src/HOL/Hilbert_Choice.thy
author wenzelm
Sun Apr 15 14:31:44 2007 +0200 (2007-04-15)
changeset 22690 0b08f218f260
parent 21999 0cf192e489e2
child 23433 c2c10abd2a1e
permissions -rw-r--r--
replaced axioms/finalconsts by proper axiomatization;
     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 axiomatization
    17   Eps :: "('a => bool) => 'a"
    18 where
    19   someI: "P x ==> P (Eps P)"
    20 
    21 syntax (epsilon)
    22   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3\<some>_./ _)" [0, 10] 10)
    23 syntax (HOL)
    24   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3@ _./ _)" [0, 10] 10)
    25 syntax
    26   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3SOME _./ _)" [0, 10] 10)
    27 translations
    28   "SOME x. P" == "CONST Eps (%x. P)"
    29 
    30 print_translation {*
    31 (* to avoid eta-contraction of body *)
    32 [(@{const_syntax Eps}, fn [Abs abs] =>
    33      let val (x,t) = atomic_abs_tr' abs
    34      in Syntax.const "_Eps" $ x $ t end)]
    35 *}
    36 
    37 constdefs
    38   inv :: "('a => 'b) => ('b => 'a)"
    39   "inv(f :: 'a => 'b) == %y. SOME x. f x = y"
    40 
    41   Inv :: "'a set => ('a => 'b) => ('b => 'a)"
    42   "Inv A f == %x. SOME y. y \<in> A & f y = x"
    43 
    44 
    45 subsection {*Hilbert's Epsilon-operator*}
    46 
    47 text{*Easier to apply than @{text someI} if the witness comes from an
    48 existential formula*}
    49 lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)"
    50 apply (erule exE)
    51 apply (erule someI)
    52 done
    53 
    54 text{*Easier to apply than @{text someI} because the conclusion has only one
    55 occurrence of @{term P}.*}
    56 lemma someI2: "[| P a;  !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
    57 by (blast intro: someI)
    58 
    59 text{*Easier to apply than @{text someI2} if the witness comes from an
    60 existential formula*}
    61 lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
    62 by (blast intro: someI2)
    63 
    64 lemma some_equality [intro]:
    65      "[| P a;  !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
    66 by (blast intro: someI2)
    67 
    68 lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
    69 by (blast intro: some_equality)
    70 
    71 lemma some_eq_ex: "P (SOME x. P x) =  (\<exists>x. P x)"
    72 by (blast intro: someI)
    73 
    74 lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
    75 apply (rule some_equality)
    76 apply (rule refl, assumption)
    77 done
    78 
    79 lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"
    80 apply (rule some_equality)
    81 apply (rule refl)
    82 apply (erule sym)
    83 done
    84 
    85 
    86 subsection{*Axiom of Choice, Proved Using the Description Operator*}
    87 
    88 text{*Used in @{text "Tools/meson.ML"}*}
    89 lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)"
    90 by (fast elim: someI)
    91 
    92 lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)"
    93 by (fast elim: someI)
    94 
    95 
    96 subsection {*Function Inverse*}
    97 
    98 lemma inv_id [simp]: "inv id = id"
    99 by (simp add: inv_def id_def)
   100 
   101 text{*A one-to-one function has an inverse.*}
   102 lemma inv_f_f [simp]: "inj f ==> inv f (f x) = x"
   103 by (simp add: inv_def inj_eq)
   104 
   105 lemma inv_f_eq: "[| inj f;  f x = y |] ==> inv f y = x"
   106 apply (erule subst)
   107 apply (erule inv_f_f)
   108 done
   109 
   110 lemma inj_imp_inv_eq: "[| inj f; \<forall>x. f(g x) = x |] ==> inv f = g"
   111 by (blast intro: ext inv_f_eq)
   112 
   113 text{*But is it useful?*}
   114 lemma inj_transfer:
   115   assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)"
   116   shows "P x"
   117 proof -
   118   have "f x \<in> range f" by auto
   119   hence "P(inv f (f x))" by (rule minor)
   120   thus "P x" by (simp add: inv_f_f [OF injf])
   121 qed
   122 
   123 
   124 lemma inj_iff: "(inj f) = (inv f o f = id)"
   125 apply (simp add: o_def expand_fun_eq)
   126 apply (blast intro: inj_on_inverseI inv_f_f)
   127 done
   128 
   129 lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
   130 by (blast intro: surjI inv_f_f)
   131 
   132 lemma f_inv_f: "y \<in> range(f) ==> f(inv f y) = y"
   133 apply (simp add: inv_def)
   134 apply (fast intro: someI)
   135 done
   136 
   137 lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
   138 by (simp add: f_inv_f surj_range)
   139 
   140 lemma inv_injective:
   141   assumes eq: "inv f x = inv f y"
   142       and x: "x: range f"
   143       and y: "y: range f"
   144   shows "x=y"
   145 proof -
   146   have "f (inv f x) = f (inv f y)" using eq by simp
   147   thus ?thesis by (simp add: f_inv_f x y) 
   148 qed
   149 
   150 lemma inj_on_inv: "A <= range(f) ==> inj_on (inv f) A"
   151 by (fast intro: inj_onI elim: inv_injective injD)
   152 
   153 lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
   154 by (simp add: inj_on_inv surj_range)
   155 
   156 lemma surj_iff: "(surj f) = (f o inv f = id)"
   157 apply (simp add: o_def expand_fun_eq)
   158 apply (blast intro: surjI surj_f_inv_f)
   159 done
   160 
   161 lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
   162 apply (rule ext)
   163 apply (drule_tac x = "inv f x" in spec)
   164 apply (simp add: surj_f_inv_f)
   165 done
   166 
   167 lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
   168 by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
   169 
   170 lemma inv_equality: "[| !!x. g (f x) = x;  !!y. f (g y) = y |] ==> inv f = g"
   171 apply (rule ext)
   172 apply (auto simp add: inv_def)
   173 done
   174 
   175 lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
   176 apply (rule inv_equality)
   177 apply (auto simp add: bij_def surj_f_inv_f)
   178 done
   179 
   180 (** bij(inv f) implies little about f.  Consider f::bool=>bool such that
   181     f(True)=f(False)=True.  Then it's consistent with axiom someI that
   182     inv f could be any function at all, including the identity function.
   183     If inv f=id then inv f is a bijection, but inj f, surj(f) and
   184     inv(inv f)=f all fail.
   185 **)
   186 
   187 lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
   188 apply (rule inv_equality)
   189 apply (auto simp add: bij_def surj_f_inv_f)
   190 done
   191 
   192 
   193 lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
   194 by (simp add: image_eq_UN surj_f_inv_f)
   195 
   196 lemma image_inv_f_f: "inj f ==> (inv f) ` (f ` A) = A"
   197 by (simp add: image_eq_UN)
   198 
   199 lemma inv_image_comp: "inj f ==> inv f ` (f`X) = X"
   200 by (auto simp add: image_def)
   201 
   202 lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
   203 apply auto
   204 apply (force simp add: bij_is_inj)
   205 apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
   206 done
   207 
   208 lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A" 
   209 apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
   210 apply (blast intro: bij_is_inj [THEN inv_f_f, symmetric])
   211 done
   212 
   213 
   214 subsection {*Inverse of a PI-function (restricted domain)*}
   215 
   216 lemma Inv_f_f: "[| inj_on f A;  x \<in> A |] ==> Inv A f (f x) = x"
   217 apply (simp add: Inv_def inj_on_def)
   218 apply (blast intro: someI2)
   219 done
   220 
   221 lemma f_Inv_f: "y \<in> f`A  ==> f (Inv A f y) = y"
   222 apply (simp add: Inv_def)
   223 apply (fast intro: someI2)
   224 done
   225 
   226 lemma Inv_injective:
   227   assumes eq: "Inv A f x = Inv A f y"
   228       and x: "x: f`A"
   229       and y: "y: f`A"
   230   shows "x=y"
   231 proof -
   232   have "f (Inv A f x) = f (Inv A f y)" using eq by simp
   233   thus ?thesis by (simp add: f_Inv_f x y) 
   234 qed
   235 
   236 lemma inj_on_Inv: "B <= f`A ==> inj_on (Inv A f) B"
   237 apply (rule inj_onI)
   238 apply (blast intro: inj_onI dest: Inv_injective injD)
   239 done
   240 
   241 lemma Inv_mem: "[| f ` A = B;  x \<in> B |] ==> Inv A f x \<in> A"
   242 apply (simp add: Inv_def)
   243 apply (fast intro: someI2)
   244 done
   245 
   246 lemma Inv_f_eq: "[| inj_on f A; f x = y; x \<in> A |] ==> Inv A f y = x"
   247   apply (erule subst)
   248   apply (erule Inv_f_f, assumption)
   249   done
   250 
   251 lemma Inv_comp:
   252   "[| inj_on f (g ` A); inj_on g A; x \<in> f ` g ` A |] ==>
   253   Inv A (f o g) x = (Inv A g o Inv (g ` A) f) x"
   254   apply simp
   255   apply (rule Inv_f_eq)
   256     apply (fast intro: comp_inj_on)
   257    apply (simp add: f_Inv_f Inv_mem)
   258   apply (simp add: Inv_mem)
   259   done
   260 
   261 
   262 subsection {*Other Consequences of Hilbert's Epsilon*}
   263 
   264 text {*Hilbert's Epsilon and the @{term split} Operator*}
   265 
   266 text{*Looping simprule*}
   267 lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
   268 by (simp add: split_Pair_apply)
   269 
   270 lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"
   271 by (simp add: split_def)
   272 
   273 lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
   274 by blast
   275 
   276 
   277 text{*A relation is wellfounded iff it has no infinite descending chain*}
   278 lemma wf_iff_no_infinite_down_chain:
   279   "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
   280 apply (simp only: wf_eq_minimal)
   281 apply (rule iffI)
   282  apply (rule notI)
   283  apply (erule exE)
   284  apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
   285 apply (erule contrapos_np, simp, clarify)
   286 apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
   287  apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI)
   288  apply (rule allI, simp)
   289  apply (rule someI2_ex, blast, blast)
   290 apply (rule allI)
   291 apply (induct_tac "n", simp_all)
   292 apply (rule someI2_ex, blast+)
   293 done
   294 
   295 text{*A dynamically-scoped fact for TFL *}
   296 lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
   297   by (blast intro: someI)
   298 
   299 
   300 subsection {* Least value operator *}
   301 
   302 constdefs
   303   LeastM :: "['a => 'b::ord, 'a => bool] => 'a"
   304   "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
   305 
   306 syntax
   307   "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
   308 translations
   309   "LEAST x WRT m. P" == "LeastM m (%x. P)"
   310 
   311 lemma LeastMI2:
   312   "P x ==> (!!y. P y ==> m x <= m y)
   313     ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
   314     ==> Q (LeastM m P)"
   315   apply (simp add: LeastM_def)
   316   apply (rule someI2_ex, blast, blast)
   317   done
   318 
   319 lemma LeastM_equality:
   320   "P k ==> (!!x. P x ==> m k <= m x)
   321     ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
   322   apply (rule LeastMI2, assumption, blast)
   323   apply (blast intro!: order_antisym)
   324   done
   325 
   326 lemma wf_linord_ex_has_least:
   327   "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
   328     ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
   329   apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
   330   apply (drule_tac x = "m`Collect P" in spec, force)
   331   done
   332 
   333 lemma ex_has_least_nat:
   334     "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
   335   apply (simp only: pred_nat_trancl_eq_le [symmetric])
   336   apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
   337    apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
   338   done
   339 
   340 lemma LeastM_nat_lemma:
   341     "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
   342   apply (simp add: LeastM_def)
   343   apply (rule someI_ex)
   344   apply (erule ex_has_least_nat)
   345   done
   346 
   347 lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1, standard]
   348 
   349 lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
   350 by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
   351 
   352 
   353 subsection {* Greatest value operator *}
   354 
   355 constdefs
   356   GreatestM :: "['a => 'b::ord, 'a => bool] => 'a"
   357   "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
   358 
   359   Greatest :: "('a::ord => bool) => 'a"    (binder "GREATEST " 10)
   360   "Greatest == GreatestM (%x. x)"
   361 
   362 syntax
   363   "_GreatestM" :: "[pttrn, 'a=>'b::ord, bool] => 'a"
   364       ("GREATEST _ WRT _. _" [0, 4, 10] 10)
   365 
   366 translations
   367   "GREATEST x WRT m. P" == "GreatestM m (%x. P)"
   368 
   369 lemma GreatestMI2:
   370   "P x ==> (!!y. P y ==> m y <= m x)
   371     ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
   372     ==> Q (GreatestM m P)"
   373   apply (simp add: GreatestM_def)
   374   apply (rule someI2_ex, blast, blast)
   375   done
   376 
   377 lemma GreatestM_equality:
   378  "P k ==> (!!x. P x ==> m x <= m k)
   379     ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
   380   apply (rule_tac m = m in GreatestMI2, assumption, blast)
   381   apply (blast intro!: order_antisym)
   382   done
   383 
   384 lemma Greatest_equality:
   385   "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
   386   apply (simp add: Greatest_def)
   387   apply (erule GreatestM_equality, blast)
   388   done
   389 
   390 lemma ex_has_greatest_nat_lemma:
   391   "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
   392     ==> \<exists>y. P y & ~ (m y < m k + n)"
   393   apply (induct n, force)
   394   apply (force simp add: le_Suc_eq)
   395   done
   396 
   397 lemma ex_has_greatest_nat:
   398   "P k ==> \<forall>y. P y --> m y < b
   399     ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
   400   apply (rule ccontr)
   401   apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
   402     apply (subgoal_tac [3] "m k <= b", auto)
   403   done
   404 
   405 lemma GreatestM_nat_lemma:
   406   "P k ==> \<forall>y. P y --> m y < b
   407     ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
   408   apply (simp add: GreatestM_def)
   409   apply (rule someI_ex)
   410   apply (erule ex_has_greatest_nat, assumption)
   411   done
   412 
   413 lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard]
   414 
   415 lemma GreatestM_nat_le:
   416   "P x ==> \<forall>y. P y --> m y < b
   417     ==> (m x::nat) <= m (GreatestM m P)"
   418   apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
   419   done
   420 
   421 
   422 text {* \medskip Specialization to @{text GREATEST}. *}
   423 
   424 lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
   425   apply (simp add: Greatest_def)
   426   apply (rule GreatestM_natI, auto)
   427   done
   428 
   429 lemma Greatest_le:
   430     "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
   431   apply (simp add: Greatest_def)
   432   apply (rule GreatestM_nat_le, auto)
   433   done
   434 
   435 
   436 subsection {* The Meson proof procedure *}
   437 
   438 subsubsection {* Negation Normal Form *}
   439 
   440 text {* de Morgan laws *}
   441 
   442 lemma meson_not_conjD: "~(P&Q) ==> ~P | ~Q"
   443   and meson_not_disjD: "~(P|Q) ==> ~P & ~Q"
   444   and meson_not_notD: "~~P ==> P"
   445   and meson_not_allD: "!!P. ~(\<forall>x. P(x)) ==> \<exists>x. ~P(x)"
   446   and meson_not_exD: "!!P. ~(\<exists>x. P(x)) ==> \<forall>x. ~P(x)"
   447   by fast+
   448 
   449 text {* Removal of @{text "-->"} and @{text "<->"} (positive and
   450 negative occurrences) *}
   451 
   452 lemma meson_imp_to_disjD: "P-->Q ==> ~P | Q"
   453   and meson_not_impD: "~(P-->Q) ==> P & ~Q"
   454   and meson_iff_to_disjD: "P=Q ==> (~P | Q) & (~Q | P)"
   455   and meson_not_iffD: "~(P=Q) ==> (P | Q) & (~P | ~Q)"
   456     -- {* Much more efficient than @{prop "(P & ~Q) | (Q & ~P)"} for computing CNF *}
   457   and meson_not_refl_disj_D: "x ~= x | P ==> P"
   458   by fast+
   459 
   460 
   461 subsubsection {* Pulling out the existential quantifiers *}
   462 
   463 text {* Conjunction *}
   464 
   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+
   468 
   469 
   470 text {* Disjunction *}
   471 
   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+
   478 
   479 
   480 subsubsection {* Generating clauses for the Meson Proof Procedure *}
   481 
   482 text {* Disjunctions *}
   483 
   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+
   489 
   490 
   491 subsection{*Lemmas for Meson, the Model Elimination Procedure*}
   492 
   493 text{* Generation of contrapositives *}
   494 
   495 text{*Inserts negated disjunct after removing the negation; P is a literal.
   496   Model elimination requires assuming the negation of every attempted subgoal,
   497   hence the negated disjuncts.*}
   498 lemma make_neg_rule: "~P|Q ==> ((~P==>P) ==> Q)"
   499 by blast
   500 
   501 text{*Version for Plaisted's "Postive refinement" of the Meson procedure*}
   502 lemma make_refined_neg_rule: "~P|Q ==> (P ==> Q)"
   503 by blast
   504 
   505 text{*@{term P} should be a literal*}
   506 lemma make_pos_rule: "P|Q ==> ((P==>~P) ==> Q)"
   507 by blast
   508 
   509 text{*Versions of @{text make_neg_rule} and @{text make_pos_rule} that don't
   510 insert new assumptions, for ordinary resolution.*}
   511 
   512 lemmas make_neg_rule' = make_refined_neg_rule
   513 
   514 lemma make_pos_rule': "[|P|Q; ~P|] ==> Q"
   515 by blast
   516 
   517 text{* Generation of a goal clause -- put away the final literal *}
   518 
   519 lemma make_neg_goal: "~P ==> ((~P==>P) ==> False)"
   520 by blast
   521 
   522 lemma make_pos_goal: "P ==> ((P==>~P) ==> False)"
   523 by blast
   524 
   525 
   526 subsubsection{* Lemmas for Forward Proof*}
   527 
   528 text{*There is a similarity to congruence rules*}
   529 
   530 (*NOTE: could handle conjunctions (faster?) by
   531     nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
   532 lemma conj_forward: "[| P'&Q';  P' ==> P;  Q' ==> Q |] ==> P&Q"
   533 by blast
   534 
   535 lemma disj_forward: "[| P'|Q';  P' ==> P;  Q' ==> Q |] ==> P|Q"
   536 by blast
   537 
   538 (*Version of @{text disj_forward} for removal of duplicate literals*)
   539 lemma disj_forward2:
   540     "[| P'|Q';  P' ==> P;  [| Q'; P==>False |] ==> Q |] ==> P|Q"
   541 apply blast 
   542 done
   543 
   544 lemma all_forward: "[| \<forall>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<forall>x. P(x)"
   545 by blast
   546 
   547 lemma ex_forward: "[| \<exists>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<exists>x. P(x)"
   548 by blast
   549 
   550 
   551 text{*Many of these bindings are used by the ATP linkup, and not just by
   552 legacy proof scripts.*}
   553 ML
   554 {*
   555 val inv_def = thm "inv_def";
   556 val Inv_def = thm "Inv_def";
   557 
   558 val someI = thm "someI";
   559 val someI_ex = thm "someI_ex";
   560 val someI2 = thm "someI2";
   561 val someI2_ex = thm "someI2_ex";
   562 val some_equality = thm "some_equality";
   563 val some1_equality = thm "some1_equality";
   564 val some_eq_ex = thm "some_eq_ex";
   565 val some_eq_trivial = thm "some_eq_trivial";
   566 val some_sym_eq_trivial = thm "some_sym_eq_trivial";
   567 val choice = thm "choice";
   568 val bchoice = thm "bchoice";
   569 val inv_id = thm "inv_id";
   570 val inv_f_f = thm "inv_f_f";
   571 val inv_f_eq = thm "inv_f_eq";
   572 val inj_imp_inv_eq = thm "inj_imp_inv_eq";
   573 val inj_transfer = thm "inj_transfer";
   574 val inj_iff = thm "inj_iff";
   575 val inj_imp_surj_inv = thm "inj_imp_surj_inv";
   576 val f_inv_f = thm "f_inv_f";
   577 val surj_f_inv_f = thm "surj_f_inv_f";
   578 val inv_injective = thm "inv_injective";
   579 val inj_on_inv = thm "inj_on_inv";
   580 val surj_imp_inj_inv = thm "surj_imp_inj_inv";
   581 val surj_iff = thm "surj_iff";
   582 val surj_imp_inv_eq = thm "surj_imp_inv_eq";
   583 val bij_imp_bij_inv = thm "bij_imp_bij_inv";
   584 val inv_equality = thm "inv_equality";
   585 val inv_inv_eq = thm "inv_inv_eq";
   586 val o_inv_distrib = thm "o_inv_distrib";
   587 val image_surj_f_inv_f = thm "image_surj_f_inv_f";
   588 val image_inv_f_f = thm "image_inv_f_f";
   589 val inv_image_comp = thm "inv_image_comp";
   590 val bij_image_Collect_eq = thm "bij_image_Collect_eq";
   591 val bij_vimage_eq_inv_image = thm "bij_vimage_eq_inv_image";
   592 val Inv_f_f = thm "Inv_f_f";
   593 val f_Inv_f = thm "f_Inv_f";
   594 val Inv_injective = thm "Inv_injective";
   595 val inj_on_Inv = thm "inj_on_Inv";
   596 val split_paired_Eps = thm "split_paired_Eps";
   597 val Eps_split = thm "Eps_split";
   598 val Eps_split_eq = thm "Eps_split_eq";
   599 val wf_iff_no_infinite_down_chain = thm "wf_iff_no_infinite_down_chain";
   600 val Inv_mem = thm "Inv_mem";
   601 val Inv_f_eq = thm "Inv_f_eq";
   602 val Inv_comp = thm "Inv_comp";
   603 val tfl_some = thm "tfl_some";
   604 val make_neg_rule = thm "make_neg_rule";
   605 val make_refined_neg_rule = thm "make_refined_neg_rule";
   606 val make_pos_rule = thm "make_pos_rule";
   607 val make_neg_rule' = thm "make_neg_rule'";
   608 val make_pos_rule' = thm "make_pos_rule'";
   609 val make_neg_goal = thm "make_neg_goal";
   610 val make_pos_goal = thm "make_pos_goal";
   611 val conj_forward = thm "conj_forward";
   612 val disj_forward = thm "disj_forward";
   613 val disj_forward2 = thm "disj_forward2";
   614 val all_forward = thm "all_forward";
   615 val ex_forward = thm "ex_forward";
   616 *}
   617 
   618 
   619 subsection {* Meson package *}
   620 
   621 use "Tools/meson.ML"
   622 
   623 
   624 subsection {* Specification package -- Hilbertized version *}
   625 
   626 lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
   627   by (simp only: someI_ex)
   628 
   629 use "Tools/specification_package.ML"
   630 
   631 end