src/FOL/ifol.ML
author wenzelm
Thu Aug 27 20:46:36 1998 +0200 (1998-08-27)
changeset 5400 645f46a24c72
parent 12 f17d542276b6
permissions -rw-r--r--
made tutorial first;
     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 signature IFOL_LEMMAS = 
    12   sig
    13   val allE: thm
    14   val all_cong: thm
    15   val all_dupE: thm
    16   val all_impE: thm
    17   val box_equals: thm
    18   val conjE: thm
    19   val conj_cong: thm
    20   val conj_impE: thm
    21   val contrapos: thm
    22   val disj_cong: thm
    23   val disj_impE: thm
    24   val eq_cong: thm
    25   val eq_mp_tac: int -> tactic
    26   val ex1I: thm
    27   val ex_ex1I: thm
    28   val ex1E: thm
    29   val ex1_equalsE: thm
    30   val ex1_cong: thm
    31   val ex_cong: thm
    32   val ex_impE: thm
    33   val iffD1: thm
    34   val iffD2: thm
    35   val iffE: thm
    36   val iffI: thm
    37   val iff_cong: thm
    38   val iff_impE: thm
    39   val iff_refl: thm
    40   val iff_sym: thm
    41   val iff_trans: thm
    42   val impE: thm
    43   val imp_cong: thm
    44   val imp_impE: thm
    45   val mp_tac: int -> tactic
    46   val notE: thm
    47   val notI: thm
    48   val not_cong: thm
    49   val not_impE: thm
    50   val not_sym: thm
    51   val not_to_imp: thm
    52   val pred1_cong: thm
    53   val pred2_cong: thm
    54   val pred3_cong: thm
    55   val pred_congs: thm list
    56   val rev_mp: thm
    57   val simp_equals: thm
    58   val ssubst: thm
    59   val subst_context: thm
    60   val subst_context2: thm
    61   val subst_context3: thm
    62   val sym: thm
    63   val trans: thm
    64   val TrueI: thm
    65   end;
    66 
    67 
    68 structure IFOL_Lemmas : IFOL_LEMMAS =
    69 struct
    70 
    71 val TrueI = prove_goalw IFOL.thy [True_def] "True"
    72  (fn _ => [ (REPEAT (ares_tac [impI] 1)) ]);
    73 
    74 (*** Sequent-style elimination rules for & --> and ALL ***)
    75 
    76 val conjE = prove_goal IFOL.thy 
    77     "[| P&Q; [| P; Q |] ==> R |] ==> R"
    78  (fn prems=>
    79   [ (REPEAT (resolve_tac prems 1
    80       ORELSE (resolve_tac [conjunct1, conjunct2] 1 THEN
    81               resolve_tac prems 1))) ]);
    82 
    83 val impE = prove_goal IFOL.thy 
    84     "[| P-->Q;  P;  Q ==> R |] ==> R"
    85  (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
    86 
    87 val allE = prove_goal IFOL.thy 
    88     "[| ALL x.P(x); P(x) ==> R |] ==> R"
    89  (fn prems=> [ (REPEAT (resolve_tac (prems@[spec]) 1)) ]);
    90 
    91 (*Duplicates the quantifier; for use with eresolve_tac*)
    92 val all_dupE = prove_goal IFOL.thy 
    93     "[| ALL x.P(x);  [| P(x); ALL x.P(x) |] ==> R \
    94 \    |] ==> R"
    95  (fn prems=> [ (REPEAT (resolve_tac (prems@[spec]) 1)) ]);
    96 
    97 
    98 (*** Negation rules, which translate between ~P and P-->False ***)
    99 
   100 val notI = prove_goalw IFOL.thy [not_def] "(P ==> False) ==> ~P"
   101  (fn prems=> [ (REPEAT (ares_tac (prems@[impI]) 1)) ]);
   102 
   103 val notE = prove_goalw IFOL.thy [not_def] "[| ~P;  P |] ==> R"
   104  (fn prems=>
   105   [ (resolve_tac [mp RS FalseE] 1),
   106     (REPEAT (resolve_tac prems 1)) ]);
   107 
   108 (*This is useful with the special implication rules for each kind of P. *)
   109 val not_to_imp = prove_goal IFOL.thy 
   110     "[| ~P;  (P-->False) ==> Q |] ==> Q"
   111  (fn prems=> [ (REPEAT (ares_tac (prems@[impI,notE]) 1)) ]);
   112 
   113 
   114 (* For substitution int an assumption P, reduce Q to P-->Q, substitute into
   115    this implication, then apply impI to move P back into the assumptions.
   116    To specify P use something like
   117       eres_inst_tac [ ("P","ALL y. ?S(x,y)") ] rev_mp 1   *)
   118 val rev_mp = prove_goal IFOL.thy "[| P;  P --> Q |] ==> Q"
   119  (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
   120 
   121 
   122 (*Contrapositive of an inference rule*)
   123 val contrapos = prove_goal IFOL.thy "[| ~Q;  P==>Q |] ==> ~P"
   124  (fn [major,minor]=> 
   125   [ (rtac (major RS notE RS notI) 1), 
   126     (etac minor 1) ]);
   127 
   128 
   129 (*** Modus Ponens Tactics ***)
   130 
   131 (*Finds P-->Q and P in the assumptions, replaces implication by Q *)
   132 fun mp_tac i = eresolve_tac [notE,impE] i  THEN  assume_tac i;
   133 
   134 (*Like mp_tac but instantiates no variables*)
   135 fun eq_mp_tac i = eresolve_tac [notE,impE] i  THEN  eq_assume_tac i;
   136 
   137 
   138 (*** If-and-only-if ***)
   139 
   140 val iffI = prove_goalw IFOL.thy [iff_def]
   141    "[| P ==> Q;  Q ==> P |] ==> P<->Q"
   142  (fn prems=> [ (REPEAT (ares_tac (prems@[conjI, impI]) 1)) ]);
   143 
   144 
   145 (*Observe use of rewrite_rule to unfold "<->" in meta-assumptions (prems) *)
   146 val iffE = prove_goalw IFOL.thy [iff_def]
   147     "[| P <-> Q;  [| P-->Q; Q-->P |] ==> R |] ==> R"
   148  (fn prems => [ (resolve_tac [conjE] 1), (REPEAT (ares_tac prems 1)) ]);
   149 
   150 (* Destruct rules for <-> similar to Modus Ponens *)
   151 
   152 val iffD1 = prove_goalw IFOL.thy [iff_def] "[| P <-> Q;  P |] ==> Q"
   153  (fn prems => [ (rtac (conjunct1 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
   154 
   155 val iffD2 = prove_goalw IFOL.thy [iff_def] "[| P <-> Q;  Q |] ==> P"
   156  (fn prems => [ (rtac (conjunct2 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
   157 
   158 val iff_refl = prove_goal IFOL.thy "P <-> P"
   159  (fn _ => [ (REPEAT (ares_tac [iffI] 1)) ]);
   160 
   161 val iff_sym = prove_goal IFOL.thy "Q <-> P ==> P <-> Q"
   162  (fn [major] =>
   163   [ (rtac (major RS iffE) 1),
   164     (rtac iffI 1),
   165     (REPEAT (eresolve_tac [asm_rl,mp] 1)) ]);
   166 
   167 val iff_trans = prove_goal IFOL.thy
   168     "!!P Q R. [| P <-> Q;  Q<-> R |] ==> P <-> R"
   169  (fn _ =>
   170   [ (rtac iffI 1),
   171     (REPEAT (eresolve_tac [asm_rl,iffE] 1 ORELSE mp_tac 1)) ]);
   172 
   173 
   174 (*** Unique existence.  NOTE THAT the following 2 quantifications
   175    EX!x such that [EX!y such that P(x,y)]     (sequential)
   176    EX!x,y such that P(x,y)                    (simultaneous)
   177  do NOT mean the same thing.  The parser treats EX!x y.P(x,y) as sequential.
   178 ***)
   179 
   180 val ex1I = prove_goalw IFOL.thy [ex1_def]
   181     "[| P(a);  !!x. P(x) ==> x=a |] ==> EX! x. P(x)"
   182  (fn prems => [ (REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)) ]);
   183 
   184 (*Sometimes easier to use: the premises have no shared variables*)
   185 val ex_ex1I = prove_goal IFOL.thy
   186     "[| EX x.P(x);  !!x y. [| P(x); P(y) |] ==> x=y |] ==> EX! x. P(x)"
   187  (fn [ex,eq] => [ (rtac (ex RS exE) 1),
   188 		  (REPEAT (ares_tac [ex1I,eq] 1)) ]);
   189 
   190 val ex1E = prove_goalw IFOL.thy [ex1_def]
   191     "[| EX! x.P(x);  !!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R |] ==> R"
   192  (fn prems =>
   193   [ (cut_facts_tac prems 1),
   194     (REPEAT (eresolve_tac [exE,conjE] 1 ORELSE ares_tac prems 1)) ]);
   195 
   196 
   197 (*** <-> congruence rules for simplification ***)
   198 
   199 (*Use iffE on a premise.  For conj_cong, imp_cong, all_cong, ex_cong*)
   200 fun iff_tac prems i =
   201     resolve_tac (prems RL [iffE]) i THEN
   202     REPEAT1 (eresolve_tac [asm_rl,mp] i);
   203 
   204 val conj_cong = prove_goal IFOL.thy 
   205     "[| P <-> P';  P' ==> Q <-> Q' |] ==> (P&Q) <-> (P'&Q')"
   206  (fn prems =>
   207   [ (cut_facts_tac prems 1),
   208     (REPEAT  (ares_tac [iffI,conjI] 1
   209       ORELSE  eresolve_tac [iffE,conjE,mp] 1
   210       ORELSE  iff_tac prems 1)) ]);
   211 
   212 val disj_cong = prove_goal IFOL.thy 
   213     "[| P <-> P';  Q <-> Q' |] ==> (P|Q) <-> (P'|Q')"
   214  (fn prems =>
   215   [ (cut_facts_tac prems 1),
   216     (REPEAT  (eresolve_tac [iffE,disjE,disjI1,disjI2] 1
   217       ORELSE  ares_tac [iffI] 1
   218       ORELSE  mp_tac 1)) ]);
   219 
   220 val imp_cong = prove_goal IFOL.thy 
   221     "[| P <-> P';  P' ==> Q <-> Q' |] ==> (P-->Q) <-> (P'-->Q')"
   222  (fn prems =>
   223   [ (cut_facts_tac prems 1),
   224     (REPEAT   (ares_tac [iffI,impI] 1
   225       ORELSE  eresolve_tac [iffE] 1
   226       ORELSE  mp_tac 1 ORELSE iff_tac prems 1)) ]);
   227 
   228 val iff_cong = prove_goal IFOL.thy 
   229     "[| P <-> P';  Q <-> Q' |] ==> (P<->Q) <-> (P'<->Q')"
   230  (fn prems =>
   231   [ (cut_facts_tac prems 1),
   232     (REPEAT   (eresolve_tac [iffE] 1
   233       ORELSE  ares_tac [iffI] 1
   234       ORELSE  mp_tac 1)) ]);
   235 
   236 val not_cong = prove_goal IFOL.thy 
   237     "P <-> P' ==> ~P <-> ~P'"
   238  (fn prems =>
   239   [ (cut_facts_tac prems 1),
   240     (REPEAT   (ares_tac [iffI,notI] 1
   241       ORELSE  mp_tac 1
   242       ORELSE  eresolve_tac [iffE,notE] 1)) ]);
   243 
   244 val all_cong = prove_goal IFOL.thy 
   245     "(!!x.P(x) <-> Q(x)) ==> (ALL x.P(x)) <-> (ALL x.Q(x))"
   246  (fn prems =>
   247   [ (REPEAT   (ares_tac [iffI,allI] 1
   248       ORELSE   mp_tac 1
   249       ORELSE   eresolve_tac [allE] 1 ORELSE iff_tac prems 1)) ]);
   250 
   251 val ex_cong = prove_goal IFOL.thy 
   252     "(!!x.P(x) <-> Q(x)) ==> (EX x.P(x)) <-> (EX x.Q(x))"
   253  (fn prems =>
   254   [ (REPEAT   (eresolve_tac [exE] 1 ORELSE ares_tac [iffI,exI] 1
   255       ORELSE   mp_tac 1
   256       ORELSE   iff_tac prems 1)) ]);
   257 
   258 val ex1_cong = prove_goal IFOL.thy 
   259     "(!!x.P(x) <-> Q(x)) ==> (EX! x.P(x)) <-> (EX! x.Q(x))"
   260  (fn prems =>
   261   [ (REPEAT   (eresolve_tac [ex1E, spec RS mp] 1 ORELSE ares_tac [iffI,ex1I] 1
   262       ORELSE   mp_tac 1
   263       ORELSE   iff_tac prems 1)) ]);
   264 
   265 (*** Equality rules ***)
   266 
   267 val sym = prove_goal IFOL.thy "a=b ==> b=a"
   268  (fn [major] => [ (rtac (major RS subst) 1), (rtac refl 1) ]);
   269 
   270 val trans = prove_goal IFOL.thy "[| a=b;  b=c |] ==> a=c"
   271  (fn [prem1,prem2] => [ (rtac (prem2 RS subst) 1), (rtac prem1 1) ]);
   272 
   273 (** ~ b=a ==> ~ a=b **)
   274 val [not_sym] = compose(sym,2,contrapos);
   275 
   276 (*calling "standard" reduces maxidx to 0*)
   277 val ssubst = standard (sym RS subst);
   278 
   279 (*A special case of ex1E that would otherwise need quantifier expansion*)
   280 val ex1_equalsE = prove_goal IFOL.thy
   281     "[| EX! x.P(x);  P(a);  P(b) |] ==> a=b"
   282  (fn prems =>
   283   [ (cut_facts_tac prems 1),
   284     (etac ex1E 1),
   285     (rtac trans 1),
   286     (rtac sym 2),
   287     (REPEAT (eresolve_tac [asm_rl, spec RS mp] 1)) ]);
   288 
   289 (** Polymorphic congruence rules **)
   290 
   291 val subst_context = prove_goal IFOL.thy 
   292    "[| a=b |]  ==>  t(a)=t(b)"
   293  (fn prems=>
   294   [ (resolve_tac (prems RL [ssubst]) 1),
   295     (resolve_tac [refl] 1) ]);
   296 
   297 val subst_context2 = prove_goal IFOL.thy 
   298    "[| a=b;  c=d |]  ==>  t(a,c)=t(b,d)"
   299  (fn prems=>
   300   [ (EVERY1 (map rtac ((prems RL [ssubst]) @ [refl]))) ]);
   301 
   302 val subst_context3 = prove_goal IFOL.thy 
   303    "[| a=b;  c=d;  e=f |]  ==>  t(a,c,e)=t(b,d,f)"
   304  (fn prems=>
   305   [ (EVERY1 (map rtac ((prems RL [ssubst]) @ [refl]))) ]);
   306 
   307 (*Useful with eresolve_tac for proving equalties from known equalities.
   308 	a = b
   309 	|   |
   310 	c = d	*)
   311 val box_equals = prove_goal IFOL.thy
   312     "[| a=b;  a=c;  b=d |] ==> c=d"  
   313  (fn prems=>
   314   [ (resolve_tac [trans] 1),
   315     (resolve_tac [trans] 1),
   316     (resolve_tac [sym] 1),
   317     (REPEAT (resolve_tac prems 1)) ]);
   318 
   319 (*Dual of box_equals: for proving equalities backwards*)
   320 val simp_equals = prove_goal IFOL.thy
   321     "[| a=c;  b=d;  c=d |] ==> a=b"  
   322  (fn prems=>
   323   [ (resolve_tac [trans] 1),
   324     (resolve_tac [trans] 1),
   325     (REPEAT (resolve_tac (prems @ (prems RL [sym])) 1)) ]);
   326 
   327 (** Congruence rules for predicate letters **)
   328 
   329 val pred1_cong = prove_goal IFOL.thy
   330     "a=a' ==> P(a) <-> P(a')"
   331  (fn prems =>
   332   [ (cut_facts_tac prems 1),
   333     (rtac iffI 1),
   334     (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);
   335 
   336 val pred2_cong = prove_goal IFOL.thy
   337     "[| a=a';  b=b' |] ==> P(a,b) <-> P(a',b')"
   338  (fn prems =>
   339   [ (cut_facts_tac prems 1),
   340     (rtac iffI 1),
   341     (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);
   342 
   343 val pred3_cong = prove_goal IFOL.thy
   344     "[| a=a';  b=b';  c=c' |] ==> P(a,b,c) <-> P(a',b',c')"
   345  (fn prems =>
   346   [ (cut_facts_tac prems 1),
   347     (rtac iffI 1),
   348     (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);
   349 
   350 (*special cases for free variables P, Q, R, S -- up to 3 arguments*)
   351 
   352 val pred_congs = 
   353     flat (map (fn c => 
   354 	       map (fn th => read_instantiate [("P",c)] th)
   355 		   [pred1_cong,pred2_cong,pred3_cong])
   356 	       (explode"PQRS"));
   357 
   358 (*special case for the equality predicate!*)
   359 val eq_cong = read_instantiate [("P","op =")] pred2_cong;
   360 
   361 
   362 (*** Simplifications of assumed implications.
   363      Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE
   364      used with mp_tac (restricted to atomic formulae) is COMPLETE for 
   365      intuitionistic propositional logic.  See
   366    R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic
   367     (preprint, University of St Andrews, 1991)  ***)
   368 
   369 val conj_impE = prove_goal IFOL.thy 
   370     "[| (P&Q)-->S;  P-->(Q-->S) ==> R |] ==> R"
   371  (fn major::prems=>
   372   [ (REPEAT (ares_tac ([conjI, impI, major RS mp]@prems) 1)) ]);
   373 
   374 val disj_impE = prove_goal IFOL.thy 
   375     "[| (P|Q)-->S;  [| P-->S; Q-->S |] ==> R |] ==> R"
   376  (fn major::prems=>
   377   [ (DEPTH_SOLVE (ares_tac ([disjI1, disjI2, impI, major RS mp]@prems) 1)) ]);
   378 
   379 (*Simplifies the implication.  Classical version is stronger. 
   380   Still UNSAFE since Q must be provable -- backtracking needed.  *)
   381 val imp_impE = prove_goal IFOL.thy 
   382     "[| (P-->Q)-->S;  [| P; Q-->S |] ==> Q;  S ==> R |] ==> R"
   383  (fn major::prems=>
   384   [ (REPEAT (ares_tac ([impI, major RS mp]@prems) 1)) ]);
   385 
   386 (*Simplifies the implication.  Classical version is stronger. 
   387   Still UNSAFE since ~P must be provable -- backtracking needed.  *)
   388 val not_impE = prove_goal IFOL.thy
   389     "[| ~P --> S;  P ==> False;  S ==> R |] ==> R"
   390  (fn major::prems=>
   391   [ (REPEAT (ares_tac ([notI, impI, major RS mp]@prems) 1)) ]);
   392 
   393 (*Simplifies the implication.   UNSAFE.  *)
   394 val iff_impE = prove_goal IFOL.thy 
   395     "[| (P<->Q)-->S;  [| P; Q-->S |] ==> Q;  [| Q; P-->S |] ==> P;  \
   396 \       S ==> R |] ==> R"
   397  (fn major::prems=>
   398   [ (REPEAT (ares_tac ([iffI, impI, major RS mp]@prems) 1)) ]);
   399 
   400 (*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*)
   401 val all_impE = prove_goal IFOL.thy 
   402     "[| (ALL x.P(x))-->S;  !!x.P(x);  S ==> R |] ==> R"
   403  (fn major::prems=>
   404   [ (REPEAT (ares_tac ([allI, impI, major RS mp]@prems) 1)) ]);
   405 
   406 (*Unsafe: (EX x.P(x))-->S  is equivalent to  ALL x.P(x)-->S.  *)
   407 val ex_impE = prove_goal IFOL.thy 
   408     "[| (EX x.P(x))-->S;  P(x)-->S ==> R |] ==> R"
   409  (fn major::prems=>
   410   [ (REPEAT (ares_tac ([exI, impI, major RS mp]@prems) 1)) ]);
   411 
   412 end;
   413 
   414 open IFOL_Lemmas;
   415