src/HOL/Hilbert_Choice.thy
 author haftmann Mon Mar 01 13:40:23 2010 +0100 (2010-03-01) changeset 35416 d8d7d1b785af parent 35216 7641e8d831d2 child 39036 dff91b90d74c permissions -rw-r--r--
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
1 (*  Title:      HOL/Hilbert_Choice.thy
2     Author:     Lawrence C Paulson, Tobias Nipkow
3     Copyright   2001  University of Cambridge
4 *)
6 header {* Hilbert's Epsilon-Operator and the Axiom of Choice *}
8 theory Hilbert_Choice
9 imports Nat Wellfounded Plain
10 uses ("Tools/meson.ML") ("Tools/choice_specification.ML")
11 begin
13 subsection {* Hilbert's epsilon *}
15 axiomatization Eps :: "('a => bool) => 'a" where
16   someI: "P x ==> P (Eps P)"
18 syntax (epsilon)
19   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3\<some>_./ _)" [0, 10] 10)
20 syntax (HOL)
21   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3@ _./ _)" [0, 10] 10)
22 syntax
23   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3SOME _./ _)" [0, 10] 10)
24 translations
25   "SOME x. P" == "CONST Eps (%x. P)"
27 print_translation {*
28   [(@{const_syntax Eps}, fn [Abs abs] =>
29       let val (x, t) = atomic_abs_tr' abs
30       in Syntax.const @{syntax_const "_Eps"} \$ x \$ t end)]
31 *} -- {* to avoid eta-contraction of body *}
33 definition inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
34 "inv_into A f == %x. SOME y. y : A & f y = x"
36 abbreviation inv :: "('a => 'b) => ('b => 'a)" where
37 "inv == inv_into UNIV"
40 subsection {*Hilbert's Epsilon-operator*}
42 text{*Easier to apply than @{text someI} if the witness comes from an
43 existential formula*}
44 lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)"
45 apply (erule exE)
46 apply (erule someI)
47 done
49 text{*Easier to apply than @{text someI} because the conclusion has only one
50 occurrence of @{term P}.*}
51 lemma someI2: "[| P a;  !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
52 by (blast intro: someI)
54 text{*Easier to apply than @{text someI2} if the witness comes from an
55 existential formula*}
56 lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
57 by (blast intro: someI2)
59 lemma some_equality [intro]:
60      "[| P a;  !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
61 by (blast intro: someI2)
63 lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
64 by blast
66 lemma some_eq_ex: "P (SOME x. P x) =  (\<exists>x. P x)"
67 by (blast intro: someI)
69 lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
70 apply (rule some_equality)
71 apply (rule refl, assumption)
72 done
74 lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"
75 apply (rule some_equality)
76 apply (rule refl)
77 apply (erule sym)
78 done
81 subsection{*Axiom of Choice, Proved Using the Description Operator*}
83 text{*Used in @{text "Tools/meson.ML"}*}
84 lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)"
85 by (fast elim: someI)
87 lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)"
88 by (fast elim: someI)
91 subsection {*Function Inverse*}
93 lemma inv_def: "inv f = (%y. SOME x. f x = y)"
94 by(simp add: inv_into_def)
96 lemma inv_into_into: "x : f ` A ==> inv_into A f x : A"
97 apply (simp add: inv_into_def)
98 apply (fast intro: someI2)
99 done
101 lemma inv_id [simp]: "inv id = id"
102 by (simp add: inv_into_def id_def)
104 lemma inv_into_f_f [simp]:
105   "[| inj_on f A;  x : A |] ==> inv_into A f (f x) = x"
106 apply (simp add: inv_into_def inj_on_def)
107 apply (blast intro: someI2)
108 done
110 lemma inv_f_f: "inj f ==> inv f (f x) = x"
111 by simp
113 lemma f_inv_into_f: "y : f`A  ==> f (inv_into A f y) = y"
114 apply (simp add: inv_into_def)
115 apply (fast intro: someI2)
116 done
118 lemma inv_into_f_eq: "[| inj_on f A; x : A; f x = y |] ==> inv_into A f y = x"
119 apply (erule subst)
120 apply (fast intro: inv_into_f_f)
121 done
123 lemma inv_f_eq: "[| inj f; f x = y |] ==> inv f y = x"
124 by (simp add:inv_into_f_eq)
126 lemma inj_imp_inv_eq: "[| inj f; ALL x. f(g x) = x |] ==> inv f = g"
127 by (blast intro: ext inv_into_f_eq)
129 text{*But is it useful?*}
130 lemma inj_transfer:
131   assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)"
132   shows "P x"
133 proof -
134   have "f x \<in> range f" by auto
135   hence "P(inv f (f x))" by (rule minor)
136   thus "P x" by (simp add: inv_into_f_f [OF injf])
137 qed
139 lemma inj_iff: "(inj f) = (inv f o f = id)"
140 apply (simp add: o_def expand_fun_eq)
141 apply (blast intro: inj_on_inverseI inv_into_f_f)
142 done
144 lemma inv_o_cancel[simp]: "inj f ==> inv f o f = id"
145 by (simp add: inj_iff)
147 lemma o_inv_o_cancel[simp]: "inj f ==> g o inv f o f = g"
148 by (simp add: o_assoc[symmetric])
150 lemma inv_into_image_cancel[simp]:
151   "inj_on f A ==> S <= A ==> inv_into A f ` f ` S = S"
152 by(fastsimp simp: image_def)
154 lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
155 by (blast intro: surjI inv_into_f_f)
157 lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
158 by (simp add: f_inv_into_f surj_range)
160 lemma inv_into_injective:
161   assumes eq: "inv_into A f x = inv_into A f y"
162       and x: "x: f`A"
163       and y: "y: f`A"
164   shows "x=y"
165 proof -
166   have "f (inv_into A f x) = f (inv_into A f y)" using eq by simp
167   thus ?thesis by (simp add: f_inv_into_f x y)
168 qed
170 lemma inj_on_inv_into: "B <= f`A ==> inj_on (inv_into A f) B"
171 by (blast intro: inj_onI dest: inv_into_injective injD)
173 lemma bij_betw_inv_into: "bij_betw f A B ==> bij_betw (inv_into A f) B A"
174 by (auto simp add: bij_betw_def inj_on_inv_into)
176 lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
177 by (simp add: inj_on_inv_into surj_range)
179 lemma surj_iff: "(surj f) = (f o inv f = id)"
180 apply (simp add: o_def expand_fun_eq)
181 apply (blast intro: surjI surj_f_inv_f)
182 done
184 lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
185 apply (rule ext)
186 apply (drule_tac x = "inv f x" in spec)
187 apply (simp add: surj_f_inv_f)
188 done
190 lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
191 by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
193 lemma inv_equality: "[| !!x. g (f x) = x;  !!y. f (g y) = y |] ==> inv f = g"
194 apply (rule ext)
195 apply (auto simp add: inv_into_def)
196 done
198 lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
199 apply (rule inv_equality)
200 apply (auto simp add: bij_def surj_f_inv_f)
201 done
203 (** bij(inv f) implies little about f.  Consider f::bool=>bool such that
204     f(True)=f(False)=True.  Then it's consistent with axiom someI that
205     inv f could be any function at all, including the identity function.
206     If inv f=id then inv f is a bijection, but inj f, surj(f) and
207     inv(inv f)=f all fail.
208 **)
210 lemma inv_into_comp:
211   "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
212   inv_into A (f o g) x = (inv_into A g o inv_into (g ` A) f) x"
213 apply (rule inv_into_f_eq)
214   apply (fast intro: comp_inj_on)
215  apply (simp add: inv_into_into)
216 apply (simp add: f_inv_into_f inv_into_into)
217 done
219 lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
220 apply (rule inv_equality)
221 apply (auto simp add: bij_def surj_f_inv_f)
222 done
224 lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
225 by (simp add: image_eq_UN surj_f_inv_f)
227 lemma image_inv_f_f: "inj f ==> (inv f) ` (f ` A) = A"
228 by (simp add: image_eq_UN)
230 lemma inv_image_comp: "inj f ==> inv f ` (f`X) = X"
231 by (auto simp add: image_def)
233 lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
234 apply auto
235 apply (force simp add: bij_is_inj)
236 apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
237 done
239 lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A"
240 apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
241 apply (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric])
242 done
244 lemma finite_fun_UNIVD1:
245   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
246   and card: "card (UNIV :: 'b set) \<noteq> Suc 0"
247   shows "finite (UNIV :: 'a set)"
248 proof -
249   from fin have finb: "finite (UNIV :: 'b set)" by (rule finite_fun_UNIVD2)
250   with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)"
251     by (cases "card (UNIV :: 'b set)") (auto simp add: card_eq_0_iff)
252   then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - Suc (Suc 0)" by auto
253   then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by (auto simp add: card_Suc_eq)
254   from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by (rule finite_imageI)
255   moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)"
256   proof (rule UNIV_eq_I)
257     fix x :: 'a
258     from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_into_def)
259     thus "x \<in> range (\<lambda>f\<Colon>'a \<Rightarrow> 'b. inv f b1)" by blast
260   qed
261   ultimately show "finite (UNIV :: 'a set)" by simp
262 qed
265 subsection {*Other Consequences of Hilbert's Epsilon*}
267 text {*Hilbert's Epsilon and the @{term split} Operator*}
269 text{*Looping simprule*}
270 lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
271   by simp
273 lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"
274   by (simp add: split_def)
276 lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
277   by blast
280 text{*A relation is wellfounded iff it has no infinite descending chain*}
281 lemma wf_iff_no_infinite_down_chain:
282   "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
283 apply (simp only: wf_eq_minimal)
284 apply (rule iffI)
285  apply (rule notI)
286  apply (erule exE)
287  apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
288 apply (erule contrapos_np, simp, clarify)
289 apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
290  apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI)
291  apply (rule allI, simp)
292  apply (rule someI2_ex, blast, blast)
293 apply (rule allI)
294 apply (induct_tac "n", simp_all)
295 apply (rule someI2_ex, blast+)
296 done
298 lemma wf_no_infinite_down_chainE:
299   assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r"
300 using `wf r` wf_iff_no_infinite_down_chain[of r] by blast
303 text{*A dynamically-scoped fact for TFL *}
304 lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
305   by (blast intro: someI)
308 subsection {* Least value operator *}
310 definition
311   LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where
312   "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
314 syntax
315   "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
316 translations
317   "LEAST x WRT m. P" == "CONST LeastM m (%x. P)"
319 lemma LeastMI2:
320   "P x ==> (!!y. P y ==> m x <= m y)
321     ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
322     ==> Q (LeastM m P)"
323   apply (simp add: LeastM_def)
324   apply (rule someI2_ex, blast, blast)
325   done
327 lemma LeastM_equality:
328   "P k ==> (!!x. P x ==> m k <= m x)
329     ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
330   apply (rule LeastMI2, assumption, blast)
331   apply (blast intro!: order_antisym)
332   done
334 lemma wf_linord_ex_has_least:
335   "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
336     ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
337   apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
338   apply (drule_tac x = "m`Collect P" in spec, force)
339   done
341 lemma ex_has_least_nat:
342     "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
343   apply (simp only: pred_nat_trancl_eq_le [symmetric])
344   apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
345    apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
346   done
348 lemma LeastM_nat_lemma:
349     "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
350   apply (simp add: LeastM_def)
351   apply (rule someI_ex)
352   apply (erule ex_has_least_nat)
353   done
355 lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1, standard]
357 lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
358 by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
361 subsection {* Greatest value operator *}
363 definition
364   GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where
365   "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
367 definition
368   Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where
369   "Greatest == GreatestM (%x. x)"
371 syntax
372   "_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"
373       ("GREATEST _ WRT _. _" [0, 4, 10] 10)
374 translations
375   "GREATEST x WRT m. P" == "CONST GreatestM m (%x. P)"
377 lemma GreatestMI2:
378   "P x ==> (!!y. P y ==> m y <= m x)
379     ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
380     ==> Q (GreatestM m P)"
381   apply (simp add: GreatestM_def)
382   apply (rule someI2_ex, blast, blast)
383   done
385 lemma GreatestM_equality:
386  "P k ==> (!!x. P x ==> m x <= m k)
387     ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
388   apply (rule_tac m = m in GreatestMI2, assumption, blast)
389   apply (blast intro!: order_antisym)
390   done
392 lemma Greatest_equality:
393   "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
394   apply (simp add: Greatest_def)
395   apply (erule GreatestM_equality, blast)
396   done
398 lemma ex_has_greatest_nat_lemma:
399   "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
400     ==> \<exists>y. P y & ~ (m y < m k + n)"
401   apply (induct n, force)
402   apply (force simp add: le_Suc_eq)
403   done
405 lemma ex_has_greatest_nat:
406   "P k ==> \<forall>y. P y --> m y < b
407     ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
408   apply (rule ccontr)
409   apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
410     apply (subgoal_tac  "m k <= b", auto)
411   done
413 lemma GreatestM_nat_lemma:
414   "P k ==> \<forall>y. P y --> m y < b
415     ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
416   apply (simp add: GreatestM_def)
417   apply (rule someI_ex)
418   apply (erule ex_has_greatest_nat, assumption)
419   done
421 lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard]
423 lemma GreatestM_nat_le:
424   "P x ==> \<forall>y. P y --> m y < b
425     ==> (m x::nat) <= m (GreatestM m P)"
426   apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
427   done
430 text {* \medskip Specialization to @{text GREATEST}. *}
432 lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
433   apply (simp add: Greatest_def)
434   apply (rule GreatestM_natI, auto)
435   done
437 lemma Greatest_le:
438     "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
439   apply (simp add: Greatest_def)
440   apply (rule GreatestM_nat_le, auto)
441   done
444 subsection {* The Meson proof procedure *}
446 subsubsection {* Negation Normal Form *}
448 text {* de Morgan laws *}
450 lemma meson_not_conjD: "~(P&Q) ==> ~P | ~Q"
451   and meson_not_disjD: "~(P|Q) ==> ~P & ~Q"
452   and meson_not_notD: "~~P ==> P"
453   and meson_not_allD: "!!P. ~(\<forall>x. P(x)) ==> \<exists>x. ~P(x)"
454   and meson_not_exD: "!!P. ~(\<exists>x. P(x)) ==> \<forall>x. ~P(x)"
455   by fast+
457 text {* Removal of @{text "-->"} and @{text "<->"} (positive and
458 negative occurrences) *}
460 lemma meson_imp_to_disjD: "P-->Q ==> ~P | Q"
461   and meson_not_impD: "~(P-->Q) ==> P & ~Q"
462   and meson_iff_to_disjD: "P=Q ==> (~P | Q) & (~Q | P)"
463   and meson_not_iffD: "~(P=Q) ==> (P | Q) & (~P | ~Q)"
464     -- {* Much more efficient than @{prop "(P & ~Q) | (Q & ~P)"} for computing CNF *}
465   and meson_not_refl_disj_D: "x ~= x | P ==> P"
466   by fast+
469 subsubsection {* Pulling out the existential quantifiers *}
471 text {* Conjunction *}
473 lemma meson_conj_exD1: "!!P Q. (\<exists>x. P(x)) & Q ==> \<exists>x. P(x) & Q"
474   and meson_conj_exD2: "!!P Q. P & (\<exists>x. Q(x)) ==> \<exists>x. P & Q(x)"
475   by fast+
478 text {* Disjunction *}
480 lemma meson_disj_exD: "!!P Q. (\<exists>x. P(x)) | (\<exists>x. Q(x)) ==> \<exists>x. P(x) | Q(x)"
481   -- {* DO NOT USE with forall-Skolemization: makes fewer schematic variables!! *}
482   -- {* With ex-Skolemization, makes fewer Skolem constants *}
483   and meson_disj_exD1: "!!P Q. (\<exists>x. P(x)) | Q ==> \<exists>x. P(x) | Q"
484   and meson_disj_exD2: "!!P Q. P | (\<exists>x. Q(x)) ==> \<exists>x. P | Q(x)"
485   by fast+
488 subsubsection {* Generating clauses for the Meson Proof Procedure *}
490 text {* Disjunctions *}
492 lemma meson_disj_assoc: "(P|Q)|R ==> P|(Q|R)"
493   and meson_disj_comm: "P|Q ==> Q|P"
494   and meson_disj_FalseD1: "False|P ==> P"
495   and meson_disj_FalseD2: "P|False ==> P"
496   by fast+
499 subsection{*Lemmas for Meson, the Model Elimination Procedure*}
501 text{* Generation of contrapositives *}
503 text{*Inserts negated disjunct after removing the negation; P is a literal.
504   Model elimination requires assuming the negation of every attempted subgoal,
505   hence the negated disjuncts.*}
506 lemma make_neg_rule: "~P|Q ==> ((~P==>P) ==> Q)"
507 by blast
509 text{*Version for Plaisted's "Postive refinement" of the Meson procedure*}
510 lemma make_refined_neg_rule: "~P|Q ==> (P ==> Q)"
511 by blast
513 text{*@{term P} should be a literal*}
514 lemma make_pos_rule: "P|Q ==> ((P==>~P) ==> Q)"
515 by blast
517 text{*Versions of @{text make_neg_rule} and @{text make_pos_rule} that don't
518 insert new assumptions, for ordinary resolution.*}
520 lemmas make_neg_rule' = make_refined_neg_rule
522 lemma make_pos_rule': "[|P|Q; ~P|] ==> Q"
523 by blast
525 text{* Generation of a goal clause -- put away the final literal *}
527 lemma make_neg_goal: "~P ==> ((~P==>P) ==> False)"
528 by blast
530 lemma make_pos_goal: "P ==> ((P==>~P) ==> False)"
531 by blast
534 subsubsection{* Lemmas for Forward Proof*}
536 text{*There is a similarity to congruence rules*}
538 (*NOTE: could handle conjunctions (faster?) by
539     nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
540 lemma conj_forward: "[| P'&Q';  P' ==> P;  Q' ==> Q |] ==> P&Q"
541 by blast
543 lemma disj_forward: "[| P'|Q';  P' ==> P;  Q' ==> Q |] ==> P|Q"
544 by blast
546 (*Version of @{text disj_forward} for removal of duplicate literals*)
547 lemma disj_forward2:
548     "[| P'|Q';  P' ==> P;  [| Q'; P==>False |] ==> Q |] ==> P|Q"
549 apply blast
550 done
552 lemma all_forward: "[| \<forall>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<forall>x. P(x)"
553 by blast
555 lemma ex_forward: "[| \<exists>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<exists>x. P(x)"
556 by blast
559 subsection {* Meson package *}
561 use "Tools/meson.ML"
563 setup Meson.setup
566 subsection {* Specification package -- Hilbertized version *}
568 lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
569   by (simp only: someI_ex)
571 use "Tools/choice_specification.ML"
574 end