src/HOL/HOL.ML
changeset 7357 d0e16da40ea2
parent 7030 53934985426a
child 7529 fa534e4f7e49
equal deleted inserted replaced
7356:1714c91b8729 7357:d0e16da40ea2
     1 (*  Title:      HOL/HOL.ML
       
     2     ID:         $Id$
       
     3     Author:     Tobias Nipkow
       
     4     Copyright   1991  University of Cambridge
       
     5 
     1 
     6 Derived rules from Appendix of Mike Gordons HOL Report, Cambridge TR 68 
     2 structure HOL =
     7 *)
     3 struct
     8 
     4   val thy = the_context ();
     9 
     5   val plusI = plusI;
    10 (** Equality **)
     6   val minusI = minusI;
    11 section "=";
     7   val timesI = timesI;
    12 
     8   val powerI = powerI;
    13 qed_goal "sym" HOL.thy "s=t ==> t=s"
     9   val eq_reflection = eq_reflection;
    14  (fn prems => [cut_facts_tac prems 1, etac subst 1, rtac refl 1]);
    10   val refl = refl;
    15 
    11   val subst = subst;
    16 (*calling "standard" reduces maxidx to 0*)
    12   val ext = ext;
    17 bind_thm ("ssubst", (sym RS subst));
    13   val selectI = selectI;
    18 
    14   val impI = impI;
    19 qed_goal "trans" HOL.thy "[| r=s; s=t |] ==> r=t"
    15   val mp = mp;
    20  (fn prems =>
    16   val True_def = True_def;
    21         [rtac subst 1, resolve_tac prems 1, resolve_tac prems 1]);
    17   val All_def = All_def;
    22 
    18   val Ex_def = Ex_def;
    23 val prems = goal thy "(A == B) ==> A = B";
    19   val False_def = False_def;
    24 by (rewrite_goals_tac prems);
    20   val not_def = not_def;
    25 by (rtac refl 1);
    21   val and_def = and_def;
    26 qed "def_imp_eq";
    22   val or_def = or_def;
    27 
    23   val Ex1_def = Ex1_def;
    28 (*Useful with eresolve_tac for proving equalties from known equalities.
    24   val iff = iff;
    29         a = b
    25   val True_or_False = True_or_False;
    30         |   |
    26   val Let_def = Let_def;
    31         c = d   *)
    27   val if_def = if_def;
    32 Goal "[| a=b;  a=c;  b=d |] ==> c=d";
    28   val arbitrary_def = arbitrary_def;
    33 by (rtac trans 1);
       
    34 by (rtac trans 1);
       
    35 by (rtac sym 1);
       
    36 by (REPEAT (assume_tac 1)) ;
       
    37 qed "box_equals";
       
    38 
       
    39 
       
    40 (** Congruence rules for meta-application **)
       
    41 section "Congruence";
       
    42 
       
    43 (*similar to AP_THM in Gordon's HOL*)
       
    44 qed_goal "fun_cong" HOL.thy "(f::'a=>'b) = g ==> f(x)=g(x)"
       
    45   (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);
       
    46 
       
    47 (*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
       
    48 qed_goal "arg_cong" HOL.thy "x=y ==> f(x)=f(y)"
       
    49  (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);
       
    50 
       
    51 qed_goal "cong" HOL.thy
       
    52    "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
       
    53  (fn [prem1,prem2] =>
       
    54    [rtac (prem1 RS subst) 1, rtac (prem2 RS subst) 1, rtac refl 1]);
       
    55 
       
    56 
       
    57 (** Equality of booleans -- iff **)
       
    58 section "iff";
       
    59 
       
    60 val prems = Goal
       
    61    "[| P ==> Q;  Q ==> P |] ==> P=Q";
       
    62 by (REPEAT (ares_tac (prems@[impI, iff RS mp RS mp]) 1));
       
    63 qed "iffI";
       
    64 
       
    65 qed_goal "iffD2" HOL.thy "[| P=Q; Q |] ==> P"
       
    66  (fn prems =>
       
    67         [rtac ssubst 1, resolve_tac prems 1, resolve_tac prems 1]);
       
    68 
       
    69 qed_goal "rev_iffD2" HOL.thy "!!P. [| Q; P=Q |] ==> P"
       
    70  (fn _ => [etac iffD2 1, assume_tac 1]);
       
    71 
       
    72 bind_thm ("iffD1", sym RS iffD2);
       
    73 bind_thm ("rev_iffD1", sym RSN (2, rev_iffD2));
       
    74 
       
    75 qed_goal "iffE" HOL.thy
       
    76     "[| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R"
       
    77  (fn [p1,p2] => [REPEAT(ares_tac([p1 RS iffD2, p1 RS iffD1, p2, impI])1)]);
       
    78 
       
    79 
       
    80 (** True **)
       
    81 section "True";
       
    82 
       
    83 qed_goalw "TrueI" HOL.thy [True_def] "True"
       
    84   (fn _ => [(rtac refl 1)]);
       
    85 
       
    86 qed_goal "eqTrueI" HOL.thy "P ==> P=True" 
       
    87  (fn prems => [REPEAT(resolve_tac ([iffI,TrueI]@prems) 1)]);
       
    88 
       
    89 qed_goal "eqTrueE" HOL.thy "P=True ==> P" 
       
    90  (fn prems => [REPEAT(resolve_tac (prems@[TrueI,iffD2]) 1)]);
       
    91 
       
    92 
       
    93 (** Universal quantifier **)
       
    94 section "!";
       
    95 
       
    96 qed_goalw "allI" HOL.thy [All_def] "(!!x::'a. P(x)) ==> !x. P(x)"
       
    97  (fn prems => [(resolve_tac (prems RL [eqTrueI RS ext]) 1)]);
       
    98 
       
    99 qed_goalw "spec" HOL.thy [All_def] "! x::'a. P(x) ==> P(x)"
       
   100  (fn prems => [rtac eqTrueE 1, resolve_tac (prems RL [fun_cong]) 1]);
       
   101 
       
   102 val major::prems= goal HOL.thy "[| !x. P(x);  P(x) ==> R |] ==> R";
       
   103 by (REPEAT (resolve_tac (prems @ [major RS spec]) 1)) ;
       
   104 qed "allE";
       
   105 
       
   106 val prems = goal HOL.thy 
       
   107     "[| ! x. P(x);  [| P(x); ! x. P(x) |] ==> R |] ==> R";
       
   108 by (REPEAT (resolve_tac (prems @ (prems RL [spec])) 1)) ;
       
   109 qed "all_dupE";
       
   110 
       
   111 
       
   112 (** False ** Depends upon spec; it is impossible to do propositional logic
       
   113              before quantifiers! **)
       
   114 section "False";
       
   115 
       
   116 qed_goalw "FalseE" HOL.thy [False_def] "False ==> P"
       
   117  (fn [major] => [rtac (major RS spec) 1]);
       
   118 
       
   119 qed_goal "False_neq_True" HOL.thy "False=True ==> P"
       
   120  (fn [prem] => [rtac (prem RS eqTrueE RS FalseE) 1]);
       
   121 
       
   122 
       
   123 (** Negation **)
       
   124 section "~";
       
   125 
       
   126 qed_goalw "notI" HOL.thy [not_def] "(P ==> False) ==> ~P"
       
   127  (fn prems=> [rtac impI 1, eresolve_tac prems 1]);
       
   128 
       
   129 qed_goal "False_not_True" HOL.thy "False ~= True"
       
   130   (fn _ => [rtac notI 1, etac False_neq_True 1]);
       
   131 
       
   132 qed_goal "True_not_False" HOL.thy "True ~= False"
       
   133   (fn _ => [rtac notI 1, dtac sym 1, etac False_neq_True 1]);
       
   134 
       
   135 qed_goalw "notE" HOL.thy [not_def] "[| ~P;  P |] ==> R"
       
   136  (fn prems => [rtac (prems MRS mp RS FalseE) 1]);
       
   137 
       
   138 bind_thm ("classical2", notE RS notI);
       
   139 
       
   140 qed_goal "rev_notE" HOL.thy "!!P R. [| P; ~P |] ==> R"
       
   141  (fn _ => [REPEAT (ares_tac [notE] 1)]);
       
   142 
       
   143 
       
   144 (** Implication **)
       
   145 section "-->";
       
   146 
       
   147 val prems = Goal "[| P-->Q;  P;  Q ==> R |] ==> R";
       
   148 by (REPEAT (resolve_tac (prems@[mp]) 1));
       
   149 qed "impE";
       
   150 
       
   151 (* Reduces Q to P-->Q, allowing substitution in P. *)
       
   152 Goal "[| P;  P --> Q |] ==> Q";
       
   153 by (REPEAT (ares_tac [mp] 1)) ;
       
   154 qed "rev_mp";
       
   155 
       
   156 val [major,minor] = Goal "[| ~Q;  P==>Q |] ==> ~P";
       
   157 by (rtac (major RS notE RS notI) 1);
       
   158 by (etac minor 1) ;
       
   159 qed "contrapos";
       
   160 
       
   161 val [major,minor] = Goal "[| P==>Q; ~Q |] ==> ~P";
       
   162 by (rtac (minor RS contrapos) 1);
       
   163 by (etac major 1) ;
       
   164 qed "rev_contrapos";
       
   165 
       
   166 (* ~(?t = ?s) ==> ~(?s = ?t) *)
       
   167 bind_thm("not_sym", sym COMP rev_contrapos);
       
   168 
       
   169 
       
   170 (** Existential quantifier **)
       
   171 section "?";
       
   172 
       
   173 qed_goalw "exI" HOL.thy [Ex_def] "P x ==> ? x::'a. P x"
       
   174  (fn prems => [rtac selectI 1, resolve_tac prems 1]);
       
   175 
       
   176 qed_goalw "exE" HOL.thy [Ex_def]
       
   177   "[| ? x::'a. P(x); !!x. P(x) ==> Q |] ==> Q"
       
   178   (fn prems => [REPEAT(resolve_tac prems 1)]);
       
   179 
       
   180 
       
   181 (** Conjunction **)
       
   182 section "&";
       
   183 
       
   184 qed_goalw "conjI" HOL.thy [and_def] "[| P; Q |] ==> P&Q"
       
   185  (fn prems =>
       
   186   [REPEAT (resolve_tac (prems@[allI,impI]) 1 ORELSE etac (mp RS mp) 1)]);
       
   187 
       
   188 qed_goalw "conjunct1" HOL.thy [and_def] "[| P & Q |] ==> P"
       
   189  (fn prems =>
       
   190    [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
       
   191 
       
   192 qed_goalw "conjunct2" HOL.thy [and_def] "[| P & Q |] ==> Q"
       
   193  (fn prems =>
       
   194    [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
       
   195 
       
   196 qed_goal "conjE" HOL.thy "[| P&Q;  [| P; Q |] ==> R |] ==> R"
       
   197  (fn prems =>
       
   198          [cut_facts_tac prems 1, resolve_tac prems 1,
       
   199           etac conjunct1 1, etac conjunct2 1]);
       
   200 
       
   201 qed_goal "context_conjI" HOL.thy  "[| P; P ==> Q |] ==> P & Q"
       
   202  (fn prems => [REPEAT(resolve_tac (conjI::prems) 1)]);
       
   203 
       
   204 
       
   205 (** Disjunction *)
       
   206 section "|";
       
   207 
       
   208 qed_goalw "disjI1" HOL.thy [or_def] "P ==> P|Q"
       
   209  (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
       
   210 
       
   211 qed_goalw "disjI2" HOL.thy [or_def] "Q ==> P|Q"
       
   212  (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
       
   213 
       
   214 qed_goalw "disjE" HOL.thy [or_def] "[| P | Q; P ==> R; Q ==> R |] ==> R"
       
   215  (fn [a1,a2,a3] =>
       
   216         [rtac (mp RS mp) 1, rtac spec 1, rtac a1 1,
       
   217          rtac (a2 RS impI) 1, assume_tac 1, rtac (a3 RS impI) 1, assume_tac 1]);
       
   218 
       
   219 
       
   220 (** CCONTR -- classical logic **)
       
   221 section "classical logic";
       
   222 
       
   223 qed_goalw "classical" HOL.thy [not_def]  "(~P ==> P) ==> P"
       
   224  (fn [prem] =>
       
   225    [rtac (True_or_False RS (disjE RS eqTrueE)) 1,  assume_tac 1,
       
   226     rtac (impI RS prem RS eqTrueI) 1,
       
   227     etac subst 1,  assume_tac 1]);
       
   228 
       
   229 val ccontr = FalseE RS classical;
       
   230 
       
   231 (*Double negation law*)
       
   232 Goal "~~P ==> P";
       
   233 by (rtac classical 1);
       
   234 by (etac notE 1);
       
   235 by (assume_tac 1);
       
   236 qed "notnotD";
       
   237 
       
   238 val [p1,p2] = Goal "[| Q; ~ P ==> ~ Q |] ==> P";
       
   239 by (rtac classical 1);
       
   240 by (dtac p2 1);
       
   241 by (etac notE 1);
       
   242 by (rtac p1 1);
       
   243 qed "contrapos2";
       
   244 
       
   245 val [p1,p2] = Goal "[| P;  Q ==> ~ P |] ==> ~ Q";
       
   246 by (rtac notI 1);
       
   247 by (dtac p2 1);
       
   248 by (etac notE 1);
       
   249 by (rtac p1 1);
       
   250 qed "swap2";
       
   251 
       
   252 (** Unique existence **)
       
   253 section "?!";
       
   254 
       
   255 qed_goalw "ex1I" HOL.thy [Ex1_def]
       
   256             "[| P(a);  !!x. P(x) ==> x=a |] ==> ?! x. P(x)"
       
   257  (fn prems =>
       
   258   [REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)]);
       
   259 
       
   260 (*Sometimes easier to use: the premises have no shared variables.  Safe!*)
       
   261 val [ex,eq] = Goal
       
   262     "[| ? x. P(x);  !!x y. [| P(x); P(y) |] ==> x=y |] ==> ?! x. P(x)";
       
   263 by (rtac (ex RS exE) 1);
       
   264 by (REPEAT (ares_tac [ex1I,eq] 1)) ;
       
   265 qed "ex_ex1I";
       
   266 
       
   267 qed_goalw "ex1E" HOL.thy [Ex1_def]
       
   268     "[| ?! x. P(x);  !!x. [| P(x);  ! y. P(y) --> y=x |] ==> R |] ==> R"
       
   269  (fn major::prems =>
       
   270   [rtac (major RS exE) 1, REPEAT (etac conjE 1 ORELSE ares_tac prems 1)]);
       
   271 
       
   272 Goal "?! x. P x ==> ? x. P x";
       
   273 by (etac ex1E 1);
       
   274 by (rtac exI 1);
       
   275 by (assume_tac 1);
       
   276 qed "ex1_implies_ex";
       
   277 
       
   278 
       
   279 (** Select: Hilbert's Epsilon-operator **)
       
   280 section "@";
       
   281 
       
   282 (*Easier to apply than selectI: conclusion has only one occurrence of P*)
       
   283 val prems = Goal
       
   284     "[| P a;  !!x. P x ==> Q x |] ==> Q (@x. P x)";
       
   285 by (resolve_tac prems 1);
       
   286 by (rtac selectI 1);
       
   287 by (resolve_tac prems 1) ;
       
   288 qed "selectI2";
       
   289 
       
   290 (*Easier to apply than selectI2 if witness ?a comes from an EX-formula*)
       
   291 qed_goal "selectI2EX" HOL.thy
       
   292   "[| ? a. P a; !!x. P x ==> Q x |] ==> Q (Eps P)"
       
   293 (fn [major,minor] => [rtac (major RS exE) 1, etac selectI2 1, etac minor 1]);
       
   294 
       
   295 val prems = Goal
       
   296     "[| P a;  !!x. P x ==> x=a |] ==> (@x. P x) = a";
       
   297 by (rtac selectI2 1);
       
   298 by (REPEAT (ares_tac prems 1)) ;
       
   299 qed "select_equality";
       
   300 
       
   301 Goalw [Ex1_def] "[| ?!x. P x; P a |] ==> (@x. P x) = a";
       
   302 by (rtac select_equality 1);
       
   303 by (atac 1);
       
   304 by (etac exE 1);
       
   305 by (etac conjE 1);
       
   306 by (rtac allE 1);
       
   307 by (atac 1);
       
   308 by (etac impE 1);
       
   309 by (atac 1);
       
   310 by (etac ssubst 1);
       
   311 by (etac allE 1);
       
   312 by (etac mp 1);
       
   313 by (atac 1);
       
   314 qed "select1_equality";
       
   315 
       
   316 Goal "P (@ x. P x) =  (? x. P x)";
       
   317 by (rtac iffI 1);
       
   318 by (etac exI 1);
       
   319 by (etac exE 1);
       
   320 by (etac selectI 1);
       
   321 qed "select_eq_Ex";
       
   322 
       
   323 Goal "(@y. y=x) = x";
       
   324 by (rtac select_equality 1);
       
   325 by (rtac refl 1);
       
   326 by (atac 1);
       
   327 qed "Eps_eq";
       
   328 
       
   329 Goal "(Eps (op = x)) = x";
       
   330 by (rtac select_equality 1);
       
   331 by (rtac refl 1);
       
   332 by (etac sym 1);
       
   333 qed "Eps_sym_eq";
       
   334 
       
   335 (** Classical intro rules for disjunction and existential quantifiers *)
       
   336 section "classical intro rules";
       
   337 
       
   338 val prems= Goal "(~Q ==> P) ==> P|Q";
       
   339 by (rtac classical 1);
       
   340 by (REPEAT (ares_tac (prems@[disjI1,notI]) 1));
       
   341 by (REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ;
       
   342 qed "disjCI";
       
   343 
       
   344 Goal "~P | P";
       
   345 by (REPEAT (ares_tac [disjCI] 1)) ;
       
   346 qed "excluded_middle";
       
   347 
       
   348 (*For disjunctive case analysis*)
       
   349 fun excluded_middle_tac sP =
       
   350     res_inst_tac [("Q",sP)] (excluded_middle RS disjE);
       
   351 
       
   352 (*Classical implies (-->) elimination. *)
       
   353 val major::prems = Goal "[| P-->Q; ~P ==> R; Q ==> R |] ==> R";
       
   354 by (rtac (excluded_middle RS disjE) 1);
       
   355 by (REPEAT (DEPTH_SOLVE_1 (ares_tac (prems @ [major RS mp]) 1)));
       
   356 qed "impCE";
       
   357 
       
   358 (*This version of --> elimination works on Q before P.  It works best for
       
   359   those cases in which P holds "almost everywhere".  Can't install as
       
   360   default: would break old proofs.*)
       
   361 val major::prems = Goal
       
   362     "[| P-->Q;  Q ==> R;  ~P ==> R |] ==> R";
       
   363 by (resolve_tac [excluded_middle RS disjE] 1);
       
   364 by (DEPTH_SOLVE (ares_tac (prems@[major RS mp]) 1)) ;
       
   365 qed "impCE'";
       
   366 
       
   367 (*Classical <-> elimination. *)
       
   368 val major::prems = Goal
       
   369     "[| P=Q;  [| P; Q |] ==> R;  [| ~P; ~Q |] ==> R |] ==> R";
       
   370 by (rtac (major RS iffE) 1);
       
   371 by (REPEAT (DEPTH_SOLVE_1 
       
   372 	    (eresolve_tac ([asm_rl,impCE,notE]@prems) 1)));
       
   373 qed "iffCE";
       
   374 
       
   375 val prems = Goal "(! x. ~P(x) ==> P(a)) ==> ? x. P(x)";
       
   376 by (rtac ccontr 1);
       
   377 by (REPEAT (ares_tac (prems@[exI,allI,notI,notE]) 1))  ;
       
   378 qed "exCI";
       
   379 
       
   380 
       
   381 (* case distinction *)
       
   382 
       
   383 qed_goal "case_split_thm" HOL.thy "[| P ==> Q; ~P ==> Q |] ==> Q"
       
   384   (fn [p1,p2] => [rtac (excluded_middle RS disjE) 1,
       
   385                   etac p2 1, etac p1 1]);
       
   386 
       
   387 fun case_tac a = res_inst_tac [("P",a)] case_split_thm;
       
   388 
       
   389 
       
   390 (** Standard abbreviations **)
       
   391 
       
   392 (*Apply an equality or definition ONCE.
       
   393   Fails unless the substitution has an effect*)
       
   394 fun stac th = 
       
   395   let val th' = th RS def_imp_eq handle THM _ => th
       
   396   in  CHANGED_GOAL (rtac (th' RS ssubst))
       
   397   end;
       
   398 
       
   399 fun strip_tac i = REPEAT(resolve_tac [impI,allI] i); 
       
   400 
       
   401 
       
   402 (** strip ! and --> from proved goal while preserving !-bound var names **)
       
   403 
       
   404 local
       
   405 
       
   406 (* Use XXX to avoid forall_intr failing because of duplicate variable name *)
       
   407 val myspec = read_instantiate [("P","?XXX")] spec;
       
   408 val _ $ (_ $ (vx as Var(_,vxT))) = concl_of myspec;
       
   409 val cvx = cterm_of (#sign(rep_thm myspec)) vx;
       
   410 val aspec = forall_intr cvx myspec;
       
   411 
       
   412 in
       
   413 
       
   414 fun RSspec th =
       
   415   (case concl_of th of
       
   416      _ $ (Const("All",_) $ Abs(a,_,_)) =>
       
   417          let val ca = cterm_of (#sign(rep_thm th)) (Var((a,0),vxT))
       
   418          in th RS forall_elim ca aspec end
       
   419   | _ => raise THM("RSspec",0,[th]));
       
   420 
       
   421 fun RSmp th =
       
   422   (case concl_of th of
       
   423      _ $ (Const("op -->",_)$_$_) => th RS mp
       
   424   | _ => raise THM("RSmp",0,[th]));
       
   425 
       
   426 fun normalize_thm funs =
       
   427   let fun trans [] th = th
       
   428 	| trans (f::fs) th = (trans funs (f th)) handle THM _ => trans fs th
       
   429   in zero_var_indexes o trans funs end;
       
   430 
       
   431 fun qed_spec_mp name =
       
   432   let val thm = normalize_thm [RSspec,RSmp] (result())
       
   433   in ThmDatabase.ml_store_thm(name, thm) end;
       
   434 
       
   435 fun qed_goal_spec_mp name thy s p = 
       
   436 	bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goal thy s p));
       
   437 
       
   438 fun qed_goalw_spec_mp name thy defs s p = 
       
   439 	bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goalw thy defs s p));
       
   440 
       
   441 end;
    29 end;
   442 
    30 
   443 
    31 open HOL;
   444 (* attributes *)
       
   445 
       
   446 local
       
   447 
       
   448 fun gen_rulify x = Attrib.no_args (Drule.rule_attribute (fn _ => (normalize_thm [RSspec, RSmp]))) x;
       
   449 
       
   450 in
       
   451 
       
   452 val hol_setup =
       
   453  [Attrib.add_attributes
       
   454   [("rulify", (gen_rulify, gen_rulify), "put theorem into standard rule form")]];
       
   455 
       
   456 end;