(*  Title:      HOL/Hilbert_Choice.thy

     Author:     Lawrence C Paulson, Tobias Nipkow

     Copyright   2001  University of Cambridge

 *)

 header {* Hilbert's Epsilon-Operator and the Axiom of Choice *}

 theory Hilbert_Choice

 imports Nat Wellfounded Big_Operators

 keywords "specification" "ax_specification" :: thy_goal

 begin

 subsection {* Hilbert's epsilon *}

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

   someI: "P x ==> P (Eps P)"

 syntax (epsilon)

   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3\<some>_./ _)" [0, 10] 10)

 syntax (HOL)

   "_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 {*

   [(@{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)]

 *} -- {* to avoid eta-contraction of body *}

 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 {*Hilbert's Epsilon-operator*}

 text{*Easier to apply than @{text someI} if the witness comes from an

 existential formula*}

 lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)"

 apply (erule exE)

 apply (erule someI)

 done

 text{*Easier to apply than @{text someI} because the conclusion has only one

 occurrence of @{term P}.*}

 lemma someI2: "[| P a;  !!x. P x ==> Q x |] ==> Q (SOME x. P x)"

 by (blast intro: someI)

 text{*Easier to apply than @{text someI2} if the witness comes from an

 existential formula*}

 lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. 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_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{*Axiom of Choice, Proved Using the Description Operator*}

 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)

 subsection {*Function Inverse*}

 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_id [simp]: "inv id = id"

 by (simp add: inv_into_def 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{*But is it useful?*}

 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: image_eq_UN surj_f_inv_f)

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

 by (simp add: image_eq_UN)

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

 by (auto simp add: image_def)

 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\<Colon>'a \<Rightarrow> 'b. inv f b1)" by blast

   qed

   ultimately show "finite (UNIV :: 'a set)" by simp

 qed

 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 `a \<in> A` unfolding bij_betw_def by force

   hence "?f'' a = a'"

     using `a \<in> A` 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

 subsection {* The Cantor-Bernstein Theorem *}

 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 {*Other Consequences of Hilbert's Epsilon*}

 text {*Hilbert's Epsilon and the @{term split} Operator*}

 text{*Looping simprule*}

 lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"

   by simp

 lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"

   by (simp add: split_def)

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

   by blast

 text{*A relation is wellfounded iff it has no infinite descending chain*}

 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. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")

  apply (rule_tac x = "nat_rec 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 `wf r` wf_iff_no_infinite_down_chain[of r] by blast

 text{*A dynamically-scoped fact for TFL *}

 lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"

   by (blast intro: someI)

 subsection {* Least value operator *}

 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 {* Greatest value operator *}

 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 {* \medskip Specialization to @{text GREATEST}. *}

 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 {* An aside: bounded accessible part *}

 text {* Finite monotone eventually stable sequences *}

 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  `mono f` 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

     then have "inj f"

       by (auto simp: inj_on_def) (metis linorder_less_linear order_less_imp_not_eq)

     with `finite (range f)` 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)" ..

   def N \<equiv> "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] `N \<le> n` 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[consumes 1])

       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 `mono f` 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

 primrec bacc :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> 'a set" 

 where

   "bacc r 0 = {x. \<forall> y. (y, x) \<notin> r}"

 | "bacc r (Suc n) = (bacc r n \<union> {x. \<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r n})"

 lemma bacc_subseteq_acc:

   "bacc r n \<subseteq> Wellfounded.acc r"

   by (induct n) (auto intro: acc.intros)

 lemma bacc_mono:

   "n \<le> m \<Longrightarrow> bacc r n \<subseteq> bacc r m"

   by (induct rule: dec_induct) auto

 lemma bacc_upper_bound:

   "bacc (r :: ('a \<times> 'a) set)  (card (UNIV :: 'a::finite set)) = (\<Union>n. bacc r n)"

 proof -

   have "mono (bacc r)" unfolding mono_def by (simp add: bacc_mono)

   moreover have "\<forall>n. bacc r n = bacc r (Suc n) \<longrightarrow> bacc r (Suc n) = bacc r (Suc (Suc n))" by auto

   moreover have "finite (range (bacc r))" by auto

   ultimately show ?thesis

    by (intro finite_mono_strict_prefix_implies_finite_fixpoint)

      (auto intro: finite_mono_remains_stable_implies_strict_prefix)

 qed

 lemma acc_subseteq_bacc:

   assumes "finite r"

   shows "Wellfounded.acc r \<subseteq> (\<Union>n. bacc r n)"

 proof

   fix x

   assume "x : Wellfounded.acc r"

   then have "\<exists> n. x : bacc r n"

   proof (induct x arbitrary: rule: acc.induct)

     case (accI x)

     then have "\<forall>y. \<exists> n. (y, x) \<in> r --> y : bacc r n" by simp

     from choice[OF this] obtain n where n: "\<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r (n y)" ..

     obtain n where "\<And>y. (y, x) : r \<Longrightarrow> y : bacc r n"

     proof

       fix y assume y: "(y, x) : r"

       with n have "y : bacc r (n y)" by auto

       moreover have "n y <= Max ((%(y, x). n y) ` r)"

         using y `finite r` by (auto intro!: Max_ge)

       note bacc_mono[OF this, of r]

       ultimately show "y : bacc r (Max ((%(y, x). n y) ` r))" by auto

     qed

     then show ?case

       by (auto simp add: Let_def intro!: exI[of _ "Suc n"])

   qed

   then show "x : (UN n. bacc r n)" by auto

 qed

 lemma acc_bacc_eq:

   fixes A :: "('a :: finite \<times> 'a) set"

   assumes "finite A"

   shows "Wellfounded.acc A = bacc A (card (UNIV :: 'a set))"

   using assms by (metis acc_subseteq_bacc bacc_subseteq_acc bacc_upper_bound order_eq_iff)

 subsection {* Specification package -- Hilbertized version *}

 lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"

   by (simp only: someI_ex)

 ML_file "Tools/choice_specification.ML"

 end