src/HOL/Hilbert_Choice.thy
 author nipkow Mon Sep 13 11:13:15 2010 +0200 (2010-09-13) changeset 39302 d7728f65b353 parent 39198 f967a16dfcdd child 39900 549c00e0e89b permissions -rw-r--r--
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
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")
11      ("Tools/choice_specification.ML")
12 begin
14 subsection {* Hilbert's epsilon *}
16 axiomatization Eps :: "('a => bool) => 'a" where
17   someI: "P x ==> P (Eps P)"
19 syntax (epsilon)
20   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3\<some>_./ _)" [0, 10] 10)
21 syntax (HOL)
22   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3@ _./ _)" [0, 10] 10)
23 syntax
24   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3SOME _./ _)" [0, 10] 10)
25 translations
26   "SOME x. P" == "CONST Eps (%x. P)"
28 print_translation {*
29   [(@{const_syntax Eps}, fn [Abs abs] =>
30       let val (x, t) = atomic_abs_tr' abs
31       in Syntax.const @{syntax_const "_Eps"} \$ x \$ t end)]
32 *} -- {* to avoid eta-contraction of body *}
34 definition inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
35 "inv_into A f == %x. SOME y. y : A & f y = x"
37 abbreviation inv :: "('a => 'b) => ('b => 'a)" where
38 "inv == inv_into UNIV"
41 subsection {*Hilbert's Epsilon-operator*}
43 text{*Easier to apply than @{text someI} if the witness comes from an
44 existential formula*}
45 lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)"
46 apply (erule exE)
47 apply (erule someI)
48 done
50 text{*Easier to apply than @{text someI} because the conclusion has only one
51 occurrence of @{term P}.*}
52 lemma someI2: "[| P a;  !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
53 by (blast intro: someI)
55 text{*Easier to apply than @{text someI2} if the witness comes from an
56 existential formula*}
57 lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
58 by (blast intro: someI2)
60 lemma some_equality [intro]:
61      "[| P a;  !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
62 by (blast intro: someI2)
64 lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
65 by blast
67 lemma some_eq_ex: "P (SOME x. P x) =  (\<exists>x. P x)"
68 by (blast intro: someI)
70 lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
71 apply (rule some_equality)
72 apply (rule refl, assumption)
73 done
75 lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"
76 apply (rule some_equality)
77 apply (rule refl)
78 apply (erule sym)
79 done
82 subsection{*Axiom of Choice, Proved Using the Description Operator*}
84 text{*Used in @{text "Tools/meson.ML"}*}
85 lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)"
86 by (fast elim: someI)
88 lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)"
89 by (fast elim: someI)
92 subsection {*Function Inverse*}
94 lemma inv_def: "inv f = (%y. SOME x. f x = y)"
95 by(simp add: inv_into_def)
97 lemma inv_into_into: "x : f ` A ==> inv_into A f x : A"
98 apply (simp add: inv_into_def)
99 apply (fast intro: someI2)
100 done
102 lemma inv_id [simp]: "inv id = id"
103 by (simp add: inv_into_def id_def)
105 lemma inv_into_f_f [simp]:
106   "[| inj_on f A;  x : A |] ==> inv_into A f (f x) = x"
107 apply (simp add: inv_into_def inj_on_def)
108 apply (blast intro: someI2)
109 done
111 lemma inv_f_f: "inj f ==> inv f (f x) = x"
112 by simp
114 lemma f_inv_into_f: "y : f`A  ==> f (inv_into A f y) = y"
115 apply (simp add: inv_into_def)
116 apply (fast intro: someI2)
117 done
119 lemma inv_into_f_eq: "[| inj_on f A; x : A; f x = y |] ==> inv_into A f y = x"
120 apply (erule subst)
121 apply (fast intro: inv_into_f_f)
122 done
124 lemma inv_f_eq: "[| inj f; f x = y |] ==> inv f y = x"
125 by (simp add:inv_into_f_eq)
127 lemma inj_imp_inv_eq: "[| inj f; ALL x. f(g x) = x |] ==> inv f = g"
128 by (blast intro: ext inv_into_f_eq)
130 text{*But is it useful?*}
131 lemma inj_transfer:
132   assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)"
133   shows "P x"
134 proof -
135   have "f x \<in> range f" by auto
136   hence "P(inv f (f x))" by (rule minor)
137   thus "P x" by (simp add: inv_into_f_f [OF injf])
138 qed
140 lemma inj_iff: "(inj f) = (inv f o f = id)"
141 apply (simp add: o_def fun_eq_iff)
142 apply (blast intro: inj_on_inverseI inv_into_f_f)
143 done
145 lemma inv_o_cancel[simp]: "inj f ==> inv f o f = id"
146 by (simp add: inj_iff)
148 lemma o_inv_o_cancel[simp]: "inj f ==> g o inv f o f = g"
149 by (simp add: o_assoc[symmetric])
151 lemma inv_into_image_cancel[simp]:
152   "inj_on f A ==> S <= A ==> inv_into A f ` f ` S = S"
153 by(fastsimp simp: image_def)
155 lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
156 by (blast intro: surjI inv_into_f_f)
158 lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
159 by (simp add: f_inv_into_f surj_range)
161 lemma inv_into_injective:
162   assumes eq: "inv_into A f x = inv_into A f y"
163       and x: "x: f`A"
164       and y: "y: f`A"
165   shows "x=y"
166 proof -
167   have "f (inv_into A f x) = f (inv_into A f y)" using eq by simp
168   thus ?thesis by (simp add: f_inv_into_f x y)
169 qed
171 lemma inj_on_inv_into: "B <= f`A ==> inj_on (inv_into A f) B"
172 by (blast intro: inj_onI dest: inv_into_injective injD)
174 lemma bij_betw_inv_into: "bij_betw f A B ==> bij_betw (inv_into A f) B A"
175 by (auto simp add: bij_betw_def inj_on_inv_into)
177 lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
178 by (simp add: inj_on_inv_into surj_range)
180 lemma surj_iff: "(surj f) = (f o inv f = id)"
181 apply (simp add: o_def fun_eq_iff)
182 apply (blast intro: surjI surj_f_inv_f)
183 done
185 lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
186 apply (rule ext)
187 apply (drule_tac x = "inv f x" in spec)
188 apply (simp add: surj_f_inv_f)
189 done
191 lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
192 by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
194 lemma inv_equality: "[| !!x. g (f x) = x;  !!y. f (g y) = y |] ==> inv f = g"
195 apply (rule ext)
196 apply (auto simp add: inv_into_def)
197 done
199 lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
200 apply (rule inv_equality)
201 apply (auto simp add: bij_def surj_f_inv_f)
202 done
204 (** bij(inv f) implies little about f.  Consider f::bool=>bool such that
205     f(True)=f(False)=True.  Then it's consistent with axiom someI that
206     inv f could be any function at all, including the identity function.
207     If inv f=id then inv f is a bijection, but inj f, surj(f) and
208     inv(inv f)=f all fail.
209 **)
211 lemma inv_into_comp:
212   "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
213   inv_into A (f o g) x = (inv_into A g o inv_into (g ` A) f) x"
214 apply (rule inv_into_f_eq)
215   apply (fast intro: comp_inj_on)
216  apply (simp add: inv_into_into)
217 apply (simp add: f_inv_into_f inv_into_into)
218 done
220 lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
221 apply (rule inv_equality)
222 apply (auto simp add: bij_def surj_f_inv_f)
223 done
225 lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
226 by (simp add: image_eq_UN surj_f_inv_f)
228 lemma image_inv_f_f: "inj f ==> (inv f) ` (f ` A) = A"
229 by (simp add: image_eq_UN)
231 lemma inv_image_comp: "inj f ==> inv f ` (f`X) = X"
232 by (auto simp add: image_def)
234 lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
235 apply auto
236 apply (force simp add: bij_is_inj)
237 apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
238 done
240 lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A"
241 apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
242 apply (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric])
243 done
245 lemma finite_fun_UNIVD1:
246   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
247   and card: "card (UNIV :: 'b set) \<noteq> Suc 0"
248   shows "finite (UNIV :: 'a set)"
249 proof -
250   from fin have finb: "finite (UNIV :: 'b set)" by (rule finite_fun_UNIVD2)
251   with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)"
252     by (cases "card (UNIV :: 'b set)") (auto simp add: card_eq_0_iff)
253   then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - Suc (Suc 0)" by auto
254   then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by (auto simp add: card_Suc_eq)
255   from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by (rule finite_imageI)
256   moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)"
257   proof (rule UNIV_eq_I)
258     fix x :: 'a
259     from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_into_def)
260     thus "x \<in> range (\<lambda>f\<Colon>'a \<Rightarrow> 'b. inv f b1)" by blast
261   qed
262   ultimately show "finite (UNIV :: 'a set)" by simp
263 qed
266 subsection {*Other Consequences of Hilbert's Epsilon*}
268 text {*Hilbert's Epsilon and the @{term split} Operator*}
270 text{*Looping simprule*}
271 lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
272   by simp
274 lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"
275   by (simp add: split_def)
277 lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
278   by blast
281 text{*A relation is wellfounded iff it has no infinite descending chain*}
282 lemma wf_iff_no_infinite_down_chain:
283   "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
284 apply (simp only: wf_eq_minimal)
285 apply (rule iffI)
286  apply (rule notI)
287  apply (erule exE)
288  apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
289 apply (erule contrapos_np, simp, clarify)
290 apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
291  apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI)
292  apply (rule allI, simp)
293  apply (rule someI2_ex, blast, blast)
294 apply (rule allI)
295 apply (induct_tac "n", simp_all)
296 apply (rule someI2_ex, blast+)
297 done
299 lemma wf_no_infinite_down_chainE:
300   assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r"
301 using `wf r` wf_iff_no_infinite_down_chain[of r] by blast
304 text{*A dynamically-scoped fact for TFL *}
305 lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
306   by (blast intro: someI)
309 subsection {* Least value operator *}
311 definition
312   LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where
313   "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
315 syntax
316   "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
317 translations
318   "LEAST x WRT m. P" == "CONST LeastM m (%x. P)"
320 lemma LeastMI2:
321   "P x ==> (!!y. P y ==> m x <= m y)
322     ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
323     ==> Q (LeastM m P)"
324   apply (simp add: LeastM_def)
325   apply (rule someI2_ex, blast, blast)
326   done
328 lemma LeastM_equality:
329   "P k ==> (!!x. P x ==> m k <= m x)
330     ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
331   apply (rule LeastMI2, assumption, blast)
332   apply (blast intro!: order_antisym)
333   done
335 lemma wf_linord_ex_has_least:
336   "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
337     ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
338   apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
339   apply (drule_tac x = "m`Collect P" in spec, force)
340   done
342 lemma ex_has_least_nat:
343     "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
344   apply (simp only: pred_nat_trancl_eq_le [symmetric])
345   apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
346    apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
347   done
349 lemma LeastM_nat_lemma:
350     "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
351   apply (simp add: LeastM_def)
352   apply (rule someI_ex)
353   apply (erule ex_has_least_nat)
354   done
356 lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1, standard]
358 lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
359 by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
362 subsection {* Greatest value operator *}
364 definition
365   GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where
366   "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
368 definition
369   Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where
370   "Greatest == GreatestM (%x. x)"
372 syntax
373   "_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"
374       ("GREATEST _ WRT _. _" [0, 4, 10] 10)
375 translations
376   "GREATEST x WRT m. P" == "CONST GreatestM m (%x. P)"
378 lemma GreatestMI2:
379   "P x ==> (!!y. P y ==> m y <= m x)
380     ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
381     ==> Q (GreatestM m P)"
382   apply (simp add: GreatestM_def)
383   apply (rule someI2_ex, blast, blast)
384   done
386 lemma GreatestM_equality:
387  "P k ==> (!!x. P x ==> m x <= m k)
388     ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
389   apply (rule_tac m = m in GreatestMI2, assumption, blast)
390   apply (blast intro!: order_antisym)
391   done
393 lemma Greatest_equality:
394   "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
395   apply (simp add: Greatest_def)
396   apply (erule GreatestM_equality, blast)
397   done
399 lemma ex_has_greatest_nat_lemma:
400   "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
401     ==> \<exists>y. P y & ~ (m y < m k + n)"
402   apply (induct n, force)
403   apply (force simp add: le_Suc_eq)
404   done
406 lemma ex_has_greatest_nat:
407   "P k ==> \<forall>y. P y --> m y < b
408     ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
409   apply (rule ccontr)
410   apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
411     apply (subgoal_tac  "m k <= b", auto)
412   done
414 lemma GreatestM_nat_lemma:
415   "P k ==> \<forall>y. P y --> m y < b
416     ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
417   apply (simp add: GreatestM_def)
418   apply (rule someI_ex)
419   apply (erule ex_has_greatest_nat, assumption)
420   done
422 lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard]
424 lemma GreatestM_nat_le:
425   "P x ==> \<forall>y. P y --> m y < b
426     ==> (m x::nat) <= m (GreatestM m P)"
427   apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
428   done
431 text {* \medskip Specialization to @{text GREATEST}. *}
433 lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
434   apply (simp add: Greatest_def)
435   apply (rule GreatestM_natI, auto)
436   done
438 lemma Greatest_le:
439     "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
440   apply (simp add: Greatest_def)
441   apply (rule GreatestM_nat_le, auto)
442   done
445 subsection {* The Meson proof procedure *}
447 subsubsection {* Negation Normal Form *}
449 text {* de Morgan laws *}
451 lemma meson_not_conjD: "~(P&Q) ==> ~P | ~Q"
452   and meson_not_disjD: "~(P|Q) ==> ~P & ~Q"
453   and meson_not_notD: "~~P ==> P"
454   and meson_not_allD: "!!P. ~(\<forall>x. P(x)) ==> \<exists>x. ~P(x)"
455   and meson_not_exD: "!!P. ~(\<exists>x. P(x)) ==> \<forall>x. ~P(x)"
456   by fast+
458 text {* Removal of @{text "-->"} and @{text "<->"} (positive and
459 negative occurrences) *}
461 lemma meson_imp_to_disjD: "P-->Q ==> ~P | Q"
462   and meson_not_impD: "~(P-->Q) ==> P & ~Q"
463   and meson_iff_to_disjD: "P=Q ==> (~P | Q) & (~Q | P)"
464   and meson_not_iffD: "~(P=Q) ==> (P | Q) & (~P | ~Q)"
465     -- {* Much more efficient than @{prop "(P & ~Q) | (Q & ~P)"} for computing CNF *}
466   and meson_not_refl_disj_D: "x ~= x | P ==> P"
467   by fast+
470 subsubsection {* Pulling out the existential quantifiers *}
472 text {* Conjunction *}
474 lemma meson_conj_exD1: "!!P Q. (\<exists>x. P(x)) & Q ==> \<exists>x. P(x) & Q"
475   and meson_conj_exD2: "!!P Q. P & (\<exists>x. Q(x)) ==> \<exists>x. P & Q(x)"
476   by fast+
479 text {* Disjunction *}
481 lemma meson_disj_exD: "!!P Q. (\<exists>x. P(x)) | (\<exists>x. Q(x)) ==> \<exists>x. P(x) | Q(x)"
482   -- {* DO NOT USE with forall-Skolemization: makes fewer schematic variables!! *}
483   -- {* With ex-Skolemization, makes fewer Skolem constants *}
484   and meson_disj_exD1: "!!P Q. (\<exists>x. P(x)) | Q ==> \<exists>x. P(x) | Q"
485   and meson_disj_exD2: "!!P Q. P | (\<exists>x. Q(x)) ==> \<exists>x. P | Q(x)"
486   by fast+
489 subsubsection {* Generating clauses for the Meson Proof Procedure *}
491 text {* Disjunctions *}
493 lemma meson_disj_assoc: "(P|Q)|R ==> P|(Q|R)"
494   and meson_disj_comm: "P|Q ==> Q|P"
495   and meson_disj_FalseD1: "False|P ==> P"
496   and meson_disj_FalseD2: "P|False ==> P"
497   by fast+
500 subsection{*Lemmas for Meson, the Model Elimination Procedure*}
502 text{* Generation of contrapositives *}
504 text{*Inserts negated disjunct after removing the negation; P is a literal.
505   Model elimination requires assuming the negation of every attempted subgoal,
506   hence the negated disjuncts.*}
507 lemma make_neg_rule: "~P|Q ==> ((~P==>P) ==> Q)"
508 by blast
510 text{*Version for Plaisted's "Postive refinement" of the Meson procedure*}
511 lemma make_refined_neg_rule: "~P|Q ==> (P ==> Q)"
512 by blast
514 text{*@{term P} should be a literal*}
515 lemma make_pos_rule: "P|Q ==> ((P==>~P) ==> Q)"
516 by blast
518 text{*Versions of @{text make_neg_rule} and @{text make_pos_rule} that don't
519 insert new assumptions, for ordinary resolution.*}
521 lemmas make_neg_rule' = make_refined_neg_rule
523 lemma make_pos_rule': "[|P|Q; ~P|] ==> Q"
524 by blast
526 text{* Generation of a goal clause -- put away the final literal *}
528 lemma make_neg_goal: "~P ==> ((~P==>P) ==> False)"
529 by blast
531 lemma make_pos_goal: "P ==> ((P==>~P) ==> False)"
532 by blast
535 subsubsection{* Lemmas for Forward Proof*}
537 text{*There is a similarity to congruence rules*}
539 (*NOTE: could handle conjunctions (faster?) by
540     nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
541 lemma conj_forward: "[| P'&Q';  P' ==> P;  Q' ==> Q |] ==> P&Q"
542 by blast
544 lemma disj_forward: "[| P'|Q';  P' ==> P;  Q' ==> Q |] ==> P|Q"
545 by blast
547 (*Version of @{text disj_forward} for removal of duplicate literals*)
548 lemma disj_forward2:
549     "[| P'|Q';  P' ==> P;  [| Q'; P==>False |] ==> Q |] ==> P|Q"
550 apply blast
551 done
553 lemma all_forward: "[| \<forall>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<forall>x. P(x)"
554 by blast
556 lemma ex_forward: "[| \<exists>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<exists>x. P(x)"
557 by blast
560 subsection {* Meson package *}
562 use "Tools/meson.ML"
564 setup Meson.setup
567 subsection {* Specification package -- Hilbertized version *}
569 lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
570   by (simp only: someI_ex)
572 use "Tools/choice_specification.ML"
575 end