src/HOL/HOL.ML
author paulson
Mon Jul 19 15:19:11 1999 +0200 (1999-07-19)
changeset 7030 53934985426a
parent 6968 7f2977e96a5c
child 7357 d0e16da40ea2
permissions -rw-r--r--
getting rid of qed_goal
     1 (*  Title:      HOL/HOL.ML
     2     ID:         $Id$
     3     Author:     Tobias Nipkow
     4     Copyright   1991  University of Cambridge
     5 
     6 Derived rules from Appendix of Mike Gordons HOL Report, Cambridge TR 68 
     7 *)
     8 
     9 
    10 (** Equality **)
    11 section "=";
    12 
    13 qed_goal "sym" HOL.thy "s=t ==> t=s"
    14  (fn prems => [cut_facts_tac prems 1, etac subst 1, rtac refl 1]);
    15 
    16 (*calling "standard" reduces maxidx to 0*)
    17 bind_thm ("ssubst", (sym RS subst));
    18 
    19 qed_goal "trans" HOL.thy "[| r=s; s=t |] ==> r=t"
    20  (fn prems =>
    21         [rtac subst 1, resolve_tac prems 1, resolve_tac prems 1]);
    22 
    23 val prems = goal thy "(A == B) ==> A = B";
    24 by (rewrite_goals_tac prems);
    25 by (rtac refl 1);
    26 qed "def_imp_eq";
    27 
    28 (*Useful with eresolve_tac for proving equalties from known equalities.
    29         a = b
    30         |   |
    31         c = d   *)
    32 Goal "[| a=b;  a=c;  b=d |] ==> c=d";
    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;
   442 
   443 
   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;