src/FOL/IFOL.ML
author clasohm
Thu Sep 16 12:20:38 1993 +0200 (1993-09-16)
changeset 0 a5a9c433f639
child 12 f17d542276b6
permissions -rw-r--r--
Initial revision
     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 ex1E: thm
    28   val ex1_equalsE: thm
    29   val ex1_cong: thm
    30   val ex_cong: thm
    31   val ex_impE: thm
    32   val iffD1: thm
    33   val iffD2: thm
    34   val iffE: thm
    35   val iffI: thm
    36   val iff_cong: thm
    37   val iff_impE: thm
    38   val iff_refl: thm
    39   val iff_sym: thm
    40   val iff_trans: thm
    41   val impE: thm
    42   val imp_cong: thm
    43   val imp_impE: thm
    44   val mp_tac: int -> tactic
    45   val notE: thm
    46   val notI: thm
    47   val not_cong: thm
    48   val not_impE: thm
    49   val not_sym: thm
    50   val not_to_imp: thm
    51   val pred1_cong: thm
    52   val pred2_cong: thm
    53   val pred3_cong: thm
    54   val pred_congs: thm list
    55   val rev_mp: thm
    56   val simp_equals: thm
    57   val ssubst: thm
    58   val subst_context: thm
    59   val subst_context2: thm
    60   val subst_context3: thm
    61   val sym: thm
    62   val trans: thm
    63   val TrueI: thm
    64   end;
    65 
    66 
    67 structure IFOL_Lemmas : IFOL_LEMMAS =
    68 struct
    69 
    70 val TrueI = prove_goalw IFOL.thy [True_def] "True"
    71  (fn _ => [ (REPEAT (ares_tac [impI] 1)) ]);
    72 
    73 (*** Sequent-style elimination rules for & --> and ALL ***)
    74 
    75 val conjE = prove_goal IFOL.thy 
    76     "[| P&Q; [| P; Q |] ==> R |] ==> R"
    77  (fn prems=>
    78   [ (REPEAT (resolve_tac prems 1
    79       ORELSE (resolve_tac [conjunct1, conjunct2] 1 THEN
    80               resolve_tac prems 1))) ]);
    81 
    82 val impE = prove_goal IFOL.thy 
    83     "[| P-->Q;  P;  Q ==> R |] ==> R"
    84  (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
    85 
    86 val allE = prove_goal IFOL.thy 
    87     "[| ALL x.P(x); P(x) ==> R |] ==> R"
    88  (fn prems=> [ (REPEAT (resolve_tac (prems@[spec]) 1)) ]);
    89 
    90 (*Duplicates the quantifier; for use with eresolve_tac*)
    91 val all_dupE = prove_goal IFOL.thy 
    92     "[| ALL x.P(x);  [| P(x); ALL x.P(x) |] ==> R \
    93 \    |] ==> R"
    94  (fn prems=> [ (REPEAT (resolve_tac (prems@[spec]) 1)) ]);
    95 
    96 
    97 (*** Negation rules, which translate between ~P and P-->False ***)
    98 
    99 val notI = prove_goalw IFOL.thy [not_def] "(P ==> False) ==> ~P"
   100  (fn prems=> [ (REPEAT (ares_tac (prems@[impI]) 1)) ]);
   101 
   102 val notE = prove_goalw IFOL.thy [not_def] "[| ~P;  P |] ==> R"
   103  (fn prems=>
   104   [ (resolve_tac [mp RS FalseE] 1),
   105     (REPEAT (resolve_tac prems 1)) ]);
   106 
   107 (*This is useful with the special implication rules for each kind of P. *)
   108 val not_to_imp = prove_goal IFOL.thy 
   109     "[| ~P;  (P-->False) ==> Q |] ==> Q"
   110  (fn prems=> [ (REPEAT (ares_tac (prems@[impI,notE]) 1)) ]);
   111 
   112 
   113 (* For substitution int an assumption P, reduce Q to P-->Q, substitute into
   114    this implication, then apply impI to move P back into the assumptions.
   115    To specify P use something like
   116       eres_inst_tac [ ("P","ALL y. ?S(x,y)") ] rev_mp 1   *)
   117 val rev_mp = prove_goal IFOL.thy "[| P;  P --> Q |] ==> Q"
   118  (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
   119 
   120 
   121 (*Contrapositive of an inference rule*)
   122 val contrapos = prove_goal IFOL.thy "[| ~Q;  P==>Q |] ==> ~P"
   123  (fn [major,minor]=> 
   124   [ (rtac (major RS notE RS notI) 1), 
   125     (etac minor 1) ]);
   126 
   127 
   128 (*** Modus Ponens Tactics ***)
   129 
   130 (*Finds P-->Q and P in the assumptions, replaces implication by Q *)
   131 fun mp_tac i = eresolve_tac [notE,impE] i  THEN  assume_tac i;
   132 
   133 (*Like mp_tac but instantiates no variables*)
   134 fun eq_mp_tac i = eresolve_tac [notE,impE] i  THEN  eq_assume_tac i;
   135 
   136 
   137 (*** If-and-only-if ***)
   138 
   139 val iffI = prove_goalw IFOL.thy [iff_def]
   140    "[| P ==> Q;  Q ==> P |] ==> P<->Q"
   141  (fn prems=> [ (REPEAT (ares_tac (prems@[conjI, impI]) 1)) ]);
   142 
   143 
   144 (*Observe use of rewrite_rule to unfold "<->" in meta-assumptions (prems) *)
   145 val iffE = prove_goalw IFOL.thy [iff_def]
   146     "[| P <-> Q;  [| P-->Q; Q-->P |] ==> R |] ==> R"
   147  (fn prems => [ (resolve_tac [conjE] 1), (REPEAT (ares_tac prems 1)) ]);
   148 
   149 (* Destruct rules for <-> similar to Modus Ponens *)
   150 
   151 val iffD1 = prove_goalw IFOL.thy [iff_def] "[| P <-> Q;  P |] ==> Q"
   152  (fn prems => [ (rtac (conjunct1 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
   153 
   154 val iffD2 = prove_goalw IFOL.thy [iff_def] "[| P <-> Q;  Q |] ==> P"
   155  (fn prems => [ (rtac (conjunct2 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
   156 
   157 val iff_refl = prove_goal IFOL.thy "P <-> P"
   158  (fn _ => [ (REPEAT (ares_tac [iffI] 1)) ]);
   159 
   160 val iff_sym = prove_goal IFOL.thy "Q <-> P ==> P <-> Q"
   161  (fn [major] =>
   162   [ (rtac (major RS iffE) 1),
   163     (rtac iffI 1),
   164     (REPEAT (eresolve_tac [asm_rl,mp] 1)) ]);
   165 
   166 val iff_trans = prove_goal IFOL.thy
   167     "!!P Q R. [| P <-> Q;  Q<-> R |] ==> P <-> R"
   168  (fn _ =>
   169   [ (rtac iffI 1),
   170     (REPEAT (eresolve_tac [asm_rl,iffE] 1 ORELSE mp_tac 1)) ]);
   171 
   172 
   173 (*** Unique existence.  NOTE THAT the following 2 quantifications
   174    EX!x such that [EX!y such that P(x,y)]     (sequential)
   175    EX!x,y such that P(x,y)                    (simultaneous)
   176  do NOT mean the same thing.  The parser treats EX!x y.P(x,y) as sequential.
   177 ***)
   178 
   179 val ex1I = prove_goalw IFOL.thy [ex1_def]
   180     "[| P(a);  !!x. P(x) ==> x=a |] ==> EX! x. P(x)"
   181  (fn prems => [ (REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)) ]);
   182 
   183 val ex1E = prove_goalw IFOL.thy [ex1_def]
   184     "[| EX! x.P(x);  !!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R |] ==> R"
   185  (fn prems =>
   186   [ (cut_facts_tac prems 1),
   187     (REPEAT (eresolve_tac [exE,conjE] 1 ORELSE ares_tac prems 1)) ]);
   188 
   189 
   190 (*** <-> congruence rules for simplification ***)
   191 
   192 (*Use iffE on a premise.  For conj_cong, imp_cong, all_cong, ex_cong*)
   193 fun iff_tac prems i =
   194     resolve_tac (prems RL [iffE]) i THEN
   195     REPEAT1 (eresolve_tac [asm_rl,mp] i);
   196 
   197 val conj_cong = prove_goal IFOL.thy 
   198     "[| P <-> P';  P' ==> Q <-> Q' |] ==> (P&Q) <-> (P'&Q')"
   199  (fn prems =>
   200   [ (cut_facts_tac prems 1),
   201     (REPEAT  (ares_tac [iffI,conjI] 1
   202       ORELSE  eresolve_tac [iffE,conjE,mp] 1
   203       ORELSE  iff_tac prems 1)) ]);
   204 
   205 val disj_cong = prove_goal IFOL.thy 
   206     "[| P <-> P';  Q <-> Q' |] ==> (P|Q) <-> (P'|Q')"
   207  (fn prems =>
   208   [ (cut_facts_tac prems 1),
   209     (REPEAT  (eresolve_tac [iffE,disjE,disjI1,disjI2] 1
   210       ORELSE  ares_tac [iffI] 1
   211       ORELSE  mp_tac 1)) ]);
   212 
   213 val imp_cong = prove_goal IFOL.thy 
   214     "[| P <-> P';  P' ==> Q <-> Q' |] ==> (P-->Q) <-> (P'-->Q')"
   215  (fn prems =>
   216   [ (cut_facts_tac prems 1),
   217     (REPEAT   (ares_tac [iffI,impI] 1
   218       ORELSE  eresolve_tac [iffE] 1
   219       ORELSE  mp_tac 1 ORELSE iff_tac prems 1)) ]);
   220 
   221 val iff_cong = prove_goal IFOL.thy 
   222     "[| P <-> P';  Q <-> Q' |] ==> (P<->Q) <-> (P'<->Q')"
   223  (fn prems =>
   224   [ (cut_facts_tac prems 1),
   225     (REPEAT   (eresolve_tac [iffE] 1
   226       ORELSE  ares_tac [iffI] 1
   227       ORELSE  mp_tac 1)) ]);
   228 
   229 val not_cong = prove_goal IFOL.thy 
   230     "P <-> P' ==> ~P <-> ~P'"
   231  (fn prems =>
   232   [ (cut_facts_tac prems 1),
   233     (REPEAT   (ares_tac [iffI,notI] 1
   234       ORELSE  mp_tac 1
   235       ORELSE  eresolve_tac [iffE,notE] 1)) ]);
   236 
   237 val all_cong = prove_goal IFOL.thy 
   238     "(!!x.P(x) <-> Q(x)) ==> (ALL x.P(x)) <-> (ALL x.Q(x))"
   239  (fn prems =>
   240   [ (REPEAT   (ares_tac [iffI,allI] 1
   241       ORELSE   mp_tac 1
   242       ORELSE   eresolve_tac [allE] 1 ORELSE iff_tac prems 1)) ]);
   243 
   244 val ex_cong = prove_goal IFOL.thy 
   245     "(!!x.P(x) <-> Q(x)) ==> (EX x.P(x)) <-> (EX x.Q(x))"
   246  (fn prems =>
   247   [ (REPEAT   (eresolve_tac [exE] 1 ORELSE ares_tac [iffI,exI] 1
   248       ORELSE   mp_tac 1
   249       ORELSE   iff_tac prems 1)) ]);
   250 
   251 val ex1_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 [ex1E, spec RS mp] 1 ORELSE ares_tac [iffI,ex1I] 1
   255       ORELSE   mp_tac 1
   256       ORELSE   iff_tac prems 1)) ]);
   257 
   258 (*** Equality rules ***)
   259 
   260 val sym = prove_goal IFOL.thy "a=b ==> b=a"
   261  (fn [major] => [ (rtac (major RS subst) 1), (rtac refl 1) ]);
   262 
   263 val trans = prove_goal IFOL.thy "[| a=b;  b=c |] ==> a=c"
   264  (fn [prem1,prem2] => [ (rtac (prem2 RS subst) 1), (rtac prem1 1) ]);
   265 
   266 (** ~ b=a ==> ~ a=b **)
   267 val [not_sym] = compose(sym,2,contrapos);
   268 
   269 (*calling "standard" reduces maxidx to 0*)
   270 val ssubst = standard (sym RS subst);
   271 
   272 (*A special case of ex1E that would otherwise need quantifier expansion*)
   273 val ex1_equalsE = prove_goal IFOL.thy
   274     "[| EX! x.P(x);  P(a);  P(b) |] ==> a=b"
   275  (fn prems =>
   276   [ (cut_facts_tac prems 1),
   277     (etac ex1E 1),
   278     (rtac trans 1),
   279     (rtac sym 2),
   280     (REPEAT (eresolve_tac [asm_rl, spec RS mp] 1)) ]);
   281 
   282 (** Polymorphic congruence rules **)
   283 
   284 val subst_context = prove_goal IFOL.thy 
   285    "[| a=b |]  ==>  t(a)=t(b)"
   286  (fn prems=>
   287   [ (resolve_tac (prems RL [ssubst]) 1),
   288     (resolve_tac [refl] 1) ]);
   289 
   290 val subst_context2 = prove_goal IFOL.thy 
   291    "[| a=b;  c=d |]  ==>  t(a,c)=t(b,d)"
   292  (fn prems=>
   293   [ (EVERY1 (map rtac ((prems RL [ssubst]) @ [refl]))) ]);
   294 
   295 val subst_context3 = prove_goal IFOL.thy 
   296    "[| a=b;  c=d;  e=f |]  ==>  t(a,c,e)=t(b,d,f)"
   297  (fn prems=>
   298   [ (EVERY1 (map rtac ((prems RL [ssubst]) @ [refl]))) ]);
   299 
   300 (*Useful with eresolve_tac for proving equalties from known equalities.
   301 	a = b
   302 	|   |
   303 	c = d	*)
   304 val box_equals = prove_goal IFOL.thy
   305     "[| a=b;  a=c;  b=d |] ==> c=d"  
   306  (fn prems=>
   307   [ (resolve_tac [trans] 1),
   308     (resolve_tac [trans] 1),
   309     (resolve_tac [sym] 1),
   310     (REPEAT (resolve_tac prems 1)) ]);
   311 
   312 (*Dual of box_equals: for proving equalities backwards*)
   313 val simp_equals = prove_goal IFOL.thy
   314     "[| a=c;  b=d;  c=d |] ==> a=b"  
   315  (fn prems=>
   316   [ (resolve_tac [trans] 1),
   317     (resolve_tac [trans] 1),
   318     (REPEAT (resolve_tac (prems @ (prems RL [sym])) 1)) ]);
   319 
   320 (** Congruence rules for predicate letters **)
   321 
   322 val pred1_cong = prove_goal IFOL.thy
   323     "a=a' ==> P(a) <-> P(a')"
   324  (fn prems =>
   325   [ (cut_facts_tac prems 1),
   326     (rtac iffI 1),
   327     (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);
   328 
   329 val pred2_cong = prove_goal IFOL.thy
   330     "[| a=a';  b=b' |] ==> P(a,b) <-> P(a',b')"
   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 pred3_cong = prove_goal IFOL.thy
   337     "[| a=a';  b=b';  c=c' |] ==> P(a,b,c) <-> P(a',b',c')"
   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 (*special cases for free variables P, Q, R, S -- up to 3 arguments*)
   344 
   345 val pred_congs = 
   346     flat (map (fn c => 
   347 	       map (fn th => read_instantiate [("P",c)] th)
   348 		   [pred1_cong,pred2_cong,pred3_cong])
   349 	       (explode"PQRS"));
   350 
   351 (*special case for the equality predicate!*)
   352 val eq_cong = read_instantiate [("P","op =")] pred2_cong;
   353 
   354 
   355 (*** Simplifications of assumed implications.
   356      Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE
   357      used with mp_tac (restricted to atomic formulae) is COMPLETE for 
   358      intuitionistic propositional logic.  See
   359    R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic
   360     (preprint, University of St Andrews, 1991)  ***)
   361 
   362 val conj_impE = prove_goal IFOL.thy 
   363     "[| (P&Q)-->S;  P-->(Q-->S) ==> R |] ==> R"
   364  (fn major::prems=>
   365   [ (REPEAT (ares_tac ([conjI, impI, major RS mp]@prems) 1)) ]);
   366 
   367 val disj_impE = prove_goal IFOL.thy 
   368     "[| (P|Q)-->S;  [| P-->S; Q-->S |] ==> R |] ==> R"
   369  (fn major::prems=>
   370   [ (DEPTH_SOLVE (ares_tac ([disjI1, disjI2, impI, major RS mp]@prems) 1)) ]);
   371 
   372 (*Simplifies the implication.  Classical version is stronger. 
   373   Still UNSAFE since Q must be provable -- backtracking needed.  *)
   374 val imp_impE = prove_goal IFOL.thy 
   375     "[| (P-->Q)-->S;  [| P; Q-->S |] ==> Q;  S ==> R |] ==> R"
   376  (fn major::prems=>
   377   [ (REPEAT (ares_tac ([impI, major RS mp]@prems) 1)) ]);
   378 
   379 (*Simplifies the implication.  Classical version is stronger. 
   380   Still UNSAFE since ~P must be provable -- backtracking needed.  *)
   381 val not_impE = prove_goal IFOL.thy
   382     "[| ~P --> S;  P ==> False;  S ==> R |] ==> R"
   383  (fn major::prems=>
   384   [ (REPEAT (ares_tac ([notI, impI, major RS mp]@prems) 1)) ]);
   385 
   386 (*Simplifies the implication.   UNSAFE.  *)
   387 val iff_impE = prove_goal IFOL.thy 
   388     "[| (P<->Q)-->S;  [| P; Q-->S |] ==> Q;  [| Q; P-->S |] ==> P;  \
   389 \       S ==> R |] ==> R"
   390  (fn major::prems=>
   391   [ (REPEAT (ares_tac ([iffI, impI, major RS mp]@prems) 1)) ]);
   392 
   393 (*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*)
   394 val all_impE = prove_goal IFOL.thy 
   395     "[| (ALL x.P(x))-->S;  !!x.P(x);  S ==> R |] ==> R"
   396  (fn major::prems=>
   397   [ (REPEAT (ares_tac ([allI, impI, major RS mp]@prems) 1)) ]);
   398 
   399 (*Unsafe: (EX x.P(x))-->S  is equivalent to  ALL x.P(x)-->S.  *)
   400 val ex_impE = prove_goal IFOL.thy 
   401     "[| (EX x.P(x))-->S;  P(x)-->S ==> R |] ==> R"
   402  (fn major::prems=>
   403   [ (REPEAT (ares_tac ([exI, impI, major RS mp]@prems) 1)) ]);
   404 
   405 end;
   406 
   407 open IFOL_Lemmas;
   408