src/HOL/HOL.ML
author paulson
Fri Jul 17 10:50:01 1998 +0200 (1998-07-17)
changeset 5154 40fd46f3d3a1
parent 5139 013ea0f023e3
child 5185 d1067e2c3f9f
permissions -rw-r--r--
tidying
     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 
   254 (** Select: Hilbert's Epsilon-operator **)
   255 section "@";
   256 
   257 (*Easier to apply than selectI: conclusion has only one occurrence of P*)
   258 qed_goal "selectI2" HOL.thy
   259     "[| P a;  !!x. P x ==> Q x |] ==> Q (@x. P x)"
   260  (fn prems => [ resolve_tac prems 1, 
   261                 rtac selectI 1, 
   262                 resolve_tac prems 1 ]);
   263 
   264 (*Easier to apply than selectI2 if witness ?a comes from an EX-formula*)
   265 qed_goal "selectI2EX" HOL.thy
   266   "[| ? a. P a; !!x. P x ==> Q x |] ==> Q (Eps P)"
   267 (fn [major,minor] => [rtac (major RS exE) 1, etac selectI2 1, etac minor 1]);
   268 
   269 qed_goal "select_equality" HOL.thy
   270     "[| P a;  !!x. P x ==> x=a |] ==> (@x. P x) = a"
   271  (fn prems => [ rtac selectI2 1, 
   272                 REPEAT (ares_tac prems 1) ]);
   273 
   274 qed_goalw "select1_equality" HOL.thy [Ex1_def]
   275   "!!P. [| ?!x. P x; P a |] ==> (@x. P x) = a" (K [
   276 	  rtac select_equality 1, atac 1,
   277           etac exE 1, etac conjE 1,
   278           rtac allE 1, atac 1,
   279           etac impE 1, atac 1, etac ssubst 1,
   280           etac allE 1, etac impE 1, atac 1, etac ssubst 1,
   281           rtac refl 1]);
   282 
   283 qed_goal "select_eq_Ex" HOL.thy "P (@ x. P x) =  (? x. P x)" (K [
   284         rtac iffI 1,
   285         etac exI 1,
   286         etac exE 1,
   287         etac selectI 1]);
   288 
   289 
   290 (** Classical intro rules for disjunction and existential quantifiers *)
   291 section "classical intro rules";
   292 
   293 qed_goal "disjCI" HOL.thy "(~Q ==> P) ==> P|Q"
   294  (fn prems=>
   295   [ (rtac classical 1),
   296     (REPEAT (ares_tac (prems@[disjI1,notI]) 1)),
   297     (REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ]);
   298 
   299 qed_goal "excluded_middle" HOL.thy "~P | P"
   300  (fn _ => [ (REPEAT (ares_tac [disjCI] 1)) ]);
   301 
   302 (*For disjunctive case analysis*)
   303 fun excluded_middle_tac sP =
   304     res_inst_tac [("Q",sP)] (excluded_middle RS disjE);
   305 
   306 (*Classical implies (-->) elimination. *)
   307 qed_goal "impCE" HOL.thy "[| P-->Q; ~P ==> R; Q ==> R |] ==> R" 
   308  (fn major::prems=>
   309   [ rtac (excluded_middle RS disjE) 1,
   310     REPEAT (DEPTH_SOLVE_1 (ares_tac (prems @ [major RS mp]) 1))]);
   311 
   312 (*This version of --> elimination works on Q before P.  It works best for
   313   those cases in which P holds "almost everywhere".  Can't install as
   314   default: would break old proofs.*)
   315 qed_goal "impCE'" thy 
   316     "[| P-->Q;  Q ==> R;  ~P ==> R |] ==> R"
   317  (fn major::prems=>
   318   [ (resolve_tac [excluded_middle RS disjE] 1),
   319     (DEPTH_SOLVE (ares_tac (prems@[major RS mp]) 1)) ]);
   320 
   321 (*Classical <-> elimination. *)
   322 qed_goal "iffCE" HOL.thy
   323     "[| P=Q;  [| P; Q |] ==> R;  [| ~P; ~Q |] ==> R |] ==> R"
   324  (fn major::prems =>
   325   [ (rtac (major RS iffE) 1),
   326     (REPEAT (DEPTH_SOLVE_1 
   327         (eresolve_tac ([asm_rl,impCE,notE]@prems) 1))) ]);
   328 
   329 qed_goal "exCI" HOL.thy "(! x. ~P(x) ==> P(a)) ==> ? x. P(x)"
   330  (fn prems=>
   331   [ (rtac ccontr 1),
   332     (REPEAT (ares_tac (prems@[exI,allI,notI,notE]) 1))  ]);
   333 
   334 
   335 (* case distinction *)
   336 
   337 qed_goal "case_split_thm" HOL.thy "[| P ==> Q; ~P ==> Q |] ==> Q"
   338   (fn [p1,p2] => [rtac (excluded_middle RS disjE) 1,
   339                   etac p2 1, etac p1 1]);
   340 
   341 fun case_tac a = res_inst_tac [("P",a)] case_split_thm;
   342 
   343 
   344 (** Standard abbreviations **)
   345 
   346 (*Fails unless the substitution has an effect*)
   347 fun stac th = CHANGED_GOAL (rtac (th RS ssubst));
   348 
   349 fun strip_tac i = REPEAT(resolve_tac [impI,allI] i); 
   350 
   351 
   352 (** strip ! and --> from proved goal while preserving !-bound var names **)
   353 
   354 local
   355 
   356 (* Use XXX to avoid forall_intr failing because of duplicate variable name *)
   357 val myspec = read_instantiate [("P","?XXX")] spec;
   358 val _ $ (_ $ (vx as Var(_,vxT))) = concl_of myspec;
   359 val cvx = cterm_of (#sign(rep_thm myspec)) vx;
   360 val aspec = forall_intr cvx myspec;
   361 
   362 in
   363 
   364 fun RSspec th =
   365   (case concl_of th of
   366      _ $ (Const("All",_) $ Abs(a,_,_)) =>
   367          let val ca = cterm_of (#sign(rep_thm th)) (Var((a,0),vxT))
   368          in th RS forall_elim ca aspec end
   369   | _ => raise THM("RSspec",0,[th]));
   370 
   371 fun RSmp th =
   372   (case concl_of th of
   373      _ $ (Const("op -->",_)$_$_) => th RS mp
   374   | _ => raise THM("RSmp",0,[th]));
   375 
   376 fun normalize_thm funs =
   377 let fun trans [] th = th
   378       | trans (f::fs) th = (trans funs (f th)) handle THM _ => trans fs th
   379 in trans funs end;
   380 
   381 fun qed_spec_mp name =
   382   let val thm = normalize_thm [RSspec,RSmp] (result())
   383   in bind_thm(name, thm) end;
   384 
   385 fun qed_goal_spec_mp name thy s p = 
   386 	bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goal thy s p));
   387 
   388 fun qed_goalw_spec_mp name thy defs s p = 
   389 	bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goalw thy defs s p));
   390 
   391 end;