" and 3: "?f' a' = a"
    using 1 \<open>a \<in> A\<close> unfolding bij_betw_def by force
  hence "?f'' a = a'"
    using \<open>a \<in> A\<close> 1 3 by (auto simp add: f_inv_into_f bij_betw_def)
  moreover have "f a = a'" using assms 2 3
    by (auto simp add: bij_betw_def)
  ultimately show "?f'' a = f a" by simp
qed

lemma inj_on_iff_surj:
  assumes "A \<noteq> {}"
  shows "(\<exists>f. inj_on f A \<and> f ` A \<le> A') \<longleftrightarrow> (\<exists>g. g ` A' = A)"
proof safe
  fix f assume INJ: "inj_on f A" and INCL: "f ` A \<le> A'"
  let ?phi = "\<lambda>a' a. a \<in> A \<and> f a = a'"  let ?csi = "\<lambda>a. a \<in> A"
  let ?g = "\<lambda>a'. if a' \<in> f ` A then (SOME a. ?phi a' a) else (SOME a. ?csi a)"
  have "?g ` A' = A"
  proof
    show "?g ` A' \<le> A"
    proof clarify
      fix a' assume *: "a' \<in> A'"
      show "?g a' \<in> A"
      proof cases
        assume Case1: "a' \<in> f ` A"
        then obtain a where "?phi a' a" by blast
        hence "?phi a' (SOME a. ?phi a' a)" using someI[of "?phi a'" a] by blast
        with Case1 show ?thesis by auto
      next
        assume Case2: "a' \<notin> f ` A"
        hence "?csi (SOME a. ?csi a)" using assms someI_ex[of ?csi] by blast
        with Case2 show ?thesis by auto
      qed
    qed
  next
    show "A \<le> ?g ` A'"
    proof-
      {fix a assume *: "a \<in> A"
       let ?b = "SOME aa. ?phi (f a) aa"
       have "?phi (f a) a" using * by auto
       hence 1: "?phi (f a) ?b" using someI[of "?phi(f a)" a] by blast
       hence "?g(f a) = ?b" using * by auto
       moreover have "a = ?b" using 1 INJ * by (auto simp add: inj_on_def)
       ultimately have "?g(f a) = a" by simp
       with INCL * have "?g(f a) = a \<and> f a \<in> A'" by auto
      }
      thus ?thesis by force
    qed
  qed
  thus "\<exists>g. g ` A' = A" by blast
next
  fix g  let ?f = "inv_into A' g"
  have "inj_on ?f (g ` A')"
    by (auto simp add: inj_on_inv_into)
  moreover
  {fix a' assume *: "a' \<in> A'"
   let ?phi = "\<lambda> b'. b' \<in> A' \<and> g b' = g a'"
   have "?phi a'" using * by auto
   hence "?phi(SOME b'. ?phi b')" using someI[of ?phi] by blast
   hence "?f(g a') \<in> A'" unfolding inv_into_def by auto
  }
  ultimately show "\<exists>f. inj_on f (g ` A') \<and> f ` g ` A' \<subseteq> A'" by auto
qed

lemma Ex_inj_on_UNION_Sigma:
  "\<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))"
proof
  let ?phi = "\<lambda> a i. i \<in> I \<and> a \<in> A i"
  let ?sm = "\<lambda> a. SOME i. ?phi a i"
  let ?f = "\<lambda>a. (?sm a, a)"
  have "inj_on ?f (\<Union>i \<in> I. A i)" unfolding inj_on_def by auto
  moreover
  { { fix i a assume "i \<in> I" and "a \<in> A i"
      hence "?sm a \<in> I \<and> a \<in> A(?sm a)" using someI[of "?phi a" i] by auto
    }
    hence "?f ` (\<Union>i \<in> I. A i) \<le> (SIGMA i : I. A i)" by auto
  }
  ultimately
  show "inj_on ?f (\<Union>i \<in> I. A i) \<and> ?f ` (\<Union>i \<in> I. A i) \<le> (SIGMA i : I. A i)"
  by auto
qed

lemma inv_unique_comp:
  assumes fg: "f \<circ> g = id"
    and gf: "g \<circ> f = id"
  shows "inv f = g"
  using fg gf inv_equality[of g f] by (auto simp add: fun_eq_iff)


subsection \<open>The Cantor-Bernstein Theorem\<close>

lemma Cantor_Bernstein_aux:
  shows "\<exists>A' h. A' \<le> A \<and>
                (\<forall>a \<in> A'. a \<notin> g`(B - f ` A')) \<and>
                (\<forall>a \<in> A'. h a = f a) \<and>
                (\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a))"
proof-
  obtain H where H_def: "H = (\<lambda> A'. A - (g`(B - (f ` A'))))" by blast
  have 0: "mono H" unfolding mono_def H_def by blast
  then obtain A' where 1: "H A' = A'" using lfp_unfold by blast
  hence 2: "A' = A - (g`(B - (f ` A')))" unfolding H_def by simp
  hence 3: "A' \<le> A" by blast
  have 4: "\<forall>a \<in> A'.  a \<notin> g`(B - f ` A')"
  using 2 by blast
  have 5: "\<forall>a \<in> A - A'. \<exists>b \<in> B - (f ` A'). a = g b"
  using 2 by blast
  (*  *)
  obtain h where h_def:
  "h = (\<lambda> a. if a \<in> A' then f a else (SOME b. b \<in> B - (f ` A') \<and> a = g b))" by blast
  hence "\<forall>a \<in> A'. h a = f a" by auto
  moreover
  have "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
  proof
    fix a assume *: "a \<in> A - A'"
    let ?phi = "\<lambda> b. b \<in> B - (f ` A') \<and> a = g b"
    have "h a = (SOME b. ?phi b)" using h_def * by auto
    moreover have "\<exists>b. ?phi b" using 5 *  by auto
    ultimately show  "?phi (h a)" using someI_ex[of ?phi] by auto
  qed
  ultimately show ?thesis using 3 4 by blast
qed

theorem Cantor_Bernstein:
  assumes INJ1: "inj_on f A" and SUB1: "f ` A \<le> B" and
          INJ2: "inj_on g B" and SUB2: "g ` B \<le> A"
  shows "\<exists>h. bij_betw h A B"
proof-
  obtain A' and h where 0: "A' \<le> A" and
  1: "\<forall>a \<in> A'. a \<notin> g`(B - f ` A')" and
  2: "\<forall>a \<in> A'. h a = f a" and
  3: "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
  using Cantor_Bernstein_aux[of A g B f] by blast
  have "inj_on h A"
  proof (intro inj_onI)
    fix a1 a2
    assume 4: "a1 \<in> A" and 5: "a2 \<in> A" and 6: "h a1 = h a2"
    show "a1 = a2"
    proof(cases "a1 \<in> A'")
      assume Case1: "a1 \<in> A'"
      show ?thesis
      proof(cases "a2 \<in> A'")
        assume Case11: "a2 \<in> A'"
        hence "f a1 = f a2" using Case1 2 6 by auto
        thus ?thesis using INJ1 Case1 Case11 0
        unfolding inj_on_def by blast
      next
        assume Case12: "a2 \<notin> A'"
        hence False using 3 5 2 6 Case1 by force
        thus ?thesis by simp
      qed
    next
    assume Case2: "a1 \<notin> A'"
      show ?thesis
      proof(cases "a2 \<in> A'")
        assume Case21: "a2 \<in> A'"
        hence False using 3 4 2 6 Case2 by auto
        thus ?thesis by simp
      next
        assume Case22: "a2 \<notin> A'"
        hence "a1 = g(h a1) \<and> a2 = g(h a2)" using Case2 4 5 3 by auto
        thus ?thesis using 6 by simp
      qed
    qed
  qed
  (*  *)
  moreover
  have "h ` A = B"
  proof safe
    fix a assume "a \<in> A"
    thus "h a \<in> B" using SUB1 2 3 by (cases "a \<in> A'") auto
  next
    fix b assume *: "b \<in> B"
    show "b \<in> h ` A"
    proof(cases "b \<in> f ` A'")
      assume Case1: "b \<in> f ` A'"
      then obtain a where "a \<in> A' \<and> b = f a" by blast
      thus ?thesis using 2 0 by force
    next
      assume Case2: "b \<notin> f ` A'"
      hence "g b \<notin> A'" using 1 * by auto
      hence 4: "g b \<in> A - A'" using * SUB2 by auto
      hence "h(g b) \<in> B \<and> g(h(g b)) = g b"
      using 3 by auto
      hence "h(g b) = b" using * INJ2 unfolding inj_on_def by auto
      thus ?thesis using 4 by force
    qed
  qed
  (*  *)
  ultimately show ?thesis unfolding bij_betw_def by auto
qed

subsection \<open>Other Consequences of Hilbert's Epsilon\<close>

text \<open>Hilbert's Epsilon and the @{term split} Operator\<close>

text\<open>Looping simprule\<close>
lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
  by simp

lemma Eps_case_prod: "Eps (case_prod P) = (SOME xy. P (fst xy) (snd xy))"
  by (simp add: split_def)

lemma Eps_case_prod_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
  by blast


text\<open>A relation is wellfounded iff it has no infinite descending chain\<close>
lemma wf_iff_no_infinite_down_chain:
  "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
apply (simp only: wf_eq_minimal)
apply (rule iffI)
 apply (rule notI)
 apply (erule exE)
 apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
apply (erule contrapos_np, simp, clarify)
apply (subgoal_tac "\<forall>n. rec_nat x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
 apply (rule_tac x = "rec_nat x (%i y. @z. z:Q & (z,y) :r)" in exI)
 apply (rule allI, simp)
 apply (rule someI2_ex, blast, blast)
apply (rule allI)
apply (induct_tac "n", simp_all)
apply (rule someI2_ex, blast+)
done

lemma wf_no_infinite_down_chainE:
  assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r"
using \<open>wf r\<close> wf_iff_no_infinite_down_chain[of r] by blast


text\<open>A dynamically-scoped fact for TFL\<close>
lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
  by (blast intro: someI)


subsection \<open>Least value operator\<close>

definition
  LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where
  "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"

syntax
  "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
translations
  "LEAST x WRT m. P" == "CONST LeastM m (%x. P)"

lemma LeastMI2:
  "P x ==> (!!y. P y ==> m x <= m y)
    ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
    ==> Q (LeastM m P)"
  apply (simp add: LeastM_def)
  apply (rule someI2_ex, blast, blast)
  done

lemma LeastM_equality:
  "P k ==> (!!x. P x ==> m k <= m x)
    ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
  apply (rule LeastMI2, assumption, blast)
  apply (blast intro!: order_antisym)
  done

lemma wf_linord_ex_has_least:
  "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
    ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
  apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
  apply (drule_tac x = "m`Collect P" in spec, force)
  done

lemma ex_has_least_nat:
    "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
  apply (simp only: pred_nat_trancl_eq_le [symmetric])
  apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
   apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
  done

lemma LeastM_nat_lemma:
    "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
  apply (simp add: LeastM_def)
  apply (rule someI_ex)
  apply (erule ex_has_least_nat)
  done

lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1]

lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)


subsection \<open>Greatest value operator\<close>

definition
  GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where
  "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"

definition
  Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where
  "Greatest == GreatestM (%x. x)"

syntax
  "_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"
      ("GREATEST _ WRT _. _" [0, 4, 10] 10)
translations
  "GREATEST x WRT m. P" == "CONST GreatestM m (%x. P)"

lemma GreatestMI2:
  "P x ==> (!!y. P y ==> m y <= m x)
    ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
    ==> Q (GreatestM m P)"
  apply (simp add: GreatestM_def)
  apply (rule someI2_ex, blast, blast)
  done

lemma GreatestM_equality:
 "P k ==> (!!x. P x ==> m x <= m k)
    ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
  apply (rule_tac m = m in GreatestMI2, assumption, blast)
  apply (blast intro!: order_antisym)
  done

lemma Greatest_equality:
  "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
  apply (simp add: Greatest_def)
  apply (erule GreatestM_equality, blast)
  done

lemma ex_has_greatest_nat_lemma:
  "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
    ==> \<exists>y. P y & ~ (m y < m k + n)"
  apply (induct n, force)
  apply (force simp add: le_Suc_eq)
  done

lemma ex_has_greatest_nat:
  "P k ==> \<forall>y. P y --> m y < b
    ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
  apply (rule ccontr)
  apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
    apply (subgoal_tac [3] "m k <= b", auto)
  done

lemma GreatestM_nat_lemma:
  "P k ==> \<forall>y. P y --> m y < b
    ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
  apply (simp add: GreatestM_def)
  apply (rule someI_ex)
  apply (erule ex_has_greatest_nat, assumption)
  done

lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1]

lemma GreatestM_nat_le:
  "P x ==> \<forall>y. P y --> m y < b
    ==> (m x::nat) <= m (GreatestM m P)"
  apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
  done


text \<open>\medskip Specialization to \<open>GREATEST\<close>.\<close>

lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
  apply (simp add: Greatest_def)
  apply (rule GreatestM_natI, auto)
  done

lemma Greatest_le:
    "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
  apply (simp add: Greatest_def)
  apply (rule GreatestM_nat_le, auto)
  done


subsection \<open>An aside: bounded accessible part\<close>

text \<open>Finite monotone eventually stable sequences\<close>

lemma finite_mono_remains_stable_implies_strict_prefix:
  fixes f :: "nat \<Rightarrow> 'a::order"
  assumes S: "finite (range f)" "mono f" and eq: "\<forall>n. f n = f (Suc n) \<longrightarrow> f (Suc n) = f (Suc (Suc n))"
  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)"
  using assms
proof -
  have "\<exists>n. f n = f (Suc n)"
  proof (rule ccontr)
    assume "\<not> ?thesis"
    then have "\<And>n. f n \<noteq> f (Suc n)" by auto
    then have "\<And>n. f n < f (Suc n)"
      using  \<open>mono f\<close> by (auto simp: le_less mono_iff_le_Suc)
    with lift_Suc_mono_less_iff[of f]
    have *: "\<And>n m. n < m \<Longrightarrow> f n < f m" by auto
    have "inj f"
    proof (intro injI)
      fix x y
      assume "f x = f y"
      then show "x = y" by (cases x y rule: linorder_cases) (auto dest: *)
    qed
    with \<open>finite (range f)\<close> have "finite (UNIV::nat set)"
      by (rule finite_imageD)
    then show False by simp
  qed
  then obtain n where n: "f n = f (Suc n)" ..
  define N where "N = (LEAST n. f n = f (Suc n))"
  have N: "f N = f (Suc N)"
    unfolding N_def using n by (rule LeastI)
  show ?thesis
  proof (intro exI[of _ N] conjI allI impI)
    fix n assume "N \<le> n"
    then have "\<And>m. N \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m = f N"
    proof (induct rule: dec_induct)
      case (step n) then show ?case
        using eq[rule_format, of "n - 1"] N
        by (cases n) (auto simp add: le_Suc_eq)
    qed simp
    from this[of n] \<open>N \<le> n\<close> show "f N = f n" by auto
  next
    fix n m :: nat assume "m < n" "n \<le> N"
    then show "f m < f n"
    proof (induct rule: less_Suc_induct)
      case (1 i)
      then have "i < N" by simp
      then have "f i \<noteq> f (Suc i)"
        unfolding N_def by (rule not_less_Least)
      with \<open>mono f\<close> show ?case by (simp add: mono_iff_le_Suc less_le)
    qed auto
  qed
qed

lemma finite_mono_strict_prefix_implies_finite_fixpoint:
  fixes f :: "nat \<Rightarrow> 'a set"
  assumes S: "\<And>i. f i \<subseteq> S" "finite S"
    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)"
  shows "f (card S) = (\<Union>n. f n)"
proof -
  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

  { fix i have "i \<le> N \<Longrightarrow> i \<le> card (f i)"
    proof (induct i)
      case 0 then show ?case by simp
    next
      case (Suc i)
      with inj[rule_format, of "Suc i" i]
      have "(f i) \<subset> (f (Suc i))" by auto
      moreover have "finite (f (Suc i))" using S by (rule finite_subset)
      ultimately have "card (f i) < card (f (Suc i))" by (intro psubset_card_mono)
      with Suc show ?case using inj by auto
    qed
  }
  then have "N \<le> card (f N)" by simp
  also have "\<dots> \<le> card S" using S by (intro card_mono)
  finally have "f (card S) = f N" using eq by auto
  then show ?thesis using eq inj[rule_format, of N]
    apply auto
    apply (case_tac "n < N")
    apply (auto simp: not_less)
    done
qed


subsection \<open>More on injections, bijections, and inverses\<close>

locale bijection =
  fixes f :: "'a \<Rightarrow> 'a"
  assumes bij: "bij f"
begin

lemma bij_inv:
  "bij (inv f)"
  using bij by (rule bij_imp_bij_inv)

lemma surj [simp]:
  "surj f"
  using bij by (rule bij_is_surj)

lemma inj:
  "inj f"
  using bij by (rule bij_is_inj)

lemma surj_inv [simp]:
  "surj (inv f)"
  using inj by (rule inj_imp_surj_inv)

lemma inj_inv:
  "inj (inv f)"
  using surj by (rule surj_imp_inj_inv)

lemma eqI:
  "f a = f b \<Longrightarrow> a = b"
  using inj by (rule injD)

lemma eq_iff [simp]:
  "f a = f b \<longleftrightarrow> a = b"
  by (auto intro: eqI)

lemma eq_invI:
  "inv f a = inv f b \<Longrightarrow> a = b"
  using inj_inv by (rule injD)

lemma eq_inv_iff [simp]:
  "inv f a = inv f b \<longleftrightarrow> a = b"
  by (auto intro: eq_invI)

lemma inv_left [simp]:
  "inv f (f a) = a"
  using inj by (simp add: inv_f_eq)

lemma inv_comp_left [simp]:
  "inv f \<circ> f = id"
  by (simp add: fun_eq_iff)

lemma inv_right [simp]:
  "f (inv f a) = a"
  using surj by (simp add: surj_f_inv_f)

lemma inv_comp_right [simp]:
  "f \<circ> inv f = id"
  by (simp add: fun_eq_iff)

lemma inv_left_eq_iff [simp]:
  "inv f a = b \<longleftrightarrow> f b = a"
  by auto

lemma inv_right_eq_iff [simp]:
  "b = inv f a \<longleftrightarrow> f b = a"
  by auto

end

lemma infinite_imp_bij_betw:
assumes INF: "\<not> finite A"
shows "\<exists>h. bij_betw h A (A - {a})"
proof(cases "a \<in> A")
  assume Case1: "a \<notin> A"  hence "A - {a} = A" by blast
  thus ?thesis using bij_betw_id[of A] by auto
next
  assume Case2: "a \<in> A"
  have "\<not> finite (A - {a})" using INF by auto
  with infinite_iff_countable_subset[of "A - {a}"] obtain f::"nat \<Rightarrow> 'a"
  where 1: "inj f" and 2: "f ` UNIV \<le> A - {a}" by blast
  obtain g where g_def: "g = (\<lambda> n. if n = 0 then a else f (Suc n))" by blast
  obtain A' where A'_def: "A' = g ` UNIV" by blast
  have temp: "\<forall>y. f y \<noteq> a" using 2 by blast
  have 3: "inj_on g UNIV \<and> g ` UNIV \<le> A \<and> a \<in> g ` UNIV"
  proof(auto simp add: Case2 g_def, unfold inj_on_def, intro ballI impI,
        case_tac "x = 0", auto simp add: 2)
    fix y  assume "a = (if y = 0 then a else f (Suc y))"
    thus "y = 0" using temp by (case_tac "y = 0", auto)
  next
    fix x y
    assume "f (Suc x) = (if y = 0 then a else f (Suc y))"
    thus "x = y" using 1 temp unfolding inj_on_def by (case_tac "y = 0", auto)
  next
    fix n show "f (Suc n) \<in> A" using 2 by blast
  qed
  hence 4: "bij_betw g UNIV A' \<and> a \<in> A' \<and> A' \<le> A"
  using inj_on_imp_bij_betw[of g] unfolding A'_def by auto
  hence 5: "bij_betw (inv g) A' UNIV"
  by (auto simp add: bij_betw_inv_into)
  (*  *)
  obtain n where "g n = a" using 3 by auto
  hence 6: "bij_betw g (UNIV - {n}) (A' - {a})"
  using 3 4 unfolding A'_def
  by clarify (rule bij_betw_subset, auto simp: image_set_diff)
  (*  *)
  obtain v where v_def: "v = (\<lambda> m. if m < n then m else Suc m)" by blast
  have 7: "bij_betw v UNIV (UNIV - {n})"
  proof(unfold bij_betw_def inj_on_def, intro conjI, clarify)
    fix m1 m2 assume "v m1 = v m2"
    thus "m1 = m2"
    by(case_tac "m1 < n", case_tac "m2 < n",
       auto simp add: inj_on_def v_def, case_tac "m2 < n", auto)
  next
    show "v ` UNIV = UNIV - {n}"
    proof(auto simp add: v_def)
      fix m assume *: "m \<noteq> n" and **: "m \<notin> Suc ` {m'. \<not> m' < n}"
      {assume "n \<le> m" with * have 71: "Suc n \<le> m" by auto
       then obtain m' where 72: "m = Suc m'" using Suc_le_D by auto
       with 71 have "n \<le> m'" by auto
       with 72 ** have False by auto
      }
      thus "m < n" by force
    qed
  qed
  (*  *)
  obtain h' where h'_def: "h' = g o v o (inv g)" by blast
  hence 8: "bij_betw h' A' (A' - {a})" using 5 7 6
  by (auto simp add: bij_betw_trans)
  (*  *)
  obtain h where h_def: "h = (\<lambda> b. if b \<in> A' then h' b else b)" by blast
  have "\<forall>b \<in> A'. h b = h' b" unfolding h_def by auto
  hence "bij_betw h  A' (A' - {a})" using 8 bij_betw_cong[of A' h] by auto
  moreover
  {have "\<forall>b \<in> A - A'. h b = b" unfolding h_def by auto
   hence "bij_betw h  (A - A') (A - A')"
   using bij_betw_cong[of "A - A'" h id] bij_betw_id[of "A - A'"] by auto
  }
  moreover
  have "(A' Int (A - A') = {} \<and> A' \<union> (A - A') = A) \<and>
        ((A' - {a}) Int (A - A') = {} \<and> (A' - {a}) \<union> (A - A') = A - {a})"
  using 4 by blast
  ultimately have "bij_betw h A (A - {a})"
  using bij_betw_combine[of h A' "A' - {a}" "A - A'" "A - A'"] by simp
  thus ?thesis by blast
qed

lemma infinite_imp_bij_betw2:
assumes INF: "\<not> finite A"
shows "\<exists>h. bij_betw h A (A \<union> {a})"
proof(cases "a \<in> A")
  assume Case1: "a \<in> A"  hence "A \<union> {a} = A" by blast
  thus ?thesis using bij_betw_id[of A] by auto
next
  let ?A' = "A \<union> {a}"
  assume Case2: "a \<notin> A" hence "A = ?A' - {a}" by blast
  moreover have "\<not> finite ?A'" using INF by auto
  ultimately obtain f where "bij_betw f ?A' A"
  using infinite_imp_bij_betw[of ?A' a] by auto
  hence "bij_betw(inv_into ?A' f) A ?A'" using bij_betw_inv_into by blast
  thus ?thesis by auto
qed

lemma bij_betw_inv_into_left:
assumes BIJ: "bij_betw f A A'" and IN: "a \<in> A"
shows "(inv_into A f) (f a) = a"
using assms unfolding bij_betw_def
by clarify (rule inv_into_f_f)

lemma bij_betw_inv_into_right:
assumes "bij_betw f A A'" "a' \<in> A'"
shows "f(inv_into A f a') = a'"
using assms unfolding bij_betw_def using f_inv_into_f by force

lemma bij_betw_inv_into_subset:
assumes BIJ: "bij_betw f A A'" and
        SUB: "B \<le> A" and IM: "f ` B = B'"
shows "bij_betw (inv_into A f) B' B"
using assms unfolding bij_betw_def
by (auto intro: inj_on_inv_into)


subsection \<open>Specification package -- Hilbertized version\<close>

lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
  by (simp only: someI_ex)

ML_file "Tools/choice_specification.ML"

end