src/HOL/Hilbert_Choice.thy
 author wenzelm Sat Jun 05 13:07:04 2004 +0200 (2004-06-05) changeset 14872 3f2144aebd76 parent 14760 a08e916f4946 child 15131 c69542757a4d permissions -rw-r--r--
improved symbolic syntax of Eps: \<some> for mode "epsilon";
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 = NatArith
10 files ("Tools/meson.ML") ("Tools/specification_package.ML"):
13 subsection {* Hilbert's epsilon *}
15 consts
16   Eps           :: "('a => bool) => 'a"
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" == "Eps (%x. P)"
27 print_translation {*
28 (* to avoid eta-contraction of body *)
29 [("Eps", fn [Abs abs] =>
30      let val (x,t) = atomic_abs_tr' abs
31      in Syntax.const "_Eps" \$ x \$ t end)]
32 *}
34 axioms
35   someI: "P (x::'a) ==> P (SOME x. P x)"
38 constdefs
39   inv :: "('a => 'b) => ('b => 'a)"
40   "inv(f :: 'a => 'b) == %y. SOME x. f x = y"
42   Inv :: "'a set => ('a => 'b) => ('b => 'a)"
43   "Inv A f == %x. SOME y. y \<in> A & f y = x"
46 subsection {*Hilbert's Epsilon-operator*}
48 text{*Easier to apply than @{text someI} if the witness comes from an
49 existential formula*}
50 lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)"
51 apply (erule exE)
52 apply (erule someI)
53 done
55 text{*Easier to apply than @{text someI} because the conclusion has only one
56 occurrence of @{term P}.*}
57 lemma someI2: "[| P a;  !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
58 by (blast intro: someI)
60 text{*Easier to apply than @{text someI2} if the witness comes from an
61 existential formula*}
62 lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
63 by (blast intro: someI2)
65 lemma some_equality [intro]:
66      "[| P a;  !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
67 by (blast intro: someI2)
69 lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
70 by (blast intro: some_equality)
72 lemma some_eq_ex: "P (SOME x. P x) =  (\<exists>x. P x)"
73 by (blast intro: someI)
75 lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
76 apply (rule some_equality)
77 apply (rule refl, assumption)
78 done
80 lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"
81 apply (rule some_equality)
82 apply (rule refl)
83 apply (erule sym)
84 done
87 subsection{*Axiom of Choice, Proved Using the Description Operator*}
89 text{*Used in @{text "Tools/meson.ML"}*}
90 lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)"
91 by (fast elim: someI)
93 lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)"
94 by (fast elim: someI)
97 subsection {*Function Inverse*}
99 lemma inv_id [simp]: "inv id = id"
100 by (simp add: inv_def id_def)
102 text{*A one-to-one function has an inverse.*}
103 lemma inv_f_f [simp]: "inj f ==> inv f (f x) = x"
104 by (simp add: inv_def inj_eq)
106 lemma inv_f_eq: "[| inj f;  f x = y |] ==> inv f y = x"
107 apply (erule subst)
108 apply (erule inv_f_f)
109 done
111 lemma inj_imp_inv_eq: "[| inj f; \<forall>x. f(g x) = x |] ==> inv f = g"
112 by (blast intro: ext inv_f_eq)
114 text{*But is it useful?*}
115 lemma inj_transfer:
116   assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)"
117   shows "P x"
118 proof -
119   have "f x \<in> range f" by auto
120   hence "P(inv f (f x))" by (rule minor)
121   thus "P x" by (simp add: inv_f_f [OF injf])
122 qed
125 lemma inj_iff: "(inj f) = (inv f o f = id)"
126 apply (simp add: o_def expand_fun_eq)
127 apply (blast intro: inj_on_inverseI inv_f_f)
128 done
130 lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
131 by (blast intro: surjI inv_f_f)
133 lemma f_inv_f: "y \<in> range(f) ==> f(inv f y) = y"
134 apply (simp add: inv_def)
135 apply (fast intro: someI)
136 done
138 lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
139 by (simp add: f_inv_f surj_range)
141 lemma inv_injective:
142   assumes eq: "inv f x = inv f y"
143       and x: "x: range f"
144       and y: "y: range f"
145   shows "x=y"
146 proof -
147   have "f (inv f x) = f (inv f y)" using eq by simp
148   thus ?thesis by (simp add: f_inv_f x y)
149 qed
151 lemma inj_on_inv: "A <= range(f) ==> inj_on (inv f) A"
152 by (fast intro: inj_onI elim: inv_injective injD)
154 lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
155 by (simp add: inj_on_inv surj_range)
157 lemma surj_iff: "(surj f) = (f o inv f = id)"
158 apply (simp add: o_def expand_fun_eq)
159 apply (blast intro: surjI surj_f_inv_f)
160 done
162 lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
163 apply (rule ext)
164 apply (drule_tac x = "inv f x" in spec)
165 apply (simp add: surj_f_inv_f)
166 done
168 lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
169 by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
171 lemma inv_equality: "[| !!x. g (f x) = x;  !!y. f (g y) = y |] ==> inv f = g"
172 apply (rule ext)
173 apply (auto simp add: inv_def)
174 done
176 lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
177 apply (rule inv_equality)
178 apply (auto simp add: bij_def surj_f_inv_f)
179 done
181 (** bij(inv f) implies little about f.  Consider f::bool=>bool such that
182     f(True)=f(False)=True.  Then it's consistent with axiom someI that
183     inv f could be any function at all, including the identity function.
184     If inv f=id then inv f is a bijection, but inj f, surj(f) and
185     inv(inv f)=f all fail.
186 **)
188 lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
189 apply (rule inv_equality)
190 apply (auto simp add: bij_def surj_f_inv_f)
191 done
194 lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
195 by (simp add: image_eq_UN surj_f_inv_f)
197 lemma image_inv_f_f: "inj f ==> (inv f) ` (f ` A) = A"
198 by (simp add: image_eq_UN)
200 lemma inv_image_comp: "inj f ==> inv f ` (f`X) = X"
201 by (auto simp add: image_def)
203 lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
204 apply auto
205 apply (force simp add: bij_is_inj)
206 apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
207 done
209 lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A"
210 apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
211 apply (blast intro: bij_is_inj [THEN inv_f_f, symmetric])
212 done
215 subsection {*Inverse of a PI-function (restricted domain)*}
217 lemma Inv_f_f: "[| inj_on f A;  x \<in> A |] ==> Inv A f (f x) = x"
218 apply (simp add: Inv_def inj_on_def)
219 apply (blast intro: someI2)
220 done
222 lemma f_Inv_f: "y \<in> f`A  ==> f (Inv A f y) = y"
223 apply (simp add: Inv_def)
224 apply (fast intro: someI2)
225 done
227 lemma Inv_injective:
228   assumes eq: "Inv A f x = Inv A f y"
229       and x: "x: f`A"
230       and y: "y: f`A"
231   shows "x=y"
232 proof -
233   have "f (Inv A f x) = f (Inv A f y)" using eq by simp
234   thus ?thesis by (simp add: f_Inv_f x y)
235 qed
237 lemma inj_on_Inv: "B <= f`A ==> inj_on (Inv A f) B"
238 apply (rule inj_onI)
239 apply (blast intro: inj_onI dest: Inv_injective injD)
240 done
242 lemma Inv_mem: "[| f ` A = B;  x \<in> B |] ==> Inv A f x \<in> A"
243 apply (simp add: Inv_def)
244 apply (fast intro: someI2)
245 done
247 lemma Inv_f_eq: "[| inj_on f A; f x = y; x \<in> A |] ==> Inv A f y = x"
248   apply (erule subst)
249   apply (erule Inv_f_f, assumption)
250   done
252 lemma Inv_comp:
253   "[| inj_on f (g ` A); inj_on g A; x \<in> f ` g ` A |] ==>
254   Inv A (f o g) x = (Inv A g o Inv (g ` A) f) x"
255   apply simp
256   apply (rule Inv_f_eq)
257     apply (fast intro: comp_inj_on)
258    apply (simp add: f_Inv_f Inv_mem)
259   apply (simp add: Inv_mem)
260   done
263 subsection {*Other Consequences of Hilbert's Epsilon*}
265 text {*Hilbert's Epsilon and the @{term split} Operator*}
267 text{*Looping simprule*}
268 lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
269 by (simp add: split_Pair_apply)
271 lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"
272 by (simp add: split_def)
274 lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
275 by blast
278 text{*A relation is wellfounded iff it has no infinite descending chain*}
279 lemma wf_iff_no_infinite_down_chain:
280   "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
281 apply (simp only: wf_eq_minimal)
282 apply (rule iffI)
283  apply (rule notI)
284  apply (erule exE)
285  apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
286 apply (erule contrapos_np, simp, clarify)
287 apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
288  apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI)
289  apply (rule allI, simp)
290  apply (rule someI2_ex, blast, blast)
291 apply (rule allI)
292 apply (induct_tac "n", simp_all)
293 apply (rule someI2_ex, blast+)
294 done
296 text{*A dynamically-scoped fact for TFL *}
297 lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
298   by (blast intro: someI)
301 subsection {* Least value operator *}
303 constdefs
304   LeastM :: "['a => 'b::ord, 'a => bool] => 'a"
305   "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
307 syntax
308   "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
309 translations
310   "LEAST x WRT m. P" == "LeastM m (%x. P)"
312 lemma LeastMI2:
313   "P x ==> (!!y. P y ==> m x <= m y)
314     ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
315     ==> Q (LeastM m P)"
316   apply (simp add: LeastM_def)
317   apply (rule someI2_ex, blast, blast)
318   done
320 lemma LeastM_equality:
321   "P k ==> (!!x. P x ==> m k <= m x)
322     ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
323   apply (rule LeastMI2, assumption, blast)
324   apply (blast intro!: order_antisym)
325   done
327 lemma wf_linord_ex_has_least:
328   "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
329     ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
330   apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
331   apply (drule_tac x = "m`Collect P" in spec, force)
332   done
334 lemma ex_has_least_nat:
335     "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
336   apply (simp only: pred_nat_trancl_eq_le [symmetric])
337   apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
338    apply (simp add: less_eq not_le_iff_less pred_nat_trancl_eq_le, assumption)
339   done
341 lemma LeastM_nat_lemma:
342     "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
343   apply (simp add: LeastM_def)
344   apply (rule someI_ex)
345   apply (erule ex_has_least_nat)
346   done
348 lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1, standard]
350 lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
351 by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
354 subsection {* Greatest value operator *}
356 constdefs
357   GreatestM :: "['a => 'b::ord, 'a => bool] => 'a"
358   "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
360   Greatest :: "('a::ord => bool) => 'a"    (binder "GREATEST " 10)
361   "Greatest == GreatestM (%x. x)"
363 syntax
364   "_GreatestM" :: "[pttrn, 'a=>'b::ord, bool] => 'a"
365       ("GREATEST _ WRT _. _" [0, 4, 10] 10)
367 translations
368   "GREATEST x WRT m. P" == "GreatestM m (%x. P)"
370 lemma GreatestMI2:
371   "P x ==> (!!y. P y ==> m y <= m x)
372     ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
373     ==> Q (GreatestM m P)"
374   apply (simp add: GreatestM_def)
375   apply (rule someI2_ex, blast, blast)
376   done
378 lemma GreatestM_equality:
379  "P k ==> (!!x. P x ==> m x <= m k)
380     ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
381   apply (rule_tac m = m in GreatestMI2, assumption, blast)
382   apply (blast intro!: order_antisym)
383   done
385 lemma Greatest_equality:
386   "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
387   apply (simp add: Greatest_def)
388   apply (erule GreatestM_equality, blast)
389   done
391 lemma ex_has_greatest_nat_lemma:
392   "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
393     ==> \<exists>y. P y & ~ (m y < m k + n)"
394   apply (induct_tac n, force)
395   apply (force simp add: le_Suc_eq)
396   done
398 lemma ex_has_greatest_nat:
399   "P k ==> \<forall>y. P y --> m y < b
400     ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
401   apply (rule ccontr)
402   apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
403     apply (subgoal_tac  "m k <= b", auto)
404   done
406 lemma GreatestM_nat_lemma:
407   "P k ==> \<forall>y. P y --> m y < b
408     ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
409   apply (simp add: GreatestM_def)
410   apply (rule someI_ex)
411   apply (erule ex_has_greatest_nat, assumption)
412   done
414 lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard]
416 lemma GreatestM_nat_le:
417   "P x ==> \<forall>y. P y --> m y < b
418     ==> (m x::nat) <= m (GreatestM m P)"
419   apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec])
420   done
423 text {* \medskip Specialization to @{text GREATEST}. *}
425 lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
426   apply (simp add: Greatest_def)
427   apply (rule GreatestM_natI, auto)
428   done
430 lemma Greatest_le:
431     "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
432   apply (simp add: Greatest_def)
433   apply (rule GreatestM_nat_le, auto)
434   done
437 subsection {* The Meson proof procedure *}
439 subsubsection {* Negation Normal Form *}
441 text {* de Morgan laws *}
443 lemma meson_not_conjD: "~(P&Q) ==> ~P | ~Q"
444   and meson_not_disjD: "~(P|Q) ==> ~P & ~Q"
445   and meson_not_notD: "~~P ==> P"
446   and meson_not_allD: "!!P. ~(\<forall>x. P(x)) ==> \<exists>x. ~P(x)"
447   and meson_not_exD: "!!P. ~(\<exists>x. P(x)) ==> \<forall>x. ~P(x)"
448   by fast+
450 text {* Removal of @{text "-->"} and @{text "<->"} (positive and
451 negative occurrences) *}
453 lemma meson_imp_to_disjD: "P-->Q ==> ~P | Q"
454   and meson_not_impD: "~(P-->Q) ==> P & ~Q"
455   and meson_iff_to_disjD: "P=Q ==> (~P | Q) & (~Q | P)"
456   and meson_not_iffD: "~(P=Q) ==> (P | Q) & (~P | ~Q)"
457     -- {* Much more efficient than @{prop "(P & ~Q) | (Q & ~P)"} for computing CNF *}
458   by fast+
461 subsubsection {* Pulling out the existential quantifiers *}
463 text {* Conjunction *}
465 lemma meson_conj_exD1: "!!P Q. (\<exists>x. P(x)) & Q ==> \<exists>x. P(x) & Q"
466   and meson_conj_exD2: "!!P Q. P & (\<exists>x. Q(x)) ==> \<exists>x. P & Q(x)"
467   by fast+
470 text {* Disjunction *}
472 lemma meson_disj_exD: "!!P Q. (\<exists>x. P(x)) | (\<exists>x. Q(x)) ==> \<exists>x. P(x) | Q(x)"
473   -- {* DO NOT USE with forall-Skolemization: makes fewer schematic variables!! *}
474   -- {* With ex-Skolemization, makes fewer Skolem constants *}
475   and meson_disj_exD1: "!!P Q. (\<exists>x. P(x)) | Q ==> \<exists>x. P(x) | Q"
476   and meson_disj_exD2: "!!P Q. P | (\<exists>x. Q(x)) ==> \<exists>x. P | Q(x)"
477   by fast+
480 subsubsection {* Generating clauses for the Meson Proof Procedure *}
482 text {* Disjunctions *}
484 lemma meson_disj_assoc: "(P|Q)|R ==> P|(Q|R)"
485   and meson_disj_comm: "P|Q ==> Q|P"
486   and meson_disj_FalseD1: "False|P ==> P"
487   and meson_disj_FalseD2: "P|False ==> P"
488   by fast+
491 subsection{*Lemmas for Meson, the Model Elimination Procedure*}
494 text{* Generation of contrapositives *}
496 text{*Inserts negated disjunct after removing the negation; P is a literal.
497   Model elimination requires assuming the negation of every attempted subgoal,
498   hence the negated disjuncts.*}
499 lemma make_neg_rule: "~P|Q ==> ((~P==>P) ==> Q)"
500 by blast
502 text{*Version for Plaisted's "Postive refinement" of the Meson procedure*}
503 lemma make_refined_neg_rule: "~P|Q ==> (P ==> Q)"
504 by blast
506 text{*@{term P} should be a literal*}
507 lemma make_pos_rule: "P|Q ==> ((P==>~P) ==> Q)"
508 by blast
510 text{*Versions of @{text make_neg_rule} and @{text make_pos_rule} that don't
511 insert new assumptions, for ordinary resolution.*}
513 lemmas make_neg_rule' = make_refined_neg_rule
515 lemma make_pos_rule': "[|P|Q; ~P|] ==> Q"
516 by blast
518 text{* Generation of a goal clause -- put away the final literal *}
520 lemma make_neg_goal: "~P ==> ((~P==>P) ==> False)"
521 by blast
523 lemma make_pos_goal: "P ==> ((P==>~P) ==> False)"
524 by blast
527 subsubsection{* Lemmas for Forward Proof*}
529 text{*There is a similarity to congruence rules*}
531 (*NOTE: could handle conjunctions (faster?) by
532     nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
533 lemma conj_forward: "[| P'&Q';  P' ==> P;  Q' ==> Q |] ==> P&Q"
534 by blast
536 lemma disj_forward: "[| P'|Q';  P' ==> P;  Q' ==> Q |] ==> P|Q"
537 by blast
539 (*Version of @{text disj_forward} for removal of duplicate literals*)
540 lemma disj_forward2:
541     "[| P'|Q';  P' ==> P;  [| Q'; P==>False |] ==> Q |] ==> P|Q"
542 apply blast
543 done
545 lemma all_forward: "[| \<forall>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<forall>x. P(x)"
546 by blast
548 lemma ex_forward: "[| \<exists>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<exists>x. P(x)"
549 by blast
551 ML
552 {*
553 val inv_def = thm "inv_def";
554 val Inv_def = thm "Inv_def";
556 val someI = thm "someI";
557 val someI_ex = thm "someI_ex";
558 val someI2 = thm "someI2";
559 val someI2_ex = thm "someI2_ex";
560 val some_equality = thm "some_equality";
561 val some1_equality = thm "some1_equality";
562 val some_eq_ex = thm "some_eq_ex";
563 val some_eq_trivial = thm "some_eq_trivial";
564 val some_sym_eq_trivial = thm "some_sym_eq_trivial";
565 val choice = thm "choice";
566 val bchoice = thm "bchoice";
567 val inv_id = thm "inv_id";
568 val inv_f_f = thm "inv_f_f";
569 val inv_f_eq = thm "inv_f_eq";
570 val inj_imp_inv_eq = thm "inj_imp_inv_eq";
571 val inj_transfer = thm "inj_transfer";
572 val inj_iff = thm "inj_iff";
573 val inj_imp_surj_inv = thm "inj_imp_surj_inv";
574 val f_inv_f = thm "f_inv_f";
575 val surj_f_inv_f = thm "surj_f_inv_f";
576 val inv_injective = thm "inv_injective";
577 val inj_on_inv = thm "inj_on_inv";
578 val surj_imp_inj_inv = thm "surj_imp_inj_inv";
579 val surj_iff = thm "surj_iff";
580 val surj_imp_inv_eq = thm "surj_imp_inv_eq";
581 val bij_imp_bij_inv = thm "bij_imp_bij_inv";
582 val inv_equality = thm "inv_equality";
583 val inv_inv_eq = thm "inv_inv_eq";
584 val o_inv_distrib = thm "o_inv_distrib";
585 val image_surj_f_inv_f = thm "image_surj_f_inv_f";
586 val image_inv_f_f = thm "image_inv_f_f";
587 val inv_image_comp = thm "inv_image_comp";
588 val bij_image_Collect_eq = thm "bij_image_Collect_eq";
589 val bij_vimage_eq_inv_image = thm "bij_vimage_eq_inv_image";
590 val Inv_f_f = thm "Inv_f_f";
591 val f_Inv_f = thm "f_Inv_f";
592 val Inv_injective = thm "Inv_injective";
593 val inj_on_Inv = thm "inj_on_Inv";
594 val split_paired_Eps = thm "split_paired_Eps";
595 val Eps_split = thm "Eps_split";
596 val Eps_split_eq = thm "Eps_split_eq";
597 val wf_iff_no_infinite_down_chain = thm "wf_iff_no_infinite_down_chain";
598 val Inv_mem = thm "Inv_mem";
599 val Inv_f_eq = thm "Inv_f_eq";
600 val Inv_comp = thm "Inv_comp";
601 val tfl_some = thm "tfl_some";
602 val make_neg_rule = thm "make_neg_rule";
603 val make_refined_neg_rule = thm "make_refined_neg_rule";
604 val make_pos_rule = thm "make_pos_rule";
605 val make_neg_rule' = thm "make_neg_rule'";
606 val make_pos_rule' = thm "make_pos_rule'";
607 val make_neg_goal = thm "make_neg_goal";
608 val make_pos_goal = thm "make_pos_goal";
609 val conj_forward = thm "conj_forward";
610 val disj_forward = thm "disj_forward";
611 val disj_forward2 = thm "disj_forward2";
612 val all_forward = thm "all_forward";
613 val ex_forward = thm "ex_forward";
614 *}
617 use "Tools/meson.ML"
618 setup meson_setup
620 use "Tools/specification_package.ML"
622 end