src/HOL/HOL.ML
author oheimb
Wed Aug 12 15:40:47 1998 +0200 (1998-08-12)
changeset 5299 d15a4155b96b
parent 5228 66925577cefe
child 5309 01c2b317da88
permissions -rw-r--r--
added Eps_eq
     1 (*  Title:      HOL/HOL.ML
     2     ID:         $Id$
     3     Author:     Tobias Nipkow
     4     Copyright   1991  University of Cambridge
     5 
     6 For HOL.thy
     7 Derived rules from Appendix of Mike Gordons HOL Report, Cambridge TR 68 
     8 *)
     9 
    10 open HOL;
    11 
    12 
    13 (** Equality **)
    14 section "=";
    15 
    16 qed_goal "sym" HOL.thy "s=t ==> t=s"
    17  (fn prems => [cut_facts_tac prems 1, etac subst 1, rtac refl 1]);
    18 
    19 (*calling "standard" reduces maxidx to 0*)
    20 bind_thm ("ssubst", (sym RS subst));
    21 
    22 qed_goal "trans" HOL.thy "[| r=s; s=t |] ==> r=t"
    23  (fn prems =>
    24         [rtac subst 1, resolve_tac prems 1, resolve_tac prems 1]);
    25 
    26 (*Useful with eresolve_tac for proving equalties from known equalities.
    27         a = b
    28         |   |
    29         c = d   *)
    30 qed_goal "box_equals" HOL.thy
    31     "[| a=b;  a=c;  b=d |] ==> c=d"  
    32  (fn prems=>
    33   [ (rtac trans 1),
    34     (rtac trans 1),
    35     (rtac sym 1),
    36     (REPEAT (resolve_tac prems 1)) ]);
    37 
    38 
    39 (** Congruence rules for meta-application **)
    40 section "Congruence";
    41 
    42 (*similar to AP_THM in Gordon's HOL*)
    43 qed_goal "fun_cong" HOL.thy "(f::'a=>'b) = g ==> f(x)=g(x)"
    44   (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);
    45 
    46 (*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
    47 qed_goal "arg_cong" HOL.thy "x=y ==> f(x)=f(y)"
    48  (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);
    49 
    50 qed_goal "cong" HOL.thy
    51    "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
    52  (fn [prem1,prem2] =>
    53    [rtac (prem1 RS subst) 1, rtac (prem2 RS subst) 1, rtac refl 1]);
    54 
    55 
    56 (** Equality of booleans -- iff **)
    57 section "iff";
    58 
    59 qed_goal "iffI" HOL.thy
    60    "[| P ==> Q;  Q ==> P |] ==> P=Q"
    61  (fn prems=> [ (REPEAT (ares_tac (prems@[impI, iff RS mp RS mp]) 1)) ]);
    62 
    63 qed_goal "iffD2" HOL.thy "[| P=Q; Q |] ==> P"
    64  (fn prems =>
    65         [rtac ssubst 1, resolve_tac prems 1, resolve_tac prems 1]);
    66 
    67 qed_goal "rev_iffD2" HOL.thy "!!P. [| Q; P=Q |] ==> P"
    68  (fn _ => [etac iffD2 1, assume_tac 1]);
    69 
    70 bind_thm ("iffD1", sym RS iffD2);
    71 bind_thm ("rev_iffD1", sym RSN (2, rev_iffD2));
    72 
    73 qed_goal "iffE" HOL.thy
    74     "[| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R"
    75  (fn [p1,p2] => [REPEAT(ares_tac([p1 RS iffD2, p1 RS iffD1, p2, impI])1)]);
    76 
    77 
    78 (** True **)
    79 section "True";
    80 
    81 qed_goalw "TrueI" HOL.thy [True_def] "True"
    82   (fn _ => [rtac refl 1]);
    83 
    84 qed_goal "eqTrueI" HOL.thy "P ==> P=True" 
    85  (fn prems => [REPEAT(resolve_tac ([iffI,TrueI]@prems) 1)]);
    86 
    87 qed_goal "eqTrueE" HOL.thy "P=True ==> P" 
    88  (fn prems => [REPEAT(resolve_tac (prems@[TrueI,iffD2]) 1)]);
    89 
    90 
    91 (** Universal quantifier **)
    92 section "!";
    93 
    94 qed_goalw "allI" HOL.thy [All_def] "(!!x::'a. P(x)) ==> !x. P(x)"
    95  (fn prems => [resolve_tac (prems RL [eqTrueI RS ext]) 1]);
    96 
    97 qed_goalw "spec" HOL.thy [All_def] "! x::'a. P(x) ==> P(x)"
    98  (fn prems => [rtac eqTrueE 1, resolve_tac (prems RL [fun_cong]) 1]);
    99 
   100 qed_goal "allE" HOL.thy "[| !x. P(x);  P(x) ==> R |] ==> R"
   101  (fn major::prems=>
   102   [ (REPEAT (resolve_tac (prems @ [major RS spec]) 1)) ]);
   103 
   104 qed_goal "all_dupE" HOL.thy 
   105     "[| ! x. P(x);  [| P(x); ! x. P(x) |] ==> R |] ==> R"
   106  (fn prems =>
   107   [ (REPEAT (resolve_tac (prems @ (prems RL [spec])) 1)) ]);
   108 
   109 
   110 (** False ** Depends upon spec; it is impossible to do propositional logic
   111              before quantifiers! **)
   112 section "False";
   113 
   114 qed_goalw "FalseE" HOL.thy [False_def] "False ==> P"
   115  (fn [major] => [rtac (major RS spec) 1]);
   116 
   117 qed_goal "False_neq_True" HOL.thy "False=True ==> P"
   118  (fn [prem] => [rtac (prem RS eqTrueE RS FalseE) 1]);
   119 
   120 
   121 (** Negation **)
   122 section "~";
   123 
   124 qed_goalw "notI" HOL.thy [not_def] "(P ==> False) ==> ~P"
   125  (fn prems=> [rtac impI 1, eresolve_tac prems 1]);
   126 
   127 qed_goalw "notE" HOL.thy [not_def] "[| ~P;  P |] ==> R"
   128  (fn prems => [rtac (prems MRS mp RS FalseE) 1]);
   129 
   130 bind_thm ("classical2", notE RS notI);
   131 
   132 qed_goal "rev_notE" HOL.thy "!!P R. [| P; ~P |] ==> R"
   133  (fn _ => [REPEAT (ares_tac [notE] 1)]);
   134 
   135 
   136 (** Implication **)
   137 section "-->";
   138 
   139 qed_goal "impE" HOL.thy "[| P-->Q;  P;  Q ==> R |] ==> R"
   140  (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
   141 
   142 (* Reduces Q to P-->Q, allowing substitution in P. *)
   143 qed_goal "rev_mp" HOL.thy "[| P;  P --> Q |] ==> Q"
   144  (fn prems=>  [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
   145 
   146 qed_goal "contrapos" HOL.thy "[| ~Q;  P==>Q |] ==> ~P"
   147  (fn [major,minor]=> 
   148   [ (rtac (major RS notE RS notI) 1), 
   149     (etac minor 1) ]);
   150 
   151 qed_goal "rev_contrapos" HOL.thy "[| P==>Q; ~Q |] ==> ~P"
   152  (fn [major,minor]=> 
   153   [ (rtac (minor RS contrapos) 1), (etac major 1) ]);
   154 
   155 (* ~(?t = ?s) ==> ~(?s = ?t) *)
   156 bind_thm("not_sym", sym COMP rev_contrapos);
   157 
   158 
   159 (** Existential quantifier **)
   160 section "?";
   161 
   162 qed_goalw "exI" HOL.thy [Ex_def] "P x ==> ? x::'a. P x"
   163  (fn prems => [rtac selectI 1, resolve_tac prems 1]);
   164 
   165 qed_goalw "exE" HOL.thy [Ex_def]
   166   "[| ? x::'a. P(x); !!x. P(x) ==> Q |] ==> Q"
   167   (fn prems => [REPEAT(resolve_tac prems 1)]);
   168 
   169 
   170 (** Conjunction **)
   171 section "&";
   172 
   173 qed_goalw "conjI" HOL.thy [and_def] "[| P; Q |] ==> P&Q"
   174  (fn prems =>
   175   [REPEAT (resolve_tac (prems@[allI,impI]) 1 ORELSE etac (mp RS mp) 1)]);
   176 
   177 qed_goalw "conjunct1" HOL.thy [and_def] "[| P & Q |] ==> P"
   178  (fn prems =>
   179    [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
   180 
   181 qed_goalw "conjunct2" HOL.thy [and_def] "[| P & Q |] ==> Q"
   182  (fn prems =>
   183    [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
   184 
   185 qed_goal "conjE" HOL.thy "[| P&Q;  [| P; Q |] ==> R |] ==> R"
   186  (fn prems =>
   187          [cut_facts_tac prems 1, resolve_tac prems 1,
   188           etac conjunct1 1, etac conjunct2 1]);
   189 
   190 
   191 (** Disjunction *)
   192 section "|";
   193 
   194 qed_goalw "disjI1" HOL.thy [or_def] "P ==> P|Q"
   195  (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
   196 
   197 qed_goalw "disjI2" HOL.thy [or_def] "Q ==> P|Q"
   198  (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
   199 
   200 qed_goalw "disjE" HOL.thy [or_def] "[| P | Q; P ==> R; Q ==> R |] ==> R"
   201  (fn [a1,a2,a3] =>
   202         [rtac (mp RS mp) 1, rtac spec 1, rtac a1 1,
   203          rtac (a2 RS impI) 1, assume_tac 1, rtac (a3 RS impI) 1, assume_tac 1]);
   204 
   205 
   206 (** CCONTR -- classical logic **)
   207 section "classical logic";
   208 
   209 qed_goalw "classical" HOL.thy [not_def]  "(~P ==> P) ==> P"
   210  (fn [prem] =>
   211    [rtac (True_or_False RS (disjE RS eqTrueE)) 1,  assume_tac 1,
   212     rtac (impI RS prem RS eqTrueI) 1,
   213     etac subst 1,  assume_tac 1]);
   214 
   215 val ccontr = FalseE RS classical;
   216 
   217 (*Double negation law*)
   218 qed_goal "notnotD" HOL.thy "~~P ==> P"
   219  (fn [major]=>
   220   [ (rtac classical 1), (eresolve_tac [major RS notE] 1) ]);
   221 
   222 qed_goal "contrapos2" HOL.thy "[| Q; ~ P ==> ~ Q |] ==> P" (fn [p1,p2] => [
   223         rtac classical 1,
   224         dtac p2 1,
   225         etac notE 1,
   226         rtac p1 1]);
   227 
   228 qed_goal "swap2" HOL.thy "[| P;  Q ==> ~ P |] ==> ~ Q" (fn [p1,p2] => [
   229         rtac notI 1,
   230         dtac p2 1,
   231         etac notE 1,
   232         rtac p1 1]);
   233 
   234 (** Unique existence **)
   235 section "?!";
   236 
   237 qed_goalw "ex1I" HOL.thy [Ex1_def]
   238             "[| P(a);  !!x. P(x) ==> x=a |] ==> ?! x. P(x)"
   239  (fn prems =>
   240   [REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)]);
   241 
   242 (*Sometimes easier to use: the premises have no shared variables.  Safe!*)
   243 qed_goal "ex_ex1I" HOL.thy
   244     "[| ? x. P(x);  !!x y. [| P(x); P(y) |] ==> x=y |] ==> ?! x. P(x)"
   245  (fn [ex,eq] => [ (rtac (ex RS exE) 1),
   246                   (REPEAT (ares_tac [ex1I,eq] 1)) ]);
   247 
   248 qed_goalw "ex1E" HOL.thy [Ex1_def]
   249     "[| ?! x. P(x);  !!x. [| P(x);  ! y. P(y) --> y=x |] ==> R |] ==> R"
   250  (fn major::prems =>
   251   [rtac (major RS exE) 1, REPEAT (etac conjE 1 ORELSE ares_tac prems 1)]);
   252 
   253 Goal "?! x. P x ==> ? x. P x";
   254 by (etac ex1E 1);
   255 by (rtac exI 1);
   256 by (assume_tac 1);
   257 qed "ex1_implies_ex";
   258 
   259 
   260 (** Select: Hilbert's Epsilon-operator **)
   261 section "@";
   262 
   263 (*Easier to apply than selectI: conclusion has only one occurrence of P*)
   264 qed_goal "selectI2" HOL.thy
   265     "[| P a;  !!x. P x ==> Q x |] ==> Q (@x. P x)"
   266  (fn prems => [ resolve_tac prems 1, 
   267                 rtac selectI 1, 
   268                 resolve_tac prems 1 ]);
   269 
   270 (*Easier to apply than selectI2 if witness ?a comes from an EX-formula*)
   271 qed_goal "selectI2EX" HOL.thy
   272   "[| ? a. P a; !!x. P x ==> Q x |] ==> Q (Eps P)"
   273 (fn [major,minor] => [rtac (major RS exE) 1, etac selectI2 1, etac minor 1]);
   274 
   275 qed_goal "select_equality" HOL.thy
   276     "[| P a;  !!x. P x ==> x=a |] ==> (@x. P x) = a"
   277  (fn prems => [ rtac selectI2 1, 
   278                 REPEAT (ares_tac prems 1) ]);
   279 
   280 qed_goalw "select1_equality" HOL.thy [Ex1_def]
   281   "!!P. [| ?!x. P x; P a |] ==> (@x. P x) = a" (K [
   282 	  rtac select_equality 1, atac 1,
   283           etac exE 1, etac conjE 1,
   284           rtac allE 1, atac 1,
   285           etac impE 1, atac 1, etac ssubst 1,
   286           etac allE 1, etac impE 1, atac 1, etac ssubst 1,
   287           rtac refl 1]);
   288 
   289 qed_goal "select_eq_Ex" HOL.thy "P (@ x. P x) =  (? x. P x)" (K [
   290         rtac iffI 1,
   291         etac exI 1,
   292         etac exE 1,
   293         etac selectI 1]);
   294 
   295 qed_goal "Eps_eq" HOL.thy "(Eps (op = x)) = x" (K [
   296 	rtac select_equality 1,
   297 	rtac refl 1,
   298 	etac sym 1]);
   299 
   300 (** Classical intro rules for disjunction and existential quantifiers *)
   301 section "classical intro rules";
   302 
   303 qed_goal "disjCI" HOL.thy "(~Q ==> P) ==> P|Q"
   304  (fn prems=>
   305   [ (rtac classical 1),
   306     (REPEAT (ares_tac (prems@[disjI1,notI]) 1)),
   307     (REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ]);
   308 
   309 qed_goal "excluded_middle" HOL.thy "~P | P"
   310  (fn _ => [ (REPEAT (ares_tac [disjCI] 1)) ]);
   311 
   312 (*For disjunctive case analysis*)
   313 fun excluded_middle_tac sP =
   314     res_inst_tac [("Q",sP)] (excluded_middle RS disjE);
   315 
   316 (*Classical implies (-->) elimination. *)
   317 qed_goal "impCE" HOL.thy "[| P-->Q; ~P ==> R; Q ==> R |] ==> R" 
   318  (fn major::prems=>
   319   [ rtac (excluded_middle RS disjE) 1,
   320     REPEAT (DEPTH_SOLVE_1 (ares_tac (prems @ [major RS mp]) 1))]);
   321 
   322 (*This version of --> elimination works on Q before P.  It works best for
   323   those cases in which P holds "almost everywhere".  Can't install as
   324   default: would break old proofs.*)
   325 qed_goal "impCE'" thy 
   326     "[| P-->Q;  Q ==> R;  ~P ==> R |] ==> R"
   327  (fn major::prems=>
   328   [ (resolve_tac [excluded_middle RS disjE] 1),
   329     (DEPTH_SOLVE (ares_tac (prems@[major RS mp]) 1)) ]);
   330 
   331 (*Classical <-> elimination. *)
   332 qed_goal "iffCE" HOL.thy
   333     "[| P=Q;  [| P; Q |] ==> R;  [| ~P; ~Q |] ==> R |] ==> R"
   334  (fn major::prems =>
   335   [ (rtac (major RS iffE) 1),
   336     (REPEAT (DEPTH_SOLVE_1 
   337         (eresolve_tac ([asm_rl,impCE,notE]@prems) 1))) ]);
   338 
   339 qed_goal "exCI" HOL.thy "(! x. ~P(x) ==> P(a)) ==> ? x. P(x)"
   340  (fn prems=>
   341   [ (rtac ccontr 1),
   342     (REPEAT (ares_tac (prems@[exI,allI,notI,notE]) 1))  ]);
   343 
   344 
   345 (* case distinction *)
   346 
   347 qed_goal "case_split_thm" HOL.thy "[| P ==> Q; ~P ==> Q |] ==> Q"
   348   (fn [p1,p2] => [rtac (excluded_middle RS disjE) 1,
   349                   etac p2 1, etac p1 1]);
   350 
   351 fun case_tac a = res_inst_tac [("P",a)] case_split_thm;
   352 
   353 
   354 (** Standard abbreviations **)
   355 
   356 (*Fails unless the substitution has an effect*)
   357 fun stac th = CHANGED_GOAL (rtac (th RS ssubst));
   358 
   359 fun strip_tac i = REPEAT(resolve_tac [impI,allI] i); 
   360 
   361 
   362 (** strip ! and --> from proved goal while preserving !-bound var names **)
   363 
   364 local
   365 
   366 (* Use XXX to avoid forall_intr failing because of duplicate variable name *)
   367 val myspec = read_instantiate [("P","?XXX")] spec;
   368 val _ $ (_ $ (vx as Var(_,vxT))) = concl_of myspec;
   369 val cvx = cterm_of (#sign(rep_thm myspec)) vx;
   370 val aspec = forall_intr cvx myspec;
   371 
   372 in
   373 
   374 fun RSspec th =
   375   (case concl_of th of
   376      _ $ (Const("All",_) $ Abs(a,_,_)) =>
   377          let val ca = cterm_of (#sign(rep_thm th)) (Var((a,0),vxT))
   378          in th RS forall_elim ca aspec end
   379   | _ => raise THM("RSspec",0,[th]));
   380 
   381 fun RSmp th =
   382   (case concl_of th of
   383      _ $ (Const("op -->",_)$_$_) => th RS mp
   384   | _ => raise THM("RSmp",0,[th]));
   385 
   386 fun normalize_thm funs =
   387 let fun trans [] th = th
   388       | trans (f::fs) th = (trans funs (f th)) handle THM _ => trans fs th
   389 in trans funs end;
   390 
   391 fun qed_spec_mp name =
   392   let val thm = normalize_thm [RSspec,RSmp] (result())
   393   in bind_thm(name, thm) end;
   394 
   395 fun qed_goal_spec_mp name thy s p = 
   396 	bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goal thy s p));
   397 
   398 fun qed_goalw_spec_mp name thy defs s p = 
   399 	bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goalw thy defs s p));
   400 
   401 end;