src/FOL/IFOL.ML
author lcp
Tue Mar 07 13:15:25 1995 +0100 (1995-03-07)
changeset 928 cb31a4e97f75
parent 821 650ee089809b
child 1002 280ec187f8e1
permissions -rw-r--r--
Moved declaration of ~= to a syntax section
     1 (*  Title: 	FOL/ifol.ML
     2     ID:         $Id$
     3     Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1992  University of Cambridge
     5 
     6 Tactics and lemmas for ifol.thy (intuitionistic first-order logic)
     7 *)
     8 
     9 open IFOL;
    10 
    11 
    12 qed_goalw "TrueI" IFOL.thy [True_def] "True"
    13  (fn _ => [ (REPEAT (ares_tac [impI] 1)) ]);
    14 
    15 (*** Sequent-style elimination rules for & --> and ALL ***)
    16 
    17 qed_goal "conjE" IFOL.thy 
    18     "[| P&Q; [| P; Q |] ==> R |] ==> R"
    19  (fn prems=>
    20   [ (REPEAT (resolve_tac prems 1
    21       ORELSE (resolve_tac [conjunct1, conjunct2] 1 THEN
    22               resolve_tac prems 1))) ]);
    23 
    24 qed_goal "impE" IFOL.thy 
    25     "[| P-->Q;  P;  Q ==> R |] ==> R"
    26  (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
    27 
    28 qed_goal "allE" IFOL.thy 
    29     "[| ALL x.P(x); P(x) ==> R |] ==> R"
    30  (fn prems=> [ (REPEAT (resolve_tac (prems@[spec]) 1)) ]);
    31 
    32 (*Duplicates the quantifier; for use with eresolve_tac*)
    33 qed_goal "all_dupE" IFOL.thy 
    34     "[| ALL x.P(x);  [| P(x); ALL x.P(x) |] ==> R \
    35 \    |] ==> R"
    36  (fn prems=> [ (REPEAT (resolve_tac (prems@[spec]) 1)) ]);
    37 
    38 
    39 (*** Negation rules, which translate between ~P and P-->False ***)
    40 
    41 qed_goalw "notI" IFOL.thy [not_def] "(P ==> False) ==> ~P"
    42  (fn prems=> [ (REPEAT (ares_tac (prems@[impI]) 1)) ]);
    43 
    44 qed_goalw "notE" IFOL.thy [not_def] "[| ~P;  P |] ==> R"
    45  (fn prems=>
    46   [ (resolve_tac [mp RS FalseE] 1),
    47     (REPEAT (resolve_tac prems 1)) ]);
    48 
    49 (*This is useful with the special implication rules for each kind of P. *)
    50 qed_goal "not_to_imp" IFOL.thy 
    51     "[| ~P;  (P-->False) ==> Q |] ==> Q"
    52  (fn prems=> [ (REPEAT (ares_tac (prems@[impI,notE]) 1)) ]);
    53 
    54 
    55 (* For substitution int an assumption P, reduce Q to P-->Q, substitute into
    56    this implication, then apply impI to move P back into the assumptions.
    57    To specify P use something like
    58       eres_inst_tac [ ("P","ALL y. ?S(x,y)") ] rev_mp 1   *)
    59 qed_goal "rev_mp" IFOL.thy "[| P;  P --> Q |] ==> Q"
    60  (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
    61 
    62 
    63 (*Contrapositive of an inference rule*)
    64 qed_goal "contrapos" IFOL.thy "[| ~Q;  P==>Q |] ==> ~P"
    65  (fn [major,minor]=> 
    66   [ (rtac (major RS notE RS notI) 1), 
    67     (etac minor 1) ]);
    68 
    69 
    70 (*** Modus Ponens Tactics ***)
    71 
    72 (*Finds P-->Q and P in the assumptions, replaces implication by Q *)
    73 fun mp_tac i = eresolve_tac [notE,impE] i  THEN  assume_tac i;
    74 
    75 (*Like mp_tac but instantiates no variables*)
    76 fun eq_mp_tac i = eresolve_tac [notE,impE] i  THEN  eq_assume_tac i;
    77 
    78 
    79 (*** If-and-only-if ***)
    80 
    81 qed_goalw "iffI" IFOL.thy [iff_def]
    82    "[| P ==> Q;  Q ==> P |] ==> P<->Q"
    83  (fn prems=> [ (REPEAT (ares_tac (prems@[conjI, impI]) 1)) ]);
    84 
    85 
    86 (*Observe use of rewrite_rule to unfold "<->" in meta-assumptions (prems) *)
    87 qed_goalw "iffE" IFOL.thy [iff_def]
    88     "[| P <-> Q;  [| P-->Q; Q-->P |] ==> R |] ==> R"
    89  (fn prems => [ (resolve_tac [conjE] 1), (REPEAT (ares_tac prems 1)) ]);
    90 
    91 (* Destruct rules for <-> similar to Modus Ponens *)
    92 
    93 qed_goalw "iffD1" IFOL.thy [iff_def] "[| P <-> Q;  P |] ==> Q"
    94  (fn prems => [ (rtac (conjunct1 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
    95 
    96 qed_goalw "iffD2" IFOL.thy [iff_def] "[| P <-> Q;  Q |] ==> P"
    97  (fn prems => [ (rtac (conjunct2 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
    98 
    99 qed_goal "iff_refl" IFOL.thy "P <-> P"
   100  (fn _ => [ (REPEAT (ares_tac [iffI] 1)) ]);
   101 
   102 qed_goal "iff_sym" IFOL.thy "Q <-> P ==> P <-> Q"
   103  (fn [major] =>
   104   [ (rtac (major RS iffE) 1),
   105     (rtac iffI 1),
   106     (REPEAT (eresolve_tac [asm_rl,mp] 1)) ]);
   107 
   108 qed_goal "iff_trans" IFOL.thy
   109     "!!P Q R. [| P <-> Q;  Q<-> R |] ==> P <-> R"
   110  (fn _ =>
   111   [ (rtac iffI 1),
   112     (REPEAT (eresolve_tac [asm_rl,iffE] 1 ORELSE mp_tac 1)) ]);
   113 
   114 
   115 (*** Unique existence.  NOTE THAT the following 2 quantifications
   116    EX!x such that [EX!y such that P(x,y)]     (sequential)
   117    EX!x,y such that P(x,y)                    (simultaneous)
   118  do NOT mean the same thing.  The parser treats EX!x y.P(x,y) as sequential.
   119 ***)
   120 
   121 qed_goalw "ex1I" IFOL.thy [ex1_def]
   122     "[| P(a);  !!x. P(x) ==> x=a |] ==> EX! x. P(x)"
   123  (fn prems => [ (REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)) ]);
   124 
   125 (*Sometimes easier to use: the premises have no shared variables*)
   126 qed_goal "ex_ex1I" IFOL.thy
   127     "[| EX x.P(x);  !!x y. [| P(x); P(y) |] ==> x=y |] ==> EX! x. P(x)"
   128  (fn [ex,eq] => [ (rtac (ex RS exE) 1),
   129 		  (REPEAT (ares_tac [ex1I,eq] 1)) ]);
   130 
   131 qed_goalw "ex1E" IFOL.thy [ex1_def]
   132     "[| EX! x.P(x);  !!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R |] ==> R"
   133  (fn prems =>
   134   [ (cut_facts_tac prems 1),
   135     (REPEAT (eresolve_tac [exE,conjE] 1 ORELSE ares_tac prems 1)) ]);
   136 
   137 
   138 (*** <-> congruence rules for simplification ***)
   139 
   140 (*Use iffE on a premise.  For conj_cong, imp_cong, all_cong, ex_cong*)
   141 fun iff_tac prems i =
   142     resolve_tac (prems RL [iffE]) i THEN
   143     REPEAT1 (eresolve_tac [asm_rl,mp] i);
   144 
   145 qed_goal "conj_cong" IFOL.thy 
   146     "[| P <-> P';  P' ==> Q <-> Q' |] ==> (P&Q) <-> (P'&Q')"
   147  (fn prems =>
   148   [ (cut_facts_tac prems 1),
   149     (REPEAT  (ares_tac [iffI,conjI] 1
   150       ORELSE  eresolve_tac [iffE,conjE,mp] 1
   151       ORELSE  iff_tac prems 1)) ]);
   152 
   153 (*Reversed congruence rule!   Used in ZF/Order*)
   154 qed_goal "conj_cong2" IFOL.thy 
   155     "[| P <-> P';  P' ==> Q <-> Q' |] ==> (Q&P) <-> (Q'&P')"
   156  (fn prems =>
   157   [ (cut_facts_tac prems 1),
   158     (REPEAT  (ares_tac [iffI,conjI] 1
   159       ORELSE  eresolve_tac [iffE,conjE,mp] 1
   160       ORELSE  iff_tac prems 1)) ]);
   161 
   162 qed_goal "disj_cong" IFOL.thy 
   163     "[| P <-> P';  Q <-> Q' |] ==> (P|Q) <-> (P'|Q')"
   164  (fn prems =>
   165   [ (cut_facts_tac prems 1),
   166     (REPEAT  (eresolve_tac [iffE,disjE,disjI1,disjI2] 1
   167       ORELSE  ares_tac [iffI] 1
   168       ORELSE  mp_tac 1)) ]);
   169 
   170 qed_goal "imp_cong" IFOL.thy 
   171     "[| P <-> P';  P' ==> Q <-> Q' |] ==> (P-->Q) <-> (P'-->Q')"
   172  (fn prems =>
   173   [ (cut_facts_tac prems 1),
   174     (REPEAT   (ares_tac [iffI,impI] 1
   175       ORELSE  eresolve_tac [iffE] 1
   176       ORELSE  mp_tac 1 ORELSE iff_tac prems 1)) ]);
   177 
   178 qed_goal "iff_cong" IFOL.thy 
   179     "[| P <-> P';  Q <-> Q' |] ==> (P<->Q) <-> (P'<->Q')"
   180  (fn prems =>
   181   [ (cut_facts_tac prems 1),
   182     (REPEAT   (eresolve_tac [iffE] 1
   183       ORELSE  ares_tac [iffI] 1
   184       ORELSE  mp_tac 1)) ]);
   185 
   186 qed_goal "not_cong" IFOL.thy 
   187     "P <-> P' ==> ~P <-> ~P'"
   188  (fn prems =>
   189   [ (cut_facts_tac prems 1),
   190     (REPEAT   (ares_tac [iffI,notI] 1
   191       ORELSE  mp_tac 1
   192       ORELSE  eresolve_tac [iffE,notE] 1)) ]);
   193 
   194 qed_goal "all_cong" IFOL.thy 
   195     "(!!x.P(x) <-> Q(x)) ==> (ALL x.P(x)) <-> (ALL x.Q(x))"
   196  (fn prems =>
   197   [ (REPEAT   (ares_tac [iffI,allI] 1
   198       ORELSE   mp_tac 1
   199       ORELSE   eresolve_tac [allE] 1 ORELSE iff_tac prems 1)) ]);
   200 
   201 qed_goal "ex_cong" IFOL.thy 
   202     "(!!x.P(x) <-> Q(x)) ==> (EX x.P(x)) <-> (EX x.Q(x))"
   203  (fn prems =>
   204   [ (REPEAT   (eresolve_tac [exE] 1 ORELSE ares_tac [iffI,exI] 1
   205       ORELSE   mp_tac 1
   206       ORELSE   iff_tac prems 1)) ]);
   207 
   208 qed_goal "ex1_cong" IFOL.thy 
   209     "(!!x.P(x) <-> Q(x)) ==> (EX! x.P(x)) <-> (EX! x.Q(x))"
   210  (fn prems =>
   211   [ (REPEAT   (eresolve_tac [ex1E, spec RS mp] 1 ORELSE ares_tac [iffI,ex1I] 1
   212       ORELSE   mp_tac 1
   213       ORELSE   iff_tac prems 1)) ]);
   214 
   215 (*** Equality rules ***)
   216 
   217 qed_goal "sym" IFOL.thy "a=b ==> b=a"
   218  (fn [major] => [ (rtac (major RS subst) 1), (rtac refl 1) ]);
   219 
   220 qed_goal "trans" IFOL.thy "[| a=b;  b=c |] ==> a=c"
   221  (fn [prem1,prem2] => [ (rtac (prem2 RS subst) 1), (rtac prem1 1) ]);
   222 
   223 (** ~ b=a ==> ~ a=b **)
   224 val [not_sym] = compose(sym,2,contrapos);
   225 
   226 (*calling "standard" reduces maxidx to 0*)
   227 bind_thm ("ssubst", (sym RS subst));
   228 
   229 (*A special case of ex1E that would otherwise need quantifier expansion*)
   230 qed_goal "ex1_equalsE" IFOL.thy
   231     "[| EX! x.P(x);  P(a);  P(b) |] ==> a=b"
   232  (fn prems =>
   233   [ (cut_facts_tac prems 1),
   234     (etac ex1E 1),
   235     (rtac trans 1),
   236     (rtac sym 2),
   237     (REPEAT (eresolve_tac [asm_rl, spec RS mp] 1)) ]);
   238 
   239 (** Polymorphic congruence rules **)
   240 
   241 qed_goal "subst_context" IFOL.thy 
   242    "[| a=b |]  ==>  t(a)=t(b)"
   243  (fn prems=>
   244   [ (resolve_tac (prems RL [ssubst]) 1),
   245     (resolve_tac [refl] 1) ]);
   246 
   247 qed_goal "subst_context2" IFOL.thy 
   248    "[| a=b;  c=d |]  ==>  t(a,c)=t(b,d)"
   249  (fn prems=>
   250   [ (EVERY1 (map rtac ((prems RL [ssubst]) @ [refl]))) ]);
   251 
   252 qed_goal "subst_context3" IFOL.thy 
   253    "[| a=b;  c=d;  e=f |]  ==>  t(a,c,e)=t(b,d,f)"
   254  (fn prems=>
   255   [ (EVERY1 (map rtac ((prems RL [ssubst]) @ [refl]))) ]);
   256 
   257 (*Useful with eresolve_tac for proving equalties from known equalities.
   258 	a = b
   259 	|   |
   260 	c = d	*)
   261 qed_goal "box_equals" IFOL.thy
   262     "[| a=b;  a=c;  b=d |] ==> c=d"  
   263  (fn prems=>
   264   [ (resolve_tac [trans] 1),
   265     (resolve_tac [trans] 1),
   266     (resolve_tac [sym] 1),
   267     (REPEAT (resolve_tac prems 1)) ]);
   268 
   269 (*Dual of box_equals: for proving equalities backwards*)
   270 qed_goal "simp_equals" IFOL.thy
   271     "[| a=c;  b=d;  c=d |] ==> a=b"  
   272  (fn prems=>
   273   [ (resolve_tac [trans] 1),
   274     (resolve_tac [trans] 1),
   275     (REPEAT (resolve_tac (prems @ (prems RL [sym])) 1)) ]);
   276 
   277 (** Congruence rules for predicate letters **)
   278 
   279 qed_goal "pred1_cong" IFOL.thy
   280     "a=a' ==> P(a) <-> P(a')"
   281  (fn prems =>
   282   [ (cut_facts_tac prems 1),
   283     (rtac iffI 1),
   284     (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);
   285 
   286 qed_goal "pred2_cong" IFOL.thy
   287     "[| a=a';  b=b' |] ==> P(a,b) <-> P(a',b')"
   288  (fn prems =>
   289   [ (cut_facts_tac prems 1),
   290     (rtac iffI 1),
   291     (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);
   292 
   293 qed_goal "pred3_cong" IFOL.thy
   294     "[| a=a';  b=b';  c=c' |] ==> P(a,b,c) <-> P(a',b',c')"
   295  (fn prems =>
   296   [ (cut_facts_tac prems 1),
   297     (rtac iffI 1),
   298     (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);
   299 
   300 (*special cases for free variables P, Q, R, S -- up to 3 arguments*)
   301 
   302 val pred_congs = 
   303     flat (map (fn c => 
   304 	       map (fn th => read_instantiate [("P",c)] th)
   305 		   [pred1_cong,pred2_cong,pred3_cong])
   306 	       (explode"PQRS"));
   307 
   308 (*special case for the equality predicate!*)
   309 val eq_cong = read_instantiate [("P","op =")] pred2_cong;
   310 
   311 
   312 (*** Simplifications of assumed implications.
   313      Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE
   314      used with mp_tac (restricted to atomic formulae) is COMPLETE for 
   315      intuitionistic propositional logic.  See
   316    R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic
   317     (preprint, University of St Andrews, 1991)  ***)
   318 
   319 qed_goal "conj_impE" IFOL.thy 
   320     "[| (P&Q)-->S;  P-->(Q-->S) ==> R |] ==> R"
   321  (fn major::prems=>
   322   [ (REPEAT (ares_tac ([conjI, impI, major RS mp]@prems) 1)) ]);
   323 
   324 qed_goal "disj_impE" IFOL.thy 
   325     "[| (P|Q)-->S;  [| P-->S; Q-->S |] ==> R |] ==> R"
   326  (fn major::prems=>
   327   [ (DEPTH_SOLVE (ares_tac ([disjI1, disjI2, impI, major RS mp]@prems) 1)) ]);
   328 
   329 (*Simplifies the implication.  Classical version is stronger. 
   330   Still UNSAFE since Q must be provable -- backtracking needed.  *)
   331 qed_goal "imp_impE" IFOL.thy 
   332     "[| (P-->Q)-->S;  [| P; Q-->S |] ==> Q;  S ==> R |] ==> R"
   333  (fn major::prems=>
   334   [ (REPEAT (ares_tac ([impI, major RS mp]@prems) 1)) ]);
   335 
   336 (*Simplifies the implication.  Classical version is stronger. 
   337   Still UNSAFE since ~P must be provable -- backtracking needed.  *)
   338 qed_goal "not_impE" IFOL.thy
   339     "[| ~P --> S;  P ==> False;  S ==> R |] ==> R"
   340  (fn major::prems=>
   341   [ (REPEAT (ares_tac ([notI, impI, major RS mp]@prems) 1)) ]);
   342 
   343 (*Simplifies the implication.   UNSAFE.  *)
   344 qed_goal "iff_impE" IFOL.thy 
   345     "[| (P<->Q)-->S;  [| P; Q-->S |] ==> Q;  [| Q; P-->S |] ==> P;  \
   346 \       S ==> R |] ==> R"
   347  (fn major::prems=>
   348   [ (REPEAT (ares_tac ([iffI, impI, major RS mp]@prems) 1)) ]);
   349 
   350 (*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*)
   351 qed_goal "all_impE" IFOL.thy 
   352     "[| (ALL x.P(x))-->S;  !!x.P(x);  S ==> R |] ==> R"
   353  (fn major::prems=>
   354   [ (REPEAT (ares_tac ([allI, impI, major RS mp]@prems) 1)) ]);
   355 
   356 (*Unsafe: (EX x.P(x))-->S  is equivalent to  ALL x.P(x)-->S.  *)
   357 qed_goal "ex_impE" IFOL.thy 
   358     "[| (EX x.P(x))-->S;  P(x)-->S ==> R |] ==> R"
   359  (fn major::prems=>
   360   [ (REPEAT (ares_tac ([exI, impI, major RS mp]@prems) 1)) ]);
   361 
   362 (*Courtesy Krzysztof Grabczewski*)
   363 val major::prems = goal IFOL.thy "[| P|Q;  P==>R;  Q==>S |] ==> R|S";
   364 br (major RS disjE) 1;
   365 by (REPEAT (eresolve_tac (prems RL [disjI1, disjI2]) 1));
   366 qed "disj_imp_disj";