src/HOL/Hilbert_Choice.thy
 author paulson Tue Dec 13 15:27:43 2005 +0100 (2005-12-13) changeset 18389 8352b1d3b639 parent 17893 aef5a6d11c2a child 21020 9af9ceb16d58 permissions -rw-r--r--
removal of functional reflexivity axioms
1 (*  Title:      HOL/Hilbert_Choice.thy
2     ID: \$Id\$
3     Author:     Lawrence C Paulson
4     Copyright   2001  University of Cambridge
5 *)
7 header {* Hilbert's Epsilon-Operator and the Axiom of Choice *}
9 theory Hilbert_Choice
10 imports NatArith
11 uses ("Tools/meson.ML") ("Tools/specification_package.ML")
12 begin
14 subsection {* Hilbert's epsilon *}
16 consts
17   Eps           :: "('a => bool) => 'a"
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" == "Eps (%x. P)"
28 print_translation {*
29 (* to avoid eta-contraction of body *)
30 [("Eps", fn [Abs abs] =>
31      let val (x,t) = atomic_abs_tr' abs
32      in Syntax.const "_Eps" \$ x \$ t end)]
33 *}
35 axioms
36   someI: "P (x::'a) ==> P (SOME x. P x)"
37 finalconsts
38   Eps
41 constdefs
42   inv :: "('a => 'b) => ('b => 'a)"
43   "inv(f :: 'a => 'b) == %y. SOME x. f x = y"
45   Inv :: "'a set => ('a => 'b) => ('b => 'a)"
46   "Inv A f == %x. SOME y. y \<in> A & f y = x"
49 subsection {*Hilbert's Epsilon-operator*}
51 text{*Easier to apply than @{text someI} if the witness comes from an
52 existential formula*}
53 lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)"
54 apply (erule exE)
55 apply (erule someI)
56 done
58 text{*Easier to apply than @{text someI} because the conclusion has only one
59 occurrence of @{term P}.*}
60 lemma someI2: "[| P a;  !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
61 by (blast intro: someI)
63 text{*Easier to apply than @{text someI2} if the witness comes from an
64 existential formula*}
65 lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
66 by (blast intro: someI2)
68 lemma some_equality [intro]:
69      "[| P a;  !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
70 by (blast intro: someI2)
72 lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
73 by (blast intro: some_equality)
75 lemma some_eq_ex: "P (SOME x. P x) =  (\<exists>x. P x)"
76 by (blast intro: someI)
78 lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
79 apply (rule some_equality)
80 apply (rule refl, assumption)
81 done
83 lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"
84 apply (rule some_equality)
85 apply (rule refl)
86 apply (erule sym)
87 done
90 subsection{*Axiom of Choice, Proved Using the Description Operator*}
92 text{*Used in @{text "Tools/meson.ML"}*}
93 lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)"
94 by (fast elim: someI)
96 lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)"
97 by (fast elim: someI)
100 subsection {*Function Inverse*}
102 lemma inv_id [simp]: "inv id = id"
103 by (simp add: inv_def id_def)
105 text{*A one-to-one function has an inverse.*}
106 lemma inv_f_f [simp]: "inj f ==> inv f (f x) = x"
107 by (simp add: inv_def inj_eq)
109 lemma inv_f_eq: "[| inj f;  f x = y |] ==> inv f y = x"
110 apply (erule subst)
111 apply (erule inv_f_f)
112 done
114 lemma inj_imp_inv_eq: "[| inj f; \<forall>x. f(g x) = x |] ==> inv f = g"
115 by (blast intro: ext inv_f_eq)
117 text{*But is it useful?*}
118 lemma inj_transfer:
119   assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)"
120   shows "P x"
121 proof -
122   have "f x \<in> range f" by auto
123   hence "P(inv f (f x))" by (rule minor)
124   thus "P x" by (simp add: inv_f_f [OF injf])
125 qed
128 lemma inj_iff: "(inj f) = (inv f o f = id)"
129 apply (simp add: o_def expand_fun_eq)
130 apply (blast intro: inj_on_inverseI inv_f_f)
131 done
133 lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
134 by (blast intro: surjI inv_f_f)
136 lemma f_inv_f: "y \<in> range(f) ==> f(inv f y) = y"
137 apply (simp add: inv_def)
138 apply (fast intro: someI)
139 done
141 lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
142 by (simp add: f_inv_f surj_range)
144 lemma inv_injective:
145   assumes eq: "inv f x = inv f y"
146       and x: "x: range f"
147       and y: "y: range f"
148   shows "x=y"
149 proof -
150   have "f (inv f x) = f (inv f y)" using eq by simp
151   thus ?thesis by (simp add: f_inv_f x y)
152 qed
154 lemma inj_on_inv: "A <= range(f) ==> inj_on (inv f) A"
155 by (fast intro: inj_onI elim: inv_injective injD)
157 lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
158 by (simp add: inj_on_inv surj_range)
160 lemma surj_iff: "(surj f) = (f o inv f = id)"
161 apply (simp add: o_def expand_fun_eq)
162 apply (blast intro: surjI surj_f_inv_f)
163 done
165 lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
166 apply (rule ext)
167 apply (drule_tac x = "inv f x" in spec)
168 apply (simp add: surj_f_inv_f)
169 done
171 lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
172 by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
174 lemma inv_equality: "[| !!x. g (f x) = x;  !!y. f (g y) = y |] ==> inv f = g"
175 apply (rule ext)
176 apply (auto simp add: inv_def)
177 done
179 lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
180 apply (rule inv_equality)
181 apply (auto simp add: bij_def surj_f_inv_f)
182 done
184 (** bij(inv f) implies little about f.  Consider f::bool=>bool such that
185     f(True)=f(False)=True.  Then it's consistent with axiom someI that
186     inv f could be any function at all, including the identity function.
187     If inv f=id then inv f is a bijection, but inj f, surj(f) and
188     inv(inv f)=f all fail.
189 **)
191 lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
192 apply (rule inv_equality)
193 apply (auto simp add: bij_def surj_f_inv_f)
194 done
197 lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
198 by (simp add: image_eq_UN surj_f_inv_f)
200 lemma image_inv_f_f: "inj f ==> (inv f) ` (f ` A) = A"
201 by (simp add: image_eq_UN)
203 lemma inv_image_comp: "inj f ==> inv f ` (f`X) = X"
204 by (auto simp add: image_def)
206 lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
207 apply auto
208 apply (force simp add: bij_is_inj)
209 apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
210 done
212 lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A"
213 apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
214 apply (blast intro: bij_is_inj [THEN inv_f_f, symmetric])
215 done
218 subsection {*Inverse of a PI-function (restricted domain)*}
220 lemma Inv_f_f: "[| inj_on f A;  x \<in> A |] ==> Inv A f (f x) = x"
221 apply (simp add: Inv_def inj_on_def)
222 apply (blast intro: someI2)
223 done
225 lemma f_Inv_f: "y \<in> f`A  ==> f (Inv A f y) = y"
226 apply (simp add: Inv_def)
227 apply (fast intro: someI2)
228 done
230 lemma Inv_injective:
231   assumes eq: "Inv A f x = Inv A f y"
232       and x: "x: f`A"
233       and y: "y: f`A"
234   shows "x=y"
235 proof -
236   have "f (Inv A f x) = f (Inv A f y)" using eq by simp
237   thus ?thesis by (simp add: f_Inv_f x y)
238 qed
240 lemma inj_on_Inv: "B <= f`A ==> inj_on (Inv A f) B"
241 apply (rule inj_onI)
242 apply (blast intro: inj_onI dest: Inv_injective injD)
243 done
245 lemma Inv_mem: "[| f ` A = B;  x \<in> B |] ==> Inv A f x \<in> A"
246 apply (simp add: Inv_def)
247 apply (fast intro: someI2)
248 done
250 lemma Inv_f_eq: "[| inj_on f A; f x = y; x \<in> A |] ==> Inv A f y = x"
251   apply (erule subst)
252   apply (erule Inv_f_f, assumption)
253   done
255 lemma Inv_comp:
256   "[| inj_on f (g ` A); inj_on g A; x \<in> f ` g ` A |] ==>
257   Inv A (f o g) x = (Inv A g o Inv (g ` A) f) x"
258   apply simp
259   apply (rule Inv_f_eq)
260     apply (fast intro: comp_inj_on)
261    apply (simp add: f_Inv_f Inv_mem)
262   apply (simp add: Inv_mem)
263   done
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 add: split_Pair_apply)
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 text{*A dynamically-scoped fact for TFL *}
300 lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
301   by (blast intro: someI)
304 subsection {* Least value operator *}
306 constdefs
307   LeastM :: "['a => 'b::ord, 'a => bool] => 'a"
308   "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
310 syntax
311   "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
312 translations
313   "LEAST x WRT m. P" == "LeastM m (%x. P)"
315 lemma LeastMI2:
316   "P x ==> (!!y. P y ==> m x <= m y)
317     ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
318     ==> Q (LeastM m P)"
319   apply (simp add: LeastM_def)
320   apply (rule someI2_ex, blast, blast)
321   done
323 lemma LeastM_equality:
324   "P k ==> (!!x. P x ==> m k <= m x)
325     ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
326   apply (rule LeastMI2, assumption, blast)
327   apply (blast intro!: order_antisym)
328   done
330 lemma wf_linord_ex_has_least:
331   "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
332     ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
333   apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
334   apply (drule_tac x = "m`Collect P" in spec, force)
335   done
337 lemma ex_has_least_nat:
338     "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
339   apply (simp only: pred_nat_trancl_eq_le [symmetric])
340   apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
341    apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
342   done
344 lemma LeastM_nat_lemma:
345     "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
346   apply (simp add: LeastM_def)
347   apply (rule someI_ex)
348   apply (erule ex_has_least_nat)
349   done
351 lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1, standard]
353 lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
354 by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
357 subsection {* Greatest value operator *}
359 constdefs
360   GreatestM :: "['a => 'b::ord, 'a => bool] => 'a"
361   "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
363   Greatest :: "('a::ord => bool) => 'a"    (binder "GREATEST " 10)
364   "Greatest == GreatestM (%x. x)"
366 syntax
367   "_GreatestM" :: "[pttrn, 'a=>'b::ord, bool] => 'a"
368       ("GREATEST _ WRT _. _" [0, 4, 10] 10)
370 translations
371   "GREATEST x WRT m. P" == "GreatestM m (%x. P)"
373 lemma GreatestMI2:
374   "P x ==> (!!y. P y ==> m y <= m x)
375     ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
376     ==> Q (GreatestM m P)"
377   apply (simp add: GreatestM_def)
378   apply (rule someI2_ex, blast, blast)
379   done
381 lemma GreatestM_equality:
382  "P k ==> (!!x. P x ==> m x <= m k)
383     ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
384   apply (rule_tac m = m in GreatestMI2, assumption, blast)
385   apply (blast intro!: order_antisym)
386   done
388 lemma Greatest_equality:
389   "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
390   apply (simp add: Greatest_def)
391   apply (erule GreatestM_equality, blast)
392   done
394 lemma ex_has_greatest_nat_lemma:
395   "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
396     ==> \<exists>y. P y & ~ (m y < m k + n)"
397   apply (induct n, force)
398   apply (force simp add: le_Suc_eq)
399   done
401 lemma ex_has_greatest_nat:
402   "P k ==> \<forall>y. P y --> m y < b
403     ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
404   apply (rule ccontr)
405   apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
406     apply (subgoal_tac  "m k <= b", auto)
407   done
409 lemma GreatestM_nat_lemma:
410   "P k ==> \<forall>y. P y --> m y < b
411     ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
412   apply (simp add: GreatestM_def)
413   apply (rule someI_ex)
414   apply (erule ex_has_greatest_nat, assumption)
415   done
417 lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard]
419 lemma GreatestM_nat_le:
420   "P x ==> \<forall>y. P y --> m y < b
421     ==> (m x::nat) <= m (GreatestM m P)"
422   apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec])
423   done
426 text {* \medskip Specialization to @{text GREATEST}. *}
428 lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
429   apply (simp add: Greatest_def)
430   apply (rule GreatestM_natI, auto)
431   done
433 lemma Greatest_le:
434     "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
435   apply (simp add: Greatest_def)
436   apply (rule GreatestM_nat_le, auto)
437   done
440 subsection {* The Meson proof procedure *}
442 subsubsection {* Negation Normal Form *}
444 text {* de Morgan laws *}
446 lemma meson_not_conjD: "~(P&Q) ==> ~P | ~Q"
447   and meson_not_disjD: "~(P|Q) ==> ~P & ~Q"
448   and meson_not_notD: "~~P ==> P"
449   and meson_not_allD: "!!P. ~(\<forall>x. P(x)) ==> \<exists>x. ~P(x)"
450   and meson_not_exD: "!!P. ~(\<exists>x. P(x)) ==> \<forall>x. ~P(x)"
451   by fast+
453 text {* Removal of @{text "-->"} and @{text "<->"} (positive and
454 negative occurrences) *}
456 lemma meson_imp_to_disjD: "P-->Q ==> ~P | Q"
457   and meson_not_impD: "~(P-->Q) ==> P & ~Q"
458   and meson_iff_to_disjD: "P=Q ==> (~P | Q) & (~Q | P)"
459   and meson_not_iffD: "~(P=Q) ==> (P | Q) & (~P | ~Q)"
460     -- {* Much more efficient than @{prop "(P & ~Q) | (Q & ~P)"} for computing CNF *}
461   and meson_not_refl_disj_D: "x ~= x | P ==> P"
462   by fast+
465 subsubsection {* Pulling out the existential quantifiers *}
467 text {* Conjunction *}
469 lemma meson_conj_exD1: "!!P Q. (\<exists>x. P(x)) & Q ==> \<exists>x. P(x) & Q"
470   and meson_conj_exD2: "!!P Q. P & (\<exists>x. Q(x)) ==> \<exists>x. P & Q(x)"
471   by fast+
474 text {* Disjunction *}
476 lemma meson_disj_exD: "!!P Q. (\<exists>x. P(x)) | (\<exists>x. Q(x)) ==> \<exists>x. P(x) | Q(x)"
477   -- {* DO NOT USE with forall-Skolemization: makes fewer schematic variables!! *}
478   -- {* With ex-Skolemization, makes fewer Skolem constants *}
479   and meson_disj_exD1: "!!P Q. (\<exists>x. P(x)) | Q ==> \<exists>x. P(x) | Q"
480   and meson_disj_exD2: "!!P Q. P | (\<exists>x. Q(x)) ==> \<exists>x. P | Q(x)"
481   by fast+
484 subsubsection {* Generating clauses for the Meson Proof Procedure *}
486 text {* Disjunctions *}
488 lemma meson_disj_assoc: "(P|Q)|R ==> P|(Q|R)"
489   and meson_disj_comm: "P|Q ==> Q|P"
490   and meson_disj_FalseD1: "False|P ==> P"
491   and meson_disj_FalseD2: "P|False ==> P"
492   by fast+
495 subsection{*Lemmas for Meson, the Model Elimination Procedure*}
497 text{* Generation of contrapositives *}
499 text{*Inserts negated disjunct after removing the negation; P is a literal.
500   Model elimination requires assuming the negation of every attempted subgoal,
501   hence the negated disjuncts.*}
502 lemma make_neg_rule: "~P|Q ==> ((~P==>P) ==> Q)"
503 by blast
505 text{*Version for Plaisted's "Postive refinement" of the Meson procedure*}
506 lemma make_refined_neg_rule: "~P|Q ==> (P ==> Q)"
507 by blast
509 text{*@{term P} should be a literal*}
510 lemma make_pos_rule: "P|Q ==> ((P==>~P) ==> Q)"
511 by blast
513 text{*Versions of @{text make_neg_rule} and @{text make_pos_rule} that don't
514 insert new assumptions, for ordinary resolution.*}
516 lemmas make_neg_rule' = make_refined_neg_rule
518 lemma make_pos_rule': "[|P|Q; ~P|] ==> Q"
519 by blast
521 text{* Generation of a goal clause -- put away the final literal *}
523 lemma make_neg_goal: "~P ==> ((~P==>P) ==> False)"
524 by blast
526 lemma make_pos_goal: "P ==> ((P==>~P) ==> False)"
527 by blast
530 subsubsection{* Lemmas for Forward Proof*}
532 text{*There is a similarity to congruence rules*}
534 (*NOTE: could handle conjunctions (faster?) by
535     nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
536 lemma conj_forward: "[| P'&Q';  P' ==> P;  Q' ==> Q |] ==> P&Q"
537 by blast
539 lemma disj_forward: "[| P'|Q';  P' ==> P;  Q' ==> Q |] ==> P|Q"
540 by blast
542 (*Version of @{text disj_forward} for removal of duplicate literals*)
543 lemma disj_forward2:
544     "[| P'|Q';  P' ==> P;  [| Q'; P==>False |] ==> Q |] ==> P|Q"
545 apply blast
546 done
548 lemma all_forward: "[| \<forall>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<forall>x. P(x)"
549 by blast
551 lemma ex_forward: "[| \<exists>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<exists>x. P(x)"
552 by blast
555 text{*Many of these bindings are used by the ATP linkup, and not just by
556 legacy proof scripts.*}
557 ML
558 {*
559 val inv_def = thm "inv_def";
560 val Inv_def = thm "Inv_def";
562 val someI = thm "someI";
563 val someI_ex = thm "someI_ex";
564 val someI2 = thm "someI2";
565 val someI2_ex = thm "someI2_ex";
566 val some_equality = thm "some_equality";
567 val some1_equality = thm "some1_equality";
568 val some_eq_ex = thm "some_eq_ex";
569 val some_eq_trivial = thm "some_eq_trivial";
570 val some_sym_eq_trivial = thm "some_sym_eq_trivial";
571 val choice = thm "choice";
572 val bchoice = thm "bchoice";
573 val inv_id = thm "inv_id";
574 val inv_f_f = thm "inv_f_f";
575 val inv_f_eq = thm "inv_f_eq";
576 val inj_imp_inv_eq = thm "inj_imp_inv_eq";
577 val inj_transfer = thm "inj_transfer";
578 val inj_iff = thm "inj_iff";
579 val inj_imp_surj_inv = thm "inj_imp_surj_inv";
580 val f_inv_f = thm "f_inv_f";
581 val surj_f_inv_f = thm "surj_f_inv_f";
582 val inv_injective = thm "inv_injective";
583 val inj_on_inv = thm "inj_on_inv";
584 val surj_imp_inj_inv = thm "surj_imp_inj_inv";
585 val surj_iff = thm "surj_iff";
586 val surj_imp_inv_eq = thm "surj_imp_inv_eq";
587 val bij_imp_bij_inv = thm "bij_imp_bij_inv";
588 val inv_equality = thm "inv_equality";
589 val inv_inv_eq = thm "inv_inv_eq";
590 val o_inv_distrib = thm "o_inv_distrib";
591 val image_surj_f_inv_f = thm "image_surj_f_inv_f";
592 val image_inv_f_f = thm "image_inv_f_f";
593 val inv_image_comp = thm "inv_image_comp";
594 val bij_image_Collect_eq = thm "bij_image_Collect_eq";
595 val bij_vimage_eq_inv_image = thm "bij_vimage_eq_inv_image";
596 val Inv_f_f = thm "Inv_f_f";
597 val f_Inv_f = thm "f_Inv_f";
598 val Inv_injective = thm "Inv_injective";
599 val inj_on_Inv = thm "inj_on_Inv";
600 val split_paired_Eps = thm "split_paired_Eps";
601 val Eps_split = thm "Eps_split";
602 val Eps_split_eq = thm "Eps_split_eq";
603 val wf_iff_no_infinite_down_chain = thm "wf_iff_no_infinite_down_chain";
604 val Inv_mem = thm "Inv_mem";
605 val Inv_f_eq = thm "Inv_f_eq";
606 val Inv_comp = thm "Inv_comp";
607 val tfl_some = thm "tfl_some";
608 val make_neg_rule = thm "make_neg_rule";
609 val make_refined_neg_rule = thm "make_refined_neg_rule";
610 val make_pos_rule = thm "make_pos_rule";
611 val make_neg_rule' = thm "make_neg_rule'";
612 val make_pos_rule' = thm "make_pos_rule'";
613 val make_neg_goal = thm "make_neg_goal";
614 val make_pos_goal = thm "make_pos_goal";
615 val conj_forward = thm "conj_forward";
616 val disj_forward = thm "disj_forward";
617 val disj_forward2 = thm "disj_forward2";
618 val all_forward = thm "all_forward";
619 val ex_forward = thm "ex_forward";
620 *}
623 subsection {* Meson method setup *}
625 use "Tools/meson.ML"
626 setup Meson.skolemize_setup
629 subsection {* Specification package -- Hilbertized version *}
631 lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
632   by (simp only: someI_ex)
634 use "Tools/specification_package.ML"
636 end