Theory HOL.Hilbert_Choice

(*  Title:      HOL/Hilbert_Choice.thy
    Author:     Lawrence C Paulson, Tobias Nipkow
    Author:     Viorel Preoteasa (Results about complete distributive lattices) 
    Copyright   2001  University of Cambridge
*)

section ‹Hilbert's Epsilon-Operator and the Axiom of Choice›

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

subsection ‹Hilbert's epsilon›

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

syntax (epsilon)
  "_Eps" :: "pttrn  bool  'a"  ("(3ϵ_./ _)" [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 (λx. P)"

print_translation [(const_syntaxEps, 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 = (λx. SOME y. y  A  f y = x)"

lemma inv_into_def2: "inv_into A f x = (SOME y. y  A  f y = x)"
by(simp add: inv_into_def)

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


subsection ‹Hilbert's Epsilon-operator›

lemma Eps_cong:
  assumes "x. P x = Q x"
  shows "Eps P = Eps Q"
  using ext[of P Q, OF assms] by simp

text ‹
  Easier to use than someI› if the witness comes from an
  existential formula.
›
lemma someI_ex [elim?]: "x. P x  P (SOME x. P x)"
  by (elim exE someI)

lemma some_eq_imp:
  assumes "Eps P = a" "P b" shows "P a"
  using assms someI_ex by force

text ‹
  Easier to use than someI› because the conclusion has only one
  occurrence of termP.
›
lemma someI2: "P a  (x. P x  Q x)  Q (SOME x. P x)"
  by (blast intro: someI)

text ‹
  Easier to use than someI2› if the witness comes from an
  existential formula.
›
lemma someI2_ex: "a. P a  (x. P x  Q x)  Q (SOME x. P x)"
  by (blast intro: someI2)

lemma someI2_bex: "aA. P a  (x. x  A  P x  Q x)  Q (SOME x. x  A  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: "∃!x. P x  P a  (SOME x. P x) = a"
  by blast

lemma some_eq_ex: "P (SOME x. P x)  (x. P x)"
  by (blast intro: someI)

lemma some_in_eq: "(SOME x. x  A)  A  A  {}"
  unfolding ex_in_conv[symmetric] by (rule some_eq_ex)

lemma some_eq_trivial [simp]: "(SOME y. y = x) = x"
  by (rule some_equality) (rule refl)

lemma some_sym_eq_trivial [simp]: "(SOME y. x = y) = x"
  by (iprover intro: some_equality)


subsection ‹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: "xS. y. Q x y  f. xS. 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: "(xS. y. Q x y)  (f. xS. Q x (f x))"
  by (fast elim: someI)

lemma bchoice_iff': "(xS. P x  (y. Q x y))  (f. xS. P x  Q x (f x))"
  by (fast elim: someI)

lemma dependent_nat_choice:
  assumes 1: "x. P 0 x"
    and 2: "x n. P n x  y. P (Suc n) y  Q n x y"
  shows "f. n. P n (f n)  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) (λn x. SOME y. P (Suc n) y  Q n x y)"
  then have "P 0 (f 0)" "n. P n (f n)  P (Suc n) (f (Suc n))  Q n (f n) (f (Suc n))"
    using someI_ex[OF 1] someI_ex[OF 2] by simp_all
  then show "P n (f n)" "Q n (f n) (f (Suc n))"
    by (induct n) auto
qed

lemma finite_subset_Union:
  assumes "finite A" "A  "
  obtains  where "finite " "  " "A  "
proof -
  have "xA. B. xB"
    using assms by blast
  then obtain f where f: "x. x  A  f x    x  f x"
    by (auto simp add: bchoice_iff Bex_def)
  show thesis
  proof
    show "finite (f ` A)"
      using assms by auto
  qed (use f in auto)
qed


subsection ‹Function Inverse›

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

lemma inv_into_into: "x  f ` A  inv_into A f x  A"
  by (simp add: inv_into_def) (fast intro: someI2)

lemma inv_identity [simp]: "inv (λa. a) = (λa. 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"
  by (simp add: inv_into_def inj_on_def) (blast intro: someI2)

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"
  by (simp add: inv_into_def) (fast intro: someI2)

lemma inv_into_f_eq: "inj_on f A  x  A  f x = y  inv_into A f y = x"
  by (erule subst) (fast intro: inv_into_f_f)

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  x. f (g x) = x  inv f = g"
  by (blast intro: inv_into_f_eq)

text ‹But is it useful?›
lemma inj_transfer:
  assumes inj: "inj f"
    and minor: "y. y  range f  P (inv f y)"
  shows "P x"
proof -
  have "f x  range f" by auto
  then have "P(inv f (f x))" by (rule minor)
  then show "P x" by (simp add: inv_into_f_f [OF inj])
qed

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

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

lemma o_inv_o_cancel[simp]: "inj f  g  inv f  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 bij_inv_eq_iff: "bij p  x = inv p y  p x = y"
  using surj_f_inv_f[of p] by (auto simp add: bij_def)

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 -
  from eq have "f (inv_into A f x) = f (inv_into A f y)"
    by simp
  with x y show ?thesis
    by (simp add: f_inv_into_f)
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 inj_imp_bij_betw_inv: "inj f  bij_betw (inv f) (f ` M) M"
  by (simp add: bij_betw_def image_subsetI inj_on_inv_into)

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  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)"
  by (simp add: o_def surj_iff fun_eq_iff)

lemma surj_imp_inv_eq:
  assumes "surj f" and gf: "x. g (f x) = x"
  shows "inv f = g"
proof (rule ext)
  fix x
  have "g (f (inv f x)) = inv f x"
    by (rule gf)
  then show "inv f x = g x"
    by (simp add: surj_f_inv_f surj f)
qed

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"
  by (rule ext) (auto simp add: inv_into_def)

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

text bij (inv f)› implies little about f›. Consider f :: bool ⇒ bool› such
  that f True = f False = True›. Then it ia 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  g) x = (inv_into A g  inv_into (g ` A) f) x"
  by (auto simp: f_inv_into_f inv_into_into intro: inv_into_f_eq comp_inj_on)

lemma o_inv_distrib: "bij f  bij g  inv (f  g) = inv g  inv f"
  by (rule inv_equality) (auto simp add: bij_def surj_f_inv_f)

lemma image_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 bij_image_Collect_eq:
  assumes "bij f"
  shows "f ` Collect P = {y. P (inv f y)}"
proof
  show "f ` Collect P  {y. P (inv f y)}"
    using assms by (force simp add: bij_is_inj)
  show "{y. P (inv f y)}  f ` Collect P"
    using assms by (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
qed

lemma bij_vimage_eq_inv_image:
  assumes "bij f"
  shows "f -` A = inv f ` A"
proof
  show "f -` A  inv f ` A"
    using assms by (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric])
  show "inv f ` A  f -` A"
    using assms by (auto simp add: bij_is_surj [THEN surj_f_inv_f])
qed

lemma inv_fn_o_fn_is_id:
  fixes f::"'a  'a"
  assumes "bij f"
  shows "((inv f)^^n) o (f^^n) = (λx. x)"
proof -
  have "((inv f)^^n)((f^^n) x) = x" for x n
  proof (induction n)
    case (Suc n)
    have *: "(inv f) (f y) = y" for y
      by (simp add: assms bij_is_inj)
    have "(inv f ^^ Suc n) ((f ^^ Suc n) x) = (inv f^^n) (inv f (f ((f^^n) x)))"
      by (simp add: funpow_swap1)
    also have "... = (inv f^^n) ((f^^n) x)"
      using * by auto
    also have "... = x" using Suc.IH by auto
    finally show ?case by simp
  qed (auto)
  then show ?thesis unfolding o_def by blast
qed

lemma fn_o_inv_fn_is_id:
  fixes f::"'a  'a"
  assumes "bij f"
  shows "(f^^n) o ((inv f)^^n) = (λx. x)"
proof -
  have "(f^^n) (((inv f)^^n) x) = x" for x n
  proof (induction n)
    case (Suc n)
    have *: "f(inv f y) = y" for y
      using bij_inv_eq_iff[OF assms] by auto
    have "(f ^^ Suc n) ((inv f ^^ Suc n) x) = (f^^n) (f (inv f ((inv f^^n) x)))"
      by (simp add: funpow_swap1)
    also have "... = (f^^n) ((inv f^^n) x)"
      using * by auto
    also have "... = x" using Suc.IH by auto
    finally show ?case by simp
  qed (auto)
  then show ?thesis unfolding o_def by blast
qed

lemma inv_fn:
  fixes f::"'a  'a"
  assumes "bij f"
  shows "inv (f^^n) = ((inv f)^^n)"
proof -
  have "inv (f^^n) x = ((inv f)^^n) x" for x
  proof (rule inv_into_f_eq)
    show "inj (f ^^ n)"
      by (simp add: inj_fn[OF bij_is_inj [OF assms]])
    show "(f ^^ n) ((inv f ^^ n) x) = x"
      using fn_o_inv_fn_is_id[OF assms, THEN fun_cong] by force
  qed auto
  then show ?thesis by auto
qed

lemma funpow_inj_finite: contributor ‹Lars Noschinski›
  assumes inj p finite {y. n. y = (p ^^ n) x}
  obtains n where n > 0 (p ^^ n) x = x 
proof -
  have infinite (UNIV :: nat set)
    by simp
  moreover have {y. n. y = (p ^^ n) x} = (λn. (p ^^ n) x) ` UNIV
    by auto
  with assms have finite 
    by simp
  ultimately have "n  UNIV. ¬ finite {m  UNIV. (p ^^ m) x = (p ^^ n) x}"
    by (rule pigeonhole_infinite)
  then obtain n where "infinite {m. (p ^^ m) x = (p ^^ n) x}" by auto
  then have "infinite ({m. (p ^^ m) x = (p ^^ n) x} - {n})" by auto
  then have "({m. (p ^^ m) x = (p ^^ n) x} - {n})  {}"
    by (auto simp add: subset_singleton_iff)
  then obtain m where m: "(p ^^ m) x = (p ^^ n) x" "m  n" by auto

  { fix m n assume "(p ^^ n) x = (p ^^ m) x" "m < n"
    have "(p ^^ (n - m)) x = inv (p ^^ m) ((p ^^ m) ((p ^^ (n - m)) x))"
      using inj p by (simp add: inv_f_f)
    also have "((p ^^ m) ((p ^^ (n - m)) x)) = (p ^^ n) x"
      using m < n funpow_add [of m n - m p] by simp
    also have "inv (p ^^ m)  = x"
      using inj p  by (simp add: (p ^^ n) x = _)
    finally have "(p ^^ (n - m)) x = x" "0 < n - m"
      using m < n by auto }
  note general = this

  show thesis
  proof (cases m n rule: linorder_cases)
    case less
    then have n - m > 0 (p ^^ (n - m)) x = x
      using general [of n m] m by simp_all
    then show thesis by (blast intro: that)
  next
    case equal
    then show thesis using m by simp
  next
    case greater
    then have m - n > 0 (p ^^ (m - n)) x = x
      using general [of m n] m by simp_all
    then show thesis by (blast intro: that)
  qed
qed


lemma mono_inv:
  fixes f::"'a::linorder  'b::linorder"
  assumes "mono f" "bij f"
  shows "mono (inv f)"
proof
  fix x y::'b assume "x  y"
  from bij f obtain a b where x: "x = f a" and y: "y = f b" by(fastforce simp: bij_def surj_def)
  show "inv f x  inv f y"
  proof (rule le_cases)
    assume "a  b"
    thus ?thesis using  bij f x y by(simp add: bij_def inv_f_f)
  next
    assume "b  a"
    hence "f b  f a" by(rule monoD[OF mono f])
    hence "y  x" using x y by simp
    hence "x = y" using x  y by auto
    thus ?thesis by simp
  qed
qed

lemma strict_mono_inv_on_range:
  fixes f :: "'a::linorder  'b::order"
  assumes "strict_mono f"
  shows "strict_mono_on (range f) (inv f)"
proof (clarsimp simp: strict_mono_on_def)
  fix x y
  assume "f x < f y"
  then show "inv f (f x) < inv f (f y)"
    using assms strict_mono_imp_inj_on strict_mono_less by fastforce
qed

lemma mono_bij_Inf:
  fixes f :: "'a::complete_linorder  'b::complete_linorder"
  assumes "mono f" "bij f"
  shows "f (Inf A) = Inf (f`A)"
proof -
  have "surj f" using bij f by (auto simp: bij_betw_def)
  have *: "(inv f) (Inf (f`A))  Inf ((inv f)`(f`A))"
    using mono_Inf[OF mono_inv[OF assms], of "f`A"] by simp
  have "Inf (f`A)  f (Inf ((inv f)`(f`A)))"
    using monoD[OF mono f *] by(simp add: surj_f_inv_f[OF surj f])
  also have "... = f(Inf A)"
    using assms by (simp add: bij_is_inj)
  finally show ?thesis using mono_Inf[OF assms(1), of A] by auto
qed

lemma finite_fun_UNIVD1:
  assumes fin: "finite (UNIV :: ('a  'b) set)"
    and card: "card (UNIV :: 'b set)  Suc 0"
  shows "finite (UNIV :: 'a set)"
proof -
  let ?UNIV_b = "UNIV :: 'b set"
  from fin have "finite ?UNIV_b"
    by (rule finite_fun_UNIVD2)
  with card have "card ?UNIV_b  Suc (Suc 0)"
    by (cases "card ?UNIV_b") (auto simp: card_eq_0_iff)
  then have "card ?UNIV_b = Suc (Suc (card ?UNIV_b - Suc (Suc 0)))"
    by simp
  then obtain b1 b2 :: 'b where b1b2: "b1  b2"
    by (auto simp: card_Suc_eq)
  from fin have fin': "finite (range (λf :: 'a  'b. inv f b1))"
    by (rule finite_imageI)
  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)
    then show "x  range (λf::'a  'b. inv f b1)"
      by blast
  qed
  with fin' show ?thesis
    by simp
qed

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

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

lemma infinite_countable_subset:
  assumes inf: "¬ finite S"
  shows "f::nat  'a. inj f  range f  S"
  ― ‹Courtesy of Stephan Merz›
proof -
  define Sseq where "Sseq = rec_nat S (λn T. T - {SOME e. e  T})"
  define pick where "pick n = (SOME e. e  Sseq n)" for n
  have *: "Sseq n  S" "¬ finite (Sseq n)" for n
    by (induct n) (auto simp: Sseq_def inf)
  then have **: "n. pick n  Sseq n"
    unfolding pick_def by (subst (asm) finite.simps) (auto simp add: ex_in_conv intro: someI_ex)
  with * have "range pick  S" by auto
  moreover have "pick n  pick (n + Suc m)" for m n
  proof -
    have "pick n  Sseq (n + Suc m)"
      by (induct m) (auto simp add: Sseq_def pick_def)
    with ** show ?thesis by auto
  qed
  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: "¬ finite S  (f::nat  'a. inj f  range f  S)"
  ― ‹Courtesy of Stephan Merz›
  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'  A'"
  shows "f `((inv_into A f)`B') = B'"
  using assms
proof (auto simp: f_inv_into_f)
  let ?f' = "inv_into A f"
  fix a'
  assume *: "a'  B'"
  with sub have "a'  A'" by auto
  with surj have "a' = f (?f' a')"
    by (auto simp: f_inv_into_f)
  with * show "a'  f ` (?f' ` B')" by blast
qed

lemma inv_into_inv_into_eq:
  assumes "bij_betw f A A'"
    and 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'"
  from assms have *: "bij_betw ?f' A' A"
    by (auto simp: bij_betw_inv_into)
  with a obtain a' where a': "a'  A'" "?f' a' = a"
    unfolding bij_betw_def by force
  with a * have "?f'' a = a'"
    by (auto simp: f_inv_into_f bij_betw_def)
  moreover from assms a' have "f a = a'"
    by (auto simp: bij_betw_def)
  ultimately show "?f'' a = f a" by simp
qed

lemma 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 "a'  f ` A")
        case True
        then obtain a where "?phi a' a" by blast
        then have "?phi a' (SOME a. ?phi a' a)"
          using someI[of "?phi a'" a] by blast
        with True show ?thesis by auto
      next
        case False
        with assms have "?csi (SOME a. ?csi a)"
          using someI_ex[of ?csi] by blast
        with False show ?thesis by auto
      qed
    qed
  next
    show "A  ?g ` A'"
    proof -
      have "?g (f a) = a  f a  A'" if a: "a  A" for a
      proof -
        let ?b = "SOME aa. ?phi (f a) aa"
        from a have "?phi (f a) a" by auto
        then have *: "?phi (f a) ?b"
          using someI[of "?phi(f a)" a] by blast
        then have "?g (f a) = ?b" using a by auto
        moreover from inj * a have "a = ?b"
          by (auto simp add: inj_on_def)
        ultimately have "?g(f a) = a" by simp
        with incl a show ?thesis by auto
      qed
      then show ?thesis by force
    qed
  qed
  then show "g. g ` A' = A" by blast
next
  fix g
  let ?f = "inv_into A' g"
  have "inj_on ?f (g ` A')"
    by (auto simp: inj_on_inv_into)
  moreover have "?f (g a')  A'" if a': "a'  A'" for a'
  proof -
    let ?phi = "λ b'. b'  A'  g b' = g a'"
    from a' have "?phi a'" by auto
    then have "?phi (SOME b'. ?phi b')"
      using someI[of ?phi] by blast
    then show ?thesis by (auto simp: inv_into_def)
  qed
  ultimately show "f. inj_on f (g ` A')  f ` g ` A'  A'"
    by auto
qed

lemma Ex_inj_on_UNION_Sigma:
  "f. (inj_on f (i  I. A i)  f ` (i  I. A i)  (SIGMA i : I. A i))"
proof
  let ?phi = "λa i. i  I  a  A i"
  let ?sm = "λa. SOME i. ?phi a i"
  let ?f = "λa. (?sm a, a)"
  have "inj_on ?f (i  I. A i)"
    by (auto simp: inj_on_def)
  moreover
  have "?sm a  I  a  A(?sm a)" if "i  I" and "a  A i" for i a
    using that someI[of "?phi a" i] by auto
  then have "?f ` (i  I. A i)  (SIGMA i : I. A i)"
    by auto
  ultimately show "inj_on ?f (i  I. A i)  ?f ` (i  I. A i)  (SIGMA i : I. A i)"
    by auto
qed

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

lemma subset_image_inj:
  "S  f ` T  (U. U  T  inj_on f U  S = f ` U)"
proof safe
  show "UT. inj_on f U  S = f ` U"
    if "S  f ` T"
  proof -
    from that [unfolded subset_image_iff subset_iff]
    obtain g where g: "x. x  S  g x  T  x = f (g x)"
      by (auto simp add: image_iff Bex_def choice_iff')
    show ?thesis
    proof (intro exI conjI)
      show "g ` S  T"
        by (simp add: g image_subsetI)
      show "inj_on f (g ` S)"
        using g by (auto simp: inj_on_def)
      show "S = f ` (g ` S)"
        using g image_subset_iff by auto
    qed
  qed
qed blast


subsection ‹Other Consequences of Hilbert's Epsilon›

text ‹Hilbert's Epsilon and the termsplit Operator›

text ‹Looping simprule!›
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]: "(SOME (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  (f. i. (f (Suc i), f i)  r)"
  (is "_  ¬ ?ex")
proof
  assume "wf r"
  show "¬ ?ex"
  proof
    assume ?ex
    then obtain f where f: "(f (Suc i), f i)  r" for i
      by blast
    from wf r have minimal: "x  Q  zQ. y. (y, z)  r  y  Q" for x Q
      by (auto simp: wf_eq_minimal)
    let ?Q = "{w. i. w = f i}"
    fix n
    have "f n  ?Q" by blast
    from minimal [OF this] obtain j where "(y, f j)  r  y  ?Q" for y by blast
    with this [OF (f (Suc j), f j)  r] have "f (Suc j)  ?Q" by simp
    then show False by blast
  qed
next
  assume "¬ ?ex"
  then show "wf r"
  proof (rule contrapos_np)
    assume "¬ wf r"
    then obtain Q x where x: "x  Q" and rec: "z  Q  y. (y, z)  r  y  Q" for z
      by (auto simp add: wf_eq_minimal)
    obtain descend :: "nat  'a"
      where descend_0: "descend 0 = x"
        and descend_Suc: "descend (Suc n) = (SOME y. y  Q  (y, descend n)  r)" for n
      by (rule that [of "rec_nat x (λ_ rec. (SOME y. y  Q  (y, rec)  r))"]) simp_all
    have descend_Q: "descend n  Q" for n
    proof (induct n)
      case 0
      with x show ?case by (simp only: descend_0)
    next
      case Suc
      then show ?case by (simp only: descend_Suc) (rule someI2_ex; use rec in blast)
    qed
    have "(descend (Suc i), descend i)  r" for i
      by (simp only: descend_Suc) (rule someI2_ex; use descend_Q rec in blast)
    then show "f. i. (f (Suc i), f i)  r" by blast
  qed
qed

lemma wf_no_infinite_down_chainE:
  assumes "wf r"
  obtains k where "(f (Suc k), f k)  r"
  using assms wf_iff_no_infinite_down_chain[of r] by blast


text ‹A dynamically-scoped fact for TFL›
lemma tfl_some: "P x. P x  P (Eps P)"
  by (blast intro: someI)


subsection ‹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. (nN. mN. m < n  f m < f n)  (nN. f N = f n)"
  using assms
proof -
  have "n. f n = f (Suc n)"
  proof (rule ccontr)
    assume "¬ ?thesis"
    then have "n. f n  f (Suc n)" by auto
    with mono f have "n. f n < f (Suc n)"
      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
    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 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)" ..
  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  n"
    then have "m. N  m  m  n  f m = f N"
    proof (induct rule: dec_induct)
      case base
      then show ?case by simp
    next
      case (step n)
      then show ?case
        using eq [rule_format, of "n - 1"] N
        by (cases n) (auto simp add: le_Suc_eq)
    qed
    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)
      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)
    next
      case 2
      then show ?case by simp
    qed
  qed
qed

lemma finite_mono_strict_prefix_implies_finite_fixpoint:
  fixes f :: "nat  'a set"
  assumes S: "i. f i  S" "finite S"
    and ex: "N. (nN. mN. m < n  f m  f n)  (nN. f N = f n)"
  shows "f (card S) = (n. f n)"
proof -
  from ex obtain N where inj: "n m. n  N  m  N  m < n  f m  f n"
    and eq: "nN. f N = f n"
    by atomize auto
  have "i  N  i  card (f i)" for i
  proof (induct i)
    case 0
    then show ?case by simp
  next
    case (Suc i)
    with inj [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 inj show ?case 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
  moreover have " (range f)  f N"
  proof clarify
    fix x n
    assume "x  f n"
    with eq inj [of N] show "x  f N"
      by (cases "n < N") (auto simp: not_less)
  qed
  ultimately show ?thesis
    by auto
qed


subsection ‹More on injections, bijections, and inverses›

locale bijection =
  fixes f :: "'a  '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  a = b"
  using inj by (rule injD)

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

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

lemma eq_inv_iff [simp]: "inv f a = inv f b  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  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  inv f = id"
  by (simp add: fun_eq_iff)

lemma inv_left_eq_iff [simp]: "inv f a = b  f b = a"
  by auto

lemma inv_right_eq_iff [simp]: "b = inv f a  f b = a"
  by auto

end

lemma infinite_imp_bij_betw:
  assumes infinite: "¬ finite A"
  shows "h. bij_betw h A (A - {a})"
proof (cases "a  A")
  case False
  then have "A - {a} = A" by blast
  then show ?thesis
    using bij_betw_id[of A] by auto
next
  case True
  with infinite have "¬ finite (A - {a})" by auto
  with infinite_iff_countable_subset[of "A - {a}"]
  obtain f :: "nat  'a" where "inj f" and f: "f ` UNIV  A - {a}" by blast
  define g where "g n = (if n = 0 then a else f (Suc n))" for n
  define A' where "A' = g ` UNIV"
  have *: "y. f y  a" using f by blast
  have 3: "inj_on g UNIV  g ` UNIV  A  a  g ` UNIV"
    using inj f f * unfolding inj_on_def g_def
    by (auto simp add: True image_subset_iff)
  then have 4: "bij_betw g UNIV A'  a  A'  A'  A"
    using inj_on_imp_bij_betw[of g] by (auto simp: A'_def)
  then have 5: "bij_betw (inv g) A' UNIV"
    by (auto simp add: bij_betw_inv_into)
  from 3 obtain n where n: "g n = a" by auto
  have 6: "bij_betw g (UNIV - {n}) (A' - {a})"
    by (rule bij_betw_subset) (use 3 4 n in auto simp: image_set_diff A'_def)
  define v where "v m = (if m < n then m else Suc m)" for m
  have "m < n  m = n" if "k. k < n  m  Suc k" for m
    using that [of "m-1"] by auto
  then have 7: "bij_betw v UNIV (UNIV - {n})"
    unfolding bij_betw_def inj_on_def v_def by auto
  define h' where "h' = g  v  (inv g)"
  with 5 6 7 have 8: "bij_betw h' A' (A' - {a})"
    by (auto simp add: bij_betw_trans)
  define h where "h b = (if b  A' then h' b else b)" for b
  with 8 have "bij_betw h  A' (A' - {a})"
    using bij_betw_cong[of A' h] by auto
  moreover
  have "b  A - A'. h b = b" by (auto simp: h_def)
  then have "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
  from 4 have "(A'  (A - A') = {}  A'  (A - A') = A) 
    ((A' - {a})  (A - A') = {}  (A' - {a})  (A - A') = A - {a})"
    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
  then show ?thesis by blast
qed

lemma infinite_imp_bij_betw2:
  assumes "¬ finite A"
  shows "h. bij_betw h A (A  {a})"
proof (cases "a  A")
  case True
  then have "A  {a} = A" by blast
  then show ?thesis using bij_betw_id[of A] by auto
next
  case False
  let ?A' = "A  {a}"
  from False have "A = ?A' - {a}" by blast
  moreover from assms have "¬ finite ?A'" by auto
  ultimately obtain f where "bij_betw f ?A' A"
    using infinite_imp_bij_betw[of ?A' a] by auto
  then have "bij_betw (inv_into ?A' f) A ?A'" by (rule bij_betw_inv_into)
  then show ?thesis by auto
qed

lemma bij_betw_inv_into_left: "bij_betw f A A'  a  A  inv_into A f (f a) = a"
  unfolding bij_betw_def by clarify (rule inv_into_f_f)

lemma bij_betw_inv_into_right: "bij_betw f A A'  a'  A'  f (inv_into A f a') = a'"
  unfolding bij_betw_def using f_inv_into_f by force

lemma bij_betw_inv_into_subset:
  "bij_betw f A A'  B  A  f ` B = B'  bij_betw (inv_into A f) B' B"
  by (auto simp: bij_betw_def intro: inj_on_inv_into)


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›

subsection ‹Complete Distributive Lattices -- Properties depending on Hilbert Choice›

context complete_distrib_lattice
begin

lemma Sup_Inf: " (Inf ` A) =  (Sup ` {f ` A |f. BA. f B  B})"
proof (rule order.antisym)
  show " (Inf ` A)   (Sup ` {f ` A |f. BA. f B  B})"
    using Inf_lower2 Sup_upper
    by (fastforce simp add: intro: Sup_least INF_greatest)
next
  show " (Sup ` {f ` A |f. BA. f B  B})   (Inf ` A)"
  proof (simp add:  Inf_Sup, rule SUP_least, simp, safe)
    fix f
    assume "Y. (f. Y = f ` A  (YA. f Y  Y))  f Y  Y"
    then have B: " F . ( Y  A . F Y  Y)   Z  A . f (F ` A) = F Z"
      by auto
    show "(f ` {f ` A |f. YA. f Y  Y})  (Inf ` A)"
    proof (cases " Z  A . (f ` {f ` A |f. YA. f Y  Y})  Inf Z")
      case True
      from this obtain Z where [simp]: "Z  A" and A: "(f ` {f ` A |f. YA. f Y  Y})  Inf Z"
        by blast
      have B: "...  (Inf ` A)"
        by (simp add: SUP_upper)
      from A and B show ?thesis
        by simp
    next
      case False
      then have X: " Z . Z  A   x . x  Z  ¬ (f ` {f ` A |f. YA. f Y  Y})  x"
        using Inf_greatest by blast
      define F where "F = (λ Z . SOME x . x  Z  ¬ (f ` {f ` A |f. YA. f Y  Y})  x)"
      have C: "Y. Y  A  F Y  Y"
        using X by (simp add: F_def, rule someI2_ex, auto)
      have E: "Y. Y  A  ¬ (f ` {f ` A |f. YA. f Y  Y})  F Y"
        using X by (simp add: F_def, rule someI2_ex, auto)
      from C and B obtain  Z where D: "Z  A " and Y: "f (F ` A) = F Z"
        by blast
      from E and D have W: "¬ (f ` {f ` A |f. YA. f Y  Y})  F Z"
        by simp
      have "(f ` {f ` A |f. YA. f Y  Y})  f (F ` A)"
        using C by (blast intro: INF_lower)
      with W Y show ?thesis
        by simp
    qed
  qed
qed
  
lemma dual_complete_distrib_lattice:
  "class.complete_distrib_lattice Sup Inf sup (≥) (>) inf  "
  by (simp add: class.complete_distrib_lattice.intro [OF dual_complete_lattice] 
                class.complete_distrib_lattice_axioms_def Sup_Inf)

lemma sup_Inf: "a  B = ((⊔) a ` B)"
proof (rule order.antisym)
  show "a  B  ((⊔) a ` B)"
    using Inf_lower sup.mono by (fastforce intro: INF_greatest)
next
  have "((⊔) a ` B)  (Sup ` {{f {a}, f B} |f. f {a} = a  f B  B})"
    by (rule INF_greatest, auto simp add: INF_lower)
  also have "... = (Inf ` {{a}, B})"
    by (unfold Sup_Inf, simp)
  finally show "((⊔) a ` B)  a  B"
    by simp
qed

lemma inf_Sup: "a  B = ((⊓) a ` B)"
  using dual_complete_distrib_lattice
  by (rule complete_distrib_lattice.sup_Inf)

lemma INF_SUP: "(y. x. P x y) = (f. x. P (f x) x)"
proof (rule order.antisym)
  show "(SUP x. INF y. P (x y) y)  (INF y. SUP x. P x y)"
    by (rule SUP_least, rule INF_greatest, rule SUP_upper2, simp_all, rule INF_lower2, simp, blast)
next
  have "(INF y. SUP x. ((P x y)))  Inf (Sup ` {{P x y | x . True} | y . True })" (is "?A  ?B")
  proof (rule INF_greatest, clarsimp)
    fix y
    have "?A  (SUP x. P x y)"
      by (rule INF_lower, simp)
    also have "...  Sup {uu. x. uu = P x y}"
      by (simp add: full_SetCompr_eq)
    finally show "?A  Sup {uu. x. uu = P x y}"
      by simp
  qed
  also have "...   (SUP x. INF y. P (x y) y)"
  proof (subst Inf_Sup, rule SUP_least, clarsimp)
    fix f
    assume A: "Y. (y. Y = {uu. x. uu = P x y})  f Y  Y"
      
    have " (f ` {uu. y. uu = {uu. x. uu = P x y}}) 
      (y. P (SOME x. f {P x y |x. True} = P x y) y)"
    proof (rule INF_greatest, clarsimp)
      fix y
        have "(INF x{uu. y. uu = {uu. x. uu = P x y}}. f x)  f {uu. x. uu = P x y}"
          by (rule INF_lower, blast)
        also have "...  P (SOME x. f {uu . x. uu = P x y} = P x y) y"
          by (rule someI2_ex) (use A in auto)
        finally show "(f ` {uu. y. uu = {uu. x. uu = P x y}}) 
          P (SOME x. f {uu. x. uu = P x y} = P x y) y"
          by simp
      qed
      also have "...  (SUP x. INF y. P (x y) y)"
        by (rule SUP_upper, simp)
      finally show "(f ` {uu. y. uu = {uu. x. uu = P x y}})  (x. y. P (x y) y)"
        by simp
    qed
  finally show "(INF y. SUP x. P x y)  (SUP x. INF y. P (x y) y)"
    by simp
qed

lemma INF_SUP_set: "(BA. (g ` B)) = (B{f ` A |f. CA. f C  C}. (g ` B))"
                    (is "_ = (B?F. _)")
proof (rule order.antisym)
  have " ((g  f) ` A)   (g ` B)" if "B. B  A  f B  B" "B  A" for f B
    using that by (auto intro: SUP_upper2 INF_lower2)
  then show "(x?F. ax. g a)  (xA. ax. g a)"
    by (auto intro!: SUP_least INF_greatest simp add: image_comp)
next
  show "(xA. ax. g a)  (x?F. ax. g a)"
  proof (cases "{}  A")
    case True
    then show ?thesis 
      by (rule INF_lower2) simp_all
  next
    case False
    {fix x
      have "(xA. xx. g x)  (u. if x  A then if u  x then g u else  else )"
      proof (cases "x  A")
        case True
        then show ?thesis
          by (intro INF_lower2 SUP_least SUP_upper2) auto
      qed auto
    }
    then have "(YA. aY. g a)  (Y. y. if Y  A then if y  Y then g y else  else )"
      by (rule INF_greatest)
    also have "... = (x. Y. if Y  A then if x Y  Y then g (x Y) else  else )"
      by (simp only: INF_SUP)
    also have "...  (x?F. ax. g a)"
    proof (rule SUP_least)
      show "(B. if B  A then if x B  B then g (x B) else  else )
                (x?F. xx. g x)" for x
      proof -
        define G where "G  λY. if x Y  Y then x Y else (SOME x. x Y)"
        have "YA. G Y  Y"
          using False some_in_eq G_def by auto
        then have A: "G ` A  ?F"
          by blast
        show "(Y. if Y  A then if x Y  Y then g (x Y) else  else )  (x?F. xx. g x)"
          by (fastforce simp: G_def intro: SUP_upper2 [OF A] INF_greatest INF_lower2)
      qed
    qed
    finally show ?thesis by simp
  qed
qed

lemma SUP_INF: "(y. x. P x y) = (x. y. P (x y) y)"
  using dual_complete_distrib_lattice
  by (rule complete_distrib_lattice.INF_SUP)

lemma SUP_INF_set: "(xA.  (g ` x)) = (x{f ` A |f. YA. f Y  Y}.  (g ` x))"
  using dual_complete_distrib_lattice
  by (rule complete_distrib_lattice.INF_SUP_set)

end

(*properties of the former complete_distrib_lattice*)
context complete_distrib_lattice
begin

lemma sup_INF: "a  (bB. f b) = (bB. a  f b)"
  by (simp add: sup_Inf image_comp)

lemma inf_SUP: "a  (bB. f b) = (bB. a  f b)"
  by (simp add: inf_Sup image_comp)

lemma Inf_sup: "B  a = (bB. b  a)"
  by (simp add: sup_Inf sup_commute)

lemma Sup_inf: "B  a = (bB. b  a)"
  by (simp add: inf_Sup inf_commute)

lemma INF_sup: "(bB. f b)  a = (bB. f b  a)"
  by (simp add: sup_INF sup_commute)

lemma SUP_inf: "(bB. f b)  a = (bB. f b  a)"
  by (simp add: inf_SUP inf_commute)

lemma Inf_sup_eq_top_iff: "(B  a = )  (bB. b  a = )"
  by (simp only: Inf_sup INF_top_conv)

lemma Sup_inf_eq_bot_iff: "(B  a = )  (bB. b  a = )"
  by (simp only: Sup_inf SUP_bot_conv)

lemma INF_sup_distrib2: "(aA. f a)  (bB. g b) = (aA. bB. f a  g b)"
  by (subst INF_commute) (simp add: sup_INF INF_sup)

lemma SUP_inf_distrib2: "(aA. f a)  (bB. g b) = (aA. bB. f a  g b)"
  by (subst SUP_commute) (simp add: inf_SUP SUP_inf)

end

instantiation set :: (type) complete_distrib_lattice
begin
instance proof (standard, clarsimp)
  fix A :: "(('a set) set) set"
  fix x::'a
  assume A: "𝒮A. X𝒮. x  X"
  define F where "F  λY. SOME X. Y  A  X  Y  x  X"
  have "(S  F ` A. x  S)"
    using A unfolding F_def by (fastforce intro: someI2_ex)
  moreover have "YA. F Y  Y"
    using A unfolding F_def by (fastforce intro: someI2_ex)
  then have "f. F ` A  = f ` A  (YA. f Y  Y)"
    by blast
  ultimately show "X. (f. X = f ` A  (YA. f Y  Y))  (SX. x  S)"
    by auto
qed
end

instance set :: (type) complete_boolean_algebra ..

instantiation "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice
begin
instance by standard (simp add: le_fun_def INF_SUP_set image_comp)
end

instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..

context complete_linorder
begin

subclass complete_distrib_lattice
proof (standard, rule ccontr)
  fix A :: "'a set set"
  let ?F = "{f ` A |f. YA. f Y  Y}"
  assume "¬ (Sup ` A)  (Inf ` ?F)"
  then have C: "(Sup ` A) > (Inf ` ?F)"
    by (simp add: not_le)
  show False
  proof (cases " z . (Sup ` A) > z  z > (Inf ` ?F)")
    case True
    then obtain z where A: "z < (Sup ` A)" and X: "z > (Inf ` ?F)"
      by blast
    then have B: "Y. Y  A  k Y . z < k"
      using local.less_Sup_iff by(force dest: less_INF_D)

    define G where "G  λY. SOME k . k  Y  z < k"
    have E: "Y. Y  A  G Y  Y"
      using B unfolding G_def by (fastforce intro: someI2_ex)
    have "z  Inf (G ` A)"
    proof (rule INF_greatest)
      show  "Y. Y  A  z  G Y"
        using B unfolding G_def by (fastforce intro: someI2_ex)
    qed
    also have "...  (Inf ` ?F)"
      by (rule SUP_upper) (use E in blast)
    finally have "z  (Inf ` ?F)"
      by simp

    with X show ?thesis
      using local.not_less by blast
  next
    case False
    have B: "Y. Y  A   k Y . (Inf ` ?F) < k"
      using C local.less_Sup_iff by(force dest: less_INF_D)
    define G where "G  λ Y . SOME k . k  Y  (Inf ` ?F) < k"
    have E: "Y. Y  A  G Y  Y"
      using B unfolding G_def by (fastforce intro: someI2_ex)
    have "Y. Y  A  (Sup ` A)  G Y"
      using B False local.leI unfolding G_def by (fastforce intro: someI2_ex)
    then have "(Sup ` A)  Inf (G ` A)"
      by (simp add: local.INF_greatest)
    also have "Inf (G ` A)  (Inf ` ?F)"
      by (rule SUP_upper) (use E in blast)
    finally have "(Sup ` A)  (Inf ` ?F)"
      by simp
    with C show ?thesis
      using not_less by blast
  qed
qed
end

end