src/HOL/HOL.thy
author nipkow
Fri Mar 06 17:38:47 2009 +0100 (2009-03-06)
changeset 30313 b2441b0c8d38
parent 30254 7b8afdfa2f83
child 30309 188f0658af9f
permissions -rw-r--r--
added lemmas
     1 (*  Title:      HOL/HOL.thy
     2     Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
     3 *)
     4 
     5 header {* The basis of Higher-Order Logic *}
     6 
     7 theory HOL
     8 imports Pure
     9 uses
    10   ("Tools/hologic.ML")
    11   "~~/src/Tools/IsaPlanner/zipper.ML"
    12   "~~/src/Tools/IsaPlanner/isand.ML"
    13   "~~/src/Tools/IsaPlanner/rw_tools.ML"
    14   "~~/src/Tools/IsaPlanner/rw_inst.ML"
    15   "~~/src/Tools/intuitionistic.ML"
    16   "~~/src/Tools/project_rule.ML"
    17   "~~/src/Provers/hypsubst.ML"
    18   "~~/src/Provers/splitter.ML"
    19   "~~/src/Provers/classical.ML"
    20   "~~/src/Provers/blast.ML"
    21   "~~/src/Provers/clasimp.ML"
    22   "~~/src/Tools/coherent.ML"
    23   "~~/src/Tools/eqsubst.ML"
    24   "~~/src/Provers/quantifier1.ML"
    25   ("Tools/simpdata.ML")
    26   "~~/src/Tools/random_word.ML"
    27   "~~/src/Tools/atomize_elim.ML"
    28   "~~/src/Tools/induct.ML"
    29   ("~~/src/Tools/induct_tacs.ML")
    30   "~~/src/Tools/value.ML"
    31   "~~/src/Tools/code/code_name.ML"
    32   "~~/src/Tools/code/code_funcgr.ML" (*formal dependency*)
    33   "~~/src/Tools/code/code_wellsorted.ML" 
    34   "~~/src/Tools/code/code_thingol.ML"
    35   "~~/src/Tools/code/code_printer.ML"
    36   "~~/src/Tools/code/code_target.ML"
    37   "~~/src/Tools/code/code_ml.ML"
    38   "~~/src/Tools/code/code_haskell.ML"
    39   "~~/src/Tools/nbe.ML"
    40   ("Tools/recfun_codegen.ML")
    41 begin
    42 
    43 setup {* Intuitionistic.method_setup "iprover" *}
    44 
    45 
    46 subsection {* Primitive logic *}
    47 
    48 subsubsection {* Core syntax *}
    49 
    50 classes type
    51 defaultsort type
    52 setup {* ObjectLogic.add_base_sort @{sort type} *}
    53 
    54 arities
    55   "fun" :: (type, type) type
    56   itself :: (type) type
    57 
    58 global
    59 
    60 typedecl bool
    61 
    62 judgment
    63   Trueprop      :: "bool => prop"                   ("(_)" 5)
    64 
    65 consts
    66   Not           :: "bool => bool"                   ("~ _" [40] 40)
    67   True          :: bool
    68   False         :: bool
    69 
    70   The           :: "('a => bool) => 'a"
    71   All           :: "('a => bool) => bool"           (binder "ALL " 10)
    72   Ex            :: "('a => bool) => bool"           (binder "EX " 10)
    73   Ex1           :: "('a => bool) => bool"           (binder "EX! " 10)
    74   Let           :: "['a, 'a => 'b] => 'b"
    75 
    76   "op ="        :: "['a, 'a] => bool"               (infixl "=" 50)
    77   "op &"        :: "[bool, bool] => bool"           (infixr "&" 35)
    78   "op |"        :: "[bool, bool] => bool"           (infixr "|" 30)
    79   "op -->"      :: "[bool, bool] => bool"           (infixr "-->" 25)
    80 
    81 local
    82 
    83 consts
    84   If            :: "[bool, 'a, 'a] => 'a"           ("(if (_)/ then (_)/ else (_))" 10)
    85 
    86 
    87 subsubsection {* Additional concrete syntax *}
    88 
    89 notation (output)
    90   "op ="  (infix "=" 50)
    91 
    92 abbreviation
    93   not_equal :: "['a, 'a] => bool"  (infixl "~=" 50) where
    94   "x ~= y == ~ (x = y)"
    95 
    96 notation (output)
    97   not_equal  (infix "~=" 50)
    98 
    99 notation (xsymbols)
   100   Not  ("\<not> _" [40] 40) and
   101   "op &"  (infixr "\<and>" 35) and
   102   "op |"  (infixr "\<or>" 30) and
   103   "op -->"  (infixr "\<longrightarrow>" 25) and
   104   not_equal  (infix "\<noteq>" 50)
   105 
   106 notation (HTML output)
   107   Not  ("\<not> _" [40] 40) and
   108   "op &"  (infixr "\<and>" 35) and
   109   "op |"  (infixr "\<or>" 30) and
   110   not_equal  (infix "\<noteq>" 50)
   111 
   112 abbreviation (iff)
   113   iff :: "[bool, bool] => bool"  (infixr "<->" 25) where
   114   "A <-> B == A = B"
   115 
   116 notation (xsymbols)
   117   iff  (infixr "\<longleftrightarrow>" 25)
   118 
   119 
   120 nonterminals
   121   letbinds  letbind
   122   case_syn  cases_syn
   123 
   124 syntax
   125   "_The"        :: "[pttrn, bool] => 'a"                 ("(3THE _./ _)" [0, 10] 10)
   126 
   127   "_bind"       :: "[pttrn, 'a] => letbind"              ("(2_ =/ _)" 10)
   128   ""            :: "letbind => letbinds"                 ("_")
   129   "_binds"      :: "[letbind, letbinds] => letbinds"     ("_;/ _")
   130   "_Let"        :: "[letbinds, 'a] => 'a"                ("(let (_)/ in (_))" 10)
   131 
   132   "_case_syntax":: "['a, cases_syn] => 'b"               ("(case _ of/ _)" 10)
   133   "_case1"      :: "['a, 'b] => case_syn"                ("(2_ =>/ _)" 10)
   134   ""            :: "case_syn => cases_syn"               ("_")
   135   "_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ | _")
   136 
   137 translations
   138   "THE x. P"              == "The (%x. P)"
   139   "_Let (_binds b bs) e"  == "_Let b (_Let bs e)"
   140   "let x = a in e"        == "Let a (%x. e)"
   141 
   142 print_translation {*
   143 (* To avoid eta-contraction of body: *)
   144 [("The", fn [Abs abs] =>
   145      let val (x,t) = atomic_abs_tr' abs
   146      in Syntax.const "_The" $ x $ t end)]
   147 *}
   148 
   149 syntax (xsymbols)
   150   "_case1"      :: "['a, 'b] => case_syn"                ("(2_ \<Rightarrow>/ _)" 10)
   151 
   152 notation (xsymbols)
   153   All  (binder "\<forall>" 10) and
   154   Ex  (binder "\<exists>" 10) and
   155   Ex1  (binder "\<exists>!" 10)
   156 
   157 notation (HTML output)
   158   All  (binder "\<forall>" 10) and
   159   Ex  (binder "\<exists>" 10) and
   160   Ex1  (binder "\<exists>!" 10)
   161 
   162 notation (HOL)
   163   All  (binder "! " 10) and
   164   Ex  (binder "? " 10) and
   165   Ex1  (binder "?! " 10)
   166 
   167 
   168 subsubsection {* Axioms and basic definitions *}
   169 
   170 axioms
   171   refl:           "t = (t::'a)"
   172   subst:          "s = t \<Longrightarrow> P s \<Longrightarrow> P t"
   173   ext:            "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
   174     -- {*Extensionality is built into the meta-logic, and this rule expresses
   175          a related property.  It is an eta-expanded version of the traditional
   176          rule, and similar to the ABS rule of HOL*}
   177 
   178   the_eq_trivial: "(THE x. x = a) = (a::'a)"
   179 
   180   impI:           "(P ==> Q) ==> P-->Q"
   181   mp:             "[| P-->Q;  P |] ==> Q"
   182 
   183 
   184 defs
   185   True_def:     "True      == ((%x::bool. x) = (%x. x))"
   186   All_def:      "All(P)    == (P = (%x. True))"
   187   Ex_def:       "Ex(P)     == !Q. (!x. P x --> Q) --> Q"
   188   False_def:    "False     == (!P. P)"
   189   not_def:      "~ P       == P-->False"
   190   and_def:      "P & Q     == !R. (P-->Q-->R) --> R"
   191   or_def:       "P | Q     == !R. (P-->R) --> (Q-->R) --> R"
   192   Ex1_def:      "Ex1(P)    == ? x. P(x) & (! y. P(y) --> y=x)"
   193 
   194 axioms
   195   iff:          "(P-->Q) --> (Q-->P) --> (P=Q)"
   196   True_or_False:  "(P=True) | (P=False)"
   197 
   198 defs
   199   Let_def:      "Let s f == f(s)"
   200   if_def:       "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"
   201 
   202 finalconsts
   203   "op ="
   204   "op -->"
   205   The
   206 
   207 axiomatization
   208   undefined :: 'a
   209 
   210 abbreviation (input)
   211   "arbitrary \<equiv> undefined"
   212 
   213 
   214 subsubsection {* Generic classes and algebraic operations *}
   215 
   216 class default =
   217   fixes default :: 'a
   218 
   219 class zero = 
   220   fixes zero :: 'a  ("0")
   221 
   222 class one =
   223   fixes one  :: 'a  ("1")
   224 
   225 hide (open) const zero one
   226 
   227 class plus =
   228   fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "+" 65)
   229 
   230 class minus =
   231   fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "-" 65)
   232 
   233 class uminus =
   234   fixes uminus :: "'a \<Rightarrow> 'a"  ("- _" [81] 80)
   235 
   236 class times =
   237   fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "*" 70)
   238 
   239 class inverse =
   240   fixes inverse :: "'a \<Rightarrow> 'a"
   241     and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "'/" 70)
   242 
   243 class abs =
   244   fixes abs :: "'a \<Rightarrow> 'a"
   245 begin
   246 
   247 notation (xsymbols)
   248   abs  ("\<bar>_\<bar>")
   249 
   250 notation (HTML output)
   251   abs  ("\<bar>_\<bar>")
   252 
   253 end
   254 
   255 class sgn =
   256   fixes sgn :: "'a \<Rightarrow> 'a"
   257 
   258 class ord =
   259   fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
   260     and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
   261 begin
   262 
   263 notation
   264   less_eq  ("op <=") and
   265   less_eq  ("(_/ <= _)" [51, 51] 50) and
   266   less  ("op <") and
   267   less  ("(_/ < _)"  [51, 51] 50)
   268   
   269 notation (xsymbols)
   270   less_eq  ("op \<le>") and
   271   less_eq  ("(_/ \<le> _)"  [51, 51] 50)
   272 
   273 notation (HTML output)
   274   less_eq  ("op \<le>") and
   275   less_eq  ("(_/ \<le> _)"  [51, 51] 50)
   276 
   277 abbreviation (input)
   278   greater_eq  (infix ">=" 50) where
   279   "x >= y \<equiv> y <= x"
   280 
   281 notation (input)
   282   greater_eq  (infix "\<ge>" 50)
   283 
   284 abbreviation (input)
   285   greater  (infix ">" 50) where
   286   "x > y \<equiv> y < x"
   287 
   288 end
   289 
   290 syntax
   291   "_index1"  :: index    ("\<^sub>1")
   292 translations
   293   (index) "\<^sub>1" => (index) "\<^bsub>\<struct>\<^esub>"
   294 
   295 typed_print_translation {*
   296 let
   297   fun tr' c = (c, fn show_sorts => fn T => fn ts =>
   298     if (not o null) ts orelse T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
   299     else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T);
   300 in map tr' [@{const_syntax HOL.one}, @{const_syntax HOL.zero}] end;
   301 *} -- {* show types that are presumably too general *}
   302 
   303 
   304 subsection {* Fundamental rules *}
   305 
   306 subsubsection {* Equality *}
   307 
   308 lemma sym: "s = t ==> t = s"
   309   by (erule subst) (rule refl)
   310 
   311 lemma ssubst: "t = s ==> P s ==> P t"
   312   by (drule sym) (erule subst)
   313 
   314 lemma trans: "[| r=s; s=t |] ==> r=t"
   315   by (erule subst)
   316 
   317 lemma meta_eq_to_obj_eq: 
   318   assumes meq: "A == B"
   319   shows "A = B"
   320   by (unfold meq) (rule refl)
   321 
   322 text {* Useful with @{text erule} for proving equalities from known equalities. *}
   323      (* a = b
   324         |   |
   325         c = d   *)
   326 lemma box_equals: "[| a=b;  a=c;  b=d |] ==> c=d"
   327 apply (rule trans)
   328 apply (rule trans)
   329 apply (rule sym)
   330 apply assumption+
   331 done
   332 
   333 text {* For calculational reasoning: *}
   334 
   335 lemma forw_subst: "a = b ==> P b ==> P a"
   336   by (rule ssubst)
   337 
   338 lemma back_subst: "P a ==> a = b ==> P b"
   339   by (rule subst)
   340 
   341 
   342 subsubsection {*Congruence rules for application*}
   343 
   344 (*similar to AP_THM in Gordon's HOL*)
   345 lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
   346 apply (erule subst)
   347 apply (rule refl)
   348 done
   349 
   350 (*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
   351 lemma arg_cong: "x=y ==> f(x)=f(y)"
   352 apply (erule subst)
   353 apply (rule refl)
   354 done
   355 
   356 lemma arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d"
   357 apply (erule ssubst)+
   358 apply (rule refl)
   359 done
   360 
   361 lemma cong: "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
   362 apply (erule subst)+
   363 apply (rule refl)
   364 done
   365 
   366 
   367 subsubsection {*Equality of booleans -- iff*}
   368 
   369 lemma iffI: assumes "P ==> Q" and "Q ==> P" shows "P=Q"
   370   by (iprover intro: iff [THEN mp, THEN mp] impI assms)
   371 
   372 lemma iffD2: "[| P=Q; Q |] ==> P"
   373   by (erule ssubst)
   374 
   375 lemma rev_iffD2: "[| Q; P=Q |] ==> P"
   376   by (erule iffD2)
   377 
   378 lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P"
   379   by (drule sym) (rule iffD2)
   380 
   381 lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P"
   382   by (drule sym) (rule rev_iffD2)
   383 
   384 lemma iffE:
   385   assumes major: "P=Q"
   386     and minor: "[| P --> Q; Q --> P |] ==> R"
   387   shows R
   388   by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
   389 
   390 
   391 subsubsection {*True*}
   392 
   393 lemma TrueI: "True"
   394   unfolding True_def by (rule refl)
   395 
   396 lemma eqTrueI: "P ==> P = True"
   397   by (iprover intro: iffI TrueI)
   398 
   399 lemma eqTrueE: "P = True ==> P"
   400   by (erule iffD2) (rule TrueI)
   401 
   402 
   403 subsubsection {*Universal quantifier*}
   404 
   405 lemma allI: assumes "!!x::'a. P(x)" shows "ALL x. P(x)"
   406   unfolding All_def by (iprover intro: ext eqTrueI assms)
   407 
   408 lemma spec: "ALL x::'a. P(x) ==> P(x)"
   409 apply (unfold All_def)
   410 apply (rule eqTrueE)
   411 apply (erule fun_cong)
   412 done
   413 
   414 lemma allE:
   415   assumes major: "ALL x. P(x)"
   416     and minor: "P(x) ==> R"
   417   shows R
   418   by (iprover intro: minor major [THEN spec])
   419 
   420 lemma all_dupE:
   421   assumes major: "ALL x. P(x)"
   422     and minor: "[| P(x); ALL x. P(x) |] ==> R"
   423   shows R
   424   by (iprover intro: minor major major [THEN spec])
   425 
   426 
   427 subsubsection {* False *}
   428 
   429 text {*
   430   Depends upon @{text spec}; it is impossible to do propositional
   431   logic before quantifiers!
   432 *}
   433 
   434 lemma FalseE: "False ==> P"
   435   apply (unfold False_def)
   436   apply (erule spec)
   437   done
   438 
   439 lemma False_neq_True: "False = True ==> P"
   440   by (erule eqTrueE [THEN FalseE])
   441 
   442 
   443 subsubsection {* Negation *}
   444 
   445 lemma notI:
   446   assumes "P ==> False"
   447   shows "~P"
   448   apply (unfold not_def)
   449   apply (iprover intro: impI assms)
   450   done
   451 
   452 lemma False_not_True: "False ~= True"
   453   apply (rule notI)
   454   apply (erule False_neq_True)
   455   done
   456 
   457 lemma True_not_False: "True ~= False"
   458   apply (rule notI)
   459   apply (drule sym)
   460   apply (erule False_neq_True)
   461   done
   462 
   463 lemma notE: "[| ~P;  P |] ==> R"
   464   apply (unfold not_def)
   465   apply (erule mp [THEN FalseE])
   466   apply assumption
   467   done
   468 
   469 lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P"
   470   by (erule notE [THEN notI]) (erule meta_mp)
   471 
   472 
   473 subsubsection {*Implication*}
   474 
   475 lemma impE:
   476   assumes "P-->Q" "P" "Q ==> R"
   477   shows "R"
   478 by (iprover intro: assms mp)
   479 
   480 (* Reduces Q to P-->Q, allowing substitution in P. *)
   481 lemma rev_mp: "[| P;  P --> Q |] ==> Q"
   482 by (iprover intro: mp)
   483 
   484 lemma contrapos_nn:
   485   assumes major: "~Q"
   486       and minor: "P==>Q"
   487   shows "~P"
   488 by (iprover intro: notI minor major [THEN notE])
   489 
   490 (*not used at all, but we already have the other 3 combinations *)
   491 lemma contrapos_pn:
   492   assumes major: "Q"
   493       and minor: "P ==> ~Q"
   494   shows "~P"
   495 by (iprover intro: notI minor major notE)
   496 
   497 lemma not_sym: "t ~= s ==> s ~= t"
   498   by (erule contrapos_nn) (erule sym)
   499 
   500 lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
   501   by (erule subst, erule ssubst, assumption)
   502 
   503 (*still used in HOLCF*)
   504 lemma rev_contrapos:
   505   assumes pq: "P ==> Q"
   506       and nq: "~Q"
   507   shows "~P"
   508 apply (rule nq [THEN contrapos_nn])
   509 apply (erule pq)
   510 done
   511 
   512 subsubsection {*Existential quantifier*}
   513 
   514 lemma exI: "P x ==> EX x::'a. P x"
   515 apply (unfold Ex_def)
   516 apply (iprover intro: allI allE impI mp)
   517 done
   518 
   519 lemma exE:
   520   assumes major: "EX x::'a. P(x)"
   521       and minor: "!!x. P(x) ==> Q"
   522   shows "Q"
   523 apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
   524 apply (iprover intro: impI [THEN allI] minor)
   525 done
   526 
   527 
   528 subsubsection {*Conjunction*}
   529 
   530 lemma conjI: "[| P; Q |] ==> P&Q"
   531 apply (unfold and_def)
   532 apply (iprover intro: impI [THEN allI] mp)
   533 done
   534 
   535 lemma conjunct1: "[| P & Q |] ==> P"
   536 apply (unfold and_def)
   537 apply (iprover intro: impI dest: spec mp)
   538 done
   539 
   540 lemma conjunct2: "[| P & Q |] ==> Q"
   541 apply (unfold and_def)
   542 apply (iprover intro: impI dest: spec mp)
   543 done
   544 
   545 lemma conjE:
   546   assumes major: "P&Q"
   547       and minor: "[| P; Q |] ==> R"
   548   shows "R"
   549 apply (rule minor)
   550 apply (rule major [THEN conjunct1])
   551 apply (rule major [THEN conjunct2])
   552 done
   553 
   554 lemma context_conjI:
   555   assumes "P" "P ==> Q" shows "P & Q"
   556 by (iprover intro: conjI assms)
   557 
   558 
   559 subsubsection {*Disjunction*}
   560 
   561 lemma disjI1: "P ==> P|Q"
   562 apply (unfold or_def)
   563 apply (iprover intro: allI impI mp)
   564 done
   565 
   566 lemma disjI2: "Q ==> P|Q"
   567 apply (unfold or_def)
   568 apply (iprover intro: allI impI mp)
   569 done
   570 
   571 lemma disjE:
   572   assumes major: "P|Q"
   573       and minorP: "P ==> R"
   574       and minorQ: "Q ==> R"
   575   shows "R"
   576 by (iprover intro: minorP minorQ impI
   577                  major [unfolded or_def, THEN spec, THEN mp, THEN mp])
   578 
   579 
   580 subsubsection {*Classical logic*}
   581 
   582 lemma classical:
   583   assumes prem: "~P ==> P"
   584   shows "P"
   585 apply (rule True_or_False [THEN disjE, THEN eqTrueE])
   586 apply assumption
   587 apply (rule notI [THEN prem, THEN eqTrueI])
   588 apply (erule subst)
   589 apply assumption
   590 done
   591 
   592 lemmas ccontr = FalseE [THEN classical, standard]
   593 
   594 (*notE with premises exchanged; it discharges ~R so that it can be used to
   595   make elimination rules*)
   596 lemma rev_notE:
   597   assumes premp: "P"
   598       and premnot: "~R ==> ~P"
   599   shows "R"
   600 apply (rule ccontr)
   601 apply (erule notE [OF premnot premp])
   602 done
   603 
   604 (*Double negation law*)
   605 lemma notnotD: "~~P ==> P"
   606 apply (rule classical)
   607 apply (erule notE)
   608 apply assumption
   609 done
   610 
   611 lemma contrapos_pp:
   612   assumes p1: "Q"
   613       and p2: "~P ==> ~Q"
   614   shows "P"
   615 by (iprover intro: classical p1 p2 notE)
   616 
   617 
   618 subsubsection {*Unique existence*}
   619 
   620 lemma ex1I:
   621   assumes "P a" "!!x. P(x) ==> x=a"
   622   shows "EX! x. P(x)"
   623 by (unfold Ex1_def, iprover intro: assms exI conjI allI impI)
   624 
   625 text{*Sometimes easier to use: the premises have no shared variables.  Safe!*}
   626 lemma ex_ex1I:
   627   assumes ex_prem: "EX x. P(x)"
   628       and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
   629   shows "EX! x. P(x)"
   630 by (iprover intro: ex_prem [THEN exE] ex1I eq)
   631 
   632 lemma ex1E:
   633   assumes major: "EX! x. P(x)"
   634       and minor: "!!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R"
   635   shows "R"
   636 apply (rule major [unfolded Ex1_def, THEN exE])
   637 apply (erule conjE)
   638 apply (iprover intro: minor)
   639 done
   640 
   641 lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
   642 apply (erule ex1E)
   643 apply (rule exI)
   644 apply assumption
   645 done
   646 
   647 
   648 subsubsection {*THE: definite description operator*}
   649 
   650 lemma the_equality:
   651   assumes prema: "P a"
   652       and premx: "!!x. P x ==> x=a"
   653   shows "(THE x. P x) = a"
   654 apply (rule trans [OF _ the_eq_trivial])
   655 apply (rule_tac f = "The" in arg_cong)
   656 apply (rule ext)
   657 apply (rule iffI)
   658  apply (erule premx)
   659 apply (erule ssubst, rule prema)
   660 done
   661 
   662 lemma theI:
   663   assumes "P a" and "!!x. P x ==> x=a"
   664   shows "P (THE x. P x)"
   665 by (iprover intro: assms the_equality [THEN ssubst])
   666 
   667 lemma theI': "EX! x. P x ==> P (THE x. P x)"
   668 apply (erule ex1E)
   669 apply (erule theI)
   670 apply (erule allE)
   671 apply (erule mp)
   672 apply assumption
   673 done
   674 
   675 (*Easier to apply than theI: only one occurrence of P*)
   676 lemma theI2:
   677   assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
   678   shows "Q (THE x. P x)"
   679 by (iprover intro: assms theI)
   680 
   681 lemma the1I2: assumes "EX! x. P x" "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
   682 by(iprover intro:assms(2) theI2[where P=P and Q=Q] ex1E[OF assms(1)]
   683            elim:allE impE)
   684 
   685 lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
   686 apply (rule the_equality)
   687 apply  assumption
   688 apply (erule ex1E)
   689 apply (erule all_dupE)
   690 apply (drule mp)
   691 apply  assumption
   692 apply (erule ssubst)
   693 apply (erule allE)
   694 apply (erule mp)
   695 apply assumption
   696 done
   697 
   698 lemma the_sym_eq_trivial: "(THE y. x=y) = x"
   699 apply (rule the_equality)
   700 apply (rule refl)
   701 apply (erule sym)
   702 done
   703 
   704 
   705 subsubsection {*Classical intro rules for disjunction and existential quantifiers*}
   706 
   707 lemma disjCI:
   708   assumes "~Q ==> P" shows "P|Q"
   709 apply (rule classical)
   710 apply (iprover intro: assms disjI1 disjI2 notI elim: notE)
   711 done
   712 
   713 lemma excluded_middle: "~P | P"
   714 by (iprover intro: disjCI)
   715 
   716 text {*
   717   case distinction as a natural deduction rule.
   718   Note that @{term "~P"} is the second case, not the first
   719 *}
   720 lemma case_split [case_names True False]:
   721   assumes prem1: "P ==> Q"
   722       and prem2: "~P ==> Q"
   723   shows "Q"
   724 apply (rule excluded_middle [THEN disjE])
   725 apply (erule prem2)
   726 apply (erule prem1)
   727 done
   728 
   729 (*Classical implies (-->) elimination. *)
   730 lemma impCE:
   731   assumes major: "P-->Q"
   732       and minor: "~P ==> R" "Q ==> R"
   733   shows "R"
   734 apply (rule excluded_middle [of P, THEN disjE])
   735 apply (iprover intro: minor major [THEN mp])+
   736 done
   737 
   738 (*This version of --> elimination works on Q before P.  It works best for
   739   those cases in which P holds "almost everywhere".  Can't install as
   740   default: would break old proofs.*)
   741 lemma impCE':
   742   assumes major: "P-->Q"
   743       and minor: "Q ==> R" "~P ==> R"
   744   shows "R"
   745 apply (rule excluded_middle [of P, THEN disjE])
   746 apply (iprover intro: minor major [THEN mp])+
   747 done
   748 
   749 (*Classical <-> elimination. *)
   750 lemma iffCE:
   751   assumes major: "P=Q"
   752       and minor: "[| P; Q |] ==> R"  "[| ~P; ~Q |] ==> R"
   753   shows "R"
   754 apply (rule major [THEN iffE])
   755 apply (iprover intro: minor elim: impCE notE)
   756 done
   757 
   758 lemma exCI:
   759   assumes "ALL x. ~P(x) ==> P(a)"
   760   shows "EX x. P(x)"
   761 apply (rule ccontr)
   762 apply (iprover intro: assms exI allI notI notE [of "\<exists>x. P x"])
   763 done
   764 
   765 
   766 subsubsection {* Intuitionistic Reasoning *}
   767 
   768 lemma impE':
   769   assumes 1: "P --> Q"
   770     and 2: "Q ==> R"
   771     and 3: "P --> Q ==> P"
   772   shows R
   773 proof -
   774   from 3 and 1 have P .
   775   with 1 have Q by (rule impE)
   776   with 2 show R .
   777 qed
   778 
   779 lemma allE':
   780   assumes 1: "ALL x. P x"
   781     and 2: "P x ==> ALL x. P x ==> Q"
   782   shows Q
   783 proof -
   784   from 1 have "P x" by (rule spec)
   785   from this and 1 show Q by (rule 2)
   786 qed
   787 
   788 lemma notE':
   789   assumes 1: "~ P"
   790     and 2: "~ P ==> P"
   791   shows R
   792 proof -
   793   from 2 and 1 have P .
   794   with 1 show R by (rule notE)
   795 qed
   796 
   797 lemma TrueE: "True ==> P ==> P" .
   798 lemma notFalseE: "~ False ==> P ==> P" .
   799 
   800 lemmas [Pure.elim!] = disjE iffE FalseE conjE exE TrueE notFalseE
   801   and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
   802   and [Pure.elim 2] = allE notE' impE'
   803   and [Pure.intro] = exI disjI2 disjI1
   804 
   805 lemmas [trans] = trans
   806   and [sym] = sym not_sym
   807   and [Pure.elim?] = iffD1 iffD2 impE
   808 
   809 use "Tools/hologic.ML"
   810 
   811 
   812 subsubsection {* Atomizing meta-level connectives *}
   813 
   814 axiomatization where
   815   eq_reflection: "x = y \<Longrightarrow> x \<equiv> y" (*admissible axiom*)
   816 
   817 lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
   818 proof
   819   assume "!!x. P x"
   820   then show "ALL x. P x" ..
   821 next
   822   assume "ALL x. P x"
   823   then show "!!x. P x" by (rule allE)
   824 qed
   825 
   826 lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
   827 proof
   828   assume r: "A ==> B"
   829   show "A --> B" by (rule impI) (rule r)
   830 next
   831   assume "A --> B" and A
   832   then show B by (rule mp)
   833 qed
   834 
   835 lemma atomize_not: "(A ==> False) == Trueprop (~A)"
   836 proof
   837   assume r: "A ==> False"
   838   show "~A" by (rule notI) (rule r)
   839 next
   840   assume "~A" and A
   841   then show False by (rule notE)
   842 qed
   843 
   844 lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
   845 proof
   846   assume "x == y"
   847   show "x = y" by (unfold `x == y`) (rule refl)
   848 next
   849   assume "x = y"
   850   then show "x == y" by (rule eq_reflection)
   851 qed
   852 
   853 lemma atomize_conj [atomize]: "(A &&& B) == Trueprop (A & B)"
   854 proof
   855   assume conj: "A &&& B"
   856   show "A & B"
   857   proof (rule conjI)
   858     from conj show A by (rule conjunctionD1)
   859     from conj show B by (rule conjunctionD2)
   860   qed
   861 next
   862   assume conj: "A & B"
   863   show "A &&& B"
   864   proof -
   865     from conj show A ..
   866     from conj show B ..
   867   qed
   868 qed
   869 
   870 lemmas [symmetric, rulify] = atomize_all atomize_imp
   871   and [symmetric, defn] = atomize_all atomize_imp atomize_eq
   872 
   873 
   874 subsubsection {* Atomizing elimination rules *}
   875 
   876 setup AtomizeElim.setup
   877 
   878 lemma atomize_exL[atomize_elim]: "(!!x. P x ==> Q) == ((EX x. P x) ==> Q)"
   879   by rule iprover+
   880 
   881 lemma atomize_conjL[atomize_elim]: "(A ==> B ==> C) == (A & B ==> C)"
   882   by rule iprover+
   883 
   884 lemma atomize_disjL[atomize_elim]: "((A ==> C) ==> (B ==> C) ==> C) == ((A | B ==> C) ==> C)"
   885   by rule iprover+
   886 
   887 lemma atomize_elimL[atomize_elim]: "(!!B. (A ==> B) ==> B) == Trueprop A" ..
   888 
   889 
   890 subsection {* Package setup *}
   891 
   892 subsubsection {* Classical Reasoner setup *}
   893 
   894 lemma imp_elim: "P --> Q ==> (~ R ==> P) ==> (Q ==> R) ==> R"
   895   by (rule classical) iprover
   896 
   897 lemma swap: "~ P ==> (~ R ==> P) ==> R"
   898   by (rule classical) iprover
   899 
   900 lemma thin_refl:
   901   "\<And>X. \<lbrakk> x=x; PROP W \<rbrakk> \<Longrightarrow> PROP W" .
   902 
   903 ML {*
   904 structure Hypsubst = HypsubstFun(
   905 struct
   906   structure Simplifier = Simplifier
   907   val dest_eq = HOLogic.dest_eq
   908   val dest_Trueprop = HOLogic.dest_Trueprop
   909   val dest_imp = HOLogic.dest_imp
   910   val eq_reflection = @{thm eq_reflection}
   911   val rev_eq_reflection = @{thm meta_eq_to_obj_eq}
   912   val imp_intr = @{thm impI}
   913   val rev_mp = @{thm rev_mp}
   914   val subst = @{thm subst}
   915   val sym = @{thm sym}
   916   val thin_refl = @{thm thin_refl};
   917   val prop_subst = @{lemma "PROP P t ==> PROP prop (x = t ==> PROP P x)"
   918                      by (unfold prop_def) (drule eq_reflection, unfold)}
   919 end);
   920 open Hypsubst;
   921 
   922 structure Classical = ClassicalFun(
   923 struct
   924   val imp_elim = @{thm imp_elim}
   925   val not_elim = @{thm notE}
   926   val swap = @{thm swap}
   927   val classical = @{thm classical}
   928   val sizef = Drule.size_of_thm
   929   val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
   930 end);
   931 
   932 structure BasicClassical: BASIC_CLASSICAL = Classical; 
   933 open BasicClassical;
   934 
   935 ML_Antiquote.value "claset"
   936   (Scan.succeed "Classical.local_claset_of (ML_Context.the_local_context ())");
   937 
   938 structure ResAtpset = NamedThmsFun(val name = "atp" val description = "ATP rules");
   939 
   940 structure ResBlacklist = NamedThmsFun(val name = "noatp" val description = "theorems blacklisted for ATP");
   941 *}
   942 
   943 text {*ResBlacklist holds theorems blacklisted to sledgehammer. 
   944   These theorems typically produce clauses that are prolific (match too many equality or
   945   membership literals) and relate to seldom-used facts. Some duplicate other rules.*}
   946 
   947 setup {*
   948 let
   949   (*prevent substitution on bool*)
   950   fun hyp_subst_tac' i thm = if i <= Thm.nprems_of thm andalso
   951     Term.exists_Const (fn ("op =", Type (_, [T, _])) => T <> Type ("bool", []) | _ => false)
   952       (nth (Thm.prems_of thm) (i - 1)) then Hypsubst.hyp_subst_tac i thm else no_tac thm;
   953 in
   954   Hypsubst.hypsubst_setup
   955   #> ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac)
   956   #> Classical.setup
   957   #> ResAtpset.setup
   958   #> ResBlacklist.setup
   959 end
   960 *}
   961 
   962 declare iffI [intro!]
   963   and notI [intro!]
   964   and impI [intro!]
   965   and disjCI [intro!]
   966   and conjI [intro!]
   967   and TrueI [intro!]
   968   and refl [intro!]
   969 
   970 declare iffCE [elim!]
   971   and FalseE [elim!]
   972   and impCE [elim!]
   973   and disjE [elim!]
   974   and conjE [elim!]
   975   and conjE [elim!]
   976 
   977 declare ex_ex1I [intro!]
   978   and allI [intro!]
   979   and the_equality [intro]
   980   and exI [intro]
   981 
   982 declare exE [elim!]
   983   allE [elim]
   984 
   985 ML {* val HOL_cs = @{claset} *}
   986 
   987 lemma contrapos_np: "~ Q ==> (~ P ==> Q) ==> P"
   988   apply (erule swap)
   989   apply (erule (1) meta_mp)
   990   done
   991 
   992 declare ex_ex1I [rule del, intro! 2]
   993   and ex1I [intro]
   994 
   995 lemmas [intro?] = ext
   996   and [elim?] = ex1_implies_ex
   997 
   998 (*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
   999 lemma alt_ex1E [elim!]:
  1000   assumes major: "\<exists>!x. P x"
  1001       and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R"
  1002   shows R
  1003 apply (rule ex1E [OF major])
  1004 apply (rule prem)
  1005 apply (tactic {* ares_tac @{thms allI} 1 *})+
  1006 apply (tactic {* etac (Classical.dup_elim @{thm allE}) 1 *})
  1007 apply iprover
  1008 done
  1009 
  1010 ML {*
  1011 structure Blast = BlastFun
  1012 (
  1013   type claset = Classical.claset
  1014   val equality_name = @{const_name "op ="}
  1015   val not_name = @{const_name Not}
  1016   val notE = @{thm notE}
  1017   val ccontr = @{thm ccontr}
  1018   val contr_tac = Classical.contr_tac
  1019   val dup_intr = Classical.dup_intr
  1020   val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
  1021   val claset = Classical.claset
  1022   val rep_cs = Classical.rep_cs
  1023   val cla_modifiers = Classical.cla_modifiers
  1024   val cla_meth' = Classical.cla_meth'
  1025 );
  1026 val Blast_tac = Blast.Blast_tac;
  1027 val blast_tac = Blast.blast_tac;
  1028 *}
  1029 
  1030 setup Blast.setup
  1031 
  1032 
  1033 subsubsection {* Simplifier *}
  1034 
  1035 lemma eta_contract_eq: "(%s. f s) = f" ..
  1036 
  1037 lemma simp_thms:
  1038   shows not_not: "(~ ~ P) = P"
  1039   and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
  1040   and
  1041     "(P ~= Q) = (P = (~Q))"
  1042     "(P | ~P) = True"    "(~P | P) = True"
  1043     "(x = x) = True"
  1044   and not_True_eq_False: "(\<not> True) = False"
  1045   and not_False_eq_True: "(\<not> False) = True"
  1046   and
  1047     "(~P) ~= P"  "P ~= (~P)"
  1048     "(True=P) = P"
  1049   and eq_True: "(P = True) = P"
  1050   and "(False=P) = (~P)"
  1051   and eq_False: "(P = False) = (\<not> P)"
  1052   and
  1053     "(True --> P) = P"  "(False --> P) = True"
  1054     "(P --> True) = True"  "(P --> P) = True"
  1055     "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
  1056     "(P & True) = P"  "(True & P) = P"
  1057     "(P & False) = False"  "(False & P) = False"
  1058     "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
  1059     "(P & ~P) = False"    "(~P & P) = False"
  1060     "(P | True) = True"  "(True | P) = True"
  1061     "(P | False) = P"  "(False | P) = P"
  1062     "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
  1063     "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"
  1064     -- {* needed for the one-point-rule quantifier simplification procs *}
  1065     -- {* essential for termination!! *} and
  1066     "!!P. (EX x. x=t & P(x)) = P(t)"
  1067     "!!P. (EX x. t=x & P(x)) = P(t)"
  1068     "!!P. (ALL x. x=t --> P(x)) = P(t)"
  1069     "!!P. (ALL x. t=x --> P(x)) = P(t)"
  1070   by (blast, blast, blast, blast, blast, iprover+)
  1071 
  1072 lemma disj_absorb: "(A | A) = A"
  1073   by blast
  1074 
  1075 lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
  1076   by blast
  1077 
  1078 lemma conj_absorb: "(A & A) = A"
  1079   by blast
  1080 
  1081 lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
  1082   by blast
  1083 
  1084 lemma eq_ac:
  1085   shows eq_commute: "(a=b) = (b=a)"
  1086     and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
  1087     and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (iprover, blast+)
  1088 lemma neq_commute: "(a~=b) = (b~=a)" by iprover
  1089 
  1090 lemma conj_comms:
  1091   shows conj_commute: "(P&Q) = (Q&P)"
  1092     and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by iprover+
  1093 lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by iprover
  1094 
  1095 lemmas conj_ac = conj_commute conj_left_commute conj_assoc
  1096 
  1097 lemma disj_comms:
  1098   shows disj_commute: "(P|Q) = (Q|P)"
  1099     and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by iprover+
  1100 lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by iprover
  1101 
  1102 lemmas disj_ac = disj_commute disj_left_commute disj_assoc
  1103 
  1104 lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by iprover
  1105 lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by iprover
  1106 
  1107 lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by iprover
  1108 lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by iprover
  1109 
  1110 lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by iprover
  1111 lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by iprover
  1112 lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by iprover
  1113 
  1114 text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
  1115 lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
  1116 lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
  1117 
  1118 lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
  1119 lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
  1120 
  1121 lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
  1122   by iprover
  1123 
  1124 lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by iprover
  1125 lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
  1126 lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
  1127 lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
  1128 lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
  1129 lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
  1130   by blast
  1131 lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
  1132 
  1133 lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by iprover
  1134 
  1135 
  1136 lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
  1137   -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
  1138   -- {* cases boil down to the same thing. *}
  1139   by blast
  1140 
  1141 lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
  1142 lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
  1143 lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by iprover
  1144 lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by iprover
  1145 lemma all_not_ex: "(ALL x. P x) = (~ (EX x. ~ P x ))" by blast
  1146 
  1147 declare All_def [noatp]
  1148 
  1149 lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by iprover
  1150 lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by iprover
  1151 
  1152 text {*
  1153   \medskip The @{text "&"} congruence rule: not included by default!
  1154   May slow rewrite proofs down by as much as 50\% *}
  1155 
  1156 lemma conj_cong:
  1157     "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
  1158   by iprover
  1159 
  1160 lemma rev_conj_cong:
  1161     "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
  1162   by iprover
  1163 
  1164 text {* The @{text "|"} congruence rule: not included by default! *}
  1165 
  1166 lemma disj_cong:
  1167     "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
  1168   by blast
  1169 
  1170 
  1171 text {* \medskip if-then-else rules *}
  1172 
  1173 lemma if_True: "(if True then x else y) = x"
  1174   by (unfold if_def) blast
  1175 
  1176 lemma if_False: "(if False then x else y) = y"
  1177   by (unfold if_def) blast
  1178 
  1179 lemma if_P: "P ==> (if P then x else y) = x"
  1180   by (unfold if_def) blast
  1181 
  1182 lemma if_not_P: "~P ==> (if P then x else y) = y"
  1183   by (unfold if_def) blast
  1184 
  1185 lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
  1186   apply (rule case_split [of Q])
  1187    apply (simplesubst if_P)
  1188     prefer 3 apply (simplesubst if_not_P, blast+)
  1189   done
  1190 
  1191 lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
  1192 by (simplesubst split_if, blast)
  1193 
  1194 lemmas if_splits [noatp] = split_if split_if_asm
  1195 
  1196 lemma if_cancel: "(if c then x else x) = x"
  1197 by (simplesubst split_if, blast)
  1198 
  1199 lemma if_eq_cancel: "(if x = y then y else x) = x"
  1200 by (simplesubst split_if, blast)
  1201 
  1202 lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
  1203   -- {* This form is useful for expanding @{text "if"}s on the RIGHT of the @{text "==>"} symbol. *}
  1204   by (rule split_if)
  1205 
  1206 lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
  1207   -- {* And this form is useful for expanding @{text "if"}s on the LEFT. *}
  1208   apply (simplesubst split_if, blast)
  1209   done
  1210 
  1211 lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) iprover
  1212 lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover
  1213 
  1214 text {* \medskip let rules for simproc *}
  1215 
  1216 lemma Let_folded: "f x \<equiv> g x \<Longrightarrow>  Let x f \<equiv> Let x g"
  1217   by (unfold Let_def)
  1218 
  1219 lemma Let_unfold: "f x \<equiv> g \<Longrightarrow>  Let x f \<equiv> g"
  1220   by (unfold Let_def)
  1221 
  1222 text {*
  1223   The following copy of the implication operator is useful for
  1224   fine-tuning congruence rules.  It instructs the simplifier to simplify
  1225   its premise.
  1226 *}
  1227 
  1228 constdefs
  1229   simp_implies :: "[prop, prop] => prop"  (infixr "=simp=>" 1)
  1230   [code del]: "simp_implies \<equiv> op ==>"
  1231 
  1232 lemma simp_impliesI:
  1233   assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
  1234   shows "PROP P =simp=> PROP Q"
  1235   apply (unfold simp_implies_def)
  1236   apply (rule PQ)
  1237   apply assumption
  1238   done
  1239 
  1240 lemma simp_impliesE:
  1241   assumes PQ: "PROP P =simp=> PROP Q"
  1242   and P: "PROP P"
  1243   and QR: "PROP Q \<Longrightarrow> PROP R"
  1244   shows "PROP R"
  1245   apply (rule QR)
  1246   apply (rule PQ [unfolded simp_implies_def])
  1247   apply (rule P)
  1248   done
  1249 
  1250 lemma simp_implies_cong:
  1251   assumes PP' :"PROP P == PROP P'"
  1252   and P'QQ': "PROP P' ==> (PROP Q == PROP Q')"
  1253   shows "(PROP P =simp=> PROP Q) == (PROP P' =simp=> PROP Q')"
  1254 proof (unfold simp_implies_def, rule equal_intr_rule)
  1255   assume PQ: "PROP P \<Longrightarrow> PROP Q"
  1256   and P': "PROP P'"
  1257   from PP' [symmetric] and P' have "PROP P"
  1258     by (rule equal_elim_rule1)
  1259   then have "PROP Q" by (rule PQ)
  1260   with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
  1261 next
  1262   assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
  1263   and P: "PROP P"
  1264   from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
  1265   then have "PROP Q'" by (rule P'Q')
  1266   with P'QQ' [OF P', symmetric] show "PROP Q"
  1267     by (rule equal_elim_rule1)
  1268 qed
  1269 
  1270 lemma uncurry:
  1271   assumes "P \<longrightarrow> Q \<longrightarrow> R"
  1272   shows "P \<and> Q \<longrightarrow> R"
  1273   using assms by blast
  1274 
  1275 lemma iff_allI:
  1276   assumes "\<And>x. P x = Q x"
  1277   shows "(\<forall>x. P x) = (\<forall>x. Q x)"
  1278   using assms by blast
  1279 
  1280 lemma iff_exI:
  1281   assumes "\<And>x. P x = Q x"
  1282   shows "(\<exists>x. P x) = (\<exists>x. Q x)"
  1283   using assms by blast
  1284 
  1285 lemma all_comm:
  1286   "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
  1287   by blast
  1288 
  1289 lemma ex_comm:
  1290   "(\<exists>x y. P x y) = (\<exists>y x. P x y)"
  1291   by blast
  1292 
  1293 use "Tools/simpdata.ML"
  1294 ML {* open Simpdata *}
  1295 
  1296 setup {*
  1297   Simplifier.method_setup Splitter.split_modifiers
  1298   #> Simplifier.map_simpset (K Simpdata.simpset_simprocs)
  1299   #> Splitter.setup
  1300   #> clasimp_setup
  1301   #> EqSubst.setup
  1302 *}
  1303 
  1304 text {* Simproc for proving @{text "(y = x) == False"} from premise @{text "~(x = y)"}: *}
  1305 
  1306 simproc_setup neq ("x = y") = {* fn _ =>
  1307 let
  1308   val neq_to_EQ_False = @{thm not_sym} RS @{thm Eq_FalseI};
  1309   fun is_neq eq lhs rhs thm =
  1310     (case Thm.prop_of thm of
  1311       _ $ (Not $ (eq' $ l' $ r')) =>
  1312         Not = HOLogic.Not andalso eq' = eq andalso
  1313         r' aconv lhs andalso l' aconv rhs
  1314     | _ => false);
  1315   fun proc ss ct =
  1316     (case Thm.term_of ct of
  1317       eq $ lhs $ rhs =>
  1318         (case find_first (is_neq eq lhs rhs) (Simplifier.prems_of_ss ss) of
  1319           SOME thm => SOME (thm RS neq_to_EQ_False)
  1320         | NONE => NONE)
  1321      | _ => NONE);
  1322 in proc end;
  1323 *}
  1324 
  1325 simproc_setup let_simp ("Let x f") = {*
  1326 let
  1327   val (f_Let_unfold, x_Let_unfold) =
  1328     let val [(_ $ (f $ x) $ _)] = prems_of @{thm Let_unfold}
  1329     in (cterm_of @{theory} f, cterm_of @{theory} x) end
  1330   val (f_Let_folded, x_Let_folded) =
  1331     let val [(_ $ (f $ x) $ _)] = prems_of @{thm Let_folded}
  1332     in (cterm_of @{theory} f, cterm_of @{theory} x) end;
  1333   val g_Let_folded =
  1334     let val [(_ $ _ $ (g $ _))] = prems_of @{thm Let_folded}
  1335     in cterm_of @{theory} g end;
  1336   fun count_loose (Bound i) k = if i >= k then 1 else 0
  1337     | count_loose (s $ t) k = count_loose s k + count_loose t k
  1338     | count_loose (Abs (_, _, t)) k = count_loose  t (k + 1)
  1339     | count_loose _ _ = 0;
  1340   fun is_trivial_let (Const (@{const_name Let}, _) $ x $ t) =
  1341    case t
  1342     of Abs (_, _, t') => count_loose t' 0 <= 1
  1343      | _ => true;
  1344 in fn _ => fn ss => fn ct => if is_trivial_let (Thm.term_of ct)
  1345   then SOME @{thm Let_def} (*no or one ocurrenc of bound variable*)
  1346   else let (*Norbert Schirmer's case*)
  1347     val ctxt = Simplifier.the_context ss;
  1348     val thy = ProofContext.theory_of ctxt;
  1349     val t = Thm.term_of ct;
  1350     val ([t'], ctxt') = Variable.import_terms false [t] ctxt;
  1351   in Option.map (hd o Variable.export ctxt' ctxt o single)
  1352     (case t' of Const (@{const_name Let},_) $ x $ f => (* x and f are already in normal form *)
  1353       if is_Free x orelse is_Bound x orelse is_Const x
  1354       then SOME @{thm Let_def}
  1355       else
  1356         let
  1357           val n = case f of (Abs (x, _, _)) => x | _ => "x";
  1358           val cx = cterm_of thy x;
  1359           val {T = xT, ...} = rep_cterm cx;
  1360           val cf = cterm_of thy f;
  1361           val fx_g = Simplifier.rewrite ss (Thm.capply cf cx);
  1362           val (_ $ _ $ g) = prop_of fx_g;
  1363           val g' = abstract_over (x,g);
  1364         in (if (g aconv g')
  1365              then
  1366                 let
  1367                   val rl =
  1368                     cterm_instantiate [(f_Let_unfold, cf), (x_Let_unfold, cx)] @{thm Let_unfold};
  1369                 in SOME (rl OF [fx_g]) end
  1370              else if Term.betapply (f, x) aconv g then NONE (*avoid identity conversion*)
  1371              else let
  1372                    val abs_g'= Abs (n,xT,g');
  1373                    val g'x = abs_g'$x;
  1374                    val g_g'x = symmetric (beta_conversion false (cterm_of thy g'x));
  1375                    val rl = cterm_instantiate
  1376                              [(f_Let_folded, cterm_of thy f), (x_Let_folded, cx),
  1377                               (g_Let_folded, cterm_of thy abs_g')]
  1378                              @{thm Let_folded};
  1379                  in SOME (rl OF [transitive fx_g g_g'x])
  1380                  end)
  1381         end
  1382     | _ => NONE)
  1383   end
  1384 end *}
  1385 
  1386 lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
  1387 proof
  1388   assume "True \<Longrightarrow> PROP P"
  1389   from this [OF TrueI] show "PROP P" .
  1390 next
  1391   assume "PROP P"
  1392   then show "PROP P" .
  1393 qed
  1394 
  1395 lemma ex_simps:
  1396   "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
  1397   "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
  1398   "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
  1399   "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
  1400   "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
  1401   "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
  1402   -- {* Miniscoping: pushing in existential quantifiers. *}
  1403   by (iprover | blast)+
  1404 
  1405 lemma all_simps:
  1406   "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
  1407   "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
  1408   "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
  1409   "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
  1410   "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
  1411   "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
  1412   -- {* Miniscoping: pushing in universal quantifiers. *}
  1413   by (iprover | blast)+
  1414 
  1415 lemmas [simp] =
  1416   triv_forall_equality (*prunes params*)
  1417   True_implies_equals  (*prune asms `True'*)
  1418   if_True
  1419   if_False
  1420   if_cancel
  1421   if_eq_cancel
  1422   imp_disjL
  1423   (*In general it seems wrong to add distributive laws by default: they
  1424     might cause exponential blow-up.  But imp_disjL has been in for a while
  1425     and cannot be removed without affecting existing proofs.  Moreover,
  1426     rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
  1427     grounds that it allows simplification of R in the two cases.*)
  1428   conj_assoc
  1429   disj_assoc
  1430   de_Morgan_conj
  1431   de_Morgan_disj
  1432   imp_disj1
  1433   imp_disj2
  1434   not_imp
  1435   disj_not1
  1436   not_all
  1437   not_ex
  1438   cases_simp
  1439   the_eq_trivial
  1440   the_sym_eq_trivial
  1441   ex_simps
  1442   all_simps
  1443   simp_thms
  1444 
  1445 lemmas [cong] = imp_cong simp_implies_cong
  1446 lemmas [split] = split_if
  1447 
  1448 ML {* val HOL_ss = @{simpset} *}
  1449 
  1450 text {* Simplifies x assuming c and y assuming ~c *}
  1451 lemma if_cong:
  1452   assumes "b = c"
  1453       and "c \<Longrightarrow> x = u"
  1454       and "\<not> c \<Longrightarrow> y = v"
  1455   shows "(if b then x else y) = (if c then u else v)"
  1456   unfolding if_def using assms by simp
  1457 
  1458 text {* Prevents simplification of x and y:
  1459   faster and allows the execution of functional programs. *}
  1460 lemma if_weak_cong [cong]:
  1461   assumes "b = c"
  1462   shows "(if b then x else y) = (if c then x else y)"
  1463   using assms by (rule arg_cong)
  1464 
  1465 text {* Prevents simplification of t: much faster *}
  1466 lemma let_weak_cong:
  1467   assumes "a = b"
  1468   shows "(let x = a in t x) = (let x = b in t x)"
  1469   using assms by (rule arg_cong)
  1470 
  1471 text {* To tidy up the result of a simproc.  Only the RHS will be simplified. *}
  1472 lemma eq_cong2:
  1473   assumes "u = u'"
  1474   shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
  1475   using assms by simp
  1476 
  1477 lemma if_distrib:
  1478   "f (if c then x else y) = (if c then f x else f y)"
  1479   by simp
  1480 
  1481 text {* This lemma restricts the effect of the rewrite rule u=v to the left-hand
  1482   side of an equality.  Used in @{text "{Integ,Real}/simproc.ML"} *}
  1483 lemma restrict_to_left:
  1484   assumes "x = y"
  1485   shows "(x = z) = (y = z)"
  1486   using assms by simp
  1487 
  1488 
  1489 subsubsection {* Generic cases and induction *}
  1490 
  1491 text {* Rule projections: *}
  1492 
  1493 ML {*
  1494 structure ProjectRule = ProjectRuleFun
  1495 (
  1496   val conjunct1 = @{thm conjunct1}
  1497   val conjunct2 = @{thm conjunct2}
  1498   val mp = @{thm mp}
  1499 )
  1500 *}
  1501 
  1502 constdefs
  1503   induct_forall where "induct_forall P == \<forall>x. P x"
  1504   induct_implies where "induct_implies A B == A \<longrightarrow> B"
  1505   induct_equal where "induct_equal x y == x = y"
  1506   induct_conj where "induct_conj A B == A \<and> B"
  1507 
  1508 lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
  1509   by (unfold atomize_all induct_forall_def)
  1510 
  1511 lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
  1512   by (unfold atomize_imp induct_implies_def)
  1513 
  1514 lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
  1515   by (unfold atomize_eq induct_equal_def)
  1516 
  1517 lemma induct_conj_eq: "(A &&& B) == Trueprop (induct_conj A B)"
  1518   by (unfold atomize_conj induct_conj_def)
  1519 
  1520 lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
  1521 lemmas induct_rulify [symmetric, standard] = induct_atomize
  1522 lemmas induct_rulify_fallback =
  1523   induct_forall_def induct_implies_def induct_equal_def induct_conj_def
  1524 
  1525 
  1526 lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
  1527     induct_conj (induct_forall A) (induct_forall B)"
  1528   by (unfold induct_forall_def induct_conj_def) iprover
  1529 
  1530 lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
  1531     induct_conj (induct_implies C A) (induct_implies C B)"
  1532   by (unfold induct_implies_def induct_conj_def) iprover
  1533 
  1534 lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
  1535 proof
  1536   assume r: "induct_conj A B ==> PROP C" and A B
  1537   show "PROP C" by (rule r) (simp add: induct_conj_def `A` `B`)
  1538 next
  1539   assume r: "A ==> B ==> PROP C" and "induct_conj A B"
  1540   show "PROP C" by (rule r) (simp_all add: `induct_conj A B` [unfolded induct_conj_def])
  1541 qed
  1542 
  1543 lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
  1544 
  1545 hide const induct_forall induct_implies induct_equal induct_conj
  1546 
  1547 text {* Method setup. *}
  1548 
  1549 ML {*
  1550 structure Induct = InductFun
  1551 (
  1552   val cases_default = @{thm case_split}
  1553   val atomize = @{thms induct_atomize}
  1554   val rulify = @{thms induct_rulify}
  1555   val rulify_fallback = @{thms induct_rulify_fallback}
  1556 )
  1557 *}
  1558 
  1559 setup Induct.setup
  1560 
  1561 use "~~/src/Tools/induct_tacs.ML"
  1562 setup InductTacs.setup
  1563 
  1564 
  1565 subsubsection {* Coherent logic *}
  1566 
  1567 ML {*
  1568 structure Coherent = CoherentFun
  1569 (
  1570   val atomize_elimL = @{thm atomize_elimL}
  1571   val atomize_exL = @{thm atomize_exL}
  1572   val atomize_conjL = @{thm atomize_conjL}
  1573   val atomize_disjL = @{thm atomize_disjL}
  1574   val operator_names =
  1575     [@{const_name "op |"}, @{const_name "op &"}, @{const_name "Ex"}]
  1576 );
  1577 *}
  1578 
  1579 setup Coherent.setup
  1580 
  1581 
  1582 subsection {* Other simple lemmas and lemma duplicates *}
  1583 
  1584 lemma Let_0 [simp]: "Let 0 f = f 0"
  1585   unfolding Let_def ..
  1586 
  1587 lemma Let_1 [simp]: "Let 1 f = f 1"
  1588   unfolding Let_def ..
  1589 
  1590 lemma ex1_eq [iff]: "EX! x. x = t" "EX! x. t = x"
  1591   by blast+
  1592 
  1593 lemma choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
  1594   apply (rule iffI)
  1595   apply (rule_tac a = "%x. THE y. P x y" in ex1I)
  1596   apply (fast dest!: theI')
  1597   apply (fast intro: ext the1_equality [symmetric])
  1598   apply (erule ex1E)
  1599   apply (rule allI)
  1600   apply (rule ex1I)
  1601   apply (erule spec)
  1602   apply (erule_tac x = "%z. if z = x then y else f z" in allE)
  1603   apply (erule impE)
  1604   apply (rule allI)
  1605   apply (case_tac "xa = x")
  1606   apply (drule_tac [3] x = x in fun_cong, simp_all)
  1607   done
  1608 
  1609 lemma mk_left_commute:
  1610   fixes f (infix "\<otimes>" 60)
  1611   assumes a: "\<And>x y z. (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" and
  1612           c: "\<And>x y. x \<otimes> y = y \<otimes> x"
  1613   shows "x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
  1614   by (rule trans [OF trans [OF c a] arg_cong [OF c, of "f y"]])
  1615 
  1616 lemmas eq_sym_conv = eq_commute
  1617 
  1618 lemma nnf_simps:
  1619   "(\<not>(P \<and> Q)) = (\<not> P \<or> \<not> Q)" "(\<not> (P \<or> Q)) = (\<not> P \<and> \<not>Q)" "(P \<longrightarrow> Q) = (\<not>P \<or> Q)" 
  1620   "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not>(P = Q)) = ((P \<and> \<not> Q) \<or> (\<not>P \<and> Q))" 
  1621   "(\<not> \<not>(P)) = P"
  1622 by blast+
  1623 
  1624 
  1625 subsection {* Basic ML bindings *}
  1626 
  1627 ML {*
  1628 val FalseE = @{thm FalseE}
  1629 val Let_def = @{thm Let_def}
  1630 val TrueI = @{thm TrueI}
  1631 val allE = @{thm allE}
  1632 val allI = @{thm allI}
  1633 val all_dupE = @{thm all_dupE}
  1634 val arg_cong = @{thm arg_cong}
  1635 val box_equals = @{thm box_equals}
  1636 val ccontr = @{thm ccontr}
  1637 val classical = @{thm classical}
  1638 val conjE = @{thm conjE}
  1639 val conjI = @{thm conjI}
  1640 val conjunct1 = @{thm conjunct1}
  1641 val conjunct2 = @{thm conjunct2}
  1642 val disjCI = @{thm disjCI}
  1643 val disjE = @{thm disjE}
  1644 val disjI1 = @{thm disjI1}
  1645 val disjI2 = @{thm disjI2}
  1646 val eq_reflection = @{thm eq_reflection}
  1647 val ex1E = @{thm ex1E}
  1648 val ex1I = @{thm ex1I}
  1649 val ex1_implies_ex = @{thm ex1_implies_ex}
  1650 val exE = @{thm exE}
  1651 val exI = @{thm exI}
  1652 val excluded_middle = @{thm excluded_middle}
  1653 val ext = @{thm ext}
  1654 val fun_cong = @{thm fun_cong}
  1655 val iffD1 = @{thm iffD1}
  1656 val iffD2 = @{thm iffD2}
  1657 val iffI = @{thm iffI}
  1658 val impE = @{thm impE}
  1659 val impI = @{thm impI}
  1660 val meta_eq_to_obj_eq = @{thm meta_eq_to_obj_eq}
  1661 val mp = @{thm mp}
  1662 val notE = @{thm notE}
  1663 val notI = @{thm notI}
  1664 val not_all = @{thm not_all}
  1665 val not_ex = @{thm not_ex}
  1666 val not_iff = @{thm not_iff}
  1667 val not_not = @{thm not_not}
  1668 val not_sym = @{thm not_sym}
  1669 val refl = @{thm refl}
  1670 val rev_mp = @{thm rev_mp}
  1671 val spec = @{thm spec}
  1672 val ssubst = @{thm ssubst}
  1673 val subst = @{thm subst}
  1674 val sym = @{thm sym}
  1675 val trans = @{thm trans}
  1676 *}
  1677 
  1678 
  1679 subsection {* Code generator basics -- see further theory @{text "Code_Setup"} *}
  1680 
  1681 text {* Equality *}
  1682 
  1683 class eq =
  1684   fixes eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
  1685   assumes eq_equals: "eq x y \<longleftrightarrow> x = y"
  1686 begin
  1687 
  1688 lemma eq: "eq = (op =)"
  1689   by (rule ext eq_equals)+
  1690 
  1691 lemma eq_refl: "eq x x \<longleftrightarrow> True"
  1692   unfolding eq by rule+
  1693 
  1694 end
  1695 
  1696 text {* Module setup *}
  1697 
  1698 use "Tools/recfun_codegen.ML"
  1699 
  1700 setup {*
  1701   Code_ML.setup
  1702   #> Code_Haskell.setup
  1703   #> Nbe.setup
  1704   #> Codegen.setup
  1705   #> RecfunCodegen.setup
  1706 *}
  1707 
  1708 
  1709 subsection {* Nitpick theorem store *}
  1710 
  1711 ML {*
  1712 structure Nitpick_Const_Def_Thms = NamedThmsFun
  1713 (
  1714   val name = "nitpick_const_def"
  1715   val description = "alternative definitions of constants as needed by Nitpick"
  1716 )
  1717 structure Nitpick_Const_Simp_Thms = NamedThmsFun
  1718 (
  1719   val name = "nitpick_const_simp"
  1720   val description = "equational specification of constants as needed by Nitpick"
  1721 )
  1722 structure Nitpick_Const_Psimp_Thms = NamedThmsFun
  1723 (
  1724   val name = "nitpick_const_psimp"
  1725   val description = "partial equational specification of constants as needed by Nitpick"
  1726 )
  1727 structure Nitpick_Ind_Intro_Thms = NamedThmsFun
  1728 (
  1729   val name = "nitpick_ind_intro"
  1730   val description = "introduction rules for (co)inductive predicates as needed by Nitpick"
  1731 )
  1732 *}
  1733 setup {* Nitpick_Const_Def_Thms.setup
  1734          #> Nitpick_Const_Simp_Thms.setup
  1735          #> Nitpick_Const_Psimp_Thms.setup
  1736          #> Nitpick_Ind_Intro_Thms.setup *}
  1737 
  1738 subsection {* Legacy tactics and ML bindings *}
  1739 
  1740 ML {*
  1741 fun strip_tac i = REPEAT (resolve_tac [impI, allI] i);
  1742 
  1743 (* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
  1744 local
  1745   fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t
  1746     | wrong_prem (Bound _) = true
  1747     | wrong_prem _ = false;
  1748   val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of);
  1749 in
  1750   fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp]);
  1751   fun smp_tac j = EVERY'[dresolve_tac (smp j), atac];
  1752 end;
  1753 
  1754 val all_conj_distrib = thm "all_conj_distrib";
  1755 val all_simps = thms "all_simps";
  1756 val atomize_not = thm "atomize_not";
  1757 val case_split = thm "case_split";
  1758 val cases_simp = thm "cases_simp";
  1759 val choice_eq = thm "choice_eq"
  1760 val cong = thm "cong"
  1761 val conj_comms = thms "conj_comms";
  1762 val conj_cong = thm "conj_cong";
  1763 val de_Morgan_conj = thm "de_Morgan_conj";
  1764 val de_Morgan_disj = thm "de_Morgan_disj";
  1765 val disj_assoc = thm "disj_assoc";
  1766 val disj_comms = thms "disj_comms";
  1767 val disj_cong = thm "disj_cong";
  1768 val eq_ac = thms "eq_ac";
  1769 val eq_cong2 = thm "eq_cong2"
  1770 val Eq_FalseI = thm "Eq_FalseI";
  1771 val Eq_TrueI = thm "Eq_TrueI";
  1772 val Ex1_def = thm "Ex1_def"
  1773 val ex_disj_distrib = thm "ex_disj_distrib";
  1774 val ex_simps = thms "ex_simps";
  1775 val if_cancel = thm "if_cancel";
  1776 val if_eq_cancel = thm "if_eq_cancel";
  1777 val if_False = thm "if_False";
  1778 val iff_conv_conj_imp = thm "iff_conv_conj_imp";
  1779 val iff = thm "iff"
  1780 val if_splits = thms "if_splits";
  1781 val if_True = thm "if_True";
  1782 val if_weak_cong = thm "if_weak_cong"
  1783 val imp_all = thm "imp_all";
  1784 val imp_cong = thm "imp_cong";
  1785 val imp_conjL = thm "imp_conjL";
  1786 val imp_conjR = thm "imp_conjR";
  1787 val imp_conv_disj = thm "imp_conv_disj";
  1788 val simp_implies_def = thm "simp_implies_def";
  1789 val simp_thms = thms "simp_thms";
  1790 val split_if = thm "split_if";
  1791 val the1_equality = thm "the1_equality"
  1792 val theI = thm "theI"
  1793 val theI' = thm "theI'"
  1794 val True_implies_equals = thm "True_implies_equals";
  1795 val nnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps simp_thms @ @{thms "nnf_simps"})
  1796 
  1797 *}
  1798 
  1799 end