src/FOL/simpdata.ML
author nipkow
Mon Dec 17 14:27:18 2001 +0100 (2001-12-17 ago)
changeset 12526 1b9db2581fe2
parent 12038 343a9888e875
child 12720 f8a134b9a57f
permissions -rw-r--r--
mods due to changed 1-point simprocs (quantifier1).
     1 (*  Title:      FOL/simpdata.ML
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1994  University of Cambridge
     5 
     6 Simplification data for FOL.
     7 *)
     8 
     9 
    10 (* Elimination of True from asumptions: *)
    11 
    12 bind_thm ("True_implies_equals", prove_goal IFOL.thy
    13  "(True ==> PROP P) == PROP P"
    14 (K [rtac equal_intr_rule 1, atac 2,
    15           METAHYPS (fn prems => resolve_tac prems 1) 1,
    16           rtac TrueI 1]));
    17 
    18 
    19 (*** Rewrite rules ***)
    20 
    21 fun int_prove_fun s =
    22  (writeln s;
    23   prove_goal IFOL.thy s
    24    (fn prems => [ (cut_facts_tac prems 1),
    25                   (IntPr.fast_tac 1) ]));
    26 
    27 bind_thms ("conj_simps", map int_prove_fun
    28  ["P & True <-> P",      "True & P <-> P",
    29   "P & False <-> False", "False & P <-> False",
    30   "P & P <-> P", "P & P & Q <-> P & Q",
    31   "P & ~P <-> False",    "~P & P <-> False",
    32   "(P & Q) & R <-> P & (Q & R)"]);
    33 
    34 bind_thms ("disj_simps", map int_prove_fun
    35  ["P | True <-> True",  "True | P <-> True",
    36   "P | False <-> P",    "False | P <-> P",
    37   "P | P <-> P", "P | P | Q <-> P | Q",
    38   "(P | Q) | R <-> P | (Q | R)"]);
    39 
    40 bind_thms ("not_simps", map int_prove_fun
    41  ["~(P|Q)  <-> ~P & ~Q",
    42   "~ False <-> True",   "~ True <-> False"]);
    43 
    44 bind_thms ("imp_simps", map int_prove_fun
    45  ["(P --> False) <-> ~P",       "(P --> True) <-> True",
    46   "(False --> P) <-> True",     "(True --> P) <-> P",
    47   "(P --> P) <-> True",         "(P --> ~P) <-> ~P"]);
    48 
    49 bind_thms ("iff_simps", map int_prove_fun
    50  ["(True <-> P) <-> P",         "(P <-> True) <-> P",
    51   "(P <-> P) <-> True",
    52   "(False <-> P) <-> ~P",       "(P <-> False) <-> ~P"]);
    53 
    54 (*The x=t versions are needed for the simplification procedures*)
    55 bind_thms ("quant_simps", map int_prove_fun
    56  ["(ALL x. P) <-> P",
    57   "(ALL x. x=t --> P(x)) <-> P(t)",
    58   "(ALL x. t=x --> P(x)) <-> P(t)",
    59   "(EX x. P) <-> P",
    60   "(EX x. x=t & P(x)) <-> P(t)",
    61   "(EX x. t=x & P(x)) <-> P(t)"]);
    62 
    63 (*These are NOT supplied by default!*)
    64 bind_thms ("distrib_simps", map int_prove_fun
    65  ["P & (Q | R) <-> P&Q | P&R",
    66   "(Q | R) & P <-> Q&P | R&P",
    67   "(P | Q --> R) <-> (P --> R) & (Q --> R)"]);
    68 
    69 (** Conversion into rewrite rules **)
    70 
    71 fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th;
    72 
    73 bind_thm ("P_iff_F", int_prove_fun "~P ==> (P <-> False)");
    74 bind_thm ("iff_reflection_F", P_iff_F RS iff_reflection);
    75 
    76 bind_thm ("P_iff_T", int_prove_fun "P ==> (P <-> True)");
    77 bind_thm ("iff_reflection_T", P_iff_T RS iff_reflection);
    78 
    79 (*Make meta-equalities.  The operator below is Trueprop*)
    80 
    81 fun mk_meta_eq th = case concl_of th of
    82     _ $ (Const("op =",_)$_$_)   => th RS eq_reflection
    83   | _ $ (Const("op <->",_)$_$_) => th RS iff_reflection
    84   | _                           =>
    85   error("conclusion must be a =-equality or <->");;
    86 
    87 fun mk_eq th = case concl_of th of
    88     Const("==",_)$_$_           => th
    89   | _ $ (Const("op =",_)$_$_)   => mk_meta_eq th
    90   | _ $ (Const("op <->",_)$_$_) => mk_meta_eq th
    91   | _ $ (Const("Not",_)$_)      => th RS iff_reflection_F
    92   | _                           => th RS iff_reflection_T;
    93 
    94 (*Replace premises x=y, X<->Y by X==Y*)
    95 val mk_meta_prems =
    96     rule_by_tactic
    97       (REPEAT_FIRST (resolve_tac [meta_eq_to_obj_eq, def_imp_iff]));
    98 
    99 (*Congruence rules for = or <-> (instead of ==)*)
   100 fun mk_meta_cong rl =
   101   standard(mk_meta_eq (mk_meta_prems rl))
   102   handle THM _ =>
   103   error("Premises and conclusion of congruence rules must use =-equality or <->");
   104 
   105 val mksimps_pairs =
   106   [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
   107    ("All", [spec]), ("True", []), ("False", [])];
   108 
   109 (* ###FIXME: move to Provers/simplifier.ML
   110 val mk_atomize:      (string * thm list) list -> thm -> thm list
   111 *)
   112 (* ###FIXME: move to Provers/simplifier.ML *)
   113 fun mk_atomize pairs =
   114   let fun atoms th =
   115         (case concl_of th of
   116            Const("Trueprop",_) $ p =>
   117              (case head_of p of
   118                 Const(a,_) =>
   119                   (case assoc(pairs,a) of
   120                      Some(rls) => flat (map atoms ([th] RL rls))
   121                    | None => [th])
   122               | _ => [th])
   123          | _ => [th])
   124   in atoms end;
   125 
   126 fun mksimps pairs = (map mk_eq o mk_atomize pairs o gen_all);
   127 
   128 (*** Classical laws ***)
   129 
   130 fun prove_fun s =
   131  (writeln s;
   132   prove_goal (the_context ()) s
   133    (fn prems => [ (cut_facts_tac prems 1),
   134                   (Cla.fast_tac FOL_cs 1) ]));
   135 
   136 (*Avoids duplication of subgoals after expand_if, when the true and false
   137   cases boil down to the same thing.*)
   138 bind_thm ("cases_simp", prove_fun "(P --> Q) & (~P --> Q) <-> Q");
   139 
   140 
   141 (*** Miniscoping: pushing quantifiers in
   142      We do NOT distribute of ALL over &, or dually that of EX over |
   143      Baaz and Leitsch, On Skolemization and Proof Complexity (1994)
   144      show that this step can increase proof length!
   145 ***)
   146 
   147 (*existential miniscoping*)
   148 bind_thms ("int_ex_simps", map int_prove_fun
   149  ["(EX x. P(x) & Q) <-> (EX x. P(x)) & Q",
   150   "(EX x. P & Q(x)) <-> P & (EX x. Q(x))",
   151   "(EX x. P(x) | Q) <-> (EX x. P(x)) | Q",
   152   "(EX x. P | Q(x)) <-> P | (EX x. Q(x))"]);
   153 
   154 (*classical rules*)
   155 bind_thms ("cla_ex_simps", map prove_fun
   156  ["(EX x. P(x) --> Q) <-> (ALL x. P(x)) --> Q",
   157   "(EX x. P --> Q(x)) <-> P --> (EX x. Q(x))"]);
   158 
   159 bind_thms ("ex_simps", int_ex_simps @ cla_ex_simps);
   160 
   161 (*universal miniscoping*)
   162 bind_thms ("int_all_simps", map int_prove_fun
   163  ["(ALL x. P(x) & Q) <-> (ALL x. P(x)) & Q",
   164   "(ALL x. P & Q(x)) <-> P & (ALL x. Q(x))",
   165   "(ALL x. P(x) --> Q) <-> (EX x. P(x)) --> Q",
   166   "(ALL x. P --> Q(x)) <-> P --> (ALL x. Q(x))"]);
   167 
   168 (*classical rules*)
   169 bind_thms ("cla_all_simps", map prove_fun
   170  ["(ALL x. P(x) | Q) <-> (ALL x. P(x)) | Q",
   171   "(ALL x. P | Q(x)) <-> P | (ALL x. Q(x))"]);
   172 
   173 bind_thms ("all_simps", int_all_simps @ cla_all_simps);
   174 
   175 
   176 (*** Named rewrite rules proved for IFOL ***)
   177 
   178 fun int_prove nm thm  = qed_goal nm IFOL.thy thm
   179     (fn prems => [ (cut_facts_tac prems 1),
   180                    (IntPr.fast_tac 1) ]);
   181 
   182 fun prove nm thm  = qed_goal nm (the_context ()) thm (fn _ => [Blast_tac 1]);
   183 
   184 int_prove "conj_commute" "P&Q <-> Q&P";
   185 int_prove "conj_left_commute" "P&(Q&R) <-> Q&(P&R)";
   186 bind_thms ("conj_comms", [conj_commute, conj_left_commute]);
   187 
   188 int_prove "disj_commute" "P|Q <-> Q|P";
   189 int_prove "disj_left_commute" "P|(Q|R) <-> Q|(P|R)";
   190 bind_thms ("disj_comms", [disj_commute, disj_left_commute]);
   191 
   192 int_prove "conj_disj_distribL" "P&(Q|R) <-> (P&Q | P&R)";
   193 int_prove "conj_disj_distribR" "(P|Q)&R <-> (P&R | Q&R)";
   194 
   195 int_prove "disj_conj_distribL" "P|(Q&R) <-> (P|Q) & (P|R)";
   196 int_prove "disj_conj_distribR" "(P&Q)|R <-> (P|R) & (Q|R)";
   197 
   198 int_prove "imp_conj_distrib" "(P --> (Q&R)) <-> (P-->Q) & (P-->R)";
   199 int_prove "imp_conj"         "((P&Q)-->R)   <-> (P --> (Q --> R))";
   200 int_prove "imp_disj"         "(P|Q --> R)   <-> (P-->R) & (Q-->R)";
   201 
   202 prove "imp_disj1" "(P-->Q) | R <-> (P-->Q | R)";
   203 prove "imp_disj2" "Q | (P-->R) <-> (P-->Q | R)";
   204 
   205 int_prove "de_Morgan_disj" "(~(P | Q)) <-> (~P & ~Q)";
   206 prove     "de_Morgan_conj" "(~(P & Q)) <-> (~P | ~Q)";
   207 
   208 prove     "not_iff" "~(P <-> Q) <-> (P <-> ~Q)";
   209 
   210 prove     "not_all" "(~ (ALL x. P(x))) <-> (EX x.~P(x))";
   211 prove     "imp_all" "((ALL x. P(x)) --> Q) <-> (EX x. P(x) --> Q)";
   212 int_prove "not_ex"  "(~ (EX x. P(x))) <-> (ALL x.~P(x))";
   213 int_prove "imp_ex" "((EX x. P(x)) --> Q) <-> (ALL x. P(x) --> Q)";
   214 
   215 int_prove "ex_disj_distrib"
   216     "(EX x. P(x) | Q(x)) <-> ((EX x. P(x)) | (EX x. Q(x)))";
   217 int_prove "all_conj_distrib"
   218     "(ALL x. P(x) & Q(x)) <-> ((ALL x. P(x)) & (ALL x. Q(x)))";
   219 
   220 
   221 local
   222 val uncurry = prove_goal (the_context()) "P --> Q --> R ==> P & Q --> R"
   223               (fn prems => [cut_facts_tac prems 1, Blast_tac 1]);
   224 
   225 val iff_allI = allI RS
   226     prove_goal (the_context()) "ALL x. P(x) <-> Q(x) ==> (ALL x. P(x)) <-> (ALL x. Q(x))"
   227                (fn prems => [cut_facts_tac prems 1, Blast_tac 1])
   228 val iff_exI = allI RS
   229     prove_goal (the_context()) "ALL x. P(x) <-> Q(x) ==> (EX x. P(x)) <-> (EX x. Q(x))"
   230                (fn prems => [cut_facts_tac prems 1, Blast_tac 1])
   231 
   232 val all_comm = prove_goal (the_context()) "(ALL x y. P(x,y)) <-> (ALL y x. P(x,y))"
   233                (fn _ => [Blast_tac 1])
   234 val ex_comm = prove_goal (the_context()) "(EX x y. P(x,y)) <-> (EX y x. P(x,y))"
   235                (fn _ => [Blast_tac 1])
   236 in
   237 
   238 (** make simplification procedures for quantifier elimination **)
   239 structure Quantifier1 = Quantifier1Fun(
   240 struct
   241   (*abstract syntax*)
   242   fun dest_eq((c as Const("op =",_)) $ s $ t) = Some(c,s,t)
   243     | dest_eq _ = None;
   244   fun dest_conj((c as Const("op &",_)) $ s $ t) = Some(c,s,t)
   245     | dest_conj _ = None;
   246   fun dest_imp((c as Const("op -->",_)) $ s $ t) = Some(c,s,t)
   247     | dest_imp _ = None;
   248   val conj = FOLogic.conj
   249   val imp  = FOLogic.imp
   250   (*rules*)
   251   val iff_reflection = iff_reflection
   252   val iffI = iffI
   253   val iff_trans = iff_trans
   254   val conjI= conjI
   255   val conjE= conjE
   256   val impI = impI
   257   val mp   = mp
   258   val uncurry = uncurry
   259   val exI  = exI
   260   val exE  = exE
   261   val iff_allI = iff_allI
   262   val iff_exI = iff_exI
   263   val all_comm = all_comm
   264   val ex_comm = ex_comm
   265 end);
   266 
   267 end;
   268 
   269 local
   270 
   271 val ex_pattern =
   272   read_cterm (Theory.sign_of (the_context ())) ("EX x. P(x)", FOLogic.oT)
   273 
   274 val all_pattern =
   275   read_cterm (Theory.sign_of (the_context ())) ("ALL x. P(x)", FOLogic.oT)
   276 
   277 in
   278 val defEX_regroup =
   279   mk_simproc "defined EX" [ex_pattern] Quantifier1.rearrange_ex;
   280 val defALL_regroup =
   281   mk_simproc "defined ALL" [all_pattern] Quantifier1.rearrange_all;
   282 end;
   283 
   284 
   285 (*** Case splitting ***)
   286 
   287 bind_thm ("meta_eq_to_iff", prove_goal IFOL.thy "x==y ==> x<->y"
   288   (fn [prem] => [rewtac prem, rtac iffI 1, atac 1, atac 1]));
   289 
   290 structure SplitterData =
   291   struct
   292   structure Simplifier = Simplifier
   293   val mk_eq          = mk_eq
   294   val meta_eq_to_iff = meta_eq_to_iff
   295   val iffD           = iffD2
   296   val disjE          = disjE
   297   val conjE          = conjE
   298   val exE            = exE
   299   val contrapos      = contrapos
   300   val contrapos2     = contrapos2
   301   val notnotD        = notnotD
   302   end;
   303 
   304 structure Splitter = SplitterFun(SplitterData);
   305 
   306 val split_tac        = Splitter.split_tac;
   307 val split_inside_tac = Splitter.split_inside_tac;
   308 val split_asm_tac    = Splitter.split_asm_tac;
   309 val op addsplits     = Splitter.addsplits;
   310 val op delsplits     = Splitter.delsplits;
   311 val Addsplits        = Splitter.Addsplits;
   312 val Delsplits        = Splitter.Delsplits;
   313 
   314 
   315 (*** Standard simpsets ***)
   316 
   317 structure Induction = InductionFun(struct val spec=IFOL.spec end);
   318 
   319 open Induction;
   320 
   321 
   322 bind_thms ("meta_simps",
   323  [triv_forall_equality,   (* prunes params *)
   324   True_implies_equals]);  (* prune asms `True' *)
   325 
   326 bind_thms ("IFOL_simps",
   327  [refl RS P_iff_T] @ conj_simps @ disj_simps @ not_simps @
   328   imp_simps @ iff_simps @ quant_simps);
   329 
   330 bind_thm ("notFalseI", int_prove_fun "~False");
   331 bind_thms ("triv_rls", [TrueI,refl,reflexive_thm,iff_refl,notFalseI]);
   332 
   333 fun unsafe_solver prems = FIRST'[resolve_tac (triv_rls@prems),
   334                                  atac, etac FalseE];
   335 (*No premature instantiation of variables during simplification*)
   336 fun   safe_solver prems = FIRST'[match_tac (triv_rls@prems),
   337                                  eq_assume_tac, ematch_tac [FalseE]];
   338 
   339 (*No simprules, but basic infastructure for simplification*)
   340 val FOL_basic_ss = empty_ss
   341   setsubgoaler asm_simp_tac
   342   setSSolver (mk_solver "FOL safe" safe_solver)
   343   setSolver (mk_solver "FOL unsafe" unsafe_solver)
   344   setmksimps (mksimps mksimps_pairs)
   345   setmkcong mk_meta_cong;
   346 
   347 
   348 (*intuitionistic simprules only*)
   349 val IFOL_ss = FOL_basic_ss
   350   addsimps (meta_simps @ IFOL_simps @ int_ex_simps @ int_all_simps)
   351   addsimprocs [defALL_regroup, defEX_regroup]    
   352   addcongs [imp_cong];
   353 
   354 bind_thms ("cla_simps",
   355   [de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2,
   356    not_all, not_ex, cases_simp] @
   357   map prove_fun
   358    ["~(P&Q) <-> ~P | ~Q",
   359     "P | ~P",             "~P | P",
   360     "~ ~ P <-> P",        "(~P --> P) <-> P",
   361     "(~P <-> ~Q) <-> (P<->Q)"]);
   362 
   363 (*classical simprules too*)
   364 val FOL_ss = IFOL_ss addsimps (cla_simps @ cla_ex_simps @ cla_all_simps);
   365 
   366 val simpsetup = [fn thy => (simpset_ref_of thy := FOL_ss; thy)];
   367 
   368 
   369 (*** integration of simplifier with classical reasoner ***)
   370 
   371 structure Clasimp = ClasimpFun
   372  (structure Simplifier = Simplifier and Splitter = Splitter
   373   and Classical  = Cla and Blast = Blast
   374   val iffD1 = iffD1 val iffD2 = iffD2 val notE = notE
   375   val cla_make_elim = cla_make_elim);
   376 open Clasimp;
   377 
   378 val FOL_css = (FOL_cs, FOL_ss);