src/HOL/Hilbert_Choice.thy
author krauss
Fri Nov 24 13:44:51 2006 +0100 (2006-11-24)
changeset 21512 3786eb1b69d6
parent 21243 afffe1f72143
child 21999 0cf192e489e2
permissions -rw-r--r--
Lemma "fundef_default_value" uses predicate instead of set.
     1 (*  Title:      HOL/Hilbert_Choice.thy
     2     ID: $Id$
     3     Author:     Lawrence C Paulson
     4     Copyright   2001  University of Cambridge
     5 *)
     6 
     7 header {* Hilbert's Epsilon-Operator and the Axiom of Choice *}
     8 
     9 theory Hilbert_Choice
    10 imports Nat
    11 uses ("Tools/meson.ML") ("Tools/specification_package.ML")
    12 begin
    13 
    14 subsection {* Hilbert's epsilon *}
    15 
    16 consts
    17   Eps           :: "('a => bool) => 'a"
    18 
    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)"
    27 
    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 *}
    34 
    35 axioms
    36   someI: "P (x::'a) ==> P (SOME x. P x)"
    37 finalconsts
    38   Eps
    39 
    40 
    41 constdefs
    42   inv :: "('a => 'b) => ('b => 'a)"
    43   "inv(f :: 'a => 'b) == %y. SOME x. f x = y"
    44 
    45   Inv :: "'a set => ('a => 'b) => ('b => 'a)"
    46   "Inv A f == %x. SOME y. y \<in> A & f y = x"
    47 
    48 
    49 subsection {*Hilbert's Epsilon-operator*}
    50 
    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
    57 
    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)
    62 
    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)
    67 
    68 lemma some_equality [intro]:
    69      "[| P a;  !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
    70 by (blast intro: someI2)
    71 
    72 lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
    73 by (blast intro: some_equality)
    74 
    75 lemma some_eq_ex: "P (SOME x. P x) =  (\<exists>x. P x)"
    76 by (blast intro: someI)
    77 
    78 lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
    79 apply (rule some_equality)
    80 apply (rule refl, assumption)
    81 done
    82 
    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
    88 
    89 
    90 subsection{*Axiom of Choice, Proved Using the Description Operator*}
    91 
    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)
    95 
    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)
    98 
    99 
   100 subsection {*Function Inverse*}
   101 
   102 lemma inv_id [simp]: "inv id = id"
   103 by (simp add: inv_def id_def)
   104 
   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)
   108 
   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
   113 
   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)
   116 
   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
   126 
   127 
   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
   132 
   133 lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
   134 by (blast intro: surjI inv_f_f)
   135 
   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
   140 
   141 lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
   142 by (simp add: f_inv_f surj_range)
   143 
   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
   153 
   154 lemma inj_on_inv: "A <= range(f) ==> inj_on (inv f) A"
   155 by (fast intro: inj_onI elim: inv_injective injD)
   156 
   157 lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
   158 by (simp add: inj_on_inv surj_range)
   159 
   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
   164 
   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
   170 
   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)
   173 
   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
   178 
   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
   183 
   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 **)
   190 
   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
   195 
   196 
   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)
   199 
   200 lemma image_inv_f_f: "inj f ==> (inv f) ` (f ` A) = A"
   201 by (simp add: image_eq_UN)
   202 
   203 lemma inv_image_comp: "inj f ==> inv f ` (f`X) = X"
   204 by (auto simp add: image_def)
   205 
   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
   211 
   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
   216 
   217 
   218 subsection {*Inverse of a PI-function (restricted domain)*}
   219 
   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
   224 
   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
   229 
   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
   239 
   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
   244 
   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
   249 
   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
   254 
   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
   264 
   265 
   266 subsection {*Other Consequences of Hilbert's Epsilon*}
   267 
   268 text {*Hilbert's Epsilon and the @{term split} Operator*}
   269 
   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)
   273 
   274 lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"
   275 by (simp add: split_def)
   276 
   277 lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
   278 by blast
   279 
   280 
   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
   298 
   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)
   302 
   303 
   304 subsection {* Least value operator *}
   305 
   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)"
   309 
   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)"
   314 
   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
   322 
   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
   329 
   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
   336 
   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
   343 
   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
   350 
   351 lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1, standard]
   352 
   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)
   355 
   356 
   357 subsection {* Greatest value operator *}
   358 
   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)"
   362 
   363   Greatest :: "('a::ord => bool) => 'a"    (binder "GREATEST " 10)
   364   "Greatest == GreatestM (%x. x)"
   365 
   366 syntax
   367   "_GreatestM" :: "[pttrn, 'a=>'b::ord, bool] => 'a"
   368       ("GREATEST _ WRT _. _" [0, 4, 10] 10)
   369 
   370 translations
   371   "GREATEST x WRT m. P" == "GreatestM m (%x. P)"
   372 
   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
   380 
   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
   387 
   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
   393 
   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
   400 
   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 [3] "m k <= b", auto)
   407   done
   408 
   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
   416 
   417 lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard]
   418 
   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, of P])
   423   done
   424 
   425 
   426 text {* \medskip Specialization to @{text GREATEST}. *}
   427 
   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
   432 
   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
   438 
   439 
   440 subsection {* The Meson proof procedure *}
   441 
   442 subsubsection {* Negation Normal Form *}
   443 
   444 text {* de Morgan laws *}
   445 
   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+
   452 
   453 text {* Removal of @{text "-->"} and @{text "<->"} (positive and
   454 negative occurrences) *}
   455 
   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+
   463 
   464 
   465 subsubsection {* Pulling out the existential quantifiers *}
   466 
   467 text {* Conjunction *}
   468 
   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+
   472 
   473 
   474 text {* Disjunction *}
   475 
   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+
   482 
   483 
   484 subsubsection {* Generating clauses for the Meson Proof Procedure *}
   485 
   486 text {* Disjunctions *}
   487 
   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+
   493 
   494 
   495 subsection{*Lemmas for Meson, the Model Elimination Procedure*}
   496 
   497 text{* Generation of contrapositives *}
   498 
   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
   504 
   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
   508 
   509 text{*@{term P} should be a literal*}
   510 lemma make_pos_rule: "P|Q ==> ((P==>~P) ==> Q)"
   511 by blast
   512 
   513 text{*Versions of @{text make_neg_rule} and @{text make_pos_rule} that don't
   514 insert new assumptions, for ordinary resolution.*}
   515 
   516 lemmas make_neg_rule' = make_refined_neg_rule
   517 
   518 lemma make_pos_rule': "[|P|Q; ~P|] ==> Q"
   519 by blast
   520 
   521 text{* Generation of a goal clause -- put away the final literal *}
   522 
   523 lemma make_neg_goal: "~P ==> ((~P==>P) ==> False)"
   524 by blast
   525 
   526 lemma make_pos_goal: "P ==> ((P==>~P) ==> False)"
   527 by blast
   528 
   529 
   530 subsubsection{* Lemmas for Forward Proof*}
   531 
   532 text{*There is a similarity to congruence rules*}
   533 
   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
   538 
   539 lemma disj_forward: "[| P'|Q';  P' ==> P;  Q' ==> Q |] ==> P|Q"
   540 by blast
   541 
   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
   547 
   548 lemma all_forward: "[| \<forall>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<forall>x. P(x)"
   549 by blast
   550 
   551 lemma ex_forward: "[| \<exists>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<exists>x. P(x)"
   552 by blast
   553 
   554 
   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";
   561 
   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 *}
   621 
   622 
   623 subsection {* Meson method setup *}
   624 
   625 use "Tools/meson.ML"
   626 setup Meson.skolemize_setup
   627 
   628 
   629 subsection {* Specification package -- Hilbertized version *}
   630 
   631 lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
   632   by (simp only: someI_ex)
   633 
   634 use "Tools/specification_package.ML"
   635 
   636 end