# Theory Hilbert_Choice

Up to index of Isabelle/HOL-Proofs

theory Hilbert_Choice
imports Big_Operators
`(*  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_Choiceimports Nat Wellfounded Big_Operatorskeywords "specification" "ax_specification" :: thy_goalbeginsubsection {* 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 [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 anexistential formula*}lemma someI_ex [elim?]: "∃x. P x ==> P (SOME x. P x)"apply (erule exE)apply (erule someI)donetext{*Easier to apply than @{text someI} because the conclusion has only oneoccurrence 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 anexistential formula*}lemma someI2_ex: "[| ∃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 blastlemma some_eq_ex: "P (SOME x. P x) =  (∃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)donelemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"apply (rule some_equality)apply (rule refl)apply (erule sym)donesubsection{*Axiom of Choice, Proved Using the Description Operator*}lemma choice: "∀x. ∃y. Q x y ==> ∃f. ∀x. Q x (f x)"by (fast elim: someI)lemma bchoice: "∀x∈S. ∃y. Q x y ==> ∃f. ∀x∈S. Q x (f x)"by (fast elim: someI)lemma choice_iff: "(∀x. ∃y. Q x y) <-> (∃f. ∀x. Q x (f x))"by (fast elim: someI)lemma choice_iff': "(∀x. P x --> (∃y. Q x y)) <-> (∃f. ∀x. P x --> Q x (f x))"by (fast elim: someI)lemma bchoice_iff: "(∀x∈S. ∃y. Q x y) <-> (∃f. ∀x∈S. Q x (f x))"by (fast elim: someI)lemma bchoice_iff': "(∀x∈S. P x --> (∃y. Q x y)) <-> (∃f. ∀x∈S. P x --> 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)donelemma 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)donelemma inv_f_f: "inj f ==> inv f (f x) = x"by simplemma f_inv_into_f: "y : f`A  ==> f (inv_into A f y) = y"apply (simp add: inv_into_def)apply (fast intro: someI2)donelemma 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)donelemma 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 ∈ range(f) ==> P(inv f y)"  shows "P x"proof -  have "f x ∈ 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])qedlemma 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)donelemma 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)qedlemma 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 <-> (∀x. f (inv f x) = x)"  unfolding surj_iff by (simp add: o_def fun_eq_iff)lemma surj_imp_inv_eq: "[| surj f; ∀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)donelemma 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)donelemma 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)donelemma 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)donelemma 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 autoapply (force simp add: bij_is_inj)apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])donelemma 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])donelemma finite_fun_UNIVD1:  assumes fin: "finite (UNIV :: ('a => 'b) set)"  and card: "card (UNIV :: 'b set) ≠ 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) ≥ 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) ≠ (b2 :: 'b)" by (auto simp add: card_Suc_eq)  from fin have "finite (range (λf :: 'a => 'b. inv f b1))" by (rule finite_imageI)  moreover have "UNIV = range (λf :: 'a => 'b. inv f b1)"  proof (rule UNIV_eq_I)    fix x :: 'a    from b1b2 have "x = inv (λy. if y = x then b1 else b2) b1" by (simp add: inv_into_def)    thus "x ∈ range (λf::'a => 'b. inv f b1)" by blast  qed  ultimately show "finite (UNIV :: 'a set)" by simpqedlemma image_inv_into_cancel:  assumes SURJ: "f`A=A'" and SUB: "B' ≤ A'"  shows "f `((inv_into A f)`B') = B'"  using assmsproof (auto simp add: f_inv_into_f)  let ?f' = "(inv_into A f)"  fix a' assume *: "a' ∈ B'"  then have "a' ∈ A'" using SUB by auto  then have "a' = f (?f' a')"    using SURJ by (auto simp add: f_inv_into_f)  then show "a' ∈ f ` (?f' ` B')" using * by blastqedlemma inv_into_inv_into_eq:  assumes "bij_betw f A A'" "a ∈ 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' ∈ A'" and 3: "?f' a' = a"    using 1 `a ∈ A` unfolding bij_betw_def by force  hence "?f'' a = a'"    using `a ∈ 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 simpqedlemma inj_on_iff_surj:  assumes "A ≠ {}"  shows "(∃f. inj_on f A ∧ f ` A ≤ A') <-> (∃g. g ` A' = A)"proof safe  fix f assume INJ: "inj_on f A" and INCL: "f ` A ≤ A'"  let ?phi = "λa' a. a ∈ A ∧ f a = a'"  let ?csi = "λa. a ∈ A"  let ?g = "λa'. if a' ∈ f ` A then (SOME a. ?phi a' a) else (SOME a. ?csi a)"  have "?g ` A' = A"  proof    show "?g ` A' ≤ A"    proof clarify      fix a' assume *: "a' ∈ A'"      show "?g a' ∈ A"      proof cases        assume Case1: "a' ∈ 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' ∉ 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 ≤ ?g ` A'"    proof-      {fix a assume *: "a ∈ 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 ∧ f a ∈ A'" by auto      }      thus ?thesis by force    qed  qed  thus "∃g. g ` A' = A" by blastnext  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' ∈ A'"   let ?phi = "λ b'. b' ∈ A' ∧ 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') ∈ A'" unfolding inv_into_def by auto  }  ultimately show "∃f. inj_on f (g ` A') ∧ f ` g ` A' ⊆ A'" by autoqedlemma Ex_inj_on_UNION_Sigma:  "∃f. (inj_on f (\<Union> i ∈ I. A i) ∧ f ` (\<Union> i ∈ I. A i) ≤ (SIGMA i : I. A i))"proof  let ?phi = "λ a i. i ∈ I ∧ a ∈ A i"  let ?sm = "λ a. SOME i. ?phi a i"  let ?f = "λa. (?sm a, a)"  have "inj_on ?f (\<Union> i ∈ I. A i)" unfolding inj_on_def by auto  moreover  { { fix i a assume "i ∈ I" and "a ∈ A i"      hence "?sm a ∈ I ∧ a ∈ A(?sm a)" using someI[of "?phi a" i] by auto    }    hence "?f ` (\<Union> i ∈ I. A i) ≤ (SIGMA i : I. A i)" by auto  }  ultimately  show "inj_on ?f (\<Union> i ∈ I. A i) ∧ ?f ` (\<Union> i ∈ I. A i) ≤ (SIGMA i : I. A i)"  by autoqedsubsection {* The Cantor-Bernstein Theorem *}lemma Cantor_Bernstein_aux:  shows "∃A' h. A' ≤ A ∧                (∀a ∈ A'. a ∉ g`(B - f ` A')) ∧                (∀a ∈ A'. h a = f a) ∧                (∀a ∈ A - A'. h a ∈ B - (f ` A') ∧ a = g(h a))"proof-  obtain H where H_def: "H = (λ 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' ≤ A" by blast  have 4: "∀a ∈ A'.  a ∉ g`(B - f ` A')"  using 2 by blast  have 5: "∀a ∈ A - A'. ∃b ∈ B - (f ` A'). a = g b"  using 2 by blast  (*  *)  obtain h where h_def:  "h = (λ a. if a ∈ A' then f a else (SOME b. b ∈ B - (f ` A') ∧ a = g b))" by blast  hence "∀a ∈ A'. h a = f a" by auto  moreover  have "∀a ∈ A - A'. h a ∈ B - (f ` A') ∧ a = g(h a)"  proof    fix a assume *: "a ∈ A - A'"    let ?phi = "λ b. b ∈ B - (f ` A') ∧ a = g b"    have "h a = (SOME b. ?phi b)" using h_def * by auto    moreover have "∃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 blastqedtheorem Cantor_Bernstein:  assumes INJ1: "inj_on f A" and SUB1: "f ` A ≤ B" and          INJ2: "inj_on g B" and SUB2: "g ` B ≤ A"  shows "∃h. bij_betw h A B"proof-  obtain A' and h where 0: "A' ≤ A" and  1: "∀a ∈ A'. a ∉ g`(B - f ` A')" and  2: "∀a ∈ A'. h a = f a" and  3: "∀a ∈ A - A'. h a ∈ B - (f ` A') ∧ 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 ∈ A" and 5: "a2 ∈ A" and 6: "h a1 = h a2"    show "a1 = a2"    proof(cases "a1 ∈ A'")      assume Case1: "a1 ∈ A'"      show ?thesis      proof(cases "a2 ∈ A'")        assume Case11: "a2 ∈ 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 ∉ A'"        hence False using 3 5 2 6 Case1 by force        thus ?thesis by simp      qed    next    assume Case2: "a1 ∉ A'"      show ?thesis      proof(cases "a2 ∈ A'")        assume Case21: "a2 ∈ A'"        hence False using 3 4 2 6 Case2 by auto        thus ?thesis by simp      next        assume Case22: "a2 ∉ A'"        hence "a1 = g(h a1) ∧ 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 ∈ A"    thus "h a ∈ B" using SUB1 2 3 by (cases "a ∈ A'") auto  next    fix b assume *: "b ∈ B"    show "b ∈ h ` A"    proof(cases "b ∈ f ` A'")      assume Case1: "b ∈ f ` A'"      then obtain a where "a ∈ A' ∧ b = f a" by blast      thus ?thesis using 2 0 by force    next      assume Case2: "b ∉ f ` A'"      hence "g b ∉ A'" using 1 * by auto      hence 4: "g b ∈ A - A'" using * SUB2 by auto      hence "h(g b) ∈ B ∧ 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 autoqedsubsection {*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 simplemma 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 blasttext{*A relation is wellfounded iff it has no infinite descending chain*}lemma wf_iff_no_infinite_down_chain:  "wf r = (~(∃f. ∀i. (f(Suc i),f i) ∈ r))"apply (simp only: wf_eq_minimal)apply (rule iffI) apply (rule notI) apply (erule exE) apply (erule_tac x = "{w. ∃i. w=f i}" in allE, blast)apply (erule contrapos_np, simp, clarify)apply (subgoal_tac "∀n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n ∈ 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+)donelemma wf_no_infinite_down_chainE:  assumes "wf r" obtains k where "(f (Suc k), f k) ∉ r"using `wf r` wf_iff_no_infinite_down_chain[of r] by blasttext{*A dynamically-scoped fact for TFL *}lemma tfl_some: "∀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 & (∀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 ==> ∀y. P y --> m x ≤ m y ==> Q x)    ==> Q (LeastM m P)"  apply (simp add: LeastM_def)  apply (rule someI2_ex, blast, blast)  donelemma 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)  donelemma wf_linord_ex_has_least:  "wf r ==> ∀x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k    ==> ∃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)  donelemma ex_has_least_nat:    "P k ==> ∃x. P x & (∀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)  donelemma LeastM_nat_lemma:    "P k ==> P (LeastM m P) & (∀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)  donelemmas 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 & (∀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 ==> ∀y. P y --> m y ≤ m x ==> Q x)    ==> Q (GreatestM m P)"  apply (simp add: GreatestM_def)  apply (rule someI2_ex, blast, blast)  donelemma 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)  donelemma 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)  donelemma ex_has_greatest_nat_lemma:  "P k ==> ∀x. P x --> (∃y. P y & ~ ((m y::nat) <= m x))    ==> ∃y. P y & ~ (m y < m k + n)"  apply (induct n, force)  apply (force simp add: le_Suc_eq)  donelemma ex_has_greatest_nat:  "P k ==> ∀y. P y --> m y < b    ==> ∃x. P x & (∀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)  donelemma GreatestM_nat_lemma:  "P k ==> ∀y. P y --> m y < b    ==> P (GreatestM m P) & (∀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)  donelemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1]lemma GreatestM_nat_le:  "P x ==> ∀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])  donetext {* \medskip Specialization to @{text GREATEST}. *}lemma GreatestI: "P (k::nat) ==> ∀y. P y --> y < b ==> P (GREATEST x. P x)"  apply (simp add: Greatest_def)  apply (rule GreatestM_natI, auto)  donelemma Greatest_le:    "P x ==> ∀y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"  apply (simp add: Greatest_def)  apply (rule GreatestM_nat_le, auto)  donesubsection {* An aside: bounded accessible part *}text {* Finite monotone eventually stable sequences *}lemma finite_mono_remains_stable_implies_strict_prefix:  fixes f :: "nat => 'a::order"  assumes S: "finite (range f)" "mono f" and eq: "∀n. f n = f (Suc n) --> f (Suc n) = f (Suc (Suc n))"  shows "∃N. (∀n≤N. ∀m≤N. m < n --> f m < f n) ∧ (∀n≥N. f N = f n)"  using assmsproof -  have "∃n. f n = f (Suc n)"  proof (rule ccontr)    assume "¬ ?thesis"    then have "!!n. f n ≠ f (Suc n)" by auto    then have "!!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 "!!n m. n < m ==> 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 ≡ "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 ≤ n"    then have "!!m. N ≤ m ==> m ≤ n ==> 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 ≤ n` show "f N = f n" by auto  next    fix n m :: nat assume "m < n" "n ≤ 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 ≠ 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  qedqedlemma finite_mono_strict_prefix_implies_finite_fixpoint:  fixes f :: "nat => 'a set"  assumes S: "!!i. f i ⊆ S" "finite S"    and inj: "∃N. (∀n≤N. ∀m≤N. m < n --> f m ⊂ f n) ∧ (∀n≥N. f N = f n)"  shows "f (card S) = (\<Union>n. f n)"proof -  from inj obtain N where inj: "(∀n≤N. ∀m≤N. m < n --> f m ⊂ f n)" and eq: "(∀n≥N. f N = f n)" by auto  { fix i have "i ≤ N ==> i ≤ 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) ⊂ (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 ≤ card (f N)" by simp  also have "… ≤ 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)    doneqedprimrec bacc :: "('a × 'a) set => nat => 'a set" where  "bacc r 0 = {x. ∀ y. (y, x) ∉ r}"| "bacc r (Suc n) = (bacc r n ∪ {x. ∀y. (y, x) ∈ r --> y ∈ bacc r n})"lemma bacc_subseteq_acc:  "bacc r n ⊆ acc r"  by (induct n) (auto intro: acc.intros)lemma bacc_mono:  "n ≤ m ==> bacc r n ⊆ bacc r m"  by (induct rule: dec_induct) auto  lemma bacc_upper_bound:  "bacc (r :: ('a × '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 "∀n. bacc r n = bacc r (Suc n) --> 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)qedlemma acc_subseteq_bacc:  assumes "finite r"  shows "acc r ⊆ (\<Union>n. bacc r n)"proof  fix x  assume "x : acc r"  then have "∃ n. x : bacc r n"  proof (induct x arbitrary: rule: acc.induct)    case (accI x)    then have "∀y. ∃ n. (y, x) ∈ r --> y : bacc r n" by simp    from choice[OF this] obtain n where n: "∀y. (y, x) ∈ r --> y ∈ bacc r (n y)" ..    obtain n where "!!y. (y, x) : r ==> 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 autoqedlemma acc_bacc_eq:  fixes A :: "('a :: finite × 'a) set"  assumes "finite A"  shows "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`