(*  Title:      HOL/Hilbert_Choice.thy

     Author:     Lawrence C Paulson

     Copyright   2001  University of Cambridge

 *)

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

 theory Hilbert_Choice

 imports Nat Wellfounded Plain

 uses ("Tools/meson.ML") ("Tools/specification_package.ML")

 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 {*

 (* to avoid eta-contraction of body *)

 [(@{const_syntax Eps}, fn [Abs abs] =>

      let val (x,t) = atomic_abs_tr' abs

      in Syntax.const "_Eps" $ x $ t end)]

 *}

 constdefs

   inv :: "('a => 'b) => ('b => 'a)"

   "inv(f :: 'a => 'b) == %y. SOME x. f x = y"

   Inv :: "'a set => ('a => 'b) => ('b => 'a)"

   "Inv A f == %x. SOME y. y \<in> A & f y = x"

 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 intro: some_equality)

 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*}

 text{*Used in @{text "Tools/meson.ML"}*}

 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)

 subsection {*Function Inverse*}

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

 by (simp add: inv_def id_def)

 text{*A one-to-one function has an inverse.*}

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

 by (simp add: inv_def inj_eq)

 lemma inv_f_eq: "[| inj f;  f x = y |] ==> inv f y = x"

 apply (erule subst)

 apply (erule inv_f_f)

 done

 lemma inj_imp_inv_eq: "[| inj f; \<forall>x. f(g x) = x |] ==> inv f = g"

 by (blast intro: ext inv_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_f_f [OF injf])

 qed

 lemma inj_iff: "(inj f) = (inv f o f = id)"

 apply (simp add: o_def expand_fun_eq)

 apply (blast intro: inj_on_inverseI inv_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: o_assoc[symmetric])

 lemma inv_image_cancel[simp]:

   "inj f ==> inv f ` f ` S = S"

 by (simp add: image_compose[symmetric])

 lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"

 by (blast intro: surjI inv_f_f)

 lemma f_inv_f: "y \<in> range(f) ==> f(inv f y) = y"

 apply (simp add: inv_def)

 apply (fast intro: someI)

 done

 lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"

 by (simp add: f_inv_f surj_range)

 lemma inv_injective:

   assumes eq: "inv f x = inv f y"

       and x: "x: range f"

       and y: "y: range f"

   shows "x=y"

 proof -

   have "f (inv f x) = f (inv f y)" using eq by simp

   thus ?thesis by (simp add: f_inv_f x y) 

 qed

 lemma inj_on_inv: "A <= range(f) ==> inj_on (inv f) A"

 by (fast intro: inj_onI elim: inv_injective injD)

 lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"

 by (simp add: inj_on_inv surj_range)

 lemma surj_iff: "(surj f) = (f o inv f = id)"

 apply (simp add: o_def expand_fun_eq)

 apply (blast intro: surjI surj_f_inv_f)

 done

 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_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 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_f_f, symmetric])

 done

 subsection {*Inverse of a PI-function (restricted domain)*}

 lemma Inv_f_f: "[| inj_on f A;  x \<in> A |] ==> Inv A f (f x) = x"

 apply (simp add: Inv_def inj_on_def)

 apply (blast intro: someI2)

 done

 lemma f_Inv_f: "y \<in> f`A  ==> f (Inv A f y) = y"

 apply (simp add: Inv_def)

 apply (fast intro: someI2)

 done

 lemma Inv_injective:

   assumes eq: "Inv A f x = Inv A f y"

       and x: "x: f`A"

       and y: "y: f`A"

   shows "x=y"

 proof -

   have "f (Inv A f x) = f (Inv A f y)" using eq by simp

   thus ?thesis by (simp add: f_Inv_f x y) 

 qed

 lemma inj_on_Inv: "B <= f`A ==> inj_on (Inv A f) B"

 apply (rule inj_onI)

 apply (blast intro: inj_onI dest: Inv_injective injD)

 done

 lemma Inv_mem: "[| f ` A = B;  x \<in> B |] ==> Inv A f x \<in> A"

 apply (simp add: Inv_def)

 apply (fast intro: someI2)

 done

 lemma Inv_f_eq: "[| inj_on f A; f x = y; x \<in> A |] ==> Inv A f y = x"

   apply (erule subst)

   apply (erule Inv_f_f, assumption)

   done

 lemma Inv_comp:

   "[| inj_on f (g ` A); inj_on g A; x \<in> f ` g ` A |] ==>

   Inv A (f o g) x = (Inv A g o Inv (g ` A) f) x"

   apply simp

   apply (rule Inv_f_eq)

     apply (fast intro: comp_inj_on)

    apply (simp add: f_Inv_f Inv_mem)

   apply (simp add: Inv_mem)

   done

 lemma bij_betw_Inv: "bij_betw f A B \<Longrightarrow> bij_betw (Inv A f) B A"

   apply (auto simp add: bij_betw_def inj_on_Inv Inv_mem)

   apply (simp add: image_compose [symmetric] o_def)

   apply (simp add: image_def Inv_f_f)

   done

 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 *}

 constdefs

   LeastM :: "['a => 'b::ord, 'a => bool] => 'a"

   "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" == "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, standard]

 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 *}

 constdefs

   GreatestM :: "['a => 'b::ord, 'a => bool] => 'a"

   "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"

   Greatest :: "('a::ord => bool) => 'a"    (binder "GREATEST " 10)

   "Greatest == GreatestM (%x. x)"

 syntax

   "_GreatestM" :: "[pttrn, 'a=>'b::ord, bool] => 'a"

       ("GREATEST _ WRT _. _" [0, 4, 10] 10)

 translations

   "GREATEST x WRT m. P" == "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, standard]

 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 {* The Meson proof procedure *}

 subsubsection {* Negation Normal Form *}

 text {* de Morgan laws *}

 lemma meson_not_conjD: "~(P&Q) ==> ~P | ~Q"

   and meson_not_disjD: "~(P|Q) ==> ~P & ~Q"

   and meson_not_notD: "~~P ==> P"

   and meson_not_allD: "!!P. ~(\<forall>x. P(x)) ==> \<exists>x. ~P(x)"

   and meson_not_exD: "!!P. ~(\<exists>x. P(x)) ==> \<forall>x. ~P(x)"

   by fast+

 text {* Removal of @{text "-->"} and @{text "<->"} (positive and

 negative occurrences) *}

 lemma meson_imp_to_disjD: "P-->Q ==> ~P | Q"

   and meson_not_impD: "~(P-->Q) ==> P & ~Q"

   and meson_iff_to_disjD: "P=Q ==> (~P | Q) & (~Q | P)"

   and meson_not_iffD: "~(P=Q) ==> (P | Q) & (~P | ~Q)"

     -- {* Much more efficient than @{prop "(P & ~Q) | (Q & ~P)"} for computing CNF *}

   and meson_not_refl_disj_D: "x ~= x | P ==> P"

   by fast+

 subsubsection {* Pulling out the existential quantifiers *}

 text {* Conjunction *}

 lemma meson_conj_exD1: "!!P Q. (\<exists>x. P(x)) & Q ==> \<exists>x. P(x) & Q"

   and meson_conj_exD2: "!!P Q. P & (\<exists>x. Q(x)) ==> \<exists>x. P & Q(x)"

   by fast+

 text {* Disjunction *}

 lemma meson_disj_exD: "!!P Q. (\<exists>x. P(x)) | (\<exists>x. Q(x)) ==> \<exists>x. P(x) | Q(x)"

   -- {* DO NOT USE with forall-Skolemization: makes fewer schematic variables!! *}

   -- {* With ex-Skolemization, makes fewer Skolem constants *}

   and meson_disj_exD1: "!!P Q. (\<exists>x. P(x)) | Q ==> \<exists>x. P(x) | Q"

   and meson_disj_exD2: "!!P Q. P | (\<exists>x. Q(x)) ==> \<exists>x. P | Q(x)"

   by fast+

 subsubsection {* Generating clauses for the Meson Proof Procedure *}

 text {* Disjunctions *}

 lemma meson_disj_assoc: "(P|Q)|R ==> P|(Q|R)"

   and meson_disj_comm: "P|Q ==> Q|P"

   and meson_disj_FalseD1: "False|P ==> P"

   and meson_disj_FalseD2: "P|False ==> P"

   by fast+

 subsection{*Lemmas for Meson, the Model Elimination Procedure*}

 text{* Generation of contrapositives *}

 text{*Inserts negated disjunct after removing the negation; P is a literal.

   Model elimination requires assuming the negation of every attempted subgoal,

   hence the negated disjuncts.*}

 lemma make_neg_rule: "~P|Q ==> ((~P==>P) ==> Q)"

 by blast

 text{*Version for Plaisted's "Postive refinement" of the Meson procedure*}

 lemma make_refined_neg_rule: "~P|Q ==> (P ==> Q)"

 by blast

 text{*@{term P} should be a literal*}

 lemma make_pos_rule: "P|Q ==> ((P==>~P) ==> Q)"

 by blast

 text{*Versions of @{text make_neg_rule} and @{text make_pos_rule} that don't

 insert new assumptions, for ordinary resolution.*}

 lemmas make_neg_rule' = make_refined_neg_rule

 lemma make_pos_rule': "[|P|Q; ~P|] ==> Q"

 by blast

 text{* Generation of a goal clause -- put away the final literal *}

 lemma make_neg_goal: "~P ==> ((~P==>P) ==> False)"

 by blast

 lemma make_pos_goal: "P ==> ((P==>~P) ==> False)"

 by blast

 subsubsection{* Lemmas for Forward Proof*}

 text{*There is a similarity to congruence rules*}

 (*NOTE: could handle conjunctions (faster?) by

     nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)

 lemma conj_forward: "[| P'&Q';  P' ==> P;  Q' ==> Q |] ==> P&Q"

 by blast

 lemma disj_forward: "[| P'|Q';  P' ==> P;  Q' ==> Q |] ==> P|Q"

 by blast

 (*Version of @{text disj_forward} for removal of duplicate literals*)

 lemma disj_forward2:

     "[| P'|Q';  P' ==> P;  [| Q'; P==>False |] ==> Q |] ==> P|Q"

 apply blast 

 done

 lemma all_forward: "[| \<forall>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<forall>x. P(x)"

 by blast

 lemma ex_forward: "[| \<exists>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<exists>x. P(x)"

 by blast

 text{*Many of these bindings are used by the ATP linkup, and not just by

 legacy proof scripts.*}

 ML

 {*

 val inv_def = thm "inv_def";

 val Inv_def = thm "Inv_def";

 val someI = thm "someI";

 val someI_ex = thm "someI_ex";

 val someI2 = thm "someI2";

 val someI2_ex = thm "someI2_ex";

 val some_equality = thm "some_equality";

 val some1_equality = thm "some1_equality";

 val some_eq_ex = thm "some_eq_ex";

 val some_eq_trivial = thm "some_eq_trivial";

 val some_sym_eq_trivial = thm "some_sym_eq_trivial";

 val choice = thm "choice";

 val bchoice = thm "bchoice";

 val inv_id = thm "inv_id";

 val inv_f_f = thm "inv_f_f";

 val inv_f_eq = thm "inv_f_eq";

 val inj_imp_inv_eq = thm "inj_imp_inv_eq";

 val inj_transfer = thm "inj_transfer";

 val inj_iff = thm "inj_iff";

 val inj_imp_surj_inv = thm "inj_imp_surj_inv";

 val f_inv_f = thm "f_inv_f";

 val surj_f_inv_f = thm "surj_f_inv_f";

 val inv_injective = thm "inv_injective";

 val inj_on_inv = thm "inj_on_inv";

 val surj_imp_inj_inv = thm "surj_imp_inj_inv";

 val surj_iff = thm "surj_iff";

 val surj_imp_inv_eq = thm "surj_imp_inv_eq";

 val bij_imp_bij_inv = thm "bij_imp_bij_inv";

 val inv_equality = thm "inv_equality";

 val inv_inv_eq = thm "inv_inv_eq";

 val o_inv_distrib = thm "o_inv_distrib";

 val image_surj_f_inv_f = thm "image_surj_f_inv_f";

 val image_inv_f_f = thm "image_inv_f_f";

 val inv_image_comp = thm "inv_image_comp";

 val bij_image_Collect_eq = thm "bij_image_Collect_eq";

 val bij_vimage_eq_inv_image = thm "bij_vimage_eq_inv_image";

 val Inv_f_f = thm "Inv_f_f";

 val f_Inv_f = thm "f_Inv_f";

 val Inv_injective = thm "Inv_injective";

 val inj_on_Inv = thm "inj_on_Inv";

 val split_paired_Eps = thm "split_paired_Eps";

 val Eps_split = thm "Eps_split";

 val Eps_split_eq = thm "Eps_split_eq";

 val wf_iff_no_infinite_down_chain = thm "wf_iff_no_infinite_down_chain";

 val Inv_mem = thm "Inv_mem";

 val Inv_f_eq = thm "Inv_f_eq";

 val Inv_comp = thm "Inv_comp";

 val tfl_some = thm "tfl_some";

 val make_neg_rule = thm "make_neg_rule";

 val make_refined_neg_rule = thm "make_refined_neg_rule";

 val make_pos_rule = thm "make_pos_rule";

 val make_neg_rule' = thm "make_neg_rule'";

 val make_pos_rule' = thm "make_pos_rule'";

 val make_neg_goal = thm "make_neg_goal";

 val make_pos_goal = thm "make_pos_goal";

 val conj_forward = thm "conj_forward";

 val disj_forward = thm "disj_forward";

 val disj_forward2 = thm "disj_forward2";

 val all_forward = thm "all_forward";

 val ex_forward = thm "ex_forward";

 *}

 subsection {* Meson package *}

 use "Tools/meson.ML"

 setup Meson.setup

 subsection {* Specification package -- Hilbertized version *}

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

   by (simp only: someI_ex)

 use "Tools/specification_package.ML"

 end