# Theory Hilbert_Choice

Up to index of Isabelle/HOL

theory Hilbert_Choice
imports Big_Operators
(*  Title:      HOL/Hilbert_Choice.thy
Author: Lawrence C Paulson, Tobias Nipkow
*)

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 [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?]: "∃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: "[| ∃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) = (∃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: "∀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)"

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

lemma inv_id [simp]: "inv id = id"

lemma inv_into_f_f [simp]:
"[| inj_on f A; x : A |] ==> inv_into A f (f x) = x"
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 (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"

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])
qed

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

lemma inv_o_cancel[simp]: "inj f ==> inv f o f = id"

lemma o_inv_o_cancel[simp]: "inj f ==> g o inv f o f = g"

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"

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)"

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)
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)
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)
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"

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

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

lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
apply auto
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 => '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 simp
qed

lemma image_inv_into_cancel:
assumes SURJ: "f`A=A'" and SUB: "B' ≤ A'"
shows "f `((inv_into A f)`B') = B'"
using assms
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 blast
qed

lemma 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
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
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
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 blast
next
fix g let ?f = "inv_into A' g"
have "inj_on ?f (g ` A')"
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 auto
qed

lemma 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 auto
qed

subsection {* 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 blast
qed

theorem 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 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))"

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 = (~(∃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+)
done

lemma 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 blast

text{*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 (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 ==> ∀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)
done

lemma 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)
done

lemma LeastM_nat_lemma:
"P k ==> P (LeastM m P) & (∀y. P y --> m (LeastM m P) <= (m y::nat))"
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 & (∀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 (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 (erule GreatestM_equality, blast)
done

lemma 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)
done

lemma 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)
done

lemma 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 (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 ==> ∀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) ==> ∀y. P y --> y < b ==> P (GREATEST x. P x)"
apply (rule GreatestM_natI, auto)
done

lemma Greatest_le:
"P x ==> ∀y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
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 => '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 assms
proof -
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
qed
qed

lemma 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)
done
qed

primrec 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)
qed

lemma 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 auto
qed

lemma 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